A good model of random noise, such as the celebrated Brownian motion, cannot evolve smoothly in time, otherwise the noise would be predictable on a small scale! As a result, stochastic perturbations of differential equations are intrinsically irregular and require fundamentally new methods and theories. The groundbreaking contributions of Ito, which allowed to make all this possible, are one of the great achievements of 20th century mathematics. The analysis of stochastic differential equations including their numerical treatment is nowadays of crucial importance for applications in finance, modeling of particle systems for solving nonlinear kinetic equations, and other areas in applied mathematics.
A main goal is to use rough path ideas to develop and analyze new numerical schemes for stochastic differential equations. A balance is kept between concrete applications and further development of the required theory. On the theoretical side, for instance, it seems desirable to develop a rough path point of view for stochastic partial differential equations. Turning to applied topics, market micro structure and recently the analysis of lithiumion batteries require models in the form of realistic stochastic evolutionary equations.
Highlights
Large classes of stochastic differential equations have been analyzed with rough path methods and there is a fruitful interplay with classic techniques of stochastic analysis, including Malliavin's stochastic calculus of variations. Mean field stochastic differential equations have been used for analysis and simulations in a model for lithiumion batteries.In the area of statistics for stochastic differential equations, a Monte Carlo based method for estimating densities of solutions with rootN accuracy in any dimension has been developed. In contrast, classical density estimation methods suffer from curse of dimensionality in this respect. Recently, a rootN consistent estimator for conditional or pinned diffusions has been established using similar principles.
Publications
Monographs

P. Friz, M. Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Universitext, Springer International Publishing, Basel, 2020, 346 pages, (Monograph Published), DOI 10.1007/9783030415563 .

D. Belomestny, J. Schoenmakers, Advanced SimulationBased Methods for Optimal Stopping and Control: With Applications in Finance, Macmillan Publishers Ltd., London, 2018, 364 pages, (Monograph Published), DOI 10.1057/9781137033512 .

P. Friz, J. Gatheral, A. Gulisashvili, A. Jaquier, J. Teichmann, eds., Large Deviations and Asymptotic Methods in Finance, 110 of Springer Proceedings in Mathematics & Statistics, Springer, Berlin et al., 2015, ix+590 pages, (Collection Published).

J. Diehl, P. Friz, H. Mai , H. Oberhauser, S. Riedel, W. Stannat, Chapter 8: Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs, in: Extraction of Quantifiable Information from Complex Systems, S. Dahlke, W. Dahmen, M. Griebel, W. Hackbusch, K. Ritter, R. Schneider, Ch. Schwab, H. Yserentant, eds., 102 of Lecture Notes in Computational Science and Engineering, Springer International Publishing Switzerland, Cham, 2014, pp. 161178, (Chapter Published).

P. Friz, M. Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Universitext, Springer, Berlin et al., 2014, 251 pages, (Monograph Published).

P. Friz, N.B. Victoir, Multidimensional Stochastic Processes as Rough Paths, 120 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2010, 670 pages, (Monograph Published).
Articles in Refereed Journals

P. Friz, P. Gassiat, P. Pigato, Precise asymptotics: Robust stochastic volatility models, The Annals of Applied Probability, 31 (2021), pp. 896940, DOI 10.1214/20AAP1608 .
Abstract
We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to shorttime and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of Bayer et al. (Math. Finance30 (2020) 782832) In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, InahamaKawabi on rough path space. When applied to rough volatility models, for example, in the setting of Bayer, Friz and Gatheral (Quant. Finance16 (2016) 887904) and FordeZhang (SIAM J. Financial Math.8 (2017) 114145), one obtains precise asymptotics for European options which refine known large deviation asymptotics. 
O. Butkovsky, A. Kulik, M. Scheutzow, Generalized couplings and ergodic rates for SPDEs and other Markov models, The Annals of Applied Probability, 30 (2020), pp. 139, DOI 10.1214/19AAP1485 .
Abstract
We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic NavierStokes equations. Our main tool is a new version of the generalized coupling method. 
O. Butkovsky, M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, Communications in Mathematical Physics, 379 (2020), pp. 10011034, DOI 10.1007/s0022002003834w .
Abstract
We develop a general framework for studying ergodicity of orderpreserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an orderpreserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronizationbynoise of the stochastic reaction?diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer and Mattingly (Electron J Probab 16:658?738, 2011). 
S. Athreya, O. Butkovsky, L. Mytnik, Strong existence and uniqueness for stable stochastic differential equations with distributional drift, The Annals of Probability, 48 (2020), pp. 178210, DOI 10.1214/19AOP1358 .

D. Belomestny, J.G.M. Schoenmakers, Optimal stopping of McKeanVlasov diffusions via regression on particle systems, SIAM Journal on Control and Optimization, 58 (2020), pp. 529550, DOI 10.1137/18M1195590 .
Abstract
In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKeanVlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered. 
I. Chevyrev, P. Friz, A. Korepanov, I. Melbourne, Superdiffusive limits for deterministic fastslow dynamical systems, Probability Theory and Related Fields, 178 (2020), pp. 735770, DOI 10.1007/s00440020009885 .

M. Coghi, J.D. Deuschel, P. Friz, M. Maurelli, Pathwise McKeanVlasov theory with additive noise, The Annals of Applied Probability, 30 (2020), pp. 23552392, DOI 10.1214/20AAP1560 .
Abstract
We take a pathwise approach to classical McKeanVlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [34]. Our study was prompted by some concrete problems in battery modelling [19], and also by recent progress on roughpathwise McKeanVlasov theory, notably CassLyons [9], and then Bailleul, Catellier and Delarue [4]. Such a “pathwise McKeanVlasov theory” can be traced back to Tanaka [36]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4, 9, 36]. As novel applications we discuss mean field convergence without a priori independence and exchangeability assumption; common noise and reflecting boundaries. Last not least, we generalize DawsonGärtner large deviations to a nonBrownian noise setting. 
CH. Bayer, D. Belomestny, M. Redmann, S. Riedel, J.G.M. Schoenmakers, Solving linear parabolic rough partial differential equations, Journal of Mathematical Analysis and Applications, 490 (2020), pp. 124236/1124236/45, DOI 10.1016/j.jmaa.2020.124236 .
Abstract
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity α with ⅓ < α ≤ ½ . Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatiotemporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented. 
CH. Bayer, Ch.B. Hammouda, R. Tempone, Hierarchical adaptive sparse grids and quasiMonte Carlo for option pricing under the rough Bergomi model, Quantitative Finance, published online on 20.04.2020, urlhttps://doi.org/10.1080/14697688.2020.1744700, DOI 10.1080/14697688.2020.1744700 .
Abstract
The rough Bergomi (rBergomi) model, introduced recently in Bayer et al. [Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887?904], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the nonMarkovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on: (i) adaptive sparse grids quadrature (ASGQ), and (ii) quasiMonte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method. They reach a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e. to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model. 
P. Friz, T. Nilssen, W. Stannat , Existence, uniqueness and stability of semilinear rough partial differential equations, Journal of Differential Equations, 268 (2020), pp. 16861721, DOI 10.1016/j.jde.2019.09.033 .

A. Stephan, H. Stephan, Memory equations as reduced Markov processes, Discrete and Continuous Dynamical Systems, 39 (2019), pp. 21332155, DOI 10.3934/dcds.2019089 .
Abstract
A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along oneway loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations. 
C. Améndola, P. Friz, B. Sturmfels, Varieties of signature tensors, Forum of Mathematics. Sigma, 7 (2019), pp. e10/1e10/54, DOI 10.1017/fms.2019.3 .
Abstract
The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. 
P. Goyal, M. Redmann, Timelimited H2optimal model order reduction, Applied Mathematics and Computation, 355 (2019), pp. 184197, DOI 10.1016/j.amc.2019.02.065 .

W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic manyparticle model for LFP electrodes, Continuum Mechanics and Thermodynamics, 30 (2018), pp. 593628, DOI 10.1007/s0016101806297 .
Abstract
In the framework of nonequilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithiumpoor to a lithiumrich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltagecurrent relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates. 
M. Heida, M. Röger, Large deviation principle for a stochastic AllenCahn equation, Journal of Theoretical Probability, 31 (2018), pp. 364401, DOI 10.1007/s1095901607117 .
Abstract
The AllenCahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reactiondiffusion equation. Stochastic perturbations, especially in the case of additive noise, to the AllenCahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber [Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013)]. We first provide a large deviation principle for stochastic flows in spaces of functions that are Höldercontinuous in time, which extends results by Budhiraja, Dupuis and Maroulas [Ann. Probab. 36 (2008)]. From this result and a continuity argument we deduce a large deviation principle for the AllenCahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional. 
M. Redmann, Energy estimates and model order reduction for stochastic bilinear systems, International Journal of Control, 93 (2020), pp. 19541963 (published online on 08.11.2018), DOI 10.1080/00207179.2018.1538568 .
Abstract
In this paper, we investigate a largescale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatiallydiscretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an L2error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far. 
M. Redmann, Type II singular perturbation approximation for linear systems with Lévy noise, SIAM Journal on Control and Optimization, 56 (2018), pp. 21202158, DOI 10.1137/17M113160X .
Abstract
When solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatiallydiscretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As socalled type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems 
D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKeanVlasov equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 31693195, DOI 10.1137/17M1111024 .
Abstract
We propose a novel projectionbased particle method for solving McKeanVlasov stochastic differential equations. Our approach is based on a projectiontype estimation of the marginal density of the solution in each time step. The projectionbased particle method leads in many situations to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis, particularly in the case of linearly growing coefficients, turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKeanVlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples. 
K. Chouk, P. Friz, Support theorem for a singular SPDE: The case of gPAM, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 54 (2018), pp. 202219.
Abstract
We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equation in the subcritical regime, with (renormalized) solutions only recently understood via Hairer?s regularity structures and, in some cases equivalently, paracontrollled distributions by Gubinelli, Imkeller and Perkowski. In the present paper we utilise the paracontrolled machinery and obtain a (Stroock?Varadhan) type support description for the law of gPAM. In the spirit of rough paths, the crucial step is to identify the support of the enhanced noise in a sufficiently fine topology. The renormalization is seen to affect the support description. 
CH. Bayer, H. Mai, J.G.M. Schoenmakers, Forwardreverse expectationmaximization algorithm for Markov chains: Convergence and numerical analysis, Advances in Applied Probability, 2 (2018), pp. 621644, DOI 10.1017/apr.2018.27 .
Abstract
We develop a forwardreverse expectationmaximization (FREM) algorithm for estimating parameters of a discretetime Markov chain evolving through a certain measurable statespace. For the construction of the FREM method, we develop forwardreverse representations for Markov chains conditioned on a certain terminal state. We prove almost sure convergence of our algorithm for a Markov chain model with curved exponential family structure. On the numerical side, we carry out a complexity analysis of the forwardreverse algorithm by deriving its expected cost. Two application examples are discussed. 
P. Friz, H. Zhang, Differential equations driven by rough paths with jumps, Journal of Differential Equations, 264 (2018), pp. 62266301, DOI 10.1016/j.jde.2018.01.031 .
Abstract
We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed. 
M. Redmann, P. Benner, Singular perturbation approximation for linear systems with Lévy noise, Stochastics and Dynamics, published online on 10.08.2017, urlhttps://doi.org/10.1142/S0219493718500338, DOI 10.1142/S0219493718500338 .
Abstract
To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations, we consider the singular perturbation approximation as a model order reduction technique in this paper. This approach is wellknown from deterministic control theory and here we generalize it for controlled linear systems with Lévy noise. Additionally, we discuss properties of the reduced order model, provide an error bound, and give some examples to demonstrate the quality of this model order reduction technique. 
G. Cannizzaro, P. Friz, P. Gassiat, Malliavin calculus for regularity structures: The case of gPAM, Journal of Functional Analysis, 272 (2017), pp. 363419, DOI 10.1016/j.jfa.2016.09.024 .
Abstract
Malliavin calculus is implemented in the context of Hairer (2014) [16]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust, and purely deterministic, translation operator, in directions, between ?models?. In the concrete context of the generalized parabolic Anderson model in 2D ? one of the singular SPDEs discussed in the aforementioned article ? we establish existence of a density at positive times. 
J.D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Stochastic Processes and their Applications, 128 (2018), pp. 22282269 (published online on 21.09.2017), DOI 10.1016/j.spa.2017.09.010 .
Abstract
We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (klayer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean?Vlasov type limit, as shown in two corollaries. 
A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, SIAM Journal on Applied Mathematics, 77 (2017), pp. 537564, DOI 10.1137/16M1056869 .
Abstract
In this study the flow field of a nonlocal, diffusive upper convected Maxwell (UCM) fluid with a polymer in a solvent undergoing shearing motion is investigated for pressure driven planar channel flow and the free boundary problem of a liquid layer on a solid substrate. For large ratios of the zero shear polymer viscosity to the solvent viscosity, it is shown that channel flows exhibit boundary layers at the channel walls. In addition, for increasing stress diffusion the flow field away from the boundary layers undergoes a transition from a parabolic to a plug flow. Using experimental data for the wormlike micelle solutions CTAB/NaSal and CPyCl/NaSal, it is shown that the analytic solution of the governing equations predicts these signatures of the velocity profiles. Corresponding flow structures and transitions are found for the free boundary problem of a thin layer sheared along a solid substrate. Matched asymptotic expansions are used to first derive sharpinterface models describing the bulk flow with expressions for an em apparent slip for the boundary conditions, obtained by matching to the flow in the boundary layers. For a thin film geometry several asymptotic regimes are identified in terms of the order of magnitude of the stress diffusion, and corresponding new thin film models with a slip boundary condition are derived. 
P. Friz, J. Diehl, P. Gassiat, Stochastic control with rough paths, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 75 (2017), pp. 285315, DOI 10.1007/s0024501693339 .

P. Friz, S. Gerhold, A. Pinter, Option pricing in the moderate deviations regime, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, published online on 25.08.2017, urlhttps://doi.org/10.1111/mafi.12156, DOI 10.1111/mafi.12156 .

P. Friz, A. Shekhar, General rough integration, Lévy rough paths and a LévyKintchine type formula, The Annals of Probability, 45 (2017), pp. 27072765, DOI 10.1214/16 AOP1123 .

P. Friz, A. Shekhar, On the existence of SLE trace: Finite energy drivers and nonconstant $kappa$, Probability Theory and Related Fields, 169 (2017), pp. 353376.

P. Friz, H. Tran , On the regularity of SLE trace, Forum of Mathematics. Sigma, 5 (2017), pp. e19/1e19/17, DOI 10.1017/fms.2017.18 .
Abstract
We revisit regularity of SLE trace, for all , and establish Besov regularity under the usual halfspace capacity parametrization. With an embedding theorem of Garsia?Rodemich?Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index , improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler. 
E. Meca Álvarez, A. Münch, B. Wagner, Sharpinterface formation during lithium intercalation into silicon, European Journal of Applied Mathematics, 29 (2018), pp. 118145, DOI 10.1017/S0956792517000067 .
Abstract
In this study we present a phasefield model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as aSi. The governing equations couple a viscous CahnHilliardReaction model with elasticity in the framework of the CahnLarché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharpinterface model that also takes into account the dynamics of triple points where the sharp interface in the bulk of the layer intersects the free boundary. We numerically compare the interface motion predicted by the sharpinterface model with the longtime dynamics of the phasefield model. 
Z. Brzezniak, F. Flandoli, M. Maurelli, Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity, Archive for Rational Mechanics and Analysis, 221 (2016), pp. 107142.

J. Diehl, P. Friz, H. Mai, Pathwise stability of likelihood estimators for diffusions via rough paths, The Annals of Applied Probability, 26 (2016), pp. 21692192.
Abstract
We consider the estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. The Maximum Likelihood Estimator (MLE) is analyzed with regard to its pathwise stability properties and robustness towards misspecification in volatility and even the very nature of noise. We construct a version of the estimator based on rough integrals (in the sense of T. Lyons) and present strong evidence that this construction resolves a number of stability issues inherent to the standard MLEs. 
G.N. Milstein, J.G.M. Schoenmakers, Uniform approximation of the CIR process via exact simulation at random times, Advances in Applied Probability, 48 (2016), pp. 10951116.
Abstract
In this paper we uniformly approximate the trajectories of the CoxIngersollRoss (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed method gives a better quality of approximation in a pathwise sense than standard, or even exact simulation of the CIR dynamics at some deterministic time grid. 
CH. Bayer, P. Friz, J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), pp. 887904.
Abstract
From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum [SSRN 2509457, 2014] previously showed that logvolatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash. 
CH. Bayer, P. Friz, S. Riedel, J.G.M. Schoenmakers, From rough path estimates to multilevel Monte Carlo, SIAM Journal on Numerical Analysis, 54 (2016), pp. 14491483.
Abstract
Discrete approximations to solutions of stochastic differential equations are wellknown to converge with strong rate 1/2. Such rates have played a keyrole in Giles' multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H>1/4 as special case). 
P. Friz, B. Gess, A. Gulisashvili, S. Riedel, The JainMonrad criterion for rough paths and applications to random Fourier series and nonMarkovian Hörmander theory, The Annals of Probability, 44 (2016), pp. 684738.
Abstract
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46?57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron?Martin paths and complementary Young regularity (CYR) of the Cameron?Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itôlike probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of nonMarkovian Hörmander theory. 
P. Friz, B. Gess, Stochastic scalar conservation laws driven by rough paths, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 33 (2016), pp. 933963, DOI 10.1016/j.anihpc.2015.01.009 .

R. Allez, L. Dumaz, Random matrices in nonconfining potentials, Journal of Statistical Physics, 160 (2015), pp. 681714.
Abstract
We consider invariant matrix processes diffusing in nonconfining cubic potentials of the form Va(x)=x3/3?ax,a??. We construct the trajectories of such processes for all time by restarting them whenever an explosion occurs, from a new (well chosen) initial condition, insuring continuity of the eigenvectors and of the non exploding eigenvalues. We characterize the dynamics of the spectrum in the limit of large dimension and analyze the stationary state of this evolution explicitly. We exhibit a sharp phase transition for the limiting spectral density ?a at a critical value a=a?. If a?a?, then the potential Va presents a well near x=a?? deep enough to confine all the particles inside, and the spectral density ?a is supported on a compact interval. If a 
Z. Grbac, A. Papapantoleon, J.G.M. Schoenmakers, D. Skovmand, Affine LIBOR models with multiple curves: Theory, examples and calibration, SIAM Journal on Financial Mathematics, ISSN 1945497X, 6 (2015), pp. 9841025.
Abstract
We introduce a multiple curve LIBOR framework that combines tractable dynamics and semianalytic pricing formulas with positive interest rates and basis spreads. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR rates is developed, that allows for an efficient and accurate calibration to a system of caplet prices. 
D. Belomestny, M. Ladkau, J.G.M. Schoenmakers, Simulation based policy iteration for American style derivatives  A multilevel approach, SIAM ASA J. Uncertainty Quantification, 3 (2015), pp. 460483.
Abstract
This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration. 
G.N. Milstein, J.G.M. Schoenmakers, Uniform approximation of the CoxIngersollRoss process, Advances in Applied Probability, 47 (2015), pp. 11321156.
Abstract
The DossSussmann (DS) approach is used for simulating the CoxIngersollRoss (CIR) process. The DS formalism allows for expressing trajectories of the CIR process by solutions of some ordinary differential equation (ODE) that depend on realizations of the Wiener process involved. Via simulating the firstpassage times of the increments of the Wiener process to the boundary of an interval and solving an ODE, we approximately construct the trajectories of the CIR process. From a conceptual point of view the proposed method may be considered as an exact simulation approach. 
R. Allez, L. Dumaz, TracyWidom at high temperature, Journal of Statistical Physics, 156 (2014), pp. 11461183.
Abstract
We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature ? tends to 0. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy?Widom ? law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index k. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in Dumaz and Virág (Ann Inst H Poincaré Probab Statist 49(4):915?933, 2013). We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when ??0. As an application, we investigate the maximal eigenvalues statistics of ?Nensembles when the repulsion parameter ?N?0 when N?+?. We study the double scaling limit N?+?,?N?0 and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of Edelman and Sutton (J Stat Phys 127(6):1121?1165, 2007) and Ramírez et al. (J Am Math Soc 24:919?944, 2011) from our later study of the stochastic Airy operator. 
J.D. Deuschel, P. Friz, A. Jacquier, S. Violante , Marginal density expansions for diffusions and stochastic volatility: Part I, Communications on Pure and Applied Mathematics, 67 (2014), pp. 4082.

J.D. Deuschel, P. Friz, A. Jacquier, S. Violante , Marginal density expansions for diffusions and stochastic volatility: Part II, Communications on Pure and Applied Mathematics, 67 (2014), pp. 321350.

F. Flandoli, M. Maurelli, M. Neklyudov, Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, Journal of Mathematical Fluid Mechanics, 16 (2014), pp. 805822.

A. Gloria, S. Neukamm, F. Otto, An optimal quantitative twoscale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 325346.
Abstract
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the twoscale asymptotic expansion has the same scaling as in the periodic case. In particular the L^{2}norm in probability of the H^{1}norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author. 
CH. Bayer, H. Hoel, E. VON Schwerin, R. Tempone, On nonasymptotic optimal stopping criteria in Monte Carlo simulations, SIAM Journal on Scientific Computing, 36 (2014), pp. A869A885.
Abstract
We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule. 
CH. Bayer, J.G.M. Schoenmakers, Simulation of forwardreverse stochastic representations for conditional diffusions, The Annals of Applied Probability, 24 (2014), pp. 19942032.
Abstract
In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein et al. [Bernoulli, 10(2):281312, 2004] in the context of a forwardreverse transition density estimator. The corresponding Monte Carlo estimators have essentially rootN accuracy, hence they do not suffer from the curse of dimensionality. We provide a detailed convergence analysis and give a numerical example involving the realized variance in a stochastic volatility asset model conditioned on a fixed terminal value of the asset. 
P. Friz, S. Gerhold, M. Yor, How to make Dupire's local volatility work with jumps, Quantitative Finance, 14 (2014), pp. 13271331.
Abstract
There are several (mathematical) reasons why Dupire?s formula fails in the nondiffusion setting. And yet, in practice, adhoc preconditioning of the option data works reasonably well. In this note, we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire?s local vol diffusion process recreates the correct option prices, even in manifest presence of jumps. 
P. Friz, H. Oberhauser, Rough path stability of (semi)linear SPDEs, Probability Theory and Related Fields, 158 (2014), pp. 401434.

P. Friz, S. Riedel, Convergence rates for the full Gaussian rough paths, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 50 (2014), pp. 154194.

M. Ladkau, J.G.M. Schoenmakers, J. Zhang, Libor model with expirywise stochastic volatility and displacement, International Journal of Portfolio Analysis and Management, 1 (2013), pp. 224249.
Abstract
We develop a multifactor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own squareroot stochastic volatility process. The main advantage of this approach is that, maturitywise, each squareroot process can be calibrated to the corresponding cap(let)volastrike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in volastrikematurity panels are typically observed. 
D. Belomestny, J.G.M. Schoenmakers, F. Dickmann, Multilevel dual approach for pricing American style derivatives, Finance and Stochastics, 17 (2013), pp. 717742.
Abstract
In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the wellknown nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a nonnested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example. 
D. Crisan, J. Diehl, P. Friz, H. Oberhauser, Robust filtering: Correlated noise and multidimensional observation, The Annals of Applied Probability, 23 (2013), pp. 21392160.

P. Friz, S. Riedel, Integrability of (non)linear rough differential equations and integrals, Stochastic Analysis and Applications, 31 (2013), pp. 336358.

P. Friz, A. Shekar, DoobMeyer and rough paths, Bulletin of the Institute of Mathematics. Academia Sinica. Institute of Mathematics, Academia Sinica, Taipei, Taiwan. English. English summary., 8 (2013), pp. 7384.

P. Friz, Ch. Bayer, Functional convergence of the cubature tree on Wiener space, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 67 (2013), pp. 261278.

CH. Bayer, P. Friz, Cubature on Wiener space: Pathwise convergence, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 67 (2013), pp. 261278.

P. Friz, J. Diehl, Backward stochastic differential equations with rough drivers, The Annals of Probability, 40 (2012), pp. 17151758.

A. Papapantoleon, J.G.M. Schoenmakers, D. Skovmand, Efficient and accurate logLévy approximations to Lévy driven LIBOR models, Journal of Computational Finance, 15 (2012), pp. 344.
Abstract
The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a Lévydriven LIBOR model and aim at developing accurate and efficient logLévy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps and swaptions show that the approximations perform very well. In addition, we also consider the logLévy approximation of annuities, which offers good approximations for high volatility regimes. 
M. Caruana, P. Friz, H. Oberhauser, A (rough) pathwise approach to a class of nonlinear SPDEs, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 28 (2011), pp. 2746.

TH. Cass, P. Friz, Malliavin calculus and rough paths, Bulletin des Sciences Mathematiques, 135 (2011), pp. 542556.

P. Friz, S. Riedel, Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows, Bulletin des Sciences Mathematiques, 135 (2011), pp. 613628.

P. Friz, N. Victoir, A note on higher dimensional pvariation, Electronic Journal of Probability, 16 (2011), pp. 18801899.

H. Oberhauser, P. Friz, On the splittingup method for rough (partial) differential equations, Journal of Differential Equations, 251 (2011), pp. 316338.

P. Friz, Th. Cass, Densities for rough differential equations under Hoermander's condition, Ann. Math. (2), 171 (2010), pp. 21152141.

P. Friz, H. Oberhauser, A generalized Fernique theorem and applications, Proceedings of the American Mathematical Society, 138 (2010), pp. 36793688.

P. Friz, N. Victoir, Differential equations driven by Gaussian signals, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 46 (2010), pp. 369413.

A.E. Kyprianou, R.L. Loeffen, Refracted Lévy processes, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 46 (2010), pp. 2444.

P. Friz, E. Breuillard, M. Huesmann, From random walks to rough paths, Proceedings of the American Mathematical Society, 137 (2009), pp. 34873496.

P. Friz, M. Caruana, Partial differential equations driven by rough paths, Journal of Differential Equations, 247 (2009), pp. 140173.

P. Friz, T. Cass, N. Victoir, Nondegeneracy of Wiener functionals arising from rough differential equations, Transactions of the American Mathematical Society, 361 (2009), pp. 33593371.

P. Friz, H. Oberhauser, Rough path limits of WongZakai type with modified drift term, Journal of Functional Analysis, 256 (2009), pp. 32363256.

G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Forward and reverse representations for Markov chains, Stochastic Processes and their Applications, 117 (2007), pp. 10521075.
Abstract
In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jumpdiffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have rootN accuracy in any dimension and consider some applications. 
D. Spivakovskaya, A.W. Heemink, J.G.M. Schoenmakers, Twoparticle models for the estimation of the mean and standard deviation of concentrations in coastal waters, Stochastic Environmental Research and Risk Assessment (SERRA), 21 (2007), pp. 235251.

E. VAN DEN Berg, A.W. Heemink, H.X. Lin, J.G.M. Schoenmakers, Probability density estimation in stochastic environmental models using reverse representations, Stochastic Environmental Research & Risk Assessment, 20 (2006), pp. 126139.
Abstract
The estimation of probability densities of variables described by stochastic differential equations has long been done using forward time estimators, which rely on the generation of forward in time realizations of the model. Recently, an estimator based on the combination of forward and reverse time estimators has been developed. This estimator has a higher order of convergence than the classical one. In this article, we explore the new estimator and compare the forward and forward? reverse estimators by applying them to a biochemical oxygen demand model. Finally, we show that the computational efficiency of the forward?reverse estimator is superior to the classical one, and discuss the algorithmic aspects of the estimator. 
D. Spivakovskaya, A.W. Heemink, G.N. Milstein, J.G.M. Schoenmakers, Simulation of the transport of particles in coastal waters using forward and reverse time diffusion, Advances in Water Resources, 28 (2005), pp. 927938.
Abstract
Particle models are often used to simulate the spreading of a pollutant in coastal waters in case of a calamity at sea. Here many different particle tracks starting at the point of release are generated to determine the particle concentration at some critical locations after the release. This Monte Carlo method however consumes a large CPU time. Recently, Milstein, Schoenmakers and Spokoiny (2003) introduced the concept of reversetime diffusion. They derived a reverse system from the original forward simulation model and showed that the Monte Carlo estimator can also be based on realizations of this reverse system. In this paper we apply this concept to estimate particle concentrations in coastal waters. The results for the experiments considered show that the CPU time compared with the classical method is reduced orders of magnitude. 
G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Transition density estimation for stochastic differential equations via forwardreverse representations, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 10 (2004), pp. 281312.
Abstract
The general reverse diffusion equations are derived and applied to the problem of transition density estimation of diffusion processes between two fixed states. For this problem we propose density estimation based on forward?reverse representations and show that this method allows essentially better results to be achieved than the usual kernel or projection estimation based on forward representations only. 
O. Kurbanmuradov, K. Sabelfeld, J.G.M. Schoenmakers, Lognormal approximations to LIBOR market models, Journal of Computational Finance, 6 (2002), pp. 69100.
Abstract
We study several lognormal approximations for LIBOR market models, where special attention is paid to their simulation by direct methods and lognormal random fields. In contrast to conventional numerical solution of SDE's this approach simulates the solution directly at a desired point in time and therefore may be more efficient. As such the proposed approximations provide valuable alternatives to the Euler method, in particular for long dated instruments. We carry out a pathwise comparison of the different lognormal approximations with the 'exact' SDE solution obtained by the Euler scheme using sufficiently small time steps. Also we test approximations obtained via numerical solution of the SDE by the Euler method, using larger time steps. It turns out that for typical volatilities observed in practice, improved versions of the lognormal approximation proposed by Brace, Gatarek and Musiela, citeBrace, appear to have excellent pathwise accuracy. We found out that this accuracy can also be achieved by Euler stepping the SDE using larger time steps, however, from a comparative cost analysis it follows that, particularly for long maturity options, the latter method is more time consuming than the lognormal approximation. We conclude with applications to some example LIBOR derivatives. 
J.G.M. Schoenmakers, A.W. Heemink, K. Ponnambalm, P.E. Kloeden, Variance reduction for Monte Carlo simulation of stochastic environmental models, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems. Elsevier Science Inc., New York, NY. English, English abstracts., 26 (2002), pp. 787795.
Abstract
To determine the probability of exceedence Monte Carlo simulation of stochastic models is often used. Mathematically this requires the evaluation of an expectation of some function of a solution of a stochastic model. This can be reformulated as a Kolmogorov final value problem. It can thus be calculated numerically by either solving a deterministic partial differential equation (Kolmogorov's Backwards equations) or by simulating a large number of trajectories of the stochastic differential equation. Here we discuss a composite method of variance reduced Monte Carlo simulation. The variance reduction is obtained by the Girsanov transformation to modify the stochastic model by a correction term that is obtained from an approximate solution of the partial differential equation computed by a classical numerical method. The composite method is more efficient than either the standard Monte Carlo or the classical numerical method. The approach is applied to estimate the probability of exceedence in a model for biochemicaloxygen demand.
Contributions to Collected Editions

D. Peschka, M. Thomas, T. Ahnert, A. Münch, B. Wagner, Gradient structures for flows of concentrated suspensions, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 295318, DOI 10.1007/9783030331160 .
Abstract
In this work we investigate a twophase model for concentrated suspensions. We construct a PDE formulation using a gradient flow structure featuring dissipative coupling between fluid and solid phase as well as different driving forces. Our construction is based on the concept of flow maps that also allows it to account for flows in moving domains with free boundaries. The major difference compared to similar existing approaches is the incorporation of a nonsmooth twohomogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows. 
Y. Bruned, I. Chevyrev, P. Friz, Examples of renormalized SDEs, in: Stochastic Partial Differential Equations and Related Fields, A. Eberle, M. Grothaus, W. Hoh, M. Kassmann, W. Stannat, G. Trutnau, eds., 229 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2018, pp. 303317, DOI 10.1007/9783319749297 .

P. Friz, P. Gassiat, Geometric foundations of rough paths, in: Geometry, Analysis and Dynamics on SubRiemannian Manifolds, Vol. 2, D. Barilari, U. Boscain, M. Sigalotti, eds., EMS Series of Lectures in Mathematics, European Mathematical Society, Zurich, 2016, pp. 171210.

CH. Bayer, P. Friz, P. Laurence, On the probability density function of baskets, in: Large Deviations and Asymptotic Methods in Finance, P. Friz, J. Gatheral, A. Gulisashvili, A. Jaquier, J. Teichmann, eds., 110 of Springer Proceedings in Mathematics & Statistics, Springer, Berlin et al., 2015, pp. 449472.

D. Becherer, J.G.M. Schoenmakers, E3  Stochastic simulation methods for optimal stopping and control  Towards multilevel approaches, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 317331.

J.G.M. Schoenmakers, SHOWCASE 17  Expirywise Heston LIBOR model, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 314315.

D. Spivakovskaya, A.W. Heemink, J.G.M. Schoenmakers, G.N. Milstein, Stochastic modeling of transport in coastal waters using forward and reverse time diffusion, in: Computational Methods in Water Resources, Proceedings of the 15th International Conference on Computational Methods in Water Resources (CMWR XV), June 1317 2004, Chapel Hill, North Carolina, C.T. Miller, M.W. Farthing, W.G. Gray, G.F. Pinder, eds., 55 of Developments in Water Science, Elsevier Science, Amsterdam, 2004, pp. 18131824.
Preprints, Reports, Technical Reports

CH. Bayer, S. Breneis, Markovian approximations of stochastic Volterra equations with the fractional kernel, Preprint no. 2868, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2868 .
Abstract, PDF (525 kByte)
We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölderregularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an Ndimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in N. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model. 
N. Perkowski, W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift, Preprint no. 2768, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2768 .
Abstract, PDF (308 kByte)
We consider the stochastic differential equation on ℝ ^{d} given by d X _{t} = b(t,X_{t} ) d t + d B_{t}, where B is a Brownian motion and b is considered to be a distribution of regularity >  1/2. We show that the martingale solution of the SDE has a transition kernel Γ_{t} and prove upper and lower heat kernel bounds for Γ_{t} with explicit dependence on t and the norm of b. 
W. König, N. Perkowski, W. van Zuijlen, Longtime asymptotics of the twodimensional parabolic Anderson model with whitenoise potential, Preprint no. 2765, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2765 .
Abstract, PDF (471 kByte)
We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) whitenoise potential. We prove that the almostsure largetime asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t. 
S. Riedel, Semiimplicit Taylor schemes for stiff rough differential equations, Preprint no. 2734, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2734 .
Abstract, PDF (538 kByte)
We study a class of semiimplicit Taylortype numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T. Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a onesided Lipschitz condition only. We prove wellposedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments show that our schemes are particularly useful in the case of stiff rough stochastic differential equations driven by a fractional Brownian motion. 
M. Redmann, S. Riedel, RungeKutta methods for rough differential equations, Preprint no. 2708, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2708 .
Abstract, PDF (393 kByte)
We study RungeKutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (Bseries) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying RungeKutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a RungeKutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivativefree. We provide a full characterization of this implementable RungeKutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index 1/4 < H ≤ 1/2. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence. 
M. Ghani Varzaneh, S. Riedel, A dynamical theory for singular stochastic delay differential equations II: Nonlinear equations and invariant manifolds, Preprint no. 2701, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2701 .
Abstract, PDF (383 kByte)
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semiinvertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [GVR].
Talks, Poster

CH. Bayer, A pricing BSPDE for rough volatility (online talk), MATH4UQ Seminar (Online Event), RWTH Aachen University, Mathematics for Uncertainty Quantification, April 6, 2021.

P. Friz, Unified cumulants and Magnus expansions, Noncommutative algebra, probability and analysis in action (Hybrid Event), September 20  25, 2021, Universität Greifswald, Alfried Krupp Wissenschaftskolleg, September 21, 2021.

M. Kantner, Noise in semiconductor lasers (online talk), MATH+ Spotlight Seminar, MATH+, July 14, 2021.

W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Meeting (Online Event), University of Oxford, Department of Statistics, UK, February 10, 2021.

W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability and Statistical Physics Seminar( Online Event), The University of Chicago, Department of Mathematics, Statistics, and Computer Science, USA, February 12, 2021.

W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift (online talk), 14th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis (Online Event), February 10  12, 2021, University of Oxford, Mathematical Institute, UK, February 12, 2021.

O. Butkovsky, Regularization by noise for SDEs and related systems: A tale of two approaches, Eighth BielefeldSNU joint Workshop in Mathematics, February 24  26, 2020, Universität Bielefeld, Fakultät für Mathematik, February 24, 2020.

S. Riedel, RungeKutta methods for rough differential equations (online talk), The DNA Seminar (spring 2020), Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway, June 24, 2020.

W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Seminar, Universidade Federal da Bahia, Instituto de Matematica Doutorado em Matematica (Online Event), Salvador, Brazil, October 21, 2020.

CH. Bayer, Rough volatility, Summer School 2020 and Annual Meeting of the BerlinOxford IRTG 2544 ``Stochastic Analysis in Interaction'', September 14  17, 2020, Döllnsee, September 15, 2020.

G. Dong, Integrated physicsbased method, learninginformed model and hyperbolic PDEs for imaging, Efficient Algorithms in Data Science, Learning and Computational Physics, Sanya, China, January 12  16, 2020.

M. Coghi, Mean field limit of interacting filaments for 3D Euler equations, Second Italian Meeting on Probability and Mathematical Statistics, June 17  20, 2019, Università degli Studi di Salerno, Dipartimento di Matematica, Vietri sul Mare, Italy, June 20, 2019.

A. Stephan, EDPconvergence for linear reactiondiffusion systems with different time scales, Winter School ``Gradient Flows and Variational Methods in PDEs'', November 25  29, 2019, Universität Ulm, November 29, 2019.

CH. Bayer, A regularity structure for rough volatility, Vienna Seminar in Mathematical Finance and Probability, Technische Universität Wien, Research Unit of Financial and Actuarial Mathematics, Austria, January 10, 2019.

CH. Bayer, Calibration of rough volatility models by deep learning, Rough Workshop 2019, September 4  6, 2019, Technische Universität Wien, Financial and Actuarial Mathematics, Austria.

CH. Bayer, Deep calibration of rough volatility models, New Directions in Stochastic Analysis: Rough Paths, SPDEs and Related Topics, WIAS und TU Berlin, March 18, 2019.

CH. Bayer, Learning rough volatility, Algebraic and Analytic Perspectives in the Theory of Rough Paths and Signatures, November 14  15, 2019, University of Oslo, Department of Mathematics, Norway, November 14, 2019.

CH. Bayer, Numerics for rough volatility, Stochastic Processes and Related Topics, February 21  22, 2019, Kansai University, Senriyama Campus, Osaka, Japan, February 22, 2019.

P. Friz, Multiscale systems, homogenization and rough paths, CRC 1114 Colloquium & Lectures, Collaborative Research Center CRC 1114 ``Scaling Cascades in Complex Systems'', Freie Universität Berlin, June 13, 2019.

P. Friz, On differential equations with singular forcing, Berliner Oberseminar Nichtlineare partielle Differentialgleichungen (LangenbachSeminar), WIAS Berlin, January 9, 2019.

P. Friz, Rough paths, rough volatility, regularity structures, Rough Workshop 2019, September 4  6, 2019, Technische Universität Wien, Financial and Actuarial Mathematics, Austria.

P. Friz, Rough transport, revisited, Algebraic and Analytic Perspectives in the Theory of Rough Paths and Signatures, November 14  15, 2019, University of Oslo, Department of Mathematics, Norway, November 14, 2019.

P. Friz, Some perspectives on harmonic analysis and rough paths, Harmonic Analysis and Rough Paths, November 18  19, 2019, Hausdorff Research Institute for Mathematics, Bonn, November 18, 2019.

M. Coghi, Pathwise McKeanVlasov theory, 10th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis, November 29  December 1, 2018, University of Oxford, Mathematical Institute, UK, December 1, 2018.

M. Maurelli, McKeanVlasov SDEs with irregular drift: Large deviations for particle approximation, University of Oxford, Mathematical Institute, UK, March 5, 2018.

M. Maurelli, Sanov theorem for Brownian rough paths and an application to interacting particles, Università di Roma La Sapienza, Dipartimento di Matematica Guido Castelnuovo, Italy, January 18, 2018.

M. Maurelli , A McKeanVlasov SDE with reflecting boundaries, CASA Colloquium, Eindhoven University of Technology, Department of Mathematics and Computer Science, Netherlands, January 10, 2018.

M. Redmann, Numerical approximations of parabolic rough PDEs, Harmonic Analysis for Stochastic PDEs, July 10  13, 2018, Delft University of Technology, Netherlands, July 10, 2018.

W. van Zuijlen, Eigenvalues of the Anderson Hamiltonian with white noise potential in 2D, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 1, 2018.

W. van Zuijlen, Massasymptotics for the parabolic Anderson model in 2D, Statistical Mechanics Seminar, University of Warwick, Department of Statistics, Coventry, UK, December 6, 2018.

P. Friz, From rough paths and regularity structures to short dated option pricing under rough volatility, Workshop on Mathematical Finance and Related Issues, Osaka University, Nakanoshima Center, Japan, March 15, 2018.

P. Friz, Rough path analysis of rough volatility, Stochastic Analysis Seminar, Imperial College London, Department of Mathematics, Stochastic Analysis Group, UK, February 13, 2018.

P. Friz, Rough paths, stochastics and PDE's, ECMath Colloquium, July 6, 2018, HumboldtUniversität zu Berlin, July 6, 2018.

W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4electrodes, ModVal14  14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2  3, 2017.

M. Maurelli, Regularization by noise for scalar conservation laws, Stochastic Analysis Day, February 27, 2017, Università di Pisa, Dipartimento di Matematica, Italy, February 27, 2017.

M. Maurelli, Regularization by noise for scalar conservation laws, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 30, 2017.

M. Maurelli, Regularization by noise for scalar conservation laws, Séminaire de Probabilité et Statistique, Université de Nice SophiaAntipolis, Laboratoire Jean Alexandre Dieudonné, France, September 26, 2017.

M. Maurelli, Stochastic 2D Euler equations with transport noise, Chalmers University of Technology, Department of Mathematical Sciences, Gothenburg, Sweden, November 28, 2017.

M. Redmann, A regression method to solve parabolic rough PDEs, 7th Annual ERC BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, May 18  20, 2017, WIASBerlin, May 20, 2017.

M. Redmann, A regression method to solve parabolic rough PDEs, Ninth Workshop on Random Dynamical Systems, June 14  17, 2017, Universität Bielefeld, Fakultät für Mathematik, June 15, 2017.

M. Redmann, Type II singular perturbation approximation for linear systems with Levy noise, London Mathematical Society  EPSRC Durham Symposium: Model Order Reduction, Durham University, Department of Mathematical Sciences, UK, August 14, 2017.

W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

CH. Bayer, Numerics for rough volatility models, Ninth Workshop on Random Dynamical Systems, June 14  17, 2017, University of Bielefeld, Department of Mathematics, June 14, 2017.

P. Friz, An application of regularity structures to the analysis of rough volatility, Fractional Brownian Motion and Rough Models, June 8  9, 2017, Barcelona Graduate School of Economics, Spain, June 9, 2017.

P. Friz, Aspects of rough volatility, The 5th Imperial  ETH Workshop on Mathematical Finance, March 27  29, 2017, Imperial College London, UK, March 27, 2017.

P. Friz, General semimartingales and rough paths, Durham Symposium on Stochastic Analysis, July 10  20, 2017, Durham University, Department of Mathematical Sciences, UK, July 13, 2017.

P. Friz, Multiscale systems, homogenization and rough paths, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

P. Friz, Rough differential equations with jumps and their applications, JapaneseGerman Open Conference on Stochastic Analysis 2017, September 4  8, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, September 5, 2017.

J.G.M. Schoenmakers, Projected particle methods for solving McKeanVlasov SDEs, Dynstoch 2017, April 5  7, 2017, Universität Siegen, Department Mathematik, April 6, 2017.

M. Maurelli, Enhanced Sanov theorem and large deviations for interacting particles, Workshop ``Rough Paths, Regularity Structures and Related Topics'', May 1  7, 2016, Mathematisches Forschungsinstitut Oberwolfach, May 5, 2016.

M. Maurelli, Regularization by noise for continuity equation via Young drivers, Stochastic Partial Differential Equations and Applications, May 30  June 2, 2016, Centro Internazionale per la Ricerca Matematica (CIRM), Levico, Italy, May 30, 2016.

M. Maurelli, Regularization by noise for scalar conservation laws, Mathematical Finance and Stochastic Analysis Seminars, University of York, UK, October 26, 2016.

M. Maurelli, Regularization by noise for stochastic scalar conservation laws, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 41 ``Stochastic Partial Differential Equations'', July 1  5, 2016, The American Institute of Mathematical Science, Orlando (Florida), USA, July 4, 2016.

M. Maurelli, Regularization by noise for transporttype equations via stochastic exponentials, Workshop in Stochastic Analysis, June 28  30, 2016, Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica, Campinas, Brazil, June 29, 2016.

E. Valdinoci, Nonlocal minimal surface, JustusLiebigUniversität Gießen, Fakultät für Mathematik, February 10, 2016.

W. van Zuijlen, Mean field GibbsnonGibbs transitions, 6th BerlinOxford Meeting, December 8  10, 2016, University of Oxford, Mathematics Department, UK, December 9, 2016.

CH. Bayer, Pricing under rough volatility, Stochastic Analysis and Mathematical Finance  A Fruitful Partnership, May 22  27, 2016, Banff International Research Station for Mathematical Innovation and Discovery, Oaxaca, Mexico, May 24, 2016.

CH. Bayer, SDE based regression for random PDEs, Workshop ``Rough Paths, Regularity Structures and Related Topics'', May 1  7, 2016, Mathematisches Forschungsinstitut Oberwolfach, May 3, 2016.

P. Friz, A regularity structure for rough volatility, Stochastic Analysis, Rough Paths, Geometry, January 7  9, 2016, Imperial College London, UK, January 7, 2016.

P. Friz, Signatures, rough paths and probability, Stochastics and Finance Seminar, University of Amsterdam, Kortewegde Vries Institute for Mathematics, Netherlands, October 18, 2016.

P. Friz, Support theorem for singular SPDEs: The case of gPAM, Stochastic Partial Differential Equations and Applications, May 29  June 3, 2016, Centro Internazionale per la Ricerca Matematica (CIRM), Levico, Italy, May 31, 2016.

P. Gajewski, M. Maurelli, Stochastic methods for the analysis of lithiumion batteries, Matheon Center Days, April 20  21, 2015, Technische Universität Berlin, April 21, 2015.

M. Maurelli, A large deviation principle for enhanced Brownian empirical measure, 4th Annual ERC BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, December 7  9, 2015, WIAS Berlin, December 8, 2015.

M. Maurelli, A large deviation principle for interacting particle SDEs via rough paths, 38th Conference on Stochastic Processes and their Applications, July 13  17, 2015, University of Oxford, OxfordMan Institute of Quantitative Finance, UK, July 14, 2015.

M. Maurelli, Enhanced Sanov theorem for Brownian rough paths and an application to interacting particles, Seminar Stochastic Analysis, Imperial College London, UK, October 20, 2015.

M. Maurelli, Stochastic 2D Euler equations: A poorly correlated multiplicative noise regularizes the twopoint motion, Universität Augsburg, Institut für Mathematik, March 24, 2015.

M. Maurelli, Transport equation via Young estimates (TBC), 3rd Annual ERC BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, January 27  29, 2015, WIAS Berlin, January 29, 2015.

CH. Bayer, Rough paths and rough partial differential equations, November 16  18, 2015, University of Oslo, Department of Mathematics, Norway.

R. Allez, Invariant beta ensembles and beyond, Seminar Mathematische Physik, Universität Bielefeld, Mathematical Physics Group, June 11, 2014.

R. Allez, Les ensembles beta invariants, Séminaire de l'équipe Probas/Stats, Institut Élie Cartan de Lorraine, Équipe Probabilités et Statistiques, Nancy, France, May 22, 2014.

R. Allez, Liouville Brownian motion, Oberseminar Peter Friz, Technische Universität Berlin, August 20, 2014.

R. Allez, Random matrices at high temperature, Probability series, University of Cambridge, UK, February 11, 2014.

R. Allez, Random matrices at high temperature, Stochastic Analysis Seminar Series, OxfordMan Institute, UK, January 20, 2014.

R. Allez, Random matrix theory and some applications, WIASDay, February 19, 2014.

CH. Bayer, From rough path estimates to multilevel Monte Carlo, Eleventh International Conference on Monte Carlo and QuasiMonte Carlo Methods in Scientific Computing, April 6  11, 2014, Catholic University of Leuven, Department of Computer Science, Belgium, April 7, 2014.

CH. Bayer, Multilevel Monte Carlo meets rough paths, BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, July 1  2, 2014, University of Oxford, OxfordMan Institute of Quantitative Finance, UK, July 1, 2014.

CH. Bayer, Simulation of forwardreverse stochastic representations for conditional diffusions, Foundations of Computational Mathematics Conference 2014, December 11  20, 2014, Universidad de la República, Facultad de Ingenieria, Montevideo, Uruguay, December 19, 2014.

CH. Bayer, The forwardreverse method for conditional diffusion processes, Numerical Analysis Seminar, Royal Institute of Technology Stockholm, Department of Mathematics, Sweden, October 10, 2014.

P. Friz, Basic of rough paths, Workshop ``Stochastic Analysis: Around the KPZ Universality Class '', June 1  7, 2014, Mathematisches Forschungsinstitut Oberwolfach, June 2, 2014.

P. Friz, Fullynonlinear SPDEs with rough path dependence, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 109: Stochastic Partial Differential Equations, July 7  August 11, 2014, Madrid, Spain, July 7, 2014.

P. Friz, Rough integration with jumps, BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, July 1  2, 2014, University of Oxford, OxfordMan Institute of Quantitative Finance, UK, July 2, 2014.

P. Friz, Rough path and stochastic analysis, Trimester Partial Differential Equations & Probability, Week on Rough Paths and PDE, February 10  15, 2014, Centre International de Mathématiques et Informatique de Toulouse, France.

P. Friz, Rough paths, with jumps, Probability Seminar, The University of Edinburgh, School of Mathematics, UK, October 10, 2014.

P. Friz, Signatures, rough paths and probability, BMS Days 2014, February 17  18, 2014, The Berlin Mathematical School, February 17, 2014.

H. Mai, Pathwise stability of likelihood estimators for diffusions via rough paths, 37th Conference on Stochastic Processes and their Applications, July 28  August 1, 2014, Buonos Aires, Argentina.

H. Mai, Pathwise stability of likelihood estimators for diffusions via rough paths, 37th Conference on Stochastic Processes and their Applications, Buenos Aires, Argentina, July 28  August 1, 2014.

H. Mai, Pathwise stability of likelihood estimators for diffusions via rough paths, International Workshop ``Advances in Optimization and Statistics'', May 15  16, 2014, Russian Academy of Sciences, Institute of Information Transmission Problems (Kharkevich Institute), Moscow, May 16, 2014.

H. Mai, Robust drift estimation: Pathwise stability under volatility and noise misspecification, BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, July 1  2, 2014, University of Oxford, OxfordMan Institute of Quantitative Finance, UK, July 2, 2014.

H. Mai, Robustness and pathwise stability of maximum likelihood estimators for jump diffusions, Universidad de Buenos Aires, Instituto de Calculo, Argentina, August 8, 2014.

H. Mai, Robustness of likelihood estimators for diffusions via rough paths, Advances in Stochastic Analysis, September 3  5, 2014, National Research University  Higher School of Economics, Laboratory of Stochastic Analysis and its Applications, Moscow, Russian Federation, September 3, 2014.

J.G.M. Schoenmakers, Affine LIBOR models with multiple curves: Theory, examples and calibration, 11th German Probability and Statistics Days 2014, March 5  7, 2014, Universität Ulm, March 6, 2014.

H. Stephan, Inequalities for Markov operators and applications to forward and backward PDEs, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 88: Stochastic Processes and Spectral Theory for Partial Differential Equations and Boundary Value Problems, July 7  11, 2014, Madrid, Spain, July 8, 2014.

M. Ladkau, A new multifactor stochastic volatility model with displacement, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16  17, 2013, WIASBerlin, May 16, 2013.

S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, GermanJapanese Meeting on Stochastic Analysis, September 9  13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

S. Neukamm, Quantitative results in stochastic homogenization, sc Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, June 27, 2013.

S. Neukamm, Quantitative results in stochastic homogenization, Oberseminar Analysis, Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, June 13, 2013.

P. Friz, (Rough) pathwise stochastic analysis: Old and new, Stochastic Analysis and its Applications, 60th Birthday of Terry Lyons, September 23  27, 2013, University of Oxford, OxfordMan Institute, UK, September 24, 2013.

P. Friz, Information content ot iterated integrals and applications, LUHKolloquium "Versicherungs und Finanzmathematik", Stochastic Analysis Day, June 27, 2013, Leibniz Universität Hannover, Institut für Stochastik, June 27, 2013.

P. Friz, Marginal density expansions for diffusions and stochastic volatility and related stuff, Large Deviations and Asymptotic Methods in Finance, April 9  11, 2013, Imperial College London, UK, April 10, 2013.

P. Friz, Physical Brownian motion in magnetic field as rough path, GermanJapanese Meeting on Stochastic Analysis, September 9  13, 2013, Universität Leipzig, Mathematisches Institut, September 13, 2013.

P. Friz, Rational shapes of the local volatility surface, 20th Annual Global Derivatives & Risk Management, April 15  19, 2013, The International Centre for Business Information (ICBI), Amsterdam, Netherlands, April 17, 2013.

P. Friz, Rough path analysis, Summer School ``Numerical Methods for Stochastic Differential Equations'', September 2  4, 2013, Vienna University of Technology, Institute for Analysis and Scientific Computing E 101, Austria, September 4, 2013.

P. Friz, Rough paths, 29th European Meeting of Statisticians (EMS), July 20  25, 2013, Eötvös Loránd University, Budapest, Hungary, July 20, 2013.

P. Friz, Some aspects of stochastic area, UK Probability Easter Meeting; Geometry and Analysis of Random Processes, April 8  12, 2013, University of Cambridge, Department of Pure Mathematics, UK, April 11, 2013.

P. Friz, Stochastic control with rough paths, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16  17, 2013, WIASBerlin, May 16, 2013.

CH. Bayer, Asymptotics beats Monte Carlo: The case of correlated local volatility baskets, Stochastic Methods in Finance and Physics, July 15  19, 2013, University of Crete, Department of Applied Mathematics, Heraklion, Greece, July 19, 2013.

CH. Bayer, Simulation of conditional diffusions via forwardreverse stochastic representations, Seminar in Mathematical Statistics, Linköping University, Division of Mathematical Statistics, Sweden, September 11, 2013.

CH. Bayer, Simulation of conditional diffusions via forwardreverse stochastic representations, King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering, Thuwal, Saudi Arabia, February 20, 2013.

H. Mai, Applications of rough path analysis to robust likelihood inference, Statistikseminar, HumboldtUniversität zu Berlin, Institut für Mathematik, October 18, 2013.

H. Mai, Efficient drift estimation for jump diffusion processes and jump filtering, Séminaire de Statistique du CREST, École Nationale de la Statistique et de l'Administration Économique, Centre de Recherche en Économie et Statistique, Paris, France, February 18, 2013.

J.G.M. Schoenmakers, Simulation of conditional diffusions via forwardreverse stochastic representations, DynStoch 2013, April 17  19, 2013, University of Copenhagen, Department of Mathematical Sciences, Denmark, April 19, 2013.

J.G.M. Schoenmakers, Simulation of conditional diffusions via forwardreverse stochastic representations, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16  17, 2013, WIASBerlin, May 16, 2013.

N. Willrich, Solutions of martingale problems for Lévytype operators and stochastic differential equations driven by Lévy processes with discontinous coefficients, 29th European Meeting of Statisticians (EMS), July 20  25, 2013, Eötvös Loránd University, Budapest, Hungary, July 21, 2013.

P. Friz, Applications of rough paths: Physical Brownian in a magnetic fields, Modelling of markets with infinitesimally delayed reactions, Workshop on: Rough Paths and PDEs, August 19  25, 2012, Mathematisches Forschungsinstitut Oberwolfach (MFO), August 23, 2012.

P. Friz, Generalized subRiemannian cut loci and volatility smiles, 6th European Congress of Mathematics, July 2  7, 2012, Jagiellonian University, Institute of Mathematics, Cracow, Poland, July 5, 2012.

P. Friz, Marginal density expansion with applications to Levy area and the SteinStein model, Stochastic Analysis Seminar, University of Warwick, Mathematics Institute, Coventry, UK, November 28, 2012.

P. Friz, Rough paths and control, Stochastic Systems Simulation and Control (SSSC2012), November 5  9, 2012, Universidad Autónoma de Madrid, Instituto de Ciencias Matemáticas, Spain, November 5, 2012.

CH. Bayer, Existence, uniqueness and stability of invariant distributions in continuoustime stochastic models, 12th Conference of the Society for the Advancement of Economic Theory (SAET 2012), June 30  July 3, 2012, University of Queensland, School of Economics, Australia, July 1, 2012.

CH. Bayer, Some applications of the NinomiyaVictoir scheme in the context of financial engineering, Talks in Financial and Insurance Mathematics, Eidgenössische Technische Hochschule Zürich, Switzerland, April 26, 2012.

CH. Bayer, Some applications of the NinomiyaVictoir scheme in the context of financial engineering, Stochastic Analysis Seminar Series, Oxford University, OxfordMan Institute of Quantitative Finance, UK, May 21, 2012.

H. Mai, Drift estimation for jump diffusion, Haindorf Seminar 2012 (Klausurtagung des SFB 649), February 9  12, 2012, HumboldtUniversität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, February 10, 2012.

H. Mai, Jump filtering and highfrequency data, Statistical Methods for Dynamical Stochastic Models (DynStoch2012), Paris, France, June 7  9, 2012.

H. Mai, Jump filtering for semimartingales under highfrequency observations, PreMoLab: MoscowBerlin Stochastic and Predictive Modeling, May 31  June 1, 2012, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Moscow, May 31, 2012.

H. Mai, Maximum likelihood estimation for Lévydriven SDEs, Workshop on Statistics of Lévydriven Models, Ulm, March 15  16, 2012.

H. Mai, Maximum likelihood estimation for Lévydriven SDEs, Workshop on statistics of Lévydriven models, March 15  16, 2012, Universität Ulm, Institute of Mathematical Finance.

P. Friz, Gaussian rough paths, Bonn Probability Day, Hausdorf Center for Mathematics, Universität Bonn, January 26, 2012.

P. Friz, Rough analysis applied to some classes of SPDEs and related topics, Stochastic Partial Differential Equations: Analysis, Numerics, Geometry and Modeling, September 12  17, 2011, Eidgenössische Technische Hochschule Zürich, Forschungsinstitut für Mathematik, Switzerland, September 16, 2011.

P. Friz, Rough path analysis and applications, Conference in Honor of the 70th Birthday of S. R. Srinivasa Varadhan, July 11  15, 2011, National Taiwan University, Taipeh, July 14, 2011.

M. Becker , Random walks and selfintersections, Evolving Complex Networks (ECONS) PhdStudent Meeting, WIAS, August 24, 2010.

P. Friz, A (rough) pathwise approach to SPDEs, ICM Satellite Conference on Probability and Stochastic Processes, August 13  17, 2010, Indian Statistical Institute, Bangalore, India, August 16, 2010.

P. Friz, A (rough) pathwise approach to a class of nonlinear stochastic partial diffenrential equations, Workshop ``Stochastic Partial Differential Equations (SPDEs)'', Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 4  6, 2010.

P. Friz, A new pathwise theory of SPDEs, 34th Conference on Stochastic Processes and their Applications (SPA 2010), September 6  10, 2010, Bernoulli Society for Mathematical Statistics and Probability, Osaka, Japan, September 9, 2010.

P. Friz, Ordinary, partial and backward stochastic differential equations driven by rough paths, Conference in Memory of Paul Malliavin, October 4  6, 2010, Institut de Mathématiques de Bourgogne, Dijon, France, October 6, 2010.

P. Friz, Ordinary, partial and backward stochastic differential equations driven by rough signals, Analysis, Stochastics, and Applications (AnStAp 2010), July 12  16, 2010, Universität Wien, Fakultät für Mathematik, Austria, July 13, 2010.

P. Friz, Rough path stability of SPDEs arising in nonlinear filtering and beyond, Workshop on Filtering, June 14  15, 2010, University of Cambridge, Isaac Newton Institute for Mathematical Sciences, UK, June 15, 2010.

P. Friz, Rough viscosity solutions and applications to SPDEs, Workshop on Stochastic Partial Differential Equations (SPDEs): Approximation, Asymptotics and Computation, June 28  July 2, 2010, University of Cambridge, Isaac Newton Institute for Mathematical Sciences, UK, June 29, 2010.

P. Friz, From numerical aspects of stochastic financial models to the foundations of stochastic differential equations (and back), Annual Meeting of the Deutsche MathematikerVereinigung and 17th Congress of the Österreichische Mathematische Gesellschaft, Section ``Financial and Actuarial Mathematics'', September 20  25, 2009, Technische Universität Graz, Austria, September 25, 2009.

P. Friz, Rough paths and the gap between deterministic and stochastic differential equations, Berlin Mathematical School, Friday Colloquium, December 18, 2009.

H. Stephan, Inequalities for Markov operators, Positivity VI (Sixth Edition of the International Conference on Positivity and its Applications), July 20  24, 2009, El Escorial, Madrid, Spain, July 24, 2009.

H. Stephan, Modeling of diffusion prozesses with hidden degrees of freedom, Workshop on Numerical Methods for Applications, November 5  6, 2009, Lanke, November 6, 2009.

J.G.M. Schoenmakers, Transition density estimation for stochastic differential equations via forwardreverse representations, TandemWorkshop StochastikNumerik, June 11  March 26, 2004, HumboldtUniversität zu Berlin, June 11, 2004.

J.G.M. Schoenmakers, Transition density estimation for stochastic differential equations via forward reverse representations, IV IMACS Seminar on Monte Carlo Methods (MCM 2003), September 15  19, 2003, Berlin, September 16, 2003.
External Preprints

O. Butkovsky, K. Dareiotis , M. Gerencsér, Approximation of SDEs  A stochastic sewing approach, Preprint no. arXiv:1909.07961, Cornell University, 2020.

B. Gess, M. Maurelli, Wellposedness by noise for scalar conservation laws, Preprint no. arXiv:1701.05393, Cornell University Library, arXiv.org, 2017.
Abstract
We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the corresponding deterministic scalar conservation law. We prove that perturbing the system by noise leads to wellposedness. 
J.D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Preprint no. arxiv:1602.08043, Cornell University Library, arXiv.org, 2016.

R. Allez, L. Dumaz, Random matrices in nonconfining potentials, Preprint no. arXiv:1404.5265, Cornell University Library, arXiv.org, 2014.

R. Allez, J.P. Bouchaud, Eigenvector dynamics under free addition, Preprint no. arXiv:1301.4939, Cornell University Library, arXiv.org, 2014.

R. Allez, J. Bun, J.P. Bouchaud, The eigenvectors of Gaussian matrices with an external source, Preprint no. arXiv:1412.7108, Cornell University Library, arXiv.org, 2014.

L. Beck, F. Flandoli, M. Gubinelli, M. Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift: Regularity, duality and uniqueness, Preprint no. arXiv:1401.1530, Cornell University Library, arXiv.org, 2014.

Z. Brzezniak, F. Flandoli, M. Maurelli, Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity, Preprint no. arXiv:1401.5938, Cornell University Library, arXiv.org, 2014.

F. Flandoli, M. Maurelli, M. Neklyudov, Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, Preprint no. arXiv:1403.0022, Cornell University Library, arXiv.org, 2014.

P. Friz, B. Gess, Stochastic scalar conservation laws driven by rough paths, Preprint no. arXiv:1403.6785, Cornell University Library, arXiv.org, 2014.

S. Neukamm, A. Gloria, F. Otto, An optimal quantitative twoscale expansion in stochastic homogenization of discrete elliptic equations, Preprint no. 41, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, 2013.
Abstract
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the twoscale asymptotic expansion has the same scaling as in the periodic case. In particular the L^{2}norm in probability of the H^{1}norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author. 
P. Friz, A. Shekhar, DoobMeyer for rough paths, Preprint no. arXiv:1205.2505, Cornell University Library, arXiv.org, 2012.

P. Friz, A. Shekhar, The LevyKintchine formula for rough paths, Preprint no. arXiv:1212.5888, Cornell University Library, arXiv.org, 2012.

P. Imkeller, N. Willrich, Solutions of martingale problems for Lévytype operators and stochastic differential equations driven by Lévy processes with discontinuous coefficients, Preprint no. arXiv:1208.1665, Cornell University Library, arXiv.org, 2012.
Abstract
We show the existence of Lévytype stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stablelike with stability index functions for which the closures of the discontinuity sets are countable. For this purpose, we formulate the problem in terms of a Skorokhodspace martingale problem associated with nonlocal operators with discontinuous coefficients. These operators are approximated along a sequence of smooth nonlocal operators giving rise to Feller processes with uniformly controlled symbols. They converge uniformly outside of increasingly smaller neighborhoods of a Lebesgue nullset on which the singularities of the limit operator are located. 
J. Diehl , P. Friz, H. Oberhauser, Parabolic comparison revisited and applications, Preprint no. arXiv:1102.5774, Cornell University Library, arXiv.org, 2011.

P. Friz, S. Riedel , Integrability of linear rough differential equations, Preprint no. arXiv:1104.0577, Cornell University Library, arXiv.org, 2011.

J.D. Deuschel, P. Friz, A. Jacquier, S. Violante , Marginal density expansions for diffusions and stochastic volatility, Preprint no. arXiv:1111.2462, Cornell University Library, arXiv.org, 2011.

P. Friz, S. Riedel, Convergence rates for the full Gaussian rough paths, Preprint no. arXiv:1108.1099, Cornell University Library, arXiv.org, 2011.

P. Friz, N. Victoir, A note on higher dimensional $p$ variation, Preprint no. arXiv:1102.4587, Cornell University Library, arXiv.org, 2011.

CH. Bayer, P. Friz, R.L. Loeffen, Semiclosed form cubature and applications to financial diffusion models, Preprint no. arXiv:1009.4818, Cornell University Library, arXiv.org, 2010.
Abstract
Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ. 6, 6983, 2004] and LyonsVictoir [Proc. R. Soc.
Lond. Ser. A 460, 169198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the NinomiyaVictoir algorithm [Appl. Math. Fin. 15, 107121, 2008], which corresponds to a concrete level $5$ cubature method, we study some parametric diffusion models motivated from financial applications, and exhibit structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies, by introducing a (modeldependent) variation of the NinomiyaVictoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time. 
M. Beiglboeck, P. Friz, S. Sturm, Is the minimum value of an option on variance generated by local volatility?, Preprint no. arXiv:1001.4031, Cornell University Library, arXiv.org, 2010.
Abstract
We discuss the possibility of obtaining modelfree bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counterexample to a widespread conjecture is given. 
J. Diehl, P. Friz, Backward stochastic differential equations with rough drivers, Preprint no. arXiv:1008.0290, Cornell University Library, arXiv.org, 2010.
Abstract
Backward stochastic differential equations (BSDEs) in the sense of PardouxPeng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200217, 1992] provide a nonMarkovian extension to certain classes of nonlinear partial differential equations; the nonlinearity is expressed in the socalled driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons [Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14, no. 2, 215310, 1998] and so obtain a notion of "BSDE with rough driver". Existence, uniqueness and a version of Lyons' limit theorem in this context are established. Our main tool, aside from rough path analysis, is the stability theory for quadratic BSDEs due to Kobylanski [Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab., 28(2):558602, 2000]. 
P. Friz, H. Oberhauser, A generalized Fernique theorem and applications, Preprint no. arXiv:1004.1923, Cornell University Library, arXiv.org, 2010.
Abstract
We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough path theory where one deals with iterated integrals of Gaussian processes (which are generically not Gaussian). Gaussian integrability with explicitly given constants for variation and Hölder norms of the (fractional) Brownian rough path, Gaussian rough paths and the Banach space valued Wiener process enhanced with its Lévy area [Ledoux, Lyons, Quian. "Lévy area of Wiener processes in Banach spaces". Ann. Probab., 30(2):546578, 2002] then all follow from applying our main theorem.We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough path theory where one deals with iterated integrals of Gaussian processes (which are generically not Gaussian). Gaussian integrability with explicitly given constants for variation and Hölder norms of the (fractional) Brownian rough path, Gaussian rough paths and the Banach space valued Wiener process enhanced with its Lévy area [Ledoux, Lyons, Quian. "Lévy area of Wiener processes in Banach spaces". Ann. Probab., 30(2):546578, 2002] then all follow from applying our main theorem.We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough path theory where one deals with iterated integrals of Gaussian processes (which are generically not Gaussian). Gaussian integrability with explicitly given constants for variation and Hölder norms of the (fractional) Brownian rough path, Gaussian rough paths and the Banach space valued Wiener process enhanced with its Lévy area [Ledoux, Lyons, Quian. "Lévy area of Wiener processes in Banach spaces". Ann. Probab., 30(2):546578, 2002] then all follow from applying our main theorem. 
P. Friz, H. Oberhauser, On the splittingup method for rough (partial) differential equations, Preprint no. arXiv:1008.0513, Cornell University Library, arXiv.org, 2010.

P. Friz, H. Oberhauser, Rough path stability of SPDEs arising in nonlinear filtering, Preprint no. arXiv:1005.1781, Cornell University Library, arXiv.org, 2010.

P. Friz, M. Caruana, H. Oberhauser, A (rough) pathwise approach to fully nonlinear stochastic partial differential equations, Report no. RICAM Report no. 200903, Johann Radon Institute for Computational and Applied Mathematics, 2009.

E. VAN den Berg, A.W. Heemink, H.X. Lin, J.G.M. Schoenmakers, Probability density estimation in stochastic environmental models using reverse representations, Report no. 0306, Delft University of Technology, 2003.

E. VAN DEN Berg, A.W. Heemink, H.X. Lin, J.G.M. Schoenmakers, Probability density estimation in stochastic environmental models using reverse representations, Report no. 6, TU Delft, The Netherlands, Applied Mathematical Analysis, 2003.
Contact
Contributing Groups of WIAS
Projects/Grants
 Numerical analysis of rough PDEs
 Numerics for rough paths and SPDEs
 Optimal control in energy markets using rough analysis and deep networks
 Recovery of battery ageing dynamics with multiple timescales
 Singular SPDEs  approximation and statistical properties
 Stochastic and rough aspects in deep neural networks