An ordinary differential equation is often used to model the movement of a particle. Similarly, partial differential equation can be used to describe the evolution of a total of trajectories of particles. It is natural to add randomness to such models: sometimes because this is a more realistic description which takes into account random noise, sometimes because this randomness is fundamental to the model itself as is the case for financial markets.

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 ground-breaking 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 non-linear 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 lithium-ion 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 lithium-ion batteries.

In the area of statistics for stochastic differential equations, a Monte Carlo based method for estimating densities of solutions with root-N accuracy in any dimension has been developed. In contrast, classical density estimation methods suffer from curse of dimensionality in this respect. Recently, a root-N consistent estimator for conditional or pinned diffusions has been established using similar principles.

Publications

  Monographs

  • 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. 161--178, (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

  • 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. 107--142.

  • J. Diehl, P. Friz, H. Mai, Pathwise stability of likelihood estimators for diffusions via rough paths, The Annals of Applied Probability, 26 (2016) pp. 2169--2192.
    Abstract
    We consider the estimation problem of an unknown drift parameter within classes of non-degenerate 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. Sheffield Univ. (GB). Dept. of Probability and Statistics. Applied Probability Trust, Sheffield., 48 (2016) pp. 1095--1116.
    Abstract
    In this paper we uniformly approximate the trajectories of the Cox-Ingersoll-Ross (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 path-wise 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. 887--904.
    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 log-volatility 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. 1449--1483.
    Abstract
    Discrete approximations to solutions of stochastic differential equations are well-known to converge with strong rate 1/2. Such rates have played a key-role 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 Jain--Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory, The Annals of Probability, 44 (2016) pp. 684--738.
    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 Itô-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hörmander theory.

  • R. Allez, L. Dumaz, Random matrices in non-confining potentials, Journal of Statistical Physics, 160 (2015) pp. 681--714.
    Abstract
    We consider invariant matrix processes diffusing in non-confining 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 1945-497X, 6 (2015) pp. 984--1025.
    Abstract
    We introduce a multiple curve LIBOR framework that combines tractable dynamics and semi-analytic 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. 460--483.
    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 Cox--Ingersoll--Ross process, Advances in Applied Probability. Sheffield Univ. (GB). Dept. of Probability and Statistics. Applied Probability Trust, Sheffield., 47 (2015) pp. 1132--1156.
    Abstract
    The Doss-Sussmann (DS) approach is used for simulating the Cox-Ingersoll-Ross (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 first-passage 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, Tracy--Widom at high temperature, Journal of Statistical Physics, 156 (2014) pp. 1146--1183.
    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 ?N-ensembles 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. 40--82.

  • 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. 321--350.

  • 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. 805--822.

  • A. Gloria, S. Neukamm, F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014) pp. 325--346.
    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 two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-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 non-asymptotic optimal stopping criteria in Monte Carlo simulations, SIAM Journal on Scientific Computing, 36 (2014) pp. A869--A885.
    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 forward-reverse stochastic representations for conditional diffusions, The Annals of Applied Probability, 24 (2014) pp. 1994--2032.
    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):281-312, 2004] in the context of a forward-reverse transition density estimator. The corresponding Monte Carlo estimators have essentially root-N 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. 1327--1331.
    Abstract
    There are several (mathematical) reasons why Dupire?s formula fails in the non-diffusion setting. And yet, in practice, ad-hoc 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. 401--434.

  • P. Friz, S. Riedel, Convergence rates for the full Gaussian rough paths, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 50 (2014) pp. 154--194.

  • M. Ladkau, J.G.M. Schoenmakers, J. Zhang, Libor model with expiry-wise stochastic volatility and displacement, International Journal of Portfolio Analysis and Management, 1 (2013) pp. 224--249.
    Abstract
    We develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike 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 vola-strike-maturity 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. 717-742.
    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 well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested 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. 2139--2160.

  • P. Friz, S. Riedel, Integrability of (non-)linear rough differential equations and integrals, Stochastic Analysis and Applications, 31 (2013) pp. 336--358.

  • P. Friz, A. Shekar, Doob--Meyer and rough paths, Bulletin of the Institute of Mathematics. Academia Sinica. Institute of Mathematics, Academia Sinica, Taipei, Taiwan. English. English summary., 8 (2013) pp. 73--84.

  • 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. 261--278.

  • CH. Bayer, P. Friz, Cubature on Wiener space: Pathwise convergence, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 67 (2013) pp. 261--278.

  • P. Friz, J. Diehl, Backward stochastic differential equations with rough drivers, The Annals of Probability, 40 (2012) pp. 1715--1758.

  • A. Papapantoleon, J.G.M. Schoenmakers, D. Skovmand, Efficient and accurate log-Lévy approximations to Lévy driven LIBOR models, Journal of Computational Finance, 15 (2012) pp. 3--44.
    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 Lévy-driven LIBOR model and aim at developing accurate and efficient log-Lévy 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 log-Lévy 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. 27--46.

  • TH. Cass, P. Friz, Malliavin calculus and rough paths, Bulletin des Sciences Mathematiques, 135 (2011) pp. 542--556.

  • 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. 613--628.

  • P. Friz, N. Victoir, A note on higher dimensional p-variation, Electronic Journal of Probability, 16 (2011) pp. 1880--1899.

  • H. Oberhauser, P. Friz, On the splitting-up method for rough (partial) differential equations, Journal of Differential Equations, 251 (2011) pp. 316--338.

  • P. Friz, Th. Cass, Densities for rough differential equations under Hoermander's condition, Ann. Math. (2), 171 (2010) pp. 2115--2141.

  • P. Friz, H. Oberhauser, A generalized Fernique theorem and applications, Proceedings of the American Mathematical Society, 138 (2010) pp. 3679--3688.

  • P. Friz, N. Victoir, Differential equations driven by Gaussian signals, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 46 (2010) pp. 369--413.

  • A.E. Kyprianou, R.L. Loeffen, Refracted Lévy processes, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 46 (2010) pp. 24--44.

  • P. Friz, E. Breuillard, M. Huesmann, From random walks to rough paths, Proceedings of the American Mathematical Society, 137 (2009) pp. 3487--3496.

  • P. Friz, M. Caruana, Partial differential equations driven by rough paths, Journal of Differential Equations, 247 (2009) pp. 140--173.

  • P. Friz, T. Cass, N. Victoir, Non-degeneracy of Wiener functionals arising from rough differential equations, Transactions of the American Mathematical Society, 361 (2009) pp. 3359--3371.

  • P. Friz, H. Oberhauser, Rough path limits of Wong--Zakai type with modified drift term, Journal of Functional Analysis, 256 (2009) pp. 3236--3256.

  • G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Forward and reverse representations for Markov chains, Stochastic Processes and their Applications, 117 (2007) pp. 1052--1075.
    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 jump-diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.

  • D. Spivakovskaya, A.W. Heemink, J.G.M. Schoenmakers, Two-particle models for the estimation of the mean and standard deviation of concentrations in coastal waters, Sochastic Environmental Research and Risk Assessment (SERRA), 21 (2007) pp. 235--251.

  • 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. 126--139.
    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. 927--938.
    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 reverse-time 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 forward-reverse representations, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 10 (2004) pp. 281--312.
    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. 69--100.
    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 path-wise 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 path-wise 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. 787--795.
    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 biochemical-oxygen demand.

  Contributions to Collected Editions

  • P. Friz, P. Gassiat, Geometric foundations of rough paths, in: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, Vol. 2, D. Barilari, U. Boscain, M. Sigalotti, eds., EMS Series of Lectures in Mathematics, European Mathematical Society, Zurich, 2016, pp. 171--210.

  • 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. 449-472.

  • 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. 317--331.

  • J.G.M. Schoenmakers, SHOWCASE 17 -- Expiry-wise 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. 314--315.

  • 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 13--17 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. 1813--1824.

  Preprints, Reports, Technical Reports

  • D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKean--Vlaslov equations, Preprint no. 2341, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2341 .
    Abstract, PDF (320 kByte)
    We study a novel projection-based particle method to the solution of the corresponding McKean-Vlasov equation. Our approach is based on the projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method can profit from additional smoothness of the underlying density and leads in many situation to a significant reduction of numerical complexity compared to kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The case of linearly growing coefficients of the McKean-Vlasov equation 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 McKean-Vlasov equations with affine drift.

  • M. Redmann, M.A. Freitag, Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems, Preprint no. 2339, WIAS, Berlin, 2016.
    Abstract, PDF (718 kByte)
    When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

  • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic model for LFP-electrodes, Preprint no. 2329, WIAS, Berlin, 2016.
    Abstract, PDF (1531 kByte)
    In the framework of non-equilibrium 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 lithium-poor to a lithium-rich 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 voltage-current 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.

  • E. Meca Álvarez, A. Münch, B. Wagner, Sharp-interface formation during lithium intercalation into silicon, Preprint no. 2257, WIAS, Berlin, 2016.
    Abstract, PDF (518 kByte)
    In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché 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 sharp-interface 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 sharp-interface model with the long-time dynamics of the phase-field model.

  • A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, Preprint no. 2255, WIAS, Berlin, 2016.
    Abstract, PDF (367 kByte)
    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 sharp-interface 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.

  • CH. Bayer, H. Oberhauser, Splitting methods for SPDEs: From robustness to financial engineering, optimal control and nonlinear filtering, Preprint no. 2072, WIAS, Berlin, 2015.
    Abstract, Postscript (4572 kByte), PDF (1719 kByte)
    In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control and nonlinear filtering to the viscosity theory of nonlinear SPDEs.

  Talks, Poster

  • W. Dreyer, J. Fuhrmann, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, 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 25 - March 3, 2017, Universita' di Pisa, Dipartimento di Matematica, Italy, February 27, 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 transport-type 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, Justus-Liebig-Universität Gießen, Fakultät für Mathematik, February 10, 2016.

  • W. van Zuijlen, Mean field Gibbs-non-Gibbs transitions, 6th Berlin--Oxford 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, Korteweg-de 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 lithium-ion 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 Berlin-Oxford 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, Oxford-Man 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 two-point motion, Universität Augsburg, Institut für Mathematik, March 24, 2015.

  • M. Maurelli, Transport equation via Young estimates (TBC), 3rd Annual ERC Berlin-Oxford 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, Oxford-Man Institute, UK, January 20, 2014.

  • R. Allez, Random matrix theory and some applications, WIAS-Day, February 19, 2014.

  • CH. Bayer, From rough path estimates to multilevel Monte Carlo, Eleventh International Conference on Monte Carlo and Quasi-Monte 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, Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, July 1 - 2, 2014, University of Oxford, Oxford-Man Institute of Quantitative Finance, UK, July 1, 2014.

  • CH. Bayer, Simulation of forward-reverse 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 forward-reverse 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, Fully-nonlinear 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, Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, July 1 - 2, 2014, University of Oxford, Oxford-Man 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, Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, July 1 - 2, 2014, University of Oxford, Oxford-Man 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 multi-factor stochastic volatility model with displacement, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16 - 17, 2013, WIAS-Berlin, May 16, 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.

  • S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, German-Japanese Meeting on Stochastic Analysis, September 9 - 13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 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, Oxford-Man Institute, UK, September 24, 2013.

  • P. Friz, Information content ot iterated integrals and applications, LUH-Kolloquium "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, German-Japanese 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, WIAS-Berlin, 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 forward--reverse stochastic representations, Seminar in Mathematical Statistics, Linköping University, Division of Mathematical Statistics, Sweden, September 11, 2013.

  • CH. Bayer, Simulation of conditional diffusions via forward-reverse 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, Humboldt-Universität 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 forward-reverse 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 forward-reverse stochastic representations, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16 - 17, 2013, WIAS-Berlin, May 16, 2013.

  • N. Willrich, Solutions of martingale problems for Lévy-type 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 sub-Riemannian 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 Stein--Stein 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 continuous-time 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 Ninomiya--Victoir 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 Ninomiya--Victoir scheme in the context of financial engineering, Stochastic Analysis Seminar Series, Oxford University, Oxford-Man 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, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, February 10, 2012.

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

  • H. Mai, Jump filtering for semimartingales under high-frequency observations, PreMoLab: Moscow-Berlin 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 Lévy-driven SDEs, Workshop on Statistics of Lévy-driven Models, Ulm, March 15 - 16, 2012.

  • H. Mai, Maximum likelihood estimation for Lévy-driven SDEs, Workshop on statistics of Lévy-driven 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 self-intersections, Evolving Complex Networks (ECONS) Phd-Student 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 non-linear 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 non-linear 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 Mathematiker-Vereinigung 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 forward-reverse representations, Tandem-Workshop Stochastik-Numerik, June 11 - March 26, 2004, Humboldt-Universität 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

  • B. Gess, M. Maurelli, Well-posedness 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 well-posedness.

  • 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 non-confining 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 two-scale expansion in stochastic homogenization of discrete elliptic equations, Preprint no. 41, Max-Planck-Institut 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 two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-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, Doob--Meyer for rough paths, Preprint no. arXiv:1205.2505, Cornell University Library, arXiv.org, 2012.

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

  • P. Imkeller, N. Willrich, Solutions of martingale problems for Lévy-type 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 Lévy-type stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stable-like 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 Skorokhod-space martingale problem associated with non-local operators with discontinuous coefficients. These operators are approximated along a sequence of smooth non-local 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, Semi-closed 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, 69--83, 2004] and Lyons--Victoir [Proc. R. Soc.
    Lond. Ser. A 460, 169--198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm [Appl. Math. Fin. 15, 107--121, 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 (model-dependent) variation of the Ninomiya-Victoir 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 model-free bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counter-example to a wide-spread 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 Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200--217, 1992] provide a non-Markovian extension to certain classes of non-linear partial differential equations; the non-linearity is expressed in the so-called 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, 215--310, 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):558--602, 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):546--578, 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):546--578, 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):546--578, 2002] then all follow from applying our main theorem.

  • P. Friz, H. Oberhauser, On the splitting-up 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 non-linear filtering, Preprint no. arXiv:1005.1781, Cornell University Library, arXiv.org, 2010.

  • P. Friz, M. Caruana, H. Oberhauser, A (rough) pathwise approach to fully non-linear stochastic partial differential equations, Report no. RICAM Report no. 2009-03, 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. 03-06, 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.