Research Group "Stochastic Algorithms and Nonparametric Statistics"
Seminar "Modern Methods in Applied Stochastics and Nonparametric Statistics" Winter Semester 2022/23
|11.10.2022||Thomas Wagenhofer (TU Berlin)|
|Weak error estimates for rough volatility models
We consider a rough volatility model where the volatility is a (smooth) function of a Riemann--Liouville Brownian motion with Hurst parameter H in (0,1/2). When simulating these models, one often uses a discretization of stochastic integrals as an approximation. These integrals can be interpreted as log-stock-prices. In Applications, such as in pricing, the most relevent quantities are expectations of (payoff) functions. Our main result is that moments of these integrals have a weak error rate of order 3H+1/2 if H<1/6 and order 1 otherwise. For this we first derive a moment formula for both the discretization and the true stochastic integral. We then use this formula and properties of Gaussian random variables to prove our main theorems. We furthermore show that this convergence rate also holds for slightly more general payoffs and also provide a lower bound. Note that our rate of 3H+1/2 is in stark contrast to the strong error rate which is of order H. This is a joint work with Peter Friz and William Salkeld.
|18.10.2022||Egor Gladin (Humboldt Universität zu Berlin)|
|Algorithm for constrained Markov decision process with linear convergence (hybrid talk) |
The problem of constrained Markov decision process is considered. An agent aims to maximize the expected accumulated discounted reward subject to multiple constraints on its costs (the number of constraints is relatively small). A new dual approach is proposed with the integration of two ingredients: entropy-regularized policy optimizer and Vaidya's dual optimizer, both of which are critical to achieve faster convergence. The finite-time error bound of the proposed approach is provided. Despite the challenge of the nonconcave objective subject to nonconcave constraints, the proposed approach is shown to converge (with linear rate) to the global optimum. The complexity expressed in terms of the optimality gap and the constraint violation significantly improves upon the existing primal-dual approaches.
|25.10.2022||Luca Pelizzari (WIAS Berlin)|
|Polynomial Volterra processes and rough polynomial models (hybrid talk) |
|01.11.2022||Alexandra Suvorikova (WIAS Berlin)|
|Robust k-means clustering (hybrid talk) |
In this work we investigate the theoretical properties of robust k-means clustering under assumption of adversarial data corruption. We provide non-asymptotic rates for excess distortion under weak model assumptions on the moments of the distribution.
|ESH, Mohrenstr. 39||
|22.11.2022||Dr. Pavel Dvurechensky (WIAS Berlin)|
|Generalized self-concordant analysis of Frank-Wolfe algorithms (hybrid talk)
We propose several variants of the Frank-Wolfe method for minimizing generalized self-concordant (GSC) functions over compact sets. Such problems are ill-conditioned and are motivated by machine learning applications such as inverse covariance estimation or distance-weighted discrimination problems in support vector machines. We obtain O(1/k) convergence rate guarantees in the general situation and linear convergence under strong convexity and additional assumptions.
|29.11.2022||Thomas Wagenhofer (TU Berlin)|
| Reconstructing volatility: Pricing of index options under rough volatility (hybrid talk)
In [ABOBF02, ABOBF03] Avellaneda et al. pioneered the pricing and hedging of index options - products highly sensitive to implied volatility and correlation assumptions - with large deviations methods, assuming local volatility dynamics for all components of the index. We here present an extension applicable to non-Markovian dynamics and in particular the case of rough volatility dynamics.
|06.12.2022||Alexandra Suvorikova (WIAS Berlin)|
|Anomaly detection in biometric authentication (hybrid talk)
In this work we suggest a novel framework to detect user behaviour anomalies based on transfer learning approach combined with optimal transport techniques.
|13.12.2022||Dr. Amal Alphonse (WIAS Berlin)|
|Risk-averse optimal control of random elliptic variational inequalities (hybrid talk)
In this talk, I will discuss a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. I will derive two forms of first-order stationarity conditions for the problem by passing to the limit in a penalised and smoothed approximating control problem. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to delicate analytical challenges seemingly unique to the stochastic setting. To finish, I will briefly discuss a path-following stochastic approximation algorithm and demonstrate it on an example.
|10.01.2023||Robert Gruhlke (WIAS Berlin)|
|Wasserstein polynomial chaos and Langevin dynamics (hybrid talk)
An unsupervised learning approach for the computation of an explicit functional representation of a random vector is presented, which only relies on a finite set of samples from an unknown distribution. Motivated by recent advances with computational optimal transport for estimating Wasserstein distances, we develop a generative model denoted as Wasserstein multi-element polynomial chaos expansion (WPCE). It relies on the minimization of a regularized empirical Wasserstein metric known as debiased Sinkhorn divergence. Since the used PCE grows exponentially in the number of underlying input random coordinates, we introduce an appropriate low-rank format given as stacks of tensor trains. This alleviates the curse of dimensionality, leading to only linear dependence on the input dimension. Ensemble methods nowadays are ubiquitous for the solution of Bayesian inference problems. We discuss recent advances based on state-of-the-art samplers such as Affine Invariant Langevin Dynamics (ALDI) that allow for increased convergence speed and in turn reduce the required number of forward calls encoded in the drift term of the underlying stochastic differential equation. This improvement is realised through possible adaptive ensemble enrichment and an adapted Langevin dynamics based on a homotopy formalism. Optionally, the history of particles obtained within the solution process of the Langevin dynamics then can be used to numerically construct a push forward map in terms of WPCE. Once computed, this provides functional access to the posterior without the need of further forward model evaluations.
last reviewed: November 28, 2022 by Christine Schneider