# Seminar on Stochastics and PDEs

#### Next talk:

### Mo Jan 14, 12:15-14:00 @MaA 203

Stefan Geiss:

Riemann–Liouville operators, BMO, and approximation of stochastic integrals II

#### Autumn 2018

- Tue Dec 11, 12:30-14:00 @MaA 203

Stefan Geiss:

Riemann–Liouville operators, BMO, and approximation of stochastic integrals - Tue Dec 04, 12:30-14:00 @MaA 203

Amal Attouchi:

Intrinsic scaling method and its application to the study of a class of degenerate parabolic equations II - Tue Nov 27, 12:30-14:00 @MaA 203

Karl K. Brustad (The Norwegian University of Science and Technology (NTNU)):

The dominative p-Laplacian and sublinear elliptic operators - Tue Nov 13, 12:30-14:00 @MaA 203

Ivan Yaroslavtsev: The canonical decomposition of local martingales in infinite dimensions*Abstract: The canonical decomposition of local martingales was invented by Yoeurp in 1976 as a natural extension of Lévy–Itô decomposition and it has the following form: a local martingale has the canonical decomposition if it can be decomposed into a sum of a continuous local martingale (a Wiener-like part), a purely discontinuous quasi-left continuous local martingale (a Poisson-like part, which does not jump at predictable stopping times), and a purely discontinuous local martingale with accessible jumps (a discrete-like part, which jumps only at certain predictable stopping times). Due to Yoeurp the canonical decomposition of a real-valued local martingale always exists and is unique. In this talk we will extend this result to infinite dimensions and show that the canonical decomposition of an arbitrary X-valued local martingale exists if and only if X is a UMD Banach space; moreover, if this is the case, then the corresponding L*^{p}(1<p< ∞) and weak L^{1}estimates for the decomposition terms hold. - Fri Nov 09, 08:15-10:00 @MaA 203

Peter Parczewski (University of Mannheim): Optimal approximation of Skorohod integrals and Skorohod SDEs

*Abstract: The Skorohod integral is a mean zero extension of the Itô integral to nonadapted integrands and we sketch several introductions. We consider optimal approximation with respect to the mean square error of Skorohod integrals and solutions of Skorohod SDEs given an equidistant discretization of the Brownian motion. Without the adaptedness, proof techniques from Malliavin calculus (and Wick calculus) are required. Due to reformulations in terms of Malliavin calculus, we obtain results which extend and simplify the situation for Itô integrals and Itô SDEs. Moreover, we present results beyond regular conditions.* - Tue Nov 06, 12:30-14:00 @MaA 203

Amal Attouchi: Intrinsic scaling method and its application to the study of a class of degenerate parabolic equations I

*Abstract: In this talk I will give some insights about the intrinsic scaling method. This method was first introduced in the work of DiBenedetto and it is a powerful method in the analysis of the regularity of solutions for degenerate and singular PDEs (in divergence and nondivergence form).*

We will see how to combine this method with compactness arguments in order to study the regularity of viscosity solutions of quasilinear parabolic equations modeled on the p-Laplacian. - Tue Oct 23, 12:30-14:00 @MaA 203

Antti Luoto: Random walk approximation of backward stochastic differential equations - Tue Oct 16, 12:30-14:00 @MaA 203

Angel Arroyo : Alexandroff-Bakelman-Pucci type estimates II - Tue Oct 09, 12:30-14:00 @MaA 203

Diu Tran: Statistical inference for Vasicek-type model driven by Hermite processes

*Abstract: Let (Z*_{t}^{q, H})_{t ≥ 0}denote a Hermite process of order q ≥ 1 and self-similarity parameter H ∈ (1/2, 1). This process is H-self-similar, has stationary increments and exhibits long-range dependence. When q=1, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as q ≥ 2. In this paper, we deal with the following Vasicek-type model driven by Z^{q, H}:

X_{0}=0, dX_{t}= a(b - X_{t})dt +dZ_{t}^{q, H}, t ≥ 0,

where a > 0 and b ∈ R are considered as unknown drift parameters. We provide estimators for a and b based on continuous-time observations. For all possible values of H and q, we prove strong consistency and we analyze the asymptotic fluctuations. This is joint work with Ivan Nourdin, University of Luxembourg, Luxembourg. - Tue Oct 02, 12:30-14:00 @MaA 203

Angel Arroyo : Alexandroff-Bakelman-Pucci type estimates I - Tue Sep 25, 12:20-14:00 @MaA 204

Ivan Yaroslavtsev: Weak differential subordination of martingales and its applications in Harmonic Analysis

*Abstract: Differential subordination of real-valued martingales together with its basic properties have been discovered by Burkholder in 1984. In this talk we will discuss weak differential subordination of martingales, which is a natural generalization of differential subordination to the infinite dimensional setting, and provide extension of the corresponding L*^{p}estimates for vector-valued martingales. As a corollary we extend the results of Bañuelos and Bogdan (2007) and Bañuelos, Bielaszewski, and Bogdan (2011) on sharp estimates for the norms of a broad class of even Fourier multipliers (including e.g. second order Riesz transforms) to infinite dimensions. - Tue Sep 18, 12:30-14:00 @MaA 203

Vesa Julin: The Gaussian Isoperimetric Problem for Symmetric Sets - Tue Sep 11, 12:20-14:00 @MaA 203

Diu Tran: McKean Vlasov SDEs: Existence and Uniqueness

#### Spring 2018

- We May 16, 14:15-15:15 @MaD 302 (Analysis seminar) Juan J. Manfredi (University of Pittsburgh): Some random walks in the Heisenberg group
- Tu May 8, 14:15-16:00 @MaD 302 Céline Labart (Université de Savoie Mont Blanc): Mean reflected SDEs with jumps
- We Apr 25, 14:15-15:15 @MaD 302 (Analysis seminar) Sun-Sig Byun (Department of Mathematical Sciences, Seoul National University): Regularity estimates for nonlinear parabolic problems in nonsmooth domains
- Mo Apr 16, 14:15-16:00 @MaA 203 Jarkko Siltakoski: On the comparison principle for parabolic equations involving the p-Laplacian II

Stefan Geiss: On zero-sum stochastic differential games in domains IV - Mo Apr 09, 14:15-16:00 @MaA 203 Jarkko Siltakoski: On the comparison principle for parabolic equations involving the p-Laplacian
- Mo Mar 19, 14:15-16:00 @MaA 203 Ajay Jasra (National University of Singapore): Advanced Multilevel Monte Carlo Methods

*Abstract: This talk reviews the application of some advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance by using the discretization of an associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This talk considers the case where such exact sampling is not currently possible. We consider some Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods.* - Mo Mar 12, 14:15-16:00 @MaA 203 Stefan Geiss: On zero-sum stochastic differential games in domains III
- Mo Mar 05, 14:15-16:00 @MaA 203 Thuan Nguyen: A lower bound for the approximation error of certain stochastic integrals in Lévy setting

*Abstract: We establish a lower bound for the Riemann approximation error of a stochastic integral driven by a Lévy process. The problem is considered with respect to the bmo-norm which is a variant of classical BMO-norm and we show that for any time-net, the obtained lower bound is in the term of its mesh-size.* - Mo Feb 26, 14:15-16:00 @MaA 203 Stefan Geiss: On zero-sum stochastic differential games in domains II
- Mo Feb 19, 14:15-16:00 @MaA 203

Angel Arroyo: Regularity for Tug-of-War Games with Varying Probabilities IV

Stefan Geiss: On zero-sum stochastic differential games in domains I - Mo Feb 12, 14:15-16:00 @MaA 203 Angel Arroyo: Regularity for Tug-of-War Games with Varying Probabilities III
- Mo Feb 05, 14:15-16:00 @MaA 203 Angel Arroyo: Regularity for Tug-of-War Games with Varying Probabilities II
- Mo Jan 29, 14:15-16:00 @MaA 203 Angel Arroyo: Regularity for Tug-of-War Games with Varying Probabilities I
- Mo Jan 22, 14:15-16:00 @MaA 203 Christel Geiss: A comparison result for backward SDEs and measurable sections II
- Mo Jan 15, 14:15-16:00 @MaA 203 Christel Geiss: A comparison result for backward SDEs and measurable sections I

#### Previous seminars

More information in the old seminar page.