International Seminar on SDEs and Related Topics


    This online seminar takes place every second Friday at
   12:30 UTC during European daylight saving times        11:30 UTC otherwise    

  12:30 noon       1:30 pm       2:30 pm       8:30 pm (7:30 pm from Apr 08 on)    
    London        Berlin, Paris         Helsinki         Beijing    

                    Zoom link       Meeting ID: 618 9100 7917

No registration required. To get an e-mail reminder before each event write to sde-seminar[at]jyu.fi.


Organisers

  • Stefan Ankirchner   (FSU Jena, Germany)
  • Christian Bender   (Saarland University, Germany)
  • Rainer Buckdahn   (Universite de Bretagne Occidentale, France)
  • Dan Crisan   (Imperial College London, UK)
  • Christel Geiss   (University of Jyväskylä, Finland)
  • Stefan Geiss   (University of Jyväskylä, Finland)
  • Céline Labart   (Université Savoie Mont-Blanc, France)
  • Juan Li   (Shandong University, China)
  • Andreas Neuenkirch   (University of Mannheim, Germany)
  • Shige Peng   (Shandong University, China)
  • Adrien Richou   (University of Bordeaux, France)

Schedule 2022 Autumn


Oct 28, 2022

David Nualart

Nov 11, 2022

Yaozhong Hu

Nov 25, 2022

Ying Jiao

Dec 09, 2022

TBA



Schedule 2023 Spring


Jan 13, 2023

Jan 27, 2023

Jean-François Chassagneux

Feb 10, 2023

Feb 24, 2023

Mar 10, 2023

Mar 24, 2023

Apr 14, 2023

Apr 21, 2023

May 05, 2023

May 19, 2023



Schedule 2022 Spring


Jan 14, 2022

Arturo Kohatsu-Higa                        Poster:     Slides
(Ritsumeikan University, Kusatsu):

Acceleration of convergence and regularity of the law for jump processes

Abstract: Recently, I have been interested in how acceleration of approximation methods can be used in order to derive theoretical properties about the laws of their limits. In most situations, numerical approximations do not have good properties to assure that their properties transfer to their limit laws. In this talk, I will give two examples with different acceleration techniques. In the first, we will consider a jump driven sde with purely atomic Lévy measure with infinite activity. We will apply the Asmussen-Rosinki approximation for small jumps of infinite activity Lévy processes and argue that the use of this approximation in order to derive properties of the limits is not possible. Instead, we will combine this method with the moment method in order to improve the approximation and finally obtain results on the regularity of the laws of stochastic differential equations with jumps. This approach is interesting as it only uses Malliavin calculus with respect to the Brownian motion associated to the Asmussen-Rosinki approach. This is joint work with V. Bally and L. Caramellino.
In a second part, we will discuss how to obtain optimal upper bounds for the joint density of a stable process and its supremum using Multi-level Monte Carlo method techniques. We build an ad-hoc Malliavin Calculus method based on the Chamber-Mallows-Struck/Kanter simulation approach for stable increments.
This is joint work with Jorge Gonzales-Cazares and Alex Mijatovic.


Jan 28, 2022

Denis Talay                        Poster:     Slides
(INRIA & École Polytechnique):


First hitting time Laplace transforms of solutions of SDEs are Lipschitz continuous in the Hurst parameter of the driving fractional Brownian noise

Abstract: Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this talk we consider solutions {X​tH} t ∈ R+ to stochastic differential equations driven by fractional Brownian motions. We examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of X H at a given threshold. We will present parts of the machinery necessary to prove the Lipschitz continuity w.r.t. H around the value ½. This result implies that, for applications where first hitting times are a crucial information, the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to ½.
This is joint work with Alexandre Richard (Ecole Centrale-Supelec, France).


Feb 11, 2022

Mireille Bossy                             Poster:     Slides
(INRIA, Université Côte d'Azur, France):


Weak convergence rate approximation for some SDEs with superlinear growth coefficients

Abstract: In this talk, I would like to discuss around the approximation of the solution of the following class of 1D-SDEs
                                     dXt = b(Xt )dt + σ Xt α dWt ,     X0 =x>0,
when α >1. I will first present some motivating examples where such SDEs arise in modelling approaches, as long-time limit approximation of McKean SDEs of CIR-type for the amplitude of local wind fluctuations. Of course, a priori knowledge on such SDEs about the conditions on the coefficients ensuring well-posedness and some control on the moments is required, not only for their use as models but also to study the convergence of time-integration schemes.
Here, we focus on the weak convergence rate. First, a set of conditions for the C1,4 regularity of the Feynman-Kac formula is proposed using stochastic tools. While direct derivation of the Feynman-Kac formula requires some smoothness for the drift b, we show how to avoid the costly control of the moments of the successive derivatives of the flow process by using a change of measure technique, allowing b to be taken superlinear too. We then introduce an exponential scheme for the time integration of the SDE, which reproduces well the control of the moments of the exact process and for which we prove a convergence rate of order one.
This talk is based on two recent papers with Kerlyns Martínez Rodríguez (University of Vaparaíso) and Jean Francois Jabir (HSE University Moscow).


Feb 25, 2022 @ 12:30 UTC, one hour later than usual!

Xin Guo                             Poster:     Slides
(University of California at Berkeley):


Generative adversarial models: an analytical perspective

Abstract: Recently, the popularity and successes of Generative Adversarial Networks (GANs) in computer vision and image generation have attracted intense attention from the mathematical finance community. GANs have since then been applied to financial data generation and lately shown capable of computing solutions for high dimensional mean-field games.
In this talk, we will discuss the connection of GANs with optimal transport, game structure of GANs in an SDE framework, and present our latest work on the stochastic control approach for the stability of GANs training.
Based on joint work with Haoyang Cao of Ecole Polytechnique, and Othmane Mounjid of UC Berkeley.


Mar 11, 2022

Lukasz Szpruch                             Poster:     Slides
(University of Edinburgh):


Gradient Flows for Regularized Stochastic Control Problems

Abstract: We study stochastic control problems regularized by the relative entropy, where the action space is the space of measures. By exploiting the Pontryagin optimality principle, we identify a suitable metric space on which we construct the gradient flow for the measure-valued control process along which the cost functional is guaranteed to decrease.
It is shown that under appropriate conditions, this gradient flow has an invariant measure which is the optimal control for the regularized stochastic control problem. If the problem we work with is sufficiently convex, the gradient flow converges exponentially fast. Furthermore, the optimal measure valued control admits Bayesian interpretation which means that one can incorporate prior knowledge when solving the stochastic control problem.
This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used in the reinforcement learning community to solve control problems.


Mar 25, 2022 @ 12:30 UTC, one hour later than usual!

Jianfeng Zhang                             Poster:     SlidesVideo   (expires August 2022)
(University of Southern California):


Propagation of Monotonicity for Mean Field Game Master Equations

Abstract: It is well known in the mean field game literature that a certain monotonicity condition is crucial for the uniqueness of mean field equilibria and for the wellposedness of the associated master equation. One interesting observation is that the propagation of the monotonicity (either in Lasry-Lions sense or in displacement sense) of the value function plays the key role here. We shall introduce a method to find conditions on the coefficients which ensure that any solution of the master equation will maintain the monotonicity property. This method also allows us to consider anti-monotonicity and obtain the desired wellposedness provided the coefficients are sufficient anti-monotone in appropriate sense. We finally extend our results to mean field games of controls. The talk is based on a joint work with Gangbo-Meszaros-Mou and two joint works with Mou.


Apr 08, 2022

Krzysztof Bogdan                             Poster:     SlidesVideo   (expires September 2022)
(Wrocław University of Science and Technology):


Self-similar solution for Hardy operator

Abstract: We will discuss the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup. This will be our focus. Interestingly, an Ornstein-Uhlenbeck semigroup turns out to be an important tool for the analysis. The paper is on arXiv and it is joint work with P. Kim (Seoul), T. Jakubowski, and D. Pilarczyk (Wrocław).


Apr 22, 2022

Étienne Pardoux                             Poster:     SlidesVideo   (expires September 2022)

(Aix-Marseille Université):


Epidemic models with varying infectivity and varying susceptibility

Abstract: Almost a century ago, Kermack and McKendrick suggested to take into account the fact that the infectivity of infectious individuals vary with the time elapsed since their infection, the duration of the infectious period can have a very general distribution, and in case of loss of immunity, it is progressive.
Unfortunately, almost all the literature on mathematical models of epidemics concentrate on simpler ODE models, which are the law of large numbers limits, as the size of the population tends to infinity, of finite population stochastic Markovian models.
We consider the models suggested by Kermack and McKendrick, and obtain those models (or in one case a generalization of their model), which are convolution type equations, as law of large numbers limits of general non Markov stochastic finite population models. We also derive some associated Central Limit Theorems. Our approach involves the study of a class of mean field Poisson driven SDEs.
This is joint work with Guodong Pang, Raphaël Forien and Arsene Brice Zotsa-Ngoufack.



May 06, 2022

François Delarue                             Poster:     SlidesVideo   (expires Oktober 2022)

(Université Côte d’Azur, Nice):


Weak solutions to the master equation of potential mean field games

Abstract: The talk is motivated by the theory of mean field games, initiated by Lasry and Lions and by Caines, Huang and Malhamé. The very purpose of it is to address the so-called master equation, which describes the value of the game, when equilibria may not be unique. In order to do so, we restrict ourselves to a class of mean field games that coincide with the first order condition of an optimal control problem set over McKean-Vlasov dynamics. Such games are called potential. We then introduce a notion of weak solution to the master equation and prove that existence and uniqueness hold under quite general assumptions. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. This a joint work with Alekos Cecchin (Padova, Italy), see arXiv 2204.04315



May 20, 2022

Ying Hu                             Poster:     SlidesVideo   (expires November 2022)

(Université de Rennes 1, CNRS):

Scalar valued BSDEs with sublinear, superlinear and subquadratic growth

Abstract: The aim of this talk is to give some well-posedness results for scalar valued Backward Stochastic Differential Equations (BSDEs) when the generator has a sublinear growth, superlinear growth, subquadratic growth in the second variable. In each of these cases, we give some precise conditions for terminal random variables to guarantee the existence and uniqueness of the solution. Joint works with Shengjun Fan and Shanjian Tang.







Schedule 2021


Oct 29, 2021

Peter Friz (TU Berlin and WIAS Berlin):           Poster:          

On rough SDEs

Abstract: A hybrid theory of rough stochastic analysis is built that seamlessly combines the advantages of both Itô's stochastic - and Lyons' rough differential equations. A major role is played by a new stochastic variant of controlled rough paths spaces, with norms that reflect some generalized stochastic sewing lemma, and which may prove useful whenever rough paths and Itô integration meet. We will mentioned several applications. Joint work with Antoine Hocquet and Khoa Lê (both TU Berlin).


Nov 12, 2021

Xunyu Zhou (Columbia University, New York):           Poster:          

Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space:
A Martingale Lens

Abstract: We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. Mathematically, PE is to devise a data-driven Feynman--Kac formula without knowing any coefficients of a PDE. We show that this problem is equivalent to maintaining the martingale condition of a process. From this perspective, we present two methods for designing PE algorithms. The first one, using a "martingale loss function", interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions". Solving these equations in different ways recovers various classical TD algorithms, such as TD, LSTD, and GTD. We apply these results to option pricing and portfolio selection. This is joint work with Yanwei Jia.


Nov 26, 2021

Michael Scheutzow             Poster:     Slides
(Technische Universität Berlin):

Generalized couplings and stochastic functional differential equations

Abstract: We provide an introduction to generalized couplings and present a recent result [contained in S.: Couplings, generalized couplings and uniqueness of invariant measures. ECP, 2020] which says that the existence of a generalized coupling for a Markov chain implies uniqueness of an invariant probability measure even if the state space is just a metric space without requiring separability or completeness as in previous works. The proof turns out to be rather elementary. We show how this result can be applied to show uniqueness of an invariant measure for the solution process of a stochastic functional differential equation (SFDE) and we show how generalized couplings can be employed to show weak uniqueness of solutions of an SFDE with Hölder continuous coefficients.
Parts of the talk are based on joint work with Alex Kulik (Wroclaw).


Dec 10, 2021

Nizar Touzi                 Poster:     Slides
(CMAP & Polytechnique Paris):

Entropic mean field optimal planning

Abstract: The problem of optimal planning was introduced by P.-L. Lions in the context of a mean field game, by fixing a target distribution in the Focker-Planck equation and relaxing the boundary condition in the HJB equation. We analyze an extension of this problem to the path-dependent setting which has remarkable connections with optimal transport and optimal incentive theory in economics. We provide a general characterization of mean field optimal planning solutions, and we solve explicitly the minimum entropy optimal planning problem.


Contact and links

    Stefan Geiss: stefan.geiss[at]jyu.fi or stefanfriedrich.geiss[at]gmail.com
    The online seminar is hosted by the Department of Mathematics and Statistics (University of Jyväskylä) and supported by the journal Probability, Uncertainty and Quantitative Risk.