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30.07.2019 Shige Peng (Shandong University)

Title: Spatial and temporal white noises under sublinear G-expectation

Abstract: Under the framework of sublinear expectation, we introduce a new type of G-Gaussian random fields, which contains a type of spatial white noise as a special case. Based on this result, we also introduce a spatial-temporal G-white noise. Different from the case of linear expectation, in which the probability measure needs to be known, under the uncertainty of probability measures, spatial white noises are intrinsically different from temporal cases. Joint with Ji Xiaojun.

30.07.2019 Kai Du (Dudan University)

Title: On stochastic parabolicity conditions for SPDEs

Abstract: Stochastic parabolicity conditions are a class of structural conditions for stochastic PDEs to ensure wellposedness of the equations, which, in contrast to classical parabolicity conditions, involve additionally the coefficients of leading terms in the stochastic integral part. The proper form of stochastic parabolicity condition for weak solutions of SPDEs were found long ago and also valid in constructing L^p theory and Schauder theory of SPDEs of second-order. However, things may dramatically change for other problems, such as complex-valued SPDEs, systems of SPDEs, and higher-order SPDEs. More specifically, those conditions for weak solutions (L^2 theory) is not sufficient to ensure L^p-integrability of solutions, and certain modified conditions are required in those cases. The talk will present some p-dependent parabolicity conditions imposed on systems of SPDEs and on higher-order SPDEs, which ensure us to construct stochastic Schauder theory for those equations. Examples are discussed for necessity of modified parabolicity conditions and sharpness of our modifications. The talk is based on joint works with Jiakun Liu and Fu Zhang and with Yuxing Wang.

30.07.2019 Jing Zhang (Fudan University)

Title: Systems of Reflected Stochastic PDEs in a Convex Domain: Analytical Approach

Abstract : In this paper, we establish an existence and uniqueness result for the system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in $\mathbb{R}^K$ by analytical approach. The method is based on the approximation of the penalized systems of SPDEs.

31.07.2019 Horatio Boedihardjo (University of Reading)

Title: The signature of rough paths

Abstract: Rough path theory provides the existence and uniqueness of solutions for differential equations driven by a deterministic class of paths, which include Brownian sample paths. “Path signature” is an idea coming from the development of rough path theory; Given a path X, the solution to any differential equation driven by X can be described in terms of the signature of X. Despite the rapid developments in the study of signature in the past ten years, many difficult open problems remain. In this talk we will review the motivation and developments in the study of signature as well as the open problems in the field.

31.07.2019 Yufei Zhang (Oxford University)

Title: Deep neural network approximations to high-dimensional control and games

Abstract: In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks (DNN) for the solution of high-dimensional stochastic control problems and zero-sum games arising in financial engineering. In the first part, we show that in certain cases, including those with open loop controls, the value functions can be represented by a DNN whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy, demonstrating that DNNs can break the curse of dimensionality. In the second part, we exploit policy iteration to reduce the nonlinear problem into a sequence of linear problems, which are then further approximated via a multilayer feedforward neural network ansatz. We establish that in suitable settings the numerical solutions converge globally in the $H^2$-norm. Moreover, we construct the optimal feedback controls based on the superlinear convergence of the numerical solutions. Preliminary numerical experiments are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

31.07.2019 Wenqiang Li (Yantai University/Loughborough University)

Title: Stochastic differential games with long-run average payoff

Abstract: In this talk we consider a type of zero-sum stochastic differential games with long-run average payoff in which the diffusion term of the dynamics does not need to be non-degenerate. Under some suitable conditions, the existence of a viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation is obtained. With the help of this viscosity solution we show the existence of a value under the Isaacs condition. In addition, some representation formulae for this value are also given. This talk presents a common work with Juan Li (Shandong University, Weihai) and Huaizhong Zhao (Loughborough University).