Special Session 17: New developments on nonlinear expectations

Sequential propagation of chaos: theory and algorithms

Kai Du Fudan University Peoples Rep of China Co-Author(s):

Abstract: A new class of particle systems with sequential interaction is proposed to approximate the McKean-Vlasov process that originally arises as the limit of the mean-field interacting particle system. The weighted empirical measure of this particle system is proved to converge to the law of the McKean-Vlasov process as the system grows. Numerical experiments are implemented to demonstrate the theoretical results.

Upper and lower covariance under sublinear expectation

Xinpeng Li Shandong University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we define the upper (resp. lower) covariance under sublinear expectation via a corresponding max-min-max (resp. min-max-min) problem and the related properties of covariances are obtained. In particular, we propose a fast algorithm of calculation for upper and lower covariances under a finite number of probabilities. As an application, our algorithm can be used to solve a class of quadratic programming problems which is an NP-hard problem in some cases, and we obtain a probabilistic representation of such quadratic programming problem.

Maximum principle for recursive optimal control problem of stochastic delayed evolution equations

Guomin Liu Nankai University Peoples Rep of China Co-Author(s):    Jian Song, Meng Wang

Abstract: For a class of stochastic delay evolution equations driven by cylindrical $Q$-Wiener process, we study the Pontryagin`s maximum principle for the stochastic recursive optimal control problem. The delays are given as moving averages with respect to general finite measures and appear in all the coefficients of the control system. In particular, the final cost can contain the state delay. To derive the maximum principle, we introduce a new form of anticipated backward stochastic evolution equations with a terminal acting on an interval as adjoint equations of delay state equations, and deploy a proper dual analysis between them. Under certain convex assumptions, we also show the sufficiency of the maximum principle.

Nonlinear expectation algorithm in machine learning

Shige Peng Shandong University Peoples Rep of China Co-Author(s):

Abstract: Most of machine learning algorithms are directly related to the concept of expectations. In particular, the analysis and calculation of the expected value of the loss functions or utility functions directly and deeply involve the optimization problem. But in most cases, people often make a very heavy assumption about the data samples, that is, the samples are required to be i.i.d., but it is well-known that the real world`s situation is often far from these conditions. In fact, this problem has become a well-known open and major issue in the field of machine learning. In this talk, we give more realistic assumptions: we only need to assume that the data sample is ``nonlinear iid``, e.g., iid under nonlinear expectations. Based on this more relaxed condition we obtain a very robust algorithm. The theoretical foundation is the law of large numbers and central limit theorem in the framework of sublinear expectations.

Quadratic Mean-Field Reflected BSDEs

Falei Wang Shandong University Peoples Rep of China Co-Author(s):

Abstract: In this paper, we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown z. Using a linearization technique and the BMO martingale theory, we first apply a fixed-point argument to establish the uniqueness and existence result for the case with bounded terminal condition and obstacle. Then, we develop a successive approximation procedure to remove the boundedness condition on the terminal condition and obstacle when the generator is concave (or convex) with respect to the second unknown. In a similar way, we also considermean reflected backward stochastic differential equations.Based on a joint work with Y. Hu and R. Moreau.

Value at risk model under sublinear expectation

Shuzhen Yang Shandong University Peoples Rep of China Co-Author(s):    Shige Peng

Abstract: In this paper, we review the value at risk (VaR) model under sublinear expectation. We first consider the classical VaR model, and then introduce the basic concepts under sublinear expectation. Based on sublinear expectation, we show the definition of the VaR under model uncertainty, which is called G-VaR. Furthermore, we present three methods for estimating the parameters of the G-VaR model. Those are the long-time average method, the first-order autoregressive method, and the adapted learning method. In the end, we use S&P500 index to verify the performance of G-VaR model.

A rough path approach to robust filtering

Huilin Zhang Shandong University Peoples Rep of China Co-Author(s):    Peter K. Friz, Khoa Le

Abstract: In 1978, J. M. C. Clark introduced the idea that the solution of the stochastic filtering problem should be naturally continuous in the observed signal. Such related theory is known as the robust filtering. In this talk, I would like to show the robust filtering by the rough path theory to the generality of related and non-Markovian case. Moreover, we show that the optional filter can be approximated by a discrete rough Euler scheme, and the optional convergence rate is obtained. This talk is based on an ongoing work with Peter K. Friz and Khoa Le.