Stochastic Calculus Under Nonlinear Expectation And Related Topics

The dynamic characteristics of the modern world are becoming more and more obvious,and the interaction between various parts of the system is becoming more and more close.It is often necessary to deal with large amounts of random data,and the risks that may arise are immeasurable.At the beginning of the 21st century,a nonlinear expectation theory that can robustly analyze and quantify potential risks emerged.The representative theory is the G-expectation theory established by Peng.Due to the complexity and uncertainty of G-expectation,the analytical solutions of stochastic systems under the G-expectation are difficult to obtain.Therefore,it is of great theoretical significance and application value to study numerical methods for stochastic systems under the G-expectation.In this paper,we study the related topics of stochastic calculus under nonlinear expectation,especially the numerical methods of backward stochastic differential equations and fully nonlinear parabolic partial differential equations under the G-expectation.The research content of this paper consists of three parts.In the first part,we study the numerical method for solving the backward stochastic differential equation(G-BSDE)driven by G-Brownian motion.We first consider a kind of G-BSDE with linear structure that can describe the hedging price of a contingent claim with volatility uncertainty.Due to the nonlinear structure of G-expectation,the classical method is no longer adapt.To overcome it,we construct an auxiliary G-expectation space on which a θ-scheme for solving G-BSDE is proposed.We further prove that the proposed scheme has halforder convergence for solving Y in general.In particular,the scheme can achieve first-order convergence in some special cases.Moreover,we study the numerical method for solving the general forward and backward stochastic differential equation driven by G-Brownian motion(G-FBSDE).Inspired by the G-expectation representation theorem,by introducing an approximate G-expectation,a new kind of numerical scheme for solving G-FBSDE is proposed.We further prove that the proposed scheme for solving Y is at most half-order of convergence.To sum up,the above two types of numerical methods have no inclusion relation and have their advantages:in terms of applicability,the latter is suitable for solving general G-BSDE and G-FBSDE,which has a wider scope of application;in terms of convergence,the former can reach a higher order of convergence in some special cases,which has better convergence performance.In the second part,we will introduce the numerical analysis tools of nonlinear partial differential equation into the G-expectation theory to investigate the convergence rate of two kinds of discrete schemes of fully nonlinear partial differential equations.First,we discuss the discrete approximation scheme of a class of fully nonlinear second-order Hamilton-Jacobi-Bellman(HJB)equations,which corresponds to the discrete approximation scheme for stochastic optimal control problems under the G-expectation framework.Based on the piecewise constant policy,a recursive discrete scheme is proposed to solve this problem,and the convergence rate of the discrete scheme is obtained by means of the "shaking the coefficients" method.Secondly,we discuss the monotone scheme for a class of fully nonlinear partial integro-differential equations which characterize the nonlinear α-stable processes under sublinear expectation space with α ∈(1,2).By using the monotone scheme tool of the nonlinear partial differential equation,we establish the error bounds for the monotone approximation scheme.This in turn yields an explicit Berry-Esseen bound and convergence rate of the central limit theorem for α-stable random variables under sublinear expectation.In the third part,we study the averaging problem for a class of forward and backward stochastic differential equations driven by G-Brownian motion with rapidly oscillating coefficients,which relates to the singular perturbation problem of a kind of fully nonlinear partial differential equations.With the help of the nonlinear stochastic analysis techniques and viscosity solution methods,we prove that the limit distribution of the solution is the unique viscosity solution to an averaged partial differential equation.