Showing papers by "Shige Peng published in 2009"

Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations

Abstract: This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Ito’s type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.

Mean-field backward stochastic differential equations : a limit approach

Abstract: Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

On representation theorem of G -expectations and paths of G -Brownian motion

Abstract: We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ: θ ∈ gJ} representing an important sublinear expectation— G-expectation \( \mathbb{E} \)[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cylinder functions Lip(Ω) in order to prove that Cb(Ω) belongs to the completion of Lip(Ω) under the natural norm \( \mathbb{E} \)[| · |].

Anticipated backward stochastic differential equations

Abstract: In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.

Stopping Times and Related It\^o's Calculus with G-Brownian Motion

Abstract: Under the framework of G-expectation and G-Brownian motion, we introduce Ito's integral for stochastic processes without assuming quasi-continuity. Then we can obtain Ito's integral on stopping time interval. This new formulation permits us to obtain Ito's formula for a general C^{1,2}-function, which essentially generalizes the previous results of Peng [20, 21, 22, 23, 24] as well as those of Gao [8] and Zhang et al. [26].

BSDEs with random default time and their applications to default risk

Abstract: In this paper we are concerned with backward stochastic differential equations with random default time and their applications to default risk. The equations are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. We show that these equations have unique solutions and a comparison theorem for their solutions. As an application, we get a saddle-point strategy for the related zero-sum stochastic differential game problem.

BSDEs with random default time and their applications to default risk

Abstract: In this paper we are concerned with backward stochastic differential equations with random default time and their applications to default risk. The equations are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. We show that these equations have unique solutions and a comparison theorem for their solutions. As an application, we get a saddle-point strategy for the related zero-sum stochastic differential game problem.