Backward stochastic differential equations with a uniformly continuous generator and related g-expectation

Abstract

In this paper, we will study a class of backward stochastic differential equations (BSDEs for short), for which the generator (coefficient) g(t, y, z) is Lipschitz continuous with respect to y and uniformly continuous with respect to z. We establish several properties for such BSDEs, including comparison and converse comparison theorems, a representation theorem for g and a continuous dependence theorem. Then we introduce a new class of g-expectation based on such backward stochastic differential equations, and discuss its properties.

Doubly Reflected Backward Stochastic Differential Equations Driven by G -Brownian Motion With a Kind of Non-Lipschitz Coefficients

“…Since then the theory of BSDEs and reflected BSDEs has made a rapid development because it has wide applications, for example, in stochastic control [28], mathematical finance [8,15] and PDEs [2,7,27,29]. Among others, extensive effort has been made in weakening the conditions on the generator [4,9,14,16,17,18,25,36].…”

A representation for filtration-consistent nonlinear expectations and its application

“…The theory of g-expectations has been widely applied in asset pricing, utility theory and risk measures (see [11], [19], [28], [31] and the references therein). To deal with more general cases, the notion of g-expectations was generalized by Jia [18] in the uniformly continuous case, and by Ma and Yao [22] in the quadratic growth case. One of objectives of this paper is to further develop the theory of g-expectations.…”

“…Another important property of BSDEs is the invariant representation theorem for generators, which is a powerful tool to study the generators from the solutions of BSDEs. In order to study the converse comparison problem for BSDEs, the invariant representation theorem was firstly proved in Briand et al [4], and then by generalized by [13], [18], [19] and [33], etc. To our best knowledge, all of these studies rely on some proper estimates on the solutions, which need assumptions on z.…”