Backward Stochastic Differential Equations (BSDEs) Using Infinite-Dimensional Martingales with Subdifferential Operator

Abstract

In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with subdifferential operators that are driven by infinite-dimensional martingales. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence and uniqueness of the solution are established using Yosida approximations. Furthermore, as an application of the main result, we shall show that the backward stochastic partial differential equation driven by infinite-dimensional martingales with a continuous linear operator has a unique solution under the special condition that the Ft-progressively measurable generator F of the model we proposed in this paper equals zero.

Mean-Field and Anticipated BSDEs with Time-Delayed Generator

“…where ξ is the terminal value, f is the generator related to the present time, and Bs is a standard Brownian process. After the already mentioned celebrated work of Pardoux and Peng, the interest in BSDEs has increased, mainly due to the connection of these tools with stochastic control and PDEs, a connection that will be stated clearly soon, for example, various BSDEs models and the uniqueness and existence of the solutions to these models (Bahlali et al [2]; Abdelhadiet al [3] Zhang et al [4]), the numerical solution of BSDEs (Ma et al [5]; Gobet et al [6]; Zhao et al [7]), the relationship between BSDEs and partial differential equations (PDEs) (Ren and Xia [8]; Pardoux and Rȃşcanu [9]), and the numerous applications of BSDEs in various areas including optimal control, finance, biology, and physics (for examples, refer to [11,12,10]).…”

Anticipated BSDEs Driven by Fractional Brownian Motion with a Time-Delayed Generator

“…Interest in BSDEs has increased since the work of Pardoux and Peng, primarily because of the correlation between BSDEs and stochastic control and partial differential equations (PDEs). Several publications have extensively explored these topics including research that establishes the existence and uniqueness of BSDEs under weaker conditions (He [2]; Abdelhadiet al [3]; Zhang et al [4]). Additionally, there have been studies that establish the connection between BSDEs and quasilinear parabolic PDEs (Pardoux and Rȃşcanu [5]; Ren and Xia [6]).…”