Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator

This paper discusses a new type of anticipated backward stochastic differential equation with a time-delayed generator (DABSDEs, for short) driven by fractional Brownian motion, also known as fractional BSDEs, with Hurst parameter $H\in(1/2,1)$, which extends the results of the anticipated backward stochastic differential equation to the case of the drive is fractional Brownian motion instead of a standard Brownian motion and in which the generator considers not only the present and future times but also the past time. By using the fixed point theorem, we will demonstrate the existence and uniqueness of the solutions to these equations. Moreover, we shall establish a comparison theorem for the solutions

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

In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with sub-differential operators that are driven by infinite-dimensional martingales which involve symmetry, that is, the process involves a positive definite nuclear operator Q. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence of the solution is established using Yosida approximations, and the uniqueness is proved using Fixed Point Theorem. 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 condition that the function F equals to zero

Optimal distributed control of a stochastic Cahn-Hilliard equation

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase separation, optimal control, linearized state system, adjoint state system, first-order optimality condition