Backward Stochastic Evolution Inclusions in UMD Banach Spaces

TL;DR

This paper establishes the existence of solutions for backward stochastic evolution inclusions in UMD Banach spaces, extending stochastic analysis to more general infinite-dimensional settings with set-valued dynamics.

Contribution

It proves the existence of mild solutions for backward stochastic evolution inclusions in UMD Banach spaces, including cases with martingale type 2 spaces, advancing stochastic differential inclusions theory.

Findings

Existence of mild solutions in UMD Banach spaces.

Extension to spaces with martingale type 2.

Applicable to set-valued stochastic dynamics.

Abstract

In this paper, we prove the existence of a mild $L^p$-solution for the backward stochastic evolution inclusion (BSEI for short) of the form \begin{align*}%\label{BSDI3} \begin{cases} dY_t+AY_tdt\in G(t,Y_t,Z_t)dt+Z_tdW_t,\quad t\in [0,T] Y_T =\xi, \end{cases} \end{align*} where $W=(W_t)_{t\in [0,T]}$ is a standard Brownian motion, $A$ is the generator of a $C_0$-semigroup on a UMD Banach space $E$, $\xi$ is a terminal condition from $L^p(\Omega,\mathscr{F}_T;E)$, with $p>1$ and $G$ is a set-valued function satisfying some suitable conditions. The case when the processes with values in spaces that have martingale type $2$, has been also studied.