Set-valued stochastic integrals in UMD spaces and applications

TL;DR

This paper develops set-valued stochastic integrals in UMD Banach spaces, extends martingale representation to set-valued martingales, and proves existence of solutions for set-valued backward stochastic differential equations.

Contribution

It introduces a compatible form of stochastic integrals for set-valued martingales and establishes existence results for set-valued BSDEs in UMD spaces.

Findings

Set-valued stochastic integrals are constructed in UMD spaces.

Martingale representation theorem is extended to set-valued martingales.

Existence of solutions for set-valued backward stochastic differential equations is proven.

Abstract

The purpose of this paper is to study certain set-valued integrals in UMD Banach spaces and provide a compatible form of the martingale representation theorem for set-valued martingales. Under specific conditions, these martingales can be expressed using revised set-valued stochastic integrals with respect to a real standard Brownian motion $W = (W_t)_{t\in[0,T ]}$. Moreover, we prove the existence of solutions to the following set-valued backward stochastic differential equation of the form $$ Y_t=\left(\xi+\int_t^TH_u du+\int_{[0,t]}^{\mathscr{R}}{Z}\, dW_u\right) \circleddash \int_{[0,T]}^{\mathscr{R}}{Z}\, dW_u\quad a.s.,\quad t\in [0,T], $$ where the right-hand side, of this equation, represents the Hukuhara difference of two quantities containing revised set-valued stochastic integrals, $\xi$ is a terminal set-valued function condition and $H$ is a set-valued function satisfying some suitable conditions.