Set-valued stochastic integrals in UMD spaces and applications

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 $$Y_t=\left(\hat{I}¾+{\int }_t^T{H}_{u}du+{\int }_{[0,t]}^{\mathcal{R}}Zd{W}_u\right)⊝{\int }_{[0,T]}^{\mathcal{R}}Zd{W}_{u}a.s.,t\in [0,T],$$ where the right-hand side, of this equation, represents the Hukuhara difference of two quantities containing revised set-valued stochastic integrals, $\hat{I}¾$ is a terminal set-valued function condition and $H$ is a set-valued function satisfying some suitable conditions.

35 pages