Set-valued stochastic integrals in UMD spaces and applications

2024/12/09 by El Hassan Essaky, Essaky, E. H., M. Hassani +3
Economics, Econometrics and Finance · #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2412.07001

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∈[0,T ]}. Moreover, we prove the existence of solutions to the following set-valued backward stochastic differential equation of the form Y_t=(ξ+∫_t^TH_u du+∫_[0,t]^\mathscrRZ dW_u) \circleddash ∫_[0,T]^\mathscrRZ dW_u a.s., t∈ [0,T], where the right-hand side, of this equation, represents the Hukuhara difference of two quantities containing revised set-valued stochastic integrals, ξ is a terminal set-valued function condition and H is a set-valued function satisfying some suitable conditions.

Related

• Set-Valued Backward Stochastic Differential Equations
• Path-Regularity and Martingale Properties of Set-Valued Stochastic Integrals
• Set-valued Ito's formula with an application to the general set-valued backward stochastic differential equation
• Backward Stochastic Evolution Inclusions in UMD Banach Spaces
• A representation theorem for set-valued submartingales