Set-valued Ito's formula with an application to the general set-valued backward stochastic differential equation

TL;DR

This paper develops a set-valued Itô's formula and applies it to prove existence and uniqueness of solutions for a general set-valued backward stochastic differential equation, addressing an open question in the field.

Contribution

It introduces a set-valued Itô's formula and uses it to solve an open problem on set-valued backward stochastic differential equations.

Findings

Established a set-valued Itô's formula.

Proved existence and uniqueness of solutions for the general set-valued backward stochastic differential equation.

Provided an answer to an open question in the literature.

Abstract

The overarching goal of this paper is to establish a set-valued It\^{o}'s formula. As an application, we obtain the existence and uniqueness of solutions for the general set-valued backward stochastic differential equation which gives an answer to an open question proposed by Ararat et al. (C. Ararat, J. Ma and W.Q. Wu, Set-valued backward stochastic differential equation, arXiv:2007.15073).