Path-Regularity and Martingale Properties of Set-Valued Stochastic Integrals

TL;DR

This paper investigates the path-regularity and martingale properties of set-valued stochastic integrals, establishing conditions for martingale behavior and exploring implications for set-valued backward stochastic differential equations.

Contribution

It introduces new results on the existence of continuous modifications of set-valued submartingales and characterizes when set-valued stochastic integrals are true martingales.

Findings

Existence of right- and left-continuous modifications of set-valued submartingales.

Set-valued stochastic integral is a martingale iff the terminal set is closed and decomposable.

Conditional expectation of a convex random polytope preserves vertices iff the polytope has a deterministic normal fan.

Abstract

In this paper we study the path-regularity and martingale properties of the set-valued stochastic integrals defined in our previous work Ararat et al. (2023). Such integrals have some fundamental differences from the well-known Aumann-It\^{o} stochastic integrals, and are much better suitable for representing set-valued martingales, whence potentially useful in the study of set-valued backward stochastic differential equations. However, similar to the Aumann-It\^{o} integral, the new integral is only a set-valued submartingale in general, and there is very limited knowledge about the path regularity of the related indefinite integral, much less the sufficient conditions under which the integral is a true martingale. In this paper, we first establish the existence of right- and left-continuous modifications of set-valued submartingales in continuous time, and apply the results to set-valued stochastic integrals. Moreover, we show that a set-valued stochastic integral yields a martingale if and only if the set of terminal values of the stochastic integrals associated to the integrand is closed and decomposable. Finally, as a particular example, we study the set-valued martingale in the form of the conditional expectation of a set-valued random variable. We show that when the random variable is a convex random polytope, the conditional expectation of a vertex stays as a vertex of the set-valued conditional expectation if and only if the random polytope has a deterministic normal fan.