A representation theorem for set-valued submartingales

TL;DR

This work addresses the problem of representing set-valued submartingales in a stochastic-integral framework, extending the classical martingale representation to non-degenerate set-valued processes and non-zero initial conditions. The authors establish a general representation theorem: under mild assumptions, a set-valued submartingale $F$ admits a generalized stochastic integral representation $F_t = ∫_0^t 𝒢_s \, dB_s$ with a suitable selection process $𝒢$, and they further extend the theory to an expanded space $\\mathscr{L}_{\mathbb F}^2 = 𝔯^d × L^2$ to handle non-trivial initial sets. The construction relies on Castaing representations, decomposability of selections, and a standard martingale representation applied to countable martingale selectors to derive $𝒢$, thereby unifying and broadening prior results (e.g., Kisielewicz 2014; Zhang 2020). The framework generalizes the degenerate singleton case and yields new representations for non-degenerate intervals and other non-singleton initial set-valued martingales, with potential applications in set-valued finance, economics, and control theory.

Abstract

The integral representation theorem for martingales has been widely used in probability theory. In this work, we propose and prove a general representation theorem for a class of set-valued submartingales. We also extend the stochastic integral representation for non-trivial initial set-valued martingales. Moreover, we show that this result covers the existing ones in the literature for both degenerated and non-degenerated set-valued martingales.

Theorems & Definitions (17)

• theorem 2.1
• theorem 2.2
• theorem 2.3
• remark 2.1
• theorem 2.4
• remark 2.2
• lemma 3.1
• proof
• theorem 3.1: Set-valued submartingale representation theorem
• proof