On some universal sigma finite measures and some extensions of Doob's optional stopping theorem

TL;DR

This paper constructs a universal sigma-finite measure for certain submartingales, extending Doob's optional stopping theorem, with applications in Brownian penalization and mathematical finance, solving a problem posed by Madan, Roynette, and Yor.

Contribution

It introduces a general sigma-finite measure associated with submartingales of class (Σ), extending existing measures and generalizing Doob's optional stopping theorem to stopping times.

Findings

Defined a universal measure for class (Σ) submartingales.

Extended the optional stopping theorem to bounded stopping times.

Connected the measure to applications in finance and Brownian penalization.

Abstract

In this paper, we associate, to any submartingale of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. This measure $\mathcal{Q}$ has already been defined in several particular cases, some of them are involved in the study of Brownian penalisation, and others are related with problems in mathematical finance. More precisely, the existence of $\mathcal{Q}$ in the general case solves a problem stated by D. Madan, B. Roynette and M. Yor, in a paper studying the link between Black-Scholes formula and last passage times of certain submartingales. Moreover, the equality defining $\mathcal{Q}$ remains true if one replaces the fixed time $t$ by any bounded stopping time. This generalization can be viewed as an extension of Doob's optional stopping theorem.