A new construction of the σ-finite measures associated with submartingales of class (Σ)

Abstract

In a previous paper, we proved that for any submartingale $(X_t){t \geq 0}$ 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, one can construct 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 $g$ Show more