Optional Sampling Theorem for Deformed Submartingales

Abstract

The family ${{\bf Q}}=(Q^{(n)}, {\cal F}_{n})_{n=0}^{\infty}$ of probability measures $Q^{(n)}$ defined on ${{\cal F}}_{n}$ is considered. It is called a deformation of the first kind if for all $n\in \{0,1,2,\dots\}$, $Q^{(n+1)}|_{{{\cal F}}_{n}}\ll Q^{(n)}$, and a deformation of the second kind if $Q^{(n+1)}|_{{{\cal F}}_{n}}\gg Q^{(n)}$. For finite stopping times $\tau$ measures $Q^{(\tau)}$ are introduced by the formula $Q^{(\tau)}(A)=\sum_{i=0}^\infty Q^{(i)}(A\{\tau=i\})$, where $A\in{\cal F}_{\tau}$. With the help of these measures a generalization of Doob's optional sampling theorem is formulated and proved for deformed submartingales of the first kind in the case of neighboring stopping times, and of the second kind in the case of boundedly spaced stopping times.