Multiplicative correction for summable adapted drift ($Z_n=X_n/{\prod }_{j<n}(1+Y_j)$)

Suppose $X_n\ge 0$ is an integrable adapted process, $Y_n\ge 0$ is finite and adapted, and $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]\le (1+Y_n)X_n$. The displayed correction is a nonnegative supermartingale: its next denominator is measurable at time $n$, and its reciprocal is at most one. If ${\sum }_n{Y}_n<\infty$ almost surely, the denominators converge to a finite positive limit since $\log (1+y)\le y$. The almost sure supermartingale convergence theorem then gives finite almost sure convergence of $X_n$. No integrability of the infinite product or of ${\sum }_n{Y}_n$ is needed.