= Multiplicative correction for summable adapted drift
{title2=$Z_n=X_n/\prod_{j<n}(1+Y_j)$}
Suppose $X_n\geq0$ is an <integrable> <adapted process>, $Y_n\geq0$ is finite and adapted, and $\mathbb E[X_{n+1}\mid\mathcal F_n]\leq(1+Y_n)X_n$. The displayed correction is a nonnegative <supermartingale>: its next denominator is <measurable> at time $n$, and its reciprocal is at most one. If $\sum_nY_n<\infty$ <almost surely>, the denominators converge to a finite positive limit since $\log(1+y)\leq y$. The <almost sure supermartingale convergence theorem> then gives finite <almost sure convergence> of $X_n$. No <integrability> of the infinite product or of $\sum_nY_n$ is needed.
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