 Discounted wealth martingale integrability condition

An integrable predictable strategy whose discounted gains ${\varphi }_n\Delta {\tilde{S}}_n$ are integrable preserves the martingale property of discounted prices in finite discrete time. Bounded holdings suffice. Without integrability the claim is false: let ${\tilde{S}}_0=3/2$, ${\tilde{S}}_1=1+U$, ${\tilde{S}}_2=1+U+\epsilon /2$, where $U$ is uniform on $(0,1)$ and $\epsilon$ is an independent fair sign. Holding zero stock first and $1/U$ shares in the second period, funded by the bank, gives zero-initial-cost terminal wealth $\epsilon /(2U)$, which is not integrable. Replication pricing requires the corresponding true-martingale condition, not merely formal self-financing.