= Discounted wealth martingale integrability condition
An integrable <predictable> strategy whose discounted gains $\phi_n\Delta\widetilde S_n$ are integrable preserves the <martingale> property of discounted prices in finite discrete time. Bounded holdings suffice. Without integrability the claim is false: let $\widetilde S_0=3/2$, $\widetilde S_1=1+U$, $\widetilde S_2=1+U+\varepsilon/2$, where $U$ is uniform on $(0,1)$ and $\varepsilon$ 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 $\varepsilon/(2U)$, which is not integrable. Replication pricing requires the corresponding true-martingale condition, not merely formal self-financing.
Back to article page
