Snell envelope of a multiplicative process ($U_t=Y_t{c}_t$)

Let $Y_t=Y_0{\prod }_{j=1}^t{Z}_j$, where $Y_0>0$ is deterministic and the positive factors are independent and integrable. Use the natural filtration. Its finite-horizon Snell envelope isIndeed, backward induction and independence of random variables give $\mathbb{E}[U_{t+1}\mid {\mathcal{F}}_t]=Y_t{c}_{t+1}\mathbb{E}{Z}_{t+1}$; take the maximum with immediate exercise $Y_t$. The constants are finite and at least one. Equivalently, $c_t$ is the maximum of $1$ and the successive products ${\prod }_{j=t+1}^u\mathbb{E}{Z}_j$ for $t<u\le T$. If the factors have a common mean $\lambda >1$, waiting until maturity is optimal and $c_t={\lambda }^{T-t}$. Integrability and independence of random variables from the current information are essential to the finite deterministic coefficient conclusion.