= Snell envelope of a multiplicative process
{c}
{title2=$U_t=Y_tc_t$}
Let $Y_t=Y_0\prod_{j=1}^tZ_j$, where $Y_0>0$ is deterministic and the positive factors are independent and integrable. Use the natural filtration. Its finite-horizon <Snell envelope> is
$$
U_t=Y_tc_t,\qquad c_T=1,\qquad c_t=\max\{1,c_{t+1}\mathbb E Z_{t+1}\}.
$$
Indeed, backward induction and <independence of random variables> give $\mathbb E[U_{t+1}\mid\mathcal F_t]=Y_tc_{t+1}\mathbb E Z_{t+1}$; 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.
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