The owner of a put option may exercise it on any one of the days $1, \ldots, h$, or not at all. If he exercises it on day $t$, when the share price is $x_{t}$, his profit will be $p-x_{t}$. Suppose the share price obeys $x_{t+1}=x_{t}+\epsilon_{t}$, where $\epsilon_{1}, \epsilon_{2}, \ldots$ are i.i.d. random variables for which $E\left|\epsilon_{t}\right|<\infty$. Let $F_{s}(x)$ be the maximal expected profit the owner can obtain when there are $s$ further days to go and the share price is $x$. Show that

(a) $F_{s}(x)$ is non-decreasing in $s$,

(b) $F_{s}(x)+x$ is non-decreasing in $x$, and

(c) $F_{s}(x)$ is continuous in $x$.

Deduce that there exists a non-decreasing sequence, $a_{1}, \ldots, a_{h}$, such that expected profit is maximized by exercising the option the first day that $x_{t} \leqslant a_{t}$.

Now suppose that the option never expires, so effectively $h=\infty$. Show by examples that there may or may not exist an optimal policy of the form 'exercise the option the first day that $x_{t} \leqslant a$.