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At the terminal time,so $P_T=1$ and $d_T=V_T=1/(T+1)$.

Assume $F(W_t)=P_t{\hat{x}}_t^2+d_t$. Conditional on $W_{t-1}$, the random part of the next Kalman filter estimate is $h_t$ times the innovation. Its variance isTherefore the Bellman equation givesCompleting the square givesStarting from $P_T=1$ yieldsand hence

Let $V(n,x)$ be the minimum expected remaining cost after time $n$, given $X_n=x$. The terminal condition is $V(N,x)=x^2$, and the Bellman equation is

Let the value function at time $n$ be the greatest conditional expected terminal utility attainable from wealth $x$. Independence of the innovations gives the Bellman equationfor $0\le n<N$. A policy obtained from the maximizing choices is admissible because each time-$n$ holding is a predictable process, hence is known before ${\xi }_n$ is observed.

We prove the claimed properties by backward induction. They hold at $n=N$ because $V(N,\cdot )=U$ is increasing and concave. Suppose they hold at time $n+1$. For fixed $\theta$, the mapis increasing because $1+r>0$, and taking a supremum preserves this inequality.

For concavity, let ${\theta }_x$ and ${\theta }_y$ be optimal at wealths $x$ and $y$. For $0\le t\le 1$, use the admissible portfolioThe next wealth is the same convex combination of the next wealths generated from $(x,{\theta }_x)$ and $(y,{\theta }_y)$. ThereforeThis completes the induction and proves monotonicity and concavity of a portfolio value function.

If discovery in box $i$ earns reward $R_i$ and stopping earns zero, the Bellman equation isIf ${\sum }_i{c}_i/({\alpha }_i{R}_i)<1$, every posterior state has some $i$ with ${\alpha }_i{p}_i{R}_i>c_i$, so stopping is never optimal.