Optimal Stopping for a Diffusion with Unobserved Bernoulli Drift

We solve fairly explicitly an optimal stopping problem for a Wiener process with unobserved Bernoulli drift, in the presence of a cost on terminal position which is symmetric and increases with distance from the origin, and of a fixed positive cost per unit time $c>0$. After filtering, the problem reduces to Markovian optimal stopping with complete observations for the state process centered'' by its starting position $x \in \mathbb R$. However, the solution becomes possible only after foliating by an additional state-parameter \(y \in \mathbb{R}\), representing the displacement from the initial position; this foliation lifts’’ the problem from the real line to the plane, solves the augmented problem for each fixed initial position $x$, characterizes fairly explicitly the optimal stopping region in $(x,y)$-space, and finally obtains the solution of the original problem by ``slicing’’ along $y=0$. Following this procedure, we show that, under suitable structural assumptions on the terminal cost, each fixed-$x$ continuation section is either empty or a single bounded interval, whose endpoints are determined uniquely by a balancing condition; the corresponding value function is then given in semi-explicit form. The two-dimensional continuation region is obtained by gluing these fixed-$x$ intervals over $x$; its two free boundaries satisfy natural monotonicity properties and, at regular points, can be described by a coupled system of ordinary differential equations. The resulting description yields a threshold-type solution of the original one-dimensional problem whenever the horizontal slice $y=0$ enters the two-dimensional continuation region.