Optimal Stopping for a Diffusion with Unobserved Bernoulli Drift

Abstract

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.