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.

Source: arXiv:2606.23648v1 - http://arxiv.org/abs/2606.23648v1 PDF: https://arxiv.org/pdf/2606.23648v1 Original Link: http://arxiv.org/abs/2606.23648v1