Adaptive control of a diffusion to a goal and a parabolic Monge–Ampère-type equation

Abstract

Abstract.We study the following adaptive stochastic control problem: to maximize the probability ~P[X(T) = 1] of reaching the "goal" x = 1 during the finite time-horizon [0,T], over "control" processes 7r(-) which are adapted to the natural filtration of the "observation" process Y(t) = W(t) + Bt, 0 < t < 00 and 0 < X{t) = x + Jj 7r(s)dY(s) < 1,V0 < t < T. Here W(') is standard Brownian motion, and B is an independent random variable with known distribution ^. The case B = b ^ 0 of this problem was studied by Kulldorff (1993). Modifying a martingale method due to Heath (1993), we find an optimal control process 7r(-) for the general case of this problem, and solve explicitly for its value and for the associated HamiltonJacobi-Bellman equation of Dynamic Programming. This reduces to 2QxxQs = QxxQyy ~ Qxyi ai1 apparently novel parabolic-Monge-Ampere-type equation.