Adaptive Control of a Diffusion to a Goaland

We study the following adaptive stochastic control problem: to maximize the probability PX(T) = 1] of reaching the \goal" x = 1 during the nite time-horizon 0; T], over \control" processes () which are adapted to the natural ltration of the \observation" process Y (t) = W(t) + Bt; 0 t T and satisfy almost surely R T 0 2 (t)dt < 1 and 0 X(t) = x + R t 0 (s)dY (s) 1; 80 t T. Here W() is standard Brownian motion, and B is an independent random variable with known distribution. The case B b 6 = 0 of this problem was studied by Kulldorr (1993). Modifying a martingale method due to Heath (1993), we nd an optimal control process ^ () for the general case of this problem, and solve explicitly for its value and for the associated Hamilton-Jacobi-Bellman equation of Dynamic Programming. This reduces to 2Q xx Q s = Q xx Q yy ?Q 2 xy ; an apparently novel parabolic-Monge-Amp ere-type equation.