Controlling a stopped diffusion process to reach a goal

We consider a problem of optimally controlling a two-dimensional diffusion process dxtμ,β=μ(xt)dt+β(xt)dBt1;x(0)=x,dyt=αytdt+(σxt+γ)ytdBt2;y(0)=y, initially starting in the interior of a domain Dφ={(x,y)∈R+2:φ(x)0 and θ>1 are fixed positive constants and φ(x) is a given positive strictly increasing, twice continuously differentiable function on (0,∞) such that φ(0)≥0. The goal is to maximize the probability criterion sup(μ,β)∈M(x)P(yτ=θφ(xτμ,β),τ≤T0|x(0)=x,y(0)=y),x,y∈Dφ, over a class of admissible controls M(x) consisting of bounded, Borel measurable functions. Under suitable conditions, it is shown that the maximal probability is given explicitly and the optimal process is determined explicitly by ρ(φ(x),φ′(x),φ″(x))=μ0(x)φ′(x)−(α−12(σx+γ)2)φ(x)β0(x)2.