Optimal control of the blowup time of a diffusion

If we have a controlled Markov diffusion which may explode in finite time, the problem arises regarding using the control in order to maximize the mean time to explosion, i.e. the blowup time. The maximal mean blowup time, u(x), as a function of the initial position x is an element of R(n) is characterized as the unique continuous viscosity solution of a Bellman equation, satisfying the boundary condition that u vanishes at infinity. Then we consider the problem of convergence of the maximal mean blowup time u(epsilon)(x) corresponding to a diffusion matrix sigma = root 2 epsilon I, as epsilon-->0. We establish that, in general, the stochastic mean blowup time does not converge to the deterministic blowup time. However, the certainty equivalent blowup time does converge to the deterministic blowup time.