Optimal controls for diffusion in Rd—A min-max max-min formula for the minimal cost growth rate

We study the infinite horizon optimal control problem for a system whose dynamics are described by the stochastic differential equation dx, = Wu u,) dt + 4x,) da,> x0 given. (1.1) Let e be the o-algebra generated by (j,jO G s G r; then an admissible control is an {*}t,, adapted process in R” which satisfies the constraint u, E Wx,), t 2 0. ‘In the last relation x + U(x) is a measurable set valued mapping from Rd to R”. To every control corresponds the cost flow

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