Compactification methods in the control of degenerate diffusions: existence of an optimal control

This paper concerns the control of a degenerate diffusion. We study the existence of an optimal control for a diffusion which is the solution of the equation [d] where b and σ are continuous functions and u i, is the control, for a reward of the form J(r,z,u,) = E ∫T r h(s,x,us,ds+g(x T)We do not need the non-degenerate assumption on σ. We prove the existence of an optimal Markovian relaxed control,i.e. a control which takes values in the space of probability measures on a compact space A. We extend our results to the case of Borelian coefficients, with a supplementary hypothesis, and to the case of diffusions sigh jumps