Our purpose is to study an optimal ergodic control problem where the state of the system is given by a diffusion process with jumps in the whole space. The corresponding dynamic programming (or Hamilton-Jacobi-Bellman) equation is a quasi-linear integro-differential equation of second order. A key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only locally bounded and Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic HJB equation is established.
Other Applied Mathematics | Partial Differential Equations
Menaldi JL., Robin M. (1999) On Optimal Ergodic Control of Diffusions with Jumps. In: McEneaney W.M., Yin G.G., Zhang Q. (eds) Stochastic Analysis, Control, Optimization and Applications. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA