Optimal Control of Stochastic Differential Delay Equations (SDDE) with Jumps and Markov Switching, and with an Application in Economics

This paper considers stochastic optimal control of stochastic differential delay equations (SDDEs) with jumps and Markov switching, and its economical applications. The Hamilton–Jacobi–Bellman (HJB) equation and the inverse HJB-equation are derived by applying the Dynkin’s formula and solution of the Dirichlet–Poisson problem for those SDDEs. Three cases of modified Ramsey stochastic models in economics are studied; namely, a Ramsey diffusion model with jumps, a Ramsey diffusion model with Markov switching, and a Ramsey diffusion model with jumps and Markov switching. We also present numerical examples for all three cases. The contributions of the paper are four-fold: introducing and studying three new models for SDDEs: with jumps, Markov switching and both; studying optimal control for those models; applications of these results to economics with Ramsey’s stochastic models; and numerical examples for these different models.