Publication 2003-006

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first order PDEs of Hamilton-Jacobi-Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton-Jacobi-Bellman type.