Appl Math Optim 49:159–181 (2004)
2004 Springer-Verlag New York Inc.
Max-Plus Stochastic Processes
Wendell H. Fleming
Division of Applied Mathematics and
Lefschetz Center for Dynamical Systems, Brown University,
Providence, RI 02912, USA
Abstract. This paper is concerned with processes which are max-plus counter-
parts 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–
Key Words. Max-plus probability, Stochastic differential equations, Max-plus
additive functionals, Variational inequalities.
AMS Classiﬁcation. 35F20, 60H10, 93E20.
The Maslov idempotent calculus provides a framework in which a variety of asymptotic
problems, including large deviations for stochastic processes, can be considered [MS].
The asymptotic limit is typically described through a deterministic optimization problem.
However, the limit still retains a “stochastic” interpretation, if probabilities are assigned
which are additive with respect to “max-plus” addition and expectations are deﬁned so
as to be linear with respect to max-plus addition and scalar multiplication. Instead of
“idempotent probability” we use the term “max-plus probability.” There is an exten-
sive literature on max-plus probability and max-plus stochastic processes. See [A],
[AQV], [BCOQ], [DMD], [LM], [MS], [P1], [P2], [Q], and references cited there.