Appl Math Optim 51:1–33 (2005)
2004 Springer Science+Business Media, Inc.
Limit Theorem for Controlled Backward SDEs and Homogenization
of Hamilton–Jacobi–Bellman Equations
Rainer Buckdahn and Naoyuki Ichihara
D´epartement de Math´ematiques, Universit´e de Bretagne Occidentale,
6 Avenue Victor Le Gorgeu, B.P. 809, 29285 Brest Cedex, France
Abstract. We prove a convergence theorem for a family of value functions associ-
ated with stochastic control problems whose cost functions are deﬁned by backward
stochastic differential equations. The limit function is characterized as a viscosity
solution to a fully nonlinear partial differential equation of second order. The key
assumption we use in our approach is shown to be a necessary and sufﬁcient as-
sumption for the homogenizability of the control problem. The results generalize
partially homogenization problems for Hamilton–Jacobi–Bellman equations treated
recently by Alvarez and Bardi by viscosity solution methods. In contrast to their
approach, we use mainly probabilistic arguments, and discuss a stochastic control
interpretation for the limit equation.
Key Words. Homogenization, Hamilton–Jacobi–Bellman equations, Viscosity
solutions, Backward stochastic differential equations, Stochastic optimal control,
Stochastic ergodic control.
AMS Classiﬁcation. 60H30, 35B27, 93E20, 49L25.
The asymptotic analysis for (forward) stochastic differential equations (SDEs) with
periodic structures has been largely investigated by numerous authors in connection
with the homogenization of second order partial differential equations (PDEs). Since the
nineties, motivated by their relationship to quasi-linear PDEs (generalized Feynman–Kac