Limit Theorem for Controlled Backward SDEs and Homogenization of
Hamilton–Jacobi–Bellman Equations |
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Authors: | Rainer Buckdahn Naoyuki Ichihara |
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Institution: | (1) Département de Mathématiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, B.P. 809, 29285 Brest Cedex, France |
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Abstract: | We prove a convergence theorem for a family of value functions associated 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 assumption
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. |
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Keywords: | Homogenization Hamilton– Jacobi– Bellman equations Viscosity solutions Backward stochastic differential equations Stochastic optimal control Stochastic
ergodic control |
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