Abstract
In this paper we prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a finite dimensional stochastic differential equation, driven by a multidimensional Wiener process. We drop the usual Lipschitz assumption on the drift term and substitute it with dissipativity conditions, allowing polynomial growth. The control enters both the drift and the diffusion term and takes values in a general metric space.
Highlights
Stochastic maximum principle (SMP for brevity) is a standard tool in order to provide necessary conditions for optimal control problems
In this paper we prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a finite dimensional stochastic differential equation, driven by a multidimensional Wiener process
Most of the results are only concerned with a convex control domain, or with the case in which the diffusion term does not depend on the control [2, 10]
Summary
Stochastic maximum principle (SMP for brevity) is a standard tool in order to provide necessary conditions for optimal control problems. Some works are devoted to the study of a general infinite dimensional SMP. In this paper we are interested in formulating another version of the general SMP for the optimal control of a stochastic differential equation in a finite dimensional setting, driven by a multidimensional Wiener process. We drop the lipschitzianity assumption on the drift term and we replace it with a more natural sign condition, known as dissipativity or monotonicity. This condition is widely studied in the literature both in finite and infinite dimension.
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