# American Institute of Mathematical Sciences

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December  2020, 10(4): 785-826. doi: 10.3934/mcrf.2020020

## Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach

 Department of Mathematics, University of Bordj Bou Arreridj, 34000 Algeria

Received  June 2019 Revised  December 2019 Published  March 2020

In this paper, we investigate a class of time-inconsistent stochastic control problems for stochastic differential equations with deterministic coefficients. We study these problems within the game theoretic framework, and look for open-loop Nash equilibrium controls. Under suitable conditions, we derive a verification theorem for equilibrium controls via a flow of forward-backward stochastic partial differential equations. To illustrate our results, we discuss a mean-variance problem with a state-dependent trade-off between the mean and the variance.

Citation: Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020
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