# 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  December 2020 Early access  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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