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

Ishak Alia

doi:10.3934/mcrf.2020020

Article Contents

2020, Volume 10, Issue 4: 785-826. Doi: 10.3934/mcrf.2020020

This issue Previous Article Non-exponential discounting portfolio management with habit formation Next Article On optimal

$ L^1 $

-control in coefficients for quasi-linear Dirichlet boundary value problems with

$ BMO $

-anisotropic

$ p $

-Laplacian

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

Ishak Alia

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

Received: June 2019

Revised: December 2019

Early access: March 2020

Published: October 2020

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Abstract

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.

Keywords:

Time-inconsistency,
stochastic control,
equilibrium control,
backward stochastic partial differential equations,
mean-variance problem.

Mathematics Subject Classification: Primary: 91B51, 93E20, 60H30, 93E99, 60H10; Secondary: 93E25.

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