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

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 forwardbackward 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.2010 Mathematics Subject Classification. Primary: 91B51, 93E20, 60H30, 93E99, 60H10; Secondary: 93E25.

Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

“…The author derived a verification theorem that provides a general sufficient condition for equilibriums via a coupled stochastic system of a single forward SDE and a flow of BSPDEs. The current paper differs from [2] in at least three important aspects. Firstly, the controlled system here follows a jump-diffusion process, while that of [2] is driven by Brownian motions only.…”

“…The current paper differs from [2] in at least three important aspects. Firstly, the controlled system here follows a jump-diffusion process, while that of [2] is driven by Brownian motions only. This makes the time-inconsistent problem in the current paper more general.…”

“…We then rigorously prove a verification theorem (Theorem 4.1) by which one can construct an open-loop equilibrium strategy by solving the abovementioned IPDEs. Another recent paper by Alia [2] adopts the idea of decoupling forward-backward systems to study open-loop equilibriums in a general class of time-inconsistent stochastic control problems. The author derived a verification theorem that provides a general sufficient condition for equilibriums via a coupled stochastic system of a single forward SDE and a flow of BSPDEs.…”

“…Secondly, there is a great difference in how the methods of these two papers permit us to derive an open-loop equilibrium solution. The verification theorem in [2] is a kind of a sufficient stochastic minimum principle (SMP) that involves BSPDEs instead of (ordinary) BSDEs. In contrast to [2], the proposed method in the current paper is purely deterministic in the sense that it enables us to derive an equilibrium strategy by solving deterministic IPDEs (see Remark 5 below).…”

“…The verification theorem in [2] is a kind of a sufficient stochastic minimum principle (SMP) that involves BSPDEs instead of (ordinary) BSDEs. In contrast to [2], the proposed method in the current paper is purely deterministic in the sense that it enables us to derive an equilibrium strategy by solving deterministic IPDEs (see Remark 5 below). The third aspect concerns a major advantage of the method in the present article, compared with that of [2].…”