Equilibrium strategies in time-inconsistent stochastic control problems with constraints: necessary conditions

TL;DR

This paper develops necessary conditions for equilibrium strategies in time-inconsistent stochastic control problems with constraints, using variational principles and backward stochastic differential equations, with applications to finance.

Contribution

It introduces a novel approach to derive necessary conditions for constrained, time-inconsistent stochastic control problems using second-order Hamiltonians and Ekeland's principle.

Findings

Derived necessary conditions for equilibrium strategies.

Applied framework to a constrained investment-consumption problem.

Demonstrated the approach's relevance in financial modeling.

Abstract

This paper is concerned with a time-inconsistent recursive stochastic control problems where the forward state process is constrained through an additional recursive utility system. By adapting the Ekeland variational principle, necessary conditions for equilibrium strategies are presented concerning a second-order Hamiltonian function defined by pairs of backward stochastic differential equations. At last, we consider a finite horizon state constrained investment-consumption problem with non-exponential actualisation as an example to show the application in finance. The class of constraints investigated here includes the possibility of imposing a risk bound on the terminal value of the wealth process.