Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Abstract

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which the equilibrium valued function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman-Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal values $Z(r,r)$ of $Z(\cd\,,\cd)$) is naturally introduced and the well-posedness of such kind of equations is briefly discussed.

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

“…Nash equilibria are therefore considered instead of optimal controls. This approach was adopted and further developed by Björk-Khapko-Murgoci [3], Djehiche-Huang [4], Yong [33,34], Wei-Yong-Yu [28], Yan-Yong [29], Wang-Yong [23], Hu-Jin-Zhou [13,14], Hu-Huang-Li [12], Alia [1], to mention a few. Time-inconsistent consumption-investment problems under nonexponential discounting were studied by, for example, Ekeland-Pirvu [5], Alia et al [2], and Hamaguchi [8].…”

“…Concerning this formulation, Yong [33] performed a multi-person differential game approach for a general discounting time-inconsistent stochastic control problem, and characterized the closed-loop equilibrium strategy via the so-called equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. This approach was further developed in [34,28,29,23]. (ii) An open-loop equilibrium control is an equilibrium concept for a "control process" that a controller chooses based on the initial condition.…”

“…For a stochastic control problem, a recursive cost functional, with exponential discounting, can be described by the solution of a BSDE. On the other hand, as discussed in [23], when we consider a stochastic recursive control problem with general (non-exponential) discounting, then the proper definition of the recursive cost functional is the solution of a Type-I BSVIE. In the closed-loop framework, Wang-Yong [23] and Yan-Yong [29] (Section 5) adopted the multi-person differential game approach, and studied a time-inconsistent stochastic recursive control problem where the cost functional defined by the solution of a Type-I BSVIE.…”

“…On the other hand, as discussed in [23], when we consider a stochastic recursive control problem with general (non-exponential) discounting, then the proper definition of the recursive cost functional is the solution of a Type-I BSVIE. In the closed-loop framework, Wang-Yong [23] and Yan-Yong [29] (Section 5) adopted the multi-person differential game approach, and studied a time-inconsistent stochastic recursive control problem where the cost functional defined by the solution of a Type-I BSVIE. A similar problem was studied by Wei-Yong-Yu [28] in the closed-loop framework, where the cost functional was defined by a family of parametrized BSDEs.…”

“…In this paper, we generalize his idea to a time-inconsistent setting with the cost functional defined by the solution to a BSVIE. It is also worth to mention that the paper [11] provided a necessary condition for an optimal control in a time-consistent stochastic recursive control problem, while the papers [33,28,29,23] provided sufficient conditions for closed-loop equilibrium strategies in time-inconsistent stochastic recursive control problems with general discounting. Compared with the above papers, we provide a necessary and sufficient condition for an open-loop equilibrium control in a time-inconsistent stochastic recursive control problem with general discounting.…”