Time-inconsistent consumption-investment problems in incomplete markets under general discount functions

Abstract

In this paper we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward stochastic differential equation. Moreover, we prove the uniqueness of the open-loop equilibrium pair by showing that the original time-inconsistent problem is equivalent to an associated time-consistent one.

A PDE approach for open-loop equilibriums in time-inconsistent stochastic optimal control problems

“…A large number of existing studies in the literature have examined open-loop equilibrium controls by adopting FBSDEs-approaches. For instance, in the series of papers [21], [9], [17], [37], [39], [40], [23], [35] and [50] the open-loop equilibrium control process ûx0 (•) is constructed by solving a flow of FBSDEs. Recently, the work [2] investigated open-loop equilibrium controls by using a BSPDEs-approach.…”

“…Wang et al [39] found the open-loop equilibrium strategy to a mean-variance problem under a non-markovian regime-switching model. Hamaguchi [17] performed a sophisticated FBSDEs-approach and characterized the unique openloop equilibrium solution to a general discounting Merton portfolio problem in an incomplete market with random parameters. Finally, in the work of Hamaguchi [18], the author investigated open-loop equilibrium controls in the framework of time-inconsistent stochastic recursive control problems, where the cost functional is defined by the solution to a backward stochastic Volterra integral equation.…”

“…They used a variational method in the spirit of Peng's stochastic maximum principle to characterize the existence and uniqueness of the equilibrium control in terms of the solvability of a "flow" of FBSDEs; we refer the readers to the work of Hamaguchi [16] for a detailed discussion about this new type of FBSDEs. Some recent studies devoted to the open-loop equilibrium concept can be found in [9], [1], [3], [17], [37], [39], [40], [23], [35] and [50]. Specially, Wang [37] discussed open-loop equilibrium controls and their particular closed-loop representations for a class of SLQ problems under mean-field SDEs.…”