Time-consistent investment and consumption strategies under a general discount function

Abstract

The paper [12] examines a concept of equilibrium policies instead of optimal controls in stochastic optimization to analyze a mean-variance portfolio selection problem. We follow the same approach in order to investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsistency of the decision maker. Equilibrium policies are characterized in this context by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.

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

“…This consequence generalizes the result obtained by Alia et al [3] to an incomplete market setting. The random field (p, q) satisfies a "flow" of BSDEs (3.13) with the additional condition (3.14) imposed on the diagonal terms (p s s , q s s ).…”

“…(3) 2 are bounded, and λ 1 , λ 2 are of the forms λ 1 (t, s) = λ(s − t) and λ 2 (t) = λ(T − t) for some Lipschitz continuous function λ, then our model becomes (essentially) the same one as in Alia et al [3]. They suggested that in this setting an open-loop equilibrium control is characterized by a flow of FBSDEs.…”

“…Note that the discount functions of the form as in the last example cannot be written as functions of s−t or T −t, therefore they are beyond the class of discount functions considered in the literatures [6,19,20,3].…”

“…Their model is non-Markov, while the interest rate is assumed to be deterministic. In contrast, within the class of open-loop controls, Alia et al [3] studied a time-inconsistent consumption-investment problem under a general utility function (but with some very strong assumptions) and a general (Lipschitz continuous) discount function. They derived, by using the same duality method as [12,13], a flow of FBSDEs which characterizes the open-loop equilibrium consumption-investment pair.…”

“…(ii) We provide a necessary condition (Theorem 3.1) and a verification theorem (Theorem 4.1) for an open-loop equilibrium pair. Their conditions are related to the solvability of the corresponding fully coupled FBSDE, which is more tractable than a flow of FBSDEs appearing in [6,3].…”