Characterization of stochastic equilibrium controls by the Malliavin calculus

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem.

Optimal Consumption and Investment

In this chapter we apply the general theory from Chapter 25 to the study of optimal consumption and investment problems. We solve the Merton problem and we derive the Merton mutual fund theorems.

An analytical solution for the HJB equation arising from the Merton problem

In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation that arises from the Merton problem subject to general utility functions is presented for the first time. The solution presented in this paper is written in the form of a Taylor’s series expansion and constructed through the homotopy analysis method (HAM). The fully nonlinear HJB equation is decomposed into an infinite series of linear PDEs which can be solved analytically. Four examples are presented with the first two cases showing the accuracy of our solution approach; while the last two demonstrating its versatility.