BFS 2002

Contributed Talk

In this paper we provide solutions to the optimal lifetime consumption-portfolio problem in a possibly incomplete, competitive securities market with essentially arbitrary (continuous) price dynamics, and preferences that can take any (smooth) recursive homothetic form. Homothetic recursive preferences include but are not limited to the usual HARA time-additive expected utility, and the CES specification made popular by Epstein and Zin. Moreover, we also provide a complete solution for a non-homothetic exponential recursive form, generalizing CARA time-additive utility, which can be viewed as a limiting case of homothetic specifications. The securities market considered can be incomplete, in the sense that not all sources of uncertainty are hedgeable (but we do not consider the case of a non-tradeable income stream in this paper). The solution is given in closed form in terms of the solution to a single backward stochastic differential equation (BSDE). We show how to derive the solutions using either the utility gradient (or martingale) approach, or the dynamic programming approach, without relying on any underlying Markovian structure, making clear the relationship of the two approaches. Further imposing a Markov structure on the underlying dynamics reduces the BSDE to a quasilinear partial differential equation that can be tackled numerically, and in some parametric specifications can be further reduced to a simpler system of ordinary differential equations. Applying a technique from Schroder and Skiadas (2002), the solutions also extend readily to incorporate habit formation in the preferences.