Optimal Consumption and Portfolio Selection with Stochastic Differential Utility

Abstract

We develop the utility gradient (or martingale) approach for computing portfolio and consumption plans that maximize stochastic differential utility (SDU), a continuous-time version of recursive utility due to D. Duffie and L. Epstein (1992, Econometrica 60, 353 394). We characterize the first-order conditions of optimality as a system of forward backward SDEs, which, in the Markovian case, reduces to a system of PDEs and forward only SDEs that is amenable to numerical computation. Another contribution is a proof of existence, uniqueness, and basic properties for a parametric class of homothetic SDUs that can be thought of as a continuoustime version of the CES Kreps Porteus utilities studied by L. Epstein and A. Zin (1989, Econometrica 57, 937 969). For this class, we derive closed-form solutions in terms of a single backward SDE (without imposing a Markovian structure). We conclude with several tractable concrete examples involving the type of``affine'' state price dynamics that are familiar from the term structure literature.

A Stochastic Control Approach to a Robust Utility Maximization Problem

“…As mentioned above, the BSDE (5.3) or (5.14) can be explicitly solved for δ ≡ 0. This has already been observed in [19], Appendix A; in fact, it follows immediately from Lemma 15 which gives for σ = t and τ = T the representation…”

“…We show that V is the unique solution of a backward stochastic differential equation (BSDE) with a quadratic driver, if the underlying filtration is continuous. This extends earlier work by [13,19,21]. We first recall from Section 2 the conditional cost…”

“…Our primary inspiration clearly comes from the two papers [19,21]. In [21], Skiadas studies essentially the same optimization problem as (2.1) or (2.3), and proves that its dynamic value process V can be described by the BSDE…”

“…There we treat our optimization problem by stochastic control methods and show that for a continuous filtration, the corresponding dynamic value process is characterized as the unique solution of a generalized backward stochastic differential equation (BSDE) with a quadratic term in its driver. Our BSDE is a slight generalization of an equation studied in detail by [19], but our method of attack is rather different. Like in [19], however, our BSDE involves unbounded terms in the driver and the terminal value which cannot be handled by existing general techniques from the BSDE literature.…”

“…Our BSDE is a slight generalization of an equation studied in detail by [19], but our method of attack is rather different. Like in [19], however, our BSDE involves unbounded terms in the driver and the terminal value which cannot be handled by existing general techniques from the BSDE literature. Hence our approach has to exploit the precise structure of our equation.…”