Lifetime Consumption-Portfolio Choice Under Trading Constraints, Recursive Preferences and Nontradeable Income

Abstract

We consider the lifetime consumption-portfolio problem in a competitive securities market with essentially arbitrary continuous price dynamics, a possibly nontradeable income stream, and convex constraints on the vector of market values of financial positions. (The setting extends Schroder and Skiadas, 2002, where the endowment is assumed tradeable and constraints are imposed in terms of wealth proportions.) For any utility function with a supergradient density, we develop the first-order conditions of optimality, a side-product being the characterization of a constrained notion of state-pricing. The methodology is applied to generalized continuous-time recursive utility, allowing for first and second-order risk-aversion that can depend on the risk source, reflecting the source's "ambiguity." Within this class, we isolate a more tractable formulation in which preferences exhibit no wealth effects (an example being time-additive expected discounted exponential utility), and there is unrestricted trading in a money market and a suitably defined consol bond. In this case, we derive closed-form solutions for the optimal consumption and trading strategy in terms of the solution to a single constrained backward stochastic differential equation (BSDE), which in a Markovian setting maps to a PDE. Methodologically, we develop the utility gradient approach, but for the wealth-invariant case we also verify the solution using the dynamic programming approach, without having to assume a Markovian structure. Finally, we present a class of parametric examples in which the PDE characterizing the solution simplifies to a system of ordinary differential equations (of the Riccati type).