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# Dynamic Asset Pricing Theory

By: Darrell Duffie | Book details

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Page 203

9
Portfolio and Consumption Choice
THIS CHAPTER PRESENTS basic results on optimal portfolio and consumption choice, first using dynamic programming, then using general martingale and utility-gradient methods. We begin with a review of the Hamilton-Jacobi-Bellman equation for stochastic control, and then apply it to Merton’s problem of optimal consumption and portfolio choice in finite- and infinite-horizon settings. Then, exploiting the properties of equivalent martingale measures from Chapter 6, Merton’s problem is solved once again in a non-Markovian setting. Finally, we turn to the general utility-gradient approach from Chapter 2, and show that it coincides with the approach of equivalent martingale measures.
A. Stochastic Control
Dynamic programming in continuous time is often called stochastic control and uses the same basic ideas applied in the discrete-time setting of Chapter 3. The existence of well-behaved solutions in a continuous-time setting is a delicate matter, however, and we shall focus mainly on necessary conditions. This helps us to conjecture a solution that, if correct, can often be validated.Given is a standard Brownian motion B = (B1,… , Bd) in Rd on a probability space (Ω, F, P). We fix the standard filtration F = {Ft : t 0} of B and begin with the time horizon [0, T] for some finite T > 0. The primitive objects of a stochastic-control problem are
 • a set A ⊂ Rm of actions, for some integer m ≥ 1. • a set y ⊂ RK of states, for some integer K ≥ 1. • a set c of A-valued adapted processes, called controls. • a controlled drift function g : A × y → RK.

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