## Dynamic Optimization in Continuous-Time Economic Models

**(A Guide for the Perplexed)**

Maurice Obstfeld*

University of California at Berkeley

First Draft: April 1992

*I thank the National Science Foundation for research support.

#### I. Introduction

The assumption that economic activity takes place

continuously is a convenient abstraction in many applications.

In others, such as the study of financial-market equilibrium, the

assumption of continuous trading corresponds closely to reality.

Regardless of motivation, continuous-time modeling allows

application of a powerful mathematical tool, the theory of

optimal dynamic control.

The basic idea of optimal control theory is easy to grasp--

indeed it follows from elementary principles similar to those

that underlie standard static optimization problems. The purpose

of these notes is twofold. First, I present intuitive

derivations of the first-order necessary conditions that

characterize the solutions of basic continuous-time optimization

problems. Second, I show why very similar conditions apply in

deterministic and stochastic environments alike. 1

A simple unified treatment of continuous-time deterministic

and stochastic optimization requires some restrictions on the

form that economic uncertainty takes. The stochastic models I

discuss below will assume that uncertainty evolves continuously

according to a type of process known as an Ito^ (or Gaussian

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1When the optimization is done over a finite time horizon, the

usual second-order sufficient conditions generalize immediately.

(These second-order conditions will be valid in all problems

examined here.) When the horizon is infinite, however, some

additional "terminal" conditions are needed to ensure optimality.

I make only passing reference to these conditions below, even

though I always assume (for simplicity) that horizons are

infinite. Detailed treatment of such technical questions can be

found in some of the later references. diffusion) process. Once mainly the province of finance

theorists, Ito^ processes have recently been applied to

interesting and otherwise intractable problems in other areas of

economics, for example, exchange-rate dynamics, the theory of the

firm, and endogenous growth theory. Below, I therefore include a

brief and heuristic introduction to continuous-time stochastic

processes, including the one fundamental tool needed for this

type of analysis, Ito^’s chain rule for stochastic differentials.

Don’t be intimidated: the intuition behind Ito^’s Lemma is not

hard to grasp, and the mileage one gets out of it thereafter

truly is amazing.