《1-Obstfeld-Stochastic Optimization in Continuous Time(for the perplexed)》.pdfVIP

III. Stochastic Optimization in Continuous Time The optimization principles set forth above extend directly to the stochastic case. The main difference is that to do continuous-time analysis, we will have to think about the right way to model and analyze uncertainty that evolves continuously with time. To understand the elements of continuous-time stochastic processes requires a bit of investment, but there is a large payoff in terms of the analytic simplicity that results. Let’s get our bearings by looking first at a discrete-time stochastic model. 11 Imagine now that the decision maker maximizes the von Neumann-Morgenstern expected-utility indicator 8 (19) E s edthU[c(t),k(t)]h, 0 t t=0 where E X is the expected value of random variable X conditional t on all information available up to (and including) time t. 12 Maximization is to be carried out subject to the constraint that (20) k(t+h) k(t) = G[c(t),k(t),q (t+h),h], k(0) given, 11An encyclopedic reference on discrete-time dynamic programming and its applications in economics is Nancy L. Stokey and Robert E. Lucas, Jr. (with Edward C. Prescott), Recursive Methods in Economic Dynamics (Cambridge, Mass.: Harvard University Press, 1989). The volume pays special attention to the foundations of stochastic models. 12Preferences less restrictive than those delimited by the von Neumann-Morgenstern axioms have been proposed, and can be handled by methods analogous to those sketched below. 21 8 where {q (t)} is a sequence of exogenous random variables with t=-8 a known joint distribution, and such that only realizations up to and including q (t) are known at time t. For simplicity I will assume that the q process is first-order Markov, that is, that the joint distribution of {q (t+h),