This part of the book focuses on stochastic growth models and provides a brief intro­duction to basic tools of stochastic dynamic optimization.

Stochastic growth models are useful for two related reasons. First, a range of interesting growth problems involve either aggregate uncertainty or nontrivial individual level uncertainty interacting with investment decisions and the growth process.

Some of these models will be discussed in Chapter 17. Second, the stochastic neoclassical growth model has a wide range of applications in macro­economics and in other areas of dynamic economic analysis. Various aspects of the stochastic neoclassical growth model will be discussed in the next two chapters. The study of sto­chastic models requires us to extend the dynamic optimization tools of Chapters 6 and 7 to an environment in which either returns or constraints are uncertain (governed by probabil­ity distributions).^[32] Unfortunately, dynamic optimization under uncertainty is considerably harder than the non-stochastic optimization. The generalization of continuous-time methods to stochastic optimization requires fairly advanced tools from measure theory and stochas­tic differential equations. While continuous-time stochastic optimization methods are very powerful, they are not used widely in macroeconomics and economic growth, so I have de­cided to focus on discrete-time stochastic models. Thus the next chapter will include the most straightforward generalization of the discrete-time dynamic programming techniques presented in Chapter 6 to stochastic environments. Unfortunately, a rigorous development of stochastic dynamic programming also requires further mathematical investment than is typically necessary in most macroeconomics and economic growth courses. To avoid a heavy dose of new mathematical tools, in particular a lengthy detour into measure theory at this stage of the book, the next chapter develops the basics of stochastic dynamic programming without measure theory. I will then include a few pointers about how the results in this chapter can be extended and made more rigorous.

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