Consumption and Uncertainty in Continuous Time

The intertemporal consumption model is extended to the case of uncertainty in continuous time. Stochastic variables are described by Wiener processes and Brownian motions. We study the properties of these processes and show that although they are not differentiable in time, an approximation known as Ito’s Lemma can be used. This Lemma makes it possible to solve the stochastic Hamilton-Jacobi-Bellman equation in continuous time. We derive more general implications for stochastic optimal control of consumption and savings. Finally, we conclude the chapter by applying these methods to five examples of stochastic consumption.