'Ito's Lemma' and the Bellman Equation for Poisson Processes: An Applied View
35 Pages Posted: 31 Mar 2006
Date Written: March 2006
Using the Hamilton-Jacobi-Bellman equation, we derive both a Keynes-Ramsey rule and a closed form solution for an optimal consumption-investment problem with labor income. The utility function is unbounded and uncertainty stems from a Poisson process. Our results can be derived because of the proofs presented in the accompanying paper by Sennewald (2006). Additional examples are given which highlight the correct use of the Hamilton-Jacobi-Bellman equation and the change-of-variables formula (sometimes referred to as "Ito's-Lemma") under Poisson uncertainty.
Keywords: stochastic differential equation, Poisson process, Bellman equation, portfolio optimization, consumption optimization
JEL Classification: C61, D81, D90, G11
Suggested Citation: Suggested Citation