'Ito's Lemma' and the Bellman Equation for Poisson Processes: An Applied View

Sennewald, Ken; Wälde, Klaus

doi:10.2139/ssrn.894207

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'Ito's Lemma' and the Bellman Equation for Poisson Processes: An Applied View

CESifo Working Paper Series No. 1684

35 Pages Posted: 31 Mar 2006

See all articles by Ken Sennewald

Klaus Wälde

University of Mainz; CESifo (Center for Economic Studies and Ifo Institute); UCL at Louvain la Neuve

Date Written: March 2006

Abstract

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

Sennewald, Ken and Wälde, Klaus, 'Ito's Lemma' and the Bellman Equation for Poisson Processes: An Applied View (March 2006). CESifo Working Paper Series No. 1684, Available at SSRN: https://ssrn.com/abstract=894207 or http://dx.doi.org/10.2139/ssrn.894207