Optimal control problem associated with jump processes
type: Item
instance of: scholarly article
title: Optimal control problem associated with jump processes
zbMATH Open document ID: 1052.60050
author: Yasushi Ishikawa
published in: Applied Mathematics and Optimization
publication date: 2004-10-28T00:00:00Z
review text: The paper is devoted to stochastic models in mathematical finance based on so-called geometric Lévy processes \(S_t\) which can be regarded as being driven by canonical (Marcus type) pure jump processes. The authors study the wealth process \(X_t\) which is based on \(S_t\). They control \(X_t\) and the average consumption process \(Y_t\) with some decay rate using some parameter process, corresponding to portfolio, consumption rate or disposal of the asset, so that the average utility function (value function) attains its maximum. This type of problem is called Merton's optimal investment problem. The setting is different from others in that a pair consisting of wealth process and the cumulative consumption process is chosen, in that the utility function depends only on the temporal consumption or in that the utility functions can be quite general as long as they satisfy the so-called Gossen's law. It is established, by using the Bellman principle, that the value function is a viscosity solution to some integro-differential equation with the Dirichlet boundary condition, which is similar to the HJB equation. Existence and uniqueness of the solution are proved and the Bellman principle which is necessary in the proof of existence is established.
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 49J22
Mathematics Subject Classification ID: 91B28
zbMATH DE Number: 2111598
zbMATH Keywords: stochastic control of jump type
zbMATH Keywords: mathematical finance
zbMATH Keywords: viscosity solution of partial differential equations
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
full work available at URL: s00245-004-0795-9
OpenAlex ID: W2017219760
DOI: 10.1007/S00245-004-0795-9
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description: scientific article; zbMATH DE number 2111598