Optimal control problem associated with jump processes
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.
viscosity solution of partial differential equations
scientific article; zbMATH DE number 2111598