An Optimal Control Problem Associated with SDEs Driven by Lévy-Type Processes

Abstract

In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton-JacobiBellman equation.

Stochastic differential equations with polar-decomposed Lévy measures and applications to stochastic optimization

“…As a result, we will be able to define explicitly the coefficient of the jump term, which will then allow us to apply our explicit construction to two concrete portfolio optimization problems considered in Ref. [5] 1) over R d , for the cases when d = 1 and when d 2, respectively.…”

“…Lévy-type processes associated with (general) generators of variable coefficients can be linked to Lévy processes with jumps in an abstract setting, which provides a nice way to deal with optimization problems with Lévy-type processes in financial modelling 1) [5]. In the present paper, we carry out further this approach and to concentrate our consideration to those polardecomposable Lévy measures with the concrete case of stable-like Lévy generators considered intensively in Refs.…”

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

“…In Ref. [6], we considered the optimal control problem for the wealth processes and the cumulative consumption processes driven by general Lévy-type processes. Let us also mention a very interesting recent work of Li and Peng [24] where the authors established a stochastic optimization theory of backward stochastic differential equations with jumps and investigated the viscosity solutions to the associated Hamilton-Jacobi-Bellman equations.…”