On the existence of optimal controls for a singular stochastic control problem in finance

Abstract

We prove existence of optimal investment-consumption strategies for an infinite horizon portfolio optimization problem in a Lévy market with intertemporal substitution and transaction costs. This paper complements our previous work [4], which established that the valne function can be uniquely characterized as a constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation (but [4] left open the question of existence of optimal strategies). In this paper, we also give an alternative proof of the viscosity solution property of the value function. This proof exploits the existence of optimal strategies and is consequently simpler than the one proposed in [4].

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

“…Furthermore, for the given Lévy measure ν(x, dz) to be that of a stablelike process with index α(x), it must satisfy condition (6) and have the following polar decomposition:…”

“…[6], one can show that an optimal control (π * , g * , L * ) ∈ A x,y exists, such that v(x, y) = E (X (π * ,g * ,L * ) ,Y (π * ,g * ,L * ) ) ∞ 0 e −αs u(g * s )ds holds. The reader can refer to Ref.…”

“…The reader can refer to Ref. [6] for the proof of such claim for a related problem. In the sequel, the trajectory associated with this optimal control is denoted by (X * t , Y * t ).…”

“…Moreover, optimal control problems for wealth processes and cumulative consumption processes driven by geometric Lévy processes are considered in Refs. [6][7][8][9]18]. Ref.…”

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

“…Along with this tendency, there has been considerable research interests on stochastic control for jump diffusions, see [24] and references therein. Moreover, in very interesting articles [5][6][7][8]15], the authors study an optimal control problem for a pair consisting of the wealth processes and cumulative consumption processes driven by geometric Lévy processes.…”