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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2000-44
November 2000

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


Fred E. Benth, Kenneth H. Karlsen, Kristin Reikvam


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 value 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].

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This paper has now been published in Mathematical finance (Konstanz, 2000), 79--88, Trends Math., Birkhäuser, Basel, 2001