MPS-RR 2000-44

November 2000

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

by:

### 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*