|Date: Wednesday, February 21, 2018
Location: 1360 East Hall (4:00 PM to 5:00 PM)
Title: Utility Maximization with Constant Costs
Abstract: We study the problem of maximizing expected utility of terminal wealth for an investor facing a mix of constant and proportional transaction costs. While the case of purely proportional transaction costs is by now well understood and existence of optimal strategies is known to hold for a very general class of price processes, the case of constant costs remains a challenge since the existence of optimal strategies is not even known in tractable models (such as, e.g., the Black-Scholes model). In this talk, we present a novel approach which allows us to construct optimal strategies in a multidimensional diffusion market with price processes driven by a factor process and for general lower-bounded utility functions.
One of the main challenges for the problem under consideration is that the value function turns out to be piecewise but not globally continuous. We establish this result in two steps:
(1) We apply the stochastic Perron's method to show that the value function is a discontinuous viscosity solution of the associated dynamic programming PDE (a nonlocal parabolic free boundary problem).
(2) We establish a local comparison principle for viscosity solutions of this PDE, which implies uniqueness of the value function as well as piecewise continuity.
Having established piecewise continuity, we use a characterization of the value function as the pointwise infimum of a suitable set of superharmonic functions to construct optimal trading strategies. The advantage of this approach is that the pointwise infimum (i.e. the value function) inherits the superharmonicity property, which in turn allows us to prove a verification theorem for candidate optimal strategies requiring only piecewise continuity of the value function. An application of the verification theorem entails the existence of optimal strategies.
In particular, to the best of our knowledge, our approach leads to the first uniqueness result for discontinuous viscosity solutions of nonlocal PDEs and the model is a rare example of a stochastic control with a discontinuous value function which can be solved completely.
This talk is based on joint work with Soren Christensen (University of Hamburg).
Speaker: Christoph Belak
Institution: University of Trier