Vladimir Vinogradov (Ohio University)

Properties of Levy and Geometric Levy Processes

Abstract

We generate a class of Levy processes starting from the power-variance family of probability laws. The exact asymptotics of the probabilities of large deviations and path properties for members of this class are established. The techniques of stochastic exponentials enable us to discover an additional class of Levy processes. The ordinary exponentials of its members constitute the geometric Levy processes which we utilize for describing chaotic movements of equities. Thus, we consider a self-financing portfolio comprised of one bond and k equities assuming that the returns on all k equities belong to the latter class. We demonstrate that for a particular choice of constant portfolio weights, the combined movement of k equities is governed by a geometric Levy process, which belongs to the same class. The Merton-type allocation of constant weights, which we implement, coincides with those of fund managers. Although simpler, in the discontinuous case this approach is less profitable, than portfolio weight selection using an approach that maximizes the expected logarithmic utility. We prove a converse of Merton's theorem. We derive Pythagorean-type theorems for Sharpe portfolio performance measures emphasizing their relation to Merton-type weights and the additivity of shape parameter.