Properties of certain Lévy and geometric Lévy processes

We study Lévy processes associated with the power-variance family of probability laws. Their path and structural properties as well as the exact asymptotics of the probabilities of large deviations are established. We use the techniques of DoleanMeyer exponentials to introduce an additional class of Lévy processes. The ordinary exponentials of its members constitute the geometric Lévy processes which we utilize for describing the 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 the choice of constant Merton-type portfolio weights, the combined movement of k equities is governed by a geometric Lévy process which belongs to the same class. In the continuous case, we prove a converse of Merton’s mutual fund theorem. We derive Pythagorean-type theorems for Sharpe measures emphasizing their relation to Merton-type weights and the additivity of shape parameter.