A combinatorial infinitesimal representation of Levy processes and an application to incomplete markets

摘要

Starting from an R{sup}d-valued Levy process with infinitesimal generator l, an infinitesimal mesh h and an internal hyperfinite discretisation (lattice) L of the state space Rd, the reduced lifting is constructed-this is the unique right lifting that can be written as the hyperfinite sum of a generalised Andersonian random walk for mesh h and a hyperfinite sum of independent jumps in L, each allowed to occur at any time in h·*{sup left}N{sub}0. By assigning each of these components a natural economic interpretation, a suitable internal notion of risk-neutrality is introduced. A reduced lifting for a process with solely positive jumps- modelling the stock price process of a conservatively managed company-can then be shown to describe, with respect to this notion of risk-neutrality, the logarithm of a weakly complete market model (complete in the sense that there is an essentially unique risk-neutral measure which preserves the structure of the model, that is the Markov property as well as the independence of the jumps and the lifted diffusion part), thereby circumventing the incompleteness typically entailed by a continuous Levy market model.