Replication of Wiener-Transformable Stochastic Processes with Application to Financial Markets with Memory

Abstract

We investigate Wiener-transformable markets, where the driving process is given by an adapted transformation of a Wiener process. This includes processes with long memory, like fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called "constant" and "variable depth" memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. Motivated by integral representation results in general Gaussian setting, we study the conditions under which random variables can be represented as pathwise integrals with respect to the driving process. From financial point of view, it means that we give the conditions of replication of contingent claims on such markets. As an application of our results, we consider the utility maximization problem in our specific setting. Note that the markets under consideration can be both arbitrage and arbitrage-free, and moreover, we give the representation results in terms of bounded strategies.

Gaussian processes with Volterra kernels

“…Such processes arise in finance, see e.g. [4]. They are the natural extension of fractional Brownian motion (fBm) which admits the integral representation via the Wiener process, and the Volterra kernel of its representation consists of power functions.…”