A characterization of the geometric Brownian motion in terms of infinite dimensional Laplacians

Abstract

We consider a stock price Z_t whose dynamics follows a geometric Brownian motion living on the standard Gaussian white noise space. We regard the risk-free interest rate r and volatility σ as independent variables of the stock price. We show that the partial derivatives of the stock price with respect to r and σ satisfy equations which involve the Gross Laplacian and the number operator of the stock price. Introducing an operator transferring white noise functionals to generalized functionals of square of white noise, we give equations for the stock price including the Lévy Laplacian and the Volterra Laplacian. Moreover we prove that those equations characterize the stock price up to a constant only depending on time t.