Stock exchange dynamics involving both Gaussian and Poissonian white noises: approximate solution via a symbolic stochastic calculus

Abstract

One considers the stock exchange stochastic dynamics obtained when a Poissonian white noise is added to the usual Gaussian white noise. The generalized Kolmogorov equation (GKE) defining the probability density of this process is well known, but here, by using a symbolic stochastic calculus of order n, one can obtain an approximate expression for the state of this process. As an application, a Black–Scholes equation of order n is derived without explicitly using Itô’s lemma for Poissonian stochastic differential equation.