Formulae for mixed moments of Wiener processes and a stochastic area integral

Abstract

This paper deals with the expectation of monomials with respect to the stochastic area integral $A_{1,2}(t,t+h)=\int_{t}^{t+h}\int_{t}^{s}{\rm d} W_{1}(r){\rm d} W_{2}(s) -\int_{t}^{t+h}\int_{t}^{s}{\rm d} W_{2}(r){\rm d} W_{1}(s)$ and the increments of two Wiener processes, $\Delta{W}_{i}(t,t+h)=W_{i}(t+h)-W_{i}(t),\ i=1,2$. In a monomial, if the exponent of one of the Wiener increments or the stochastic area integral is an odd number, then the expectation of the monomial is zero. However, if the exponent of any of them is an even number, then the expectation is nonzero and its exact value is not known in general. In the present paper, we derive formulae to give the value in general. As an application of the formulae, we will utilize the formulae for a careful stability analysis on a Magnus-type Milstein method. As another application, we will give some mixed moments of the increments of Wiener processes and stochastic double integrals.