Formulae for Mixed Moments of Wiener Processes and a Stochastic Area Integral

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