  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.
