Y. Komori, G. Yang and K. Burrage (2023), Formulae for mixed moments of
Wiener processes and a stochastic area integral, SIAM Journal on Numerical
Analysis, 61 (4), 1716-1736.
Abstract
This paper deals with the expectation of monomials with respect to the
stochastic area integral $A_{1,2}(t,t+h)=\int_{t}^{t+h}\int_{t}^{s}\mathrm{d}
W_{1}(r)\mathrm{d} W_{2}(s)-\int_{t}^{t+h}\int_{t}^{s}\mathrm{d} W_{2}(r)\mathrm{d}
W_{1}(s)$ 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.
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Last updated: 2023/07/16