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.
翻译：本文研究了关于随机面积积分 $A_{1,2}(t,t+h)=\int_{t}^{t+h}\int_{t}^{s}{\rm d} W_{1}(r){\rm d} W_{2}(s) -\int_{t}^{t+h}\int_{t}^{s}{\rm d} W_{2}(r){\rm d} W_{1}(s)$ 和两个Wiener过程的增量 $\Delta{W}_{i}(t,t+h)=W_{i}(t+h)-W_{i}(t),\ i=1,2$ 的单项式期望。如果单项式的指数中有一个Wiener增量或随机面积积分的指数是奇数，那么单项式的期望为零。然而，如果任何一项的指数是偶数，则期望是非零的，并且它的确切值通常不知道。在本文中，我们推导出一般情况下计算其值的公式。作为公式的应用，我们将利用公式对Magnus型Milstein方法进行谨慎的稳定性分析。作为另一个应用，我们将给出Wiener过程和随机双重积分的增量的联合矩。
相关内容
微信扫码咨询专知VIP会员