反射扩散过程的某些极限定理
SOME LIMIT THEOREMS OF REFLECTED DIFFUSION PROCESSES
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摘要: 设(Xt)是由如下随机微分方程所决定的反射扩散过程:\left\\beginarraylX_t=X_0+\int_0^t \sigma\left(X_t\right) d W_t+\int_0^t b\left(X_t\right) d s+L_t-U_t, \\ L_t=\int_0^t I_\0\\left(X_t\right) d L_t, \\ U_t=\int_0^t I_\1\\left(X_t\right) d U_t .\endarray\right.本文证明了当t→∞时,P_\Delta\left\X_, \in A\right\ \rightarrow \pi(A), \frac1t E_\infty\left(L_r\right) \rightarrow \alpha, \frac1t E_\infty\left(U_0\right) \rightarrow \beta,,其中\beginaligned & \alpha=\left(\int_0^1 \exp \left\\int_0^t \frac2 b(s)\sigma^2(s) d s\right\ d t\right)^-1 \\ & \beta=\alpha \exp \left\\int_0^1 \frac2 b(s)\sigma^2(s) d s\right\,\endaligned和\int_0^1 \sigma^\lambda y \sigma(d y)=\frac\int_0^1 \exp \left\\int_0 \frac2 b(s)\sigma^2(s) d s+\lambda t\right\ d t\int_0^1 \exp \left\\int_0^1 \frac2 b(s)\sigma^2(s) d s\right\ d t.Abstract: Let (Xt) be the reflected diffusion process governed by the S.D.E \left\\beginarraylX_t=X_0+\int_0^t \sigma\left(X_t\right) d W_t+\int_0^t b\left(X_t\right) d s+L_t-U_t, \\ L_t=\int_0^t I_\0\\left(X_t\right) d L_t, \\ U_t=\int_0^t I_\1\\left(X_t\right) d U_t .\endarray\right..In this paper we proved the limit theorem:t→∞,P_\Delta\left\X_, \in A\right\ \rightarrow \pi(A), \frac1t E_\infty\left(L_r\right) \rightarrow \alpha, \frac1t E_\infty\left(U_0\right) \rightarrow \beta,where\beginaligned & \alpha=\left(\int_0^1 \exp \left\\int_0^t \frac2 b(s)\sigma^2(s) d s\right\ d t\right)^-1 \\ & \beta=\alpha \exp \left\\int_0^1 \frac2 b(s)\sigma^2(s) d s\right\,\endaligned,and\int_0^1 \sigma^\lambda y \sigma(d y)=\frac\int_0^1 \exp \left\\int_0 \frac2 b(s)\sigma^2(s) d s+\lambda t\right\ d t\int_0^1 \exp \left\\int_0^1 \frac2 b(s)\sigma^2(s) d s\right\ d t.
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