摘要:
设(Xt)是由如下随机微分方程所决定的反射扩散过程:$\left\{\begin{array}{l}X_t=X_0+\int_0^t \sigma\left(X_t\right) d W_t+\int_0^t b\left(X_t\right) d s+L_t-U_t, \\ L_t=\int_0^t I_{\{0\}}\left(X_t\right) d L_t, \\ U_t=\int_0^t I_{\{1\}}\left(X_t\right) d U_t .\end{array}\right.$本文证明了当t→∞时,$P_{\Delta}\left\{X_{,} \in A\right\} \rightarrow \pi(A), \frac{1}{t} E_{\infty}\left(L_r\right) \rightarrow \alpha, \frac{1}{t} E_{\infty}\left(U_0\right) \rightarrow \beta$,,其中$\begin{aligned} & \alpha=\left(\int_0^1 \exp \left\{\int_0^t \frac{2 b(s)}{\sigma^2(s)} d s\right\} d t\right)^{-1} \\ & \beta=\alpha \exp \left\{\int_0^1 \frac{2 b(s)}{\sigma^2(s)} d s\right\},\end{aligned}$和$\int_0^1 \sigma^{\lambda y} \sigma(d y)=\frac{\int_0^1 \exp \left\{\int_0 \frac{2 b(s)}{\sigma^2(s)} d s+\lambda t\right\} d t}{\int_0^1 \exp \left\{\int_0^1 \frac{2 b(s)}{\sigma^2(s)} d s\right\} d t}$.
Abstract:
Let (Xt) be the reflected diffusion process governed by the S.D.E $\left\{\begin{array}{l}X_t=X_0+\int_0^t \sigma\left(X_t\right) d W_t+\int_0^t b\left(X_t\right) d s+L_t-U_t, \\ L_t=\int_0^t I_{\{0\}}\left(X_t\right) d L_t, \\ U_t=\int_0^t I_{\{1\}}\left(X_t\right) d U_t .\end{array}\right.$.In this paper we proved the limit theorem:t→∞,$P_{\Delta}\left\{X_{,} \in A\right\} \rightarrow \pi(A), \frac{1}{t} E_{\infty}\left(L_r\right) \rightarrow \alpha, \frac{1}{t} E_{\infty}\left(U_0\right) \rightarrow \beta$,where$\begin{aligned} & \alpha=\left(\int_0^1 \exp \left\{\int_0^t \frac{2 b(s)}{\sigma^2(s)} d s\right\} d t\right)^{-1} \\ & \beta=\alpha \exp \left\{\int_0^1 \frac{2 b(s)}{\sigma^2(s)} d s\right\},\end{aligned}$,and$\int_0^1 \sigma^{\lambda y} \sigma(d y)=\frac{\int_0^1 \exp \left\{\int_0 \frac{2 b(s)}{\sigma^2(s)} d s+\lambda t\right\} d t}{\int_0^1 \exp \left\{\int_0^1 \frac{2 b(s)}{\sigma^2(s)} d s\right\} d t}$.
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