Zero noise limit of a stochastic differential equation involving a local time
This paper deals with the following stochastic differential equation \(dX_{t}^{\varepsilon}=b(X_{t}^{\varepsilon})dt+\sqrt{\varepsilon}dW_{t}+\beta dL_{t}^0(X^{\varepsilon})\), \(t\in(0,T]\), \(X_{0}^{\varepsilon}=0\), where \(\varepsilon>0\), \(\beta\in(-1,1)\), \(W_{t}, t\geq0\) is a standard Brownian motion and \(L_{t}^0(X^{\varepsilon}), t\geq0\) is the symmetric local time of \(X^{\varepsilon}\) at \(0\). It is supposed that \(b:R\to R\) is continuous, locally Lipschitz continuous on \(R\setminus\{0\}\) and \(\lim_{x\downarrow 0}b(x)/x^{\gamma_1}=C_1\), \(\lim_{x\uparrow 0}b(x)/x^{\gamma_2}=C_2\), with some constants \(0<\gamma_1\leq\gamma_2<1\), \(C_1,C_2>0\). In addition, it is assumed that \(b\) is bounded and \(b(x)>0\) for \(x>0\), and \(b(x)<0\) for \(x<0\). Let \(\alpha,p_{\beta}\) be given as follows: \(\alpha=(\beta+1)/2\), \(p_{\beta}=\alpha C_1^{1/(1+\gamma_1)}/(\alpha C_1^{1/(1+\gamma_1)}+(1-\alpha)C_2^{1/(1+\gamma_2)})\). The following are the first main results of this paper. Let \(\phi_{t}\) be a solution to the ordinary differential equation \(\phi'_{t}=b(\phi_{t})\), \(t\in (0,T]\), \(\phi_0=0\). If \(\gamma_1<\gamma_2\), then for any \(\delta>0\) and \(T>0\), we have \(\lim_{\varepsilon\to 0}P(\sup_{0\leq t\leq T}| X_{t}^{\varepsilon}-\phi_{t}^{+}|\leq \delta)=1\). If \(\gamma_1=\gamma_2\), then for any \(T>0\) and sufficiently small \(\delta>0\), we have \(\lim_{\varepsilon\downarrow 0}P(\sup_{0\leq t\leq T}| X_{t}^{\varepsilon}-\phi_{t}^{+}|\leq \delta)=p_{\beta}\), \(\lim_{\varepsilon\downarrow 0}P(\sup_{0\leq t\leq T}| X_{t}^{\varepsilon}-\phi_{t}^{-}|\leq \delta)=1-p_{\beta}\). As the second main result the authors establish the Wentzel-Freidlin type large deviation principle.
stochastic differential equation with local time
scientific article; zbMATH DE number 6985313
scientific article