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Large deviation of diffusion processes with discontinuous drift and their occupation times. (English) Zbl 1044.60065

Summary: For the system of \(d\)-dim stochastic differential equations, \[ dX^\varepsilon(t) =b\bigl(X^\varepsilon(t)\bigr)d t+\varepsilon dW(t),\;t\in[0,1],\quad X^\varepsilon(0) =x^0\in \mathbb R^d, \] where \(b\) is smooth except possibly along the hyperplane \(x_1=0\), we shall consider the large deviation principle for the law of the solution diffusion process and its occupation time as \(\varepsilon\to 0\). In other words, we consider \(P(\| X^\varepsilon- \varphi\|<\delta\), \(\| u^\varepsilon- \psi\|<\delta)\) where \(u^\varepsilon (t)\) and \(\psi(t)\) are the occupation times of \(X^\varepsilon\) and \(\varphi\) in the positive half space \(\{x\in \mathbb R^d:x_1>0\}\), respectively. As a consequence, a unified approach of the lower level large deviation principle for the law of \(X^\varepsilon(\cdot)\) can be obtained.

MSC:

60J60 Diffusion processes
60F10 Large deviations
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References:

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