Mathematics – Probability
Scientific paper
2012-02-19
Mathematics
Probability
Scientific paper
When the ordinary differential equation (ODE in short) {{l} \xi ^{^{\prime}}(t) =b(\xi(t)), \xi (0) = x\in \mathbf{R}^{n},%. where $b:\mathbf{R}^{n}\rightarrow \mathbf{R}^{n}$, has not a Lipschitz right hand side, there is neither existence nor uniqueness property of the associated Cauchy problem. However, the perturbed stochastic differential equation (SDE in short) {{l} \text{d}X^{\varepsilon}(t) = b(X^{\varepsilon}(t)) \text{d}t+\varepsilon \text{d}W_{t}, X^{\varepsilon}(0) =x\in \mathbf{R}^{n},%. where $W$ is a $n$-dimensional standard Brownian motion, has a unique strong solution when $b$ is only continuous and bounded. Moreover, when $% \varepsilon \rightarrow 0,$ the solutions to the perturbed SDE converges, in a sense, to the solutions of ODE. This phenomenon has been extensively studied in the literature in the one dimensional case. The goal of present paper is to analyzes the multi-dimensional case (this needs slightly different technique that in dimension one). In the case where the ODE has infinity many solutions, one of the main outcome of our approach is to explain which solutions of the ODE can be the limits of the solutions of the perturbed SDE.
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