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Abstract: In this paper we consider the Stratonovich reflected stochastic differentialequation $dX t=\sigmaX t\circ dW t+bX tdt+dL t$ in a bounded domain $\O$which satisfies conditions, introduced by Lions and Sznitman, which arespecified below. Letting $W^N t$ be the $N$-dyadic piecewise linearinterpolation of $W t$ what we show is that one can solve the reflectedordinary differential equation $\dot X^N t=\sigmaX^N t\dotW^N t+bX^N t+\dot L^N t$ and that the distribution of the pair$X^N t,L^N t$ converges weakly to that of $X t,L t$. Hence, what we proveis a distributional version for reflected diffusions of the famous result ofWong and Zakai. Perhaps the most valuable contribution made by our procedurederives from the representation of $\dot X^N t$ in terms of a projection of$\dot W t^N$. In particular, we apply our result in hand to derive somegeometric properties of coupled reflected Brownian motion in certain domains,especially those properties which have been used in recent work on the -hotspots- conjecture for special domain.



Autor: Lawrence Christopher Evans, Daniel W. Stroock

Fuente: https://arxiv.org/







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