# Processes approximating a reflected brownian motion.

Let $W$ be a standard Brownian Motion. Let $\epsilon>0$ be given. Let $X^\epsilon$ be the process which diffuses like $W$ on $(-\epsilon,\infty)$, but when $X^\epsilon$ reaches the level $-\epsilon$, it is immediately brought back to the value zero. It then diffuses again according to $W$ until hitting $-\epsilon$, and then is brought back to zero, and so forth. Let $X^0$ be a reflected Brownian Motion (reflected at zero). Then, as $\epsilon \rightarrow 0$, in what sense does $X^\epsilon \rightarrow X^0$ Are there any references for this? I'm also interested in when $W$ is a diffusion.

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I'd be interested in knowing some (easily) readable references on this sort of thing too. One "classic" paper on diffusions with boundary conditions is Stroock and Varadhan's "Diffusions with boundary conditions." –  ShawnD Oct 23 '11 at 5:48
One thought: the reflected BM is a standard BM plus a local time term at zero. Using the interpretation of local time in terms of downcrossings should possibly do the trick. –  weakstar Oct 23 '11 at 14:28
You're saying that it's the value of the Brownian motion plus the minimal value attained rounded up to the nearest multiple of $\epsilon$. That really makes me want to take the limit by deleting the "rounded up" bit. Is that a thing? –  Will Sawin Oct 23 '11 at 18:49
The comments above have the correct idea. We have $X^\epsilon_t=W_t+\epsilon\lfloor\epsilon^{-1}\max\_{s\le t}(-W_s)\rfloor$ which converges uniformly to $W_t-\min\_{s\le t}W_s$. Convergence for this particular process is especially simple. –  George Lowther Oct 23 '11 at 21:02

It looks like it should converge in distribution in Skorokhod space D. Martingale problem approach (see the book by Ethier & Kurtz on Markov processes) should work.

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Thanks for the reference, will take a look. –  weakstar Oct 23 '11 at 14:28