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I would like to formally show that the loss of a 1-D reflected random walk (with Gaussian increment parameters $\mu$ and $\sigma$) with regulating barriers at $0$ and $b$ is bounded from above by the loss of a $(\mu,\sigma)$ reflected Brownian motion with the same barriers.

The intuition I have here is that while the RBM realizes all loss associated with the reflected RW, the RBM incurs additional loss. This is due to the fact that the RBM is continuous, allowing for excursions below $0$ and above $b$ which increase the loss between the RW steps. Any ideas on how to formalize this intuition, or any ideas of how to show the upper bound in an alternative way?

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    You're comparing Metropolis dynamics (with the proposal being $x_{k+1}=x_k+N(\mu,\sigma)$) with the SDE $dX_t=\mu dt + \sigma dW$, where each is trapped in $(0,b)$? – 2017-01-04
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    Yes -- I'm trying to show that the loss (magnitude of potential excursions) below $0$ and above $b$ by a random walk with Gaussian increments is bounded from above by the magnitude of potential excursions of a reflected Brownian motion with the same parameters – 2017-01-05
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    I would actually expect it to be the opposite: when the RW tries to make an excursion it tries to make a large one (it goes in one direction on each unit time interval). By contrast the BM turns around from one moment to the next. Can you explain why you would think that it would go the way you have proposed? – 2017-01-05

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