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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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Mean square derivatives and modifications
Suppose we have a stochastic process $X$ on $\mathbb{R}$. Suppose there exists a stochastic process $\frac{d X(t)}{d t}$ such that
$$\lim_{h\to0} \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-\frac{d X( …