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$B_t$ is said to follow a Brownian motion if i) $B_0=0$, ii) $B_t$ is continuous a.s. iii) $B_t$ has independent increments iv) $B_t-B_s\sim N(0,t-s)$. Then Wiener would go on to show that such a process does indeed exist.

What are some examples of simple and interesting stochastic processes, that can be proven to not exist?

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A simple attempt to model "white noise" would be a process $W(t)$ on $[0, \infty)$, such that $\int_0^s W(t)\; dt = B(s)$ is a Brownian motion. But that does not exist because Brownian motion is almost surely nowhere differentiable. A more sophisticated formulation of white noise must be used, using generalized functions.

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