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A stochastic process is a collection of random variables usually indexed by a totally ordered set.

1 vote

Smoothness of $g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$

I assume here the $x$ variable is the initial condition of you process? Here is a partial answer: if $(\mathcal F_t)$ is the brownian filtration, then by Itô's martingale representation theorem for an …
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2 votes
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Running supremmum of a Levy process

See Theorem 25.18 p168 in "Lévy Processes and Infinitely Divisible Distributions" by K-I Sato
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