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

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On the solution of a stochastic partial differential equation

The $\sigma-$algebra $\mathcal{F}_0$ is trivial, i.e.$\forall A\in\mathcal{F}_0$, $P(A)=0$ or $1$. So the answer is true. I think the answer is also true if $\varphi$ is a $\mathcal{G}-$measurable fun …
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