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I was wondering if it is possible to define a n-dimensional, square integrable, stochastic process taking values in a compact subset of $\mathbb{R}^{n}$.

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A non-constant one? –  Hans Engler Dec 11 '12 at 17:25
    
Let ${\cal F}_{t}$ denote the sigma algebra generated by a d- dimensional brownian motion $B(t)$. Also let the stochastic process $X(t)$ to be e.g. measurable with respect to $F_{t}$. –  Peter5 Dec 11 '12 at 17:30
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You could take $X(t) = F(B(t))$, where $F:\mathbb{R}^d \to \mathbb{R}^d $ is a suitable map that takes values in a compact set. There are a lot of choices here and therefore a lot of possible answers.

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$L^{2}$ is not compact but it is possible to determine and characterize a compact subset $K$ of $L^{2}(\mathbb{R}^{n})$. Is this the explanation ? –  Peter5 Dec 11 '12 at 17:50
    
I think you should specify your question a bit better. Compact subsets of $\mathbb{R}^n$ not the same as compact subsets of $L^2(\mathbb{R}^n)$. Which of these are you interested in? –  Hans Engler Dec 11 '12 at 18:02
    
Sorry for confusing, i was thinking about something else... Anyway, i am interested in square integrable processes taking values in a compact subset of $\mathbb{R}^{d}$. You suggested $F(B(t))$, i like that. Obviously, if $X(t)$ is a contant function the answer is clear. –  Peter5 Dec 11 '12 at 18:09
    
thanks for the help –  Peter5 Dec 11 '12 at 18:27
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