I was wondering if it is possible to define a n-dimensional, square integrable, stochastic process taking values in a compact subset of $\mathbb{R}^{n}$.
square integrable and compactness
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measure-theory
stochastic-processes
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0Let ${\cal F}_{t}$ 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 $F_{t}$. – 2012-12-11
1 Answers
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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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0Sorry 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. – 2012-12-11