3 votes
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On Tanaka's SDE

So my original answer did not take into account the fact that there could exist a transformation of the entire path $(|X_{s}|)_{0\leq s\leq t}$. Instead, I think I have found a way to prove that for a ...
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3 votes
Accepted

How can we show that the time shift $t\mapsto x(s+\;\cdot\;)$ is measurable?

A map $(F, \mathcal{F}) \to (E^{[0, \infty)}, \mathcal{E}^{\otimes [0, \infty)})$ into $E^{[0, \infty)}$ is $\mathcal{F}/\mathcal{E}^{\otimes[0, \infty)}$-measurable if and only if all of its (all $t \...
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  • 555
2 votes

What is a continuous stochastic process?

The set $\Omega$ is the sample space, which can be any set you want. Since $\Omega$ is a set, though, it doesn't make sense to say that $\omega \in \Omega$ "appears just once at time $k$". ...
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2 votes

Compute Expectation of double Ito Integral

Hint There is more or less nothing true in all what you said. You don't have Fubini with Itô integrals. Moreover, it's not clear what are $X$ and $Y$ in your question. Finally, it's not true that $...
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1 vote

Where does the randomness come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$?

I think this a legitimate question as intuition as opposed to mere manipulation of symbols is very important in mathematics in general and especially in probability theory. That said, it may be ...
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