# Tagged Questions

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### Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemannâ€“Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgueâ€“Stieltjes measure, I am looking for the corresponding results for the ...
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### Understanding a theorem from “Probability theory of Banach Spaces ” book.

I don't understand the proof after "The hypothesis of the theorem indicate...... , can someone kindly explain it for me . Thanks :)
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### Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty$, ...
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### canonical form of dyadic martingales

Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$. Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the ...
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### Random variables with the same distributions

Let $K$ and $L$ be locally compact Hausdorff spaces. Also, let $P$ be a Radon probability measure on $K$ so that $(K,P)$ is a probability space. I want to know, whether two random variables ...
Suppose $1\leq p<\infty$. Let $E$ be a Banach space. Consider a filtration $F_n$ on some probability space $\Omega$. Let $X\in L^p(\Omega,E)$ where $L^p(\Omega,E)$ denote the Bochner space. In ...