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Questions tagged [quadratic-variation]

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Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
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Operator-valued stochastic integral and quadratic variation for operator-valued processes

Let $U$ be a separable $\mathbb R$-Hilbert space and $W$ be a $Q$-Wiener process on a complete and right-continuous filtered probability space. Let $H$ be a separable $\mathbb R$-Hilbert space and $X$ ...
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How can we show that the tensor-quadratic variation has locally bounded variation?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ $U,H$ be infinite-dimensional ...
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How does the Kunita-Watanabe identity generalize to stochastic integration on Hilbert spaces?

Let $U,H$ be a separable $\mathbb R$-Hilbert spaces, $M$ be a $U$-valued square-integrable martingale on a filitered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$...
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Is there any analogous to Levy characterization theorem for purely discontinuous martingales?

Let $M_t$ and $N_t$ be two purely discontinuous martingales such that $[M]_t=[N]_t $ almost surely. Can one conclude that $M$ and $N$ have the same law?