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$X\in L^{\infty,p},Y\in L^{2,q},prove \int_{0}^{t} XY_{s}dB_{s}$ is martingale

Let $Z_t:=\int_0^t X_sY_s\,dB_s$, a continuous local martingale. By the Burkholder-Gundy-Davis inequality there is a constant $C$ such that: $$ \eqalign{ E\left[\max_{0\le t\le T}|Z_t|\right] &\le ...
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Showing that $M^T(N-N^T)$ is a continuous local martingale for all stopping times $T$ and continuous local martingales $M,N$

Your initial observation (for bounded martingales) suffices, because the map $(M,N)\mapsto M^T(N-N^T)$ commutes with stopping: If $(T_n)$ reduces both $M$ and $N$ to bounded martingales, then it does ...
John Dawkins's user avatar
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