Let $(\Omega,\mathcal F,\mathcal F_t,P)$ be a probability space and $W(t)$ be a Brwonian Motion defined on it. Let $M(t)$ be a bounded martingale orthogonal to $W(t)$, i.e., there exists a constant $C>0$ such that $\sup_{t,\omega}|M(t)|\leq C$ and $M(t)W(t)$ is a martingale. For $t\geq s$, is the following ture: $$E[(M(t)-M(s))cos(W(t)-W(s))|\mathcal F_s]=0?$$ Thanks!
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Perhaps you could remind us of the definition of an "orthogonal" martingale (and also explain what you mean by "bounded" since there are several notions of boundedness). – saz Oct 14 '18 at 06:19
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This is what I find in a book. It is said that $M\in \mathcal M_b$ which is the set of all bounded martingales and it doesnot specify the boundedness is a.s. or L^2. – SHAN Oct 14 '18 at 08:05