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Yes, $X$ is adapted to $(\mathcal{F}_t)_t$, because $\mathcal{F}^{Y}_t\subset \mathcal{F}_t$ necessarily for each $t$, otherwise $Y$ could not be adapted to $(\mathcal{F}_t)_t$. (In general $(\mathcal{F}^Y_t)_t$ is known as the natural filtration of $Y$.)


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