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What is the equivalent of product rule for stochastic differentials? I need it in the following case: Let $X_t$ be a process and $\alpha(t)$ a real function. What would be $d(\alpha(t)X_t)$?

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If $\alpha(t)$ is of finite variation, then the product rule is the same as in ordinary calculus:

$$ d(\alpha(t)X_t) = \alpha(t) dX_t + X_t d\alpha(t). $$

If you had $X_t$ and $Y_t$ as processes, you would get

$$ d(X_t Y_t) = X_t dY_t + Y_t dX_t + d [X,Y]_t. $$

If $Y$ has finite variation, the last quadratic covariation term is zero. The second equation is just applying Ito's Formula to $f(x,y) = xy$.

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  • $\begingroup$ hi! can you give the non finite product rule also? $\endgroup$
    – Nikos
    Dec 13, 2012 at 12:31
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    $\begingroup$ The second equation is the general product rule. $\endgroup$
    – quasi
    Dec 13, 2012 at 15:37

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