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 $d(\alpha(t)X_t)$?
Answer
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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