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


1 Answer 1


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 $f(x,y) = xy$.

  • $\begingroup$ hi! can you give the non finite product rule also? $\endgroup$
    – Nikos
    Commented Dec 13, 2012 at 12:31
  • 1
    $\begingroup$ The second equation is the general product rule. $\endgroup$
    – quasi
    Commented Dec 13, 2012 at 15:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.