Denote $$X_t = \int^t_0\sigma e^{-k(t-s)}dW_s$$ here $W_s$ is the Brownian motion, $k,\sigma$ are constants.

I want to calculate $d X_t$ and the variance $Var[X_t].$ I know how to take the derivatives of a integral with parameters, but don't know how to deal with this stochastic integral.

  • $\begingroup$ Please consider adding the tag "self-study" or "homework" if that is the case $\endgroup$
    – Sanjay
    Mar 5, 2018 at 2:48

1 Answer 1


You can rewrite $X_t = e^{-kt}Z_t$ and define $Z_t:=\int_{0}^{t}e^{ks}dW_s$. There is a theory (Lemma 4.15 in Björk if you use his book) which states that $$\text{Var}\left[\int_{0}^{t}f(u)dW_s\right]=\int_{0}^{t}(f(u))^2ds$$ You can use that. furthermore, You can use Ito to compute $dX_t$. By standard stochastic calculus theory the dynamics of $Z_t $ is $dZ_t=e^{kt}dW_t$


Your Answer

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

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