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 $dZ_t=e^{kt}dW_t$


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