How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\int^{t_1\wedge t_2}_0\sigma^2(t)dt.$$

If $\int^{s}_0\sigma(t)dW_t$ was a Brownian motion, then the above answer would be obvious, but unfortunately it's not. So how to calculate such covariance?



  1. bilinearity of covariance,
  2. independence of Brownian increments, and
  3. Itô's isometry,

we obtain: $$\begin{align} & \text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right) \\[6pt] & \qquad = \text{Cov}\left(\int^{t_1\wedge t_2}_0\sigma(t)dW_t,\int^{t_1\wedge t_2}_0\sigma(t)dW_t +\int^{t_1\vee t_2}_{t_1\wedge t_2}\sigma(t)dW_t\right) \\[6pt] & \qquad \overset{1}{=} V\left(\int^{t_1\wedge t_2}_0\sigma(t)dW_t\right)+\text{Cov}\left(\int^{t_1\wedge t_2}_0\sigma(t)dW_t,\int^{t_1\vee t_2}_{t_1\wedge t_2}\sigma(t)dW_t\right) \\[6pt] & \qquad \overset{2}{=} E\left(\left(\int^{t_1\wedge t_2}_0\sigma(t)dW_t\right)^2\right) \\[6pt] & \qquad \overset{3}{=} \int^{t_1\wedge t_2}_0\sigma^2(t)dt \end{align}$$

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