For $0<s<t$, only the Brownian noise accumulated through time $s$ is shared. The Itô isometry for cross terms gives
$$
\begin{aligned}
\operatorname{Cov}(X_t,X_s)
&=\int_0^se^{-\lambda(t-r)}e^{-\lambda(s-r)}\,dr\\
&=\frac{e^{-\lambda(t-s)}(1-e^{-2\lambda s})}{2\lambda}.
\end{aligned}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 6 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/6/e/solution