Exercise 4.17 (Instantaneous correlation). Let
$$
\begin{aligned}
& X_1(t)=X_1(0)+\int_0^t \Theta_1(u) d u+\int_0^t \sigma_1(u) d B_1(u), \\
& X_2(t)=X_2(0)+\int_0^t \Theta_2(u) d u+\int_0^t \sigma_2(u) d B_2(u),
\end{aligned}
$$
where $B_1(t)$ and $B_2(t)$ are Brownian motions satisfying $d B_1(t) d B_2(t)=\rho(t)$ and $\rho(t), \Theta_1(t), \Theta_2(t), \sigma_1(t)$, and $\sigma_2(t)$ are adapted processes. Then
$$
d X_1(t) d X_2(t)=\sigma_1(t) \sigma_2(t) d B_1(t) d B_2(t)=\rho(t) \sigma_1(t) \sigma_2(t) d t .
$$
We call $\rho(t)$ the instantaneous correlation between $X_1(t)$ and $X_2(t)$ for the reason explained by this exercise.
We first consider the case when $\rho, \Theta_1, \Theta_2, \sigma_1$, and $\sigma_2$ are constant. Then
$$
\begin{aligned}
& X_1(t)=X_1(0)+\Theta_1 t+\sigma_1 B_1(t), \\
& X_2(t)=X_2(0)+\Theta_2 t+\sigma_2 B_2(t) .
\end{aligned}
$$
Fix $t_0>0$, and let $\epsilon>0$ be given.
(i) Use Itô's product rule to show that
$$
\mathbf{E}\left[\left(B_1\left(t_0+\epsilon\right)-B_1\left(t_0\right)\right)\left(B_2\left(t_0+\epsilon\right)-B_2\left(t_0\right)\right) \mid \mathcal{F}\left(t_0\right)\right]=\rho \epsilon .
$$
(ii) Show that, conditioned on $\mathcal{F}\left(t_0\right)$, the pair of random variables
$$
\left(X_1\left(t_0+\epsilon\right)-X_1\left(t_0\right), X_2\left(t_0+\epsilon\right)-X_2\left(t_0\right)\right)
$$
has means, variances, and covariance
$$
\begin{aligned}
M_i(\epsilon)= & \mathbf{E}\left[X_i\left(t_0+\epsilon\right)-X_i\left(t_0\right) \mid \mathcal{F}\left(t_0\right)\right]=\Theta_i \epsilon \text { for } i=1,2 \\
V_i(\epsilon)= & \mathbf{E}\left[\left(X_i\left(t_0+\epsilon\right)-X_i\left(t_0\right)\right)^2 \mid \mathcal{F}\left(t_0\right)\right]-M_i^2(\epsilon) \\
= & \sigma_i^2 \epsilon \text { for } i=1,2 \\
C(\epsilon)= & \mathbf{E}\left[\left(X_1\left(t_0+\epsilon\right)-X_1\left(t_0\right)\right)\left(X_2\left(t_0+\epsilon\right)-X_2\left(t_0\right)\right) \mid \mathcal{F}\left(t_0\right)\right] \\
& \quad-M_1(\epsilon) M_2(\epsilon)=\rho \sigma_1 \sigma_2 \epsilon
\end{aligned}
$$
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4 Stochastic Calculus
In particular, conditioned on $\mathcal{F}\left(t_0\right)$, the correlation between the increments $X_1\left(t_0+\epsilon\right)-X_1\left(t_0\right)$ and $X_2\left(t_0+\epsilon\right)-X_2\left(t_0\right)$ is
$$
\frac{C(\epsilon)}{\sqrt{V_1(\epsilon) V_2(\epsilon)}}=\rho .
$$
We now allow $\rho(t), \Theta_1(t), \Theta_2(t), \sigma_1(t)$, and $\sigma_2(t)$ to be continuous adapted processes, assuming only that there is a constant $M$ such that
$$
\left|\Theta_1(t)\right| \leq M, \quad\left|\sigma_1(t)\right| \leq M, \quad\left|\Theta_2(t)\right| \leq M, \quad\left|\sigma_2(T)\right| \leq M, \quad|\rho(t)| \leq M
$$