# Stochastic differential equations with correlated Brownian Motions

let's consider an sde of this kind:

$$$$\label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t \\ X_0=x_0 \\ dY_t=B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\ Y_0=y_0 \end{cases}$$$$ where t $$\in [0,T]$$, $$T$$ is the time horizon , $$x_0 \in \mathbb{R}^d$$, $$y_0 \in \mathbb{R}^l$$, b, $$\sigma$$, B, C are Borel functions and $$(\Omega, \mathcal{F}, \mathcal{F}_t,W_t , \mathbb{P})_{t \geq 0}$$ is a continuous standard $$\mathbb{R}^d$$-Brownian motion and $$\{ W_t^2 \}_{t \geq 0}$$ is a continuous standard $$\mathbb{R}^l$$-Brownian motion on the same filtration having instantaneous correlation $$\rho \in (-1,1)$$ \begin{align*} \mathbb{E}[dW_t^i dW_t^{2,j}]=\rho dt && \forall i \leq d, j \leq l \end{align*} Do the classic theorem of existence and uniqueness of strong solution holds also for this system? Do you have any reference? Thank you all in advance!

• When you say "correlated", do you mean they are still jointly Gaussian, i.e. they form a two-dimensional Brownian motion with covariance $\left(\begin{smallmatrix} 1 & \rho \\ \rho & 1 \end{smallmatrix} \right)$? If yes then by a linear transformation you can rewrite the system as driven by a standard 2-D BM and everything is fine. If no then things get much harder and I don't know what can be said. May 7, 2020 at 15:26
• yes, i mean jointly gaussian. But the BM are not 1D but d-dimensional and l-dimensional respectively. Does it work also in this sense? May 7, 2020 at 15:40
• I missed that. In general it will work whenever they are a linear transformation of a standard BM, which would be true for any covariance between them. But are you sure the condition you state is possible? If I compute correctly, when $d=l=3$ and $\rho = 1/2$ the resulting covariance matrix is not positive definite. May 7, 2020 at 15:51

You replace your SDE by the above. Then as usual you need to have regularity for the coefficients (see What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$? for references).