# Simultaneous Stochastic Differential Equations

I was thinking about cointegrated time series and came up with the following simultaneous equations model:

$$dY_t = \alpha (Y_t - \gamma X_t)dt + \sigma dB_t$$

$$dX_t = \beta (Y_t - \delta X_t)dt + \tau dW_t$$

$$dW_t dB_t = \rho dt$$

With greek letters constants. $$\alpha$$ and $$\beta$$ with opposite signs.

Is it possible to find analytical solutions for $$Y_t$$ and $$X_t$$ (maybe allowing for $$\rho = 0$$) ?

I looked in Oksendal and Shreve (and obviously google) for technics to solve it, but couldn't find a clue. Any references would be appreciated.

• This is just a multi-dimensional version of the Ornstein–Uhlenbeck process, the solution is on wikipedia. Commented Feb 4, 2022 at 7:18
• Indeed! A haven't realized that the Wikipedia formulation could retrieve my formulation with appropriate parameters matrixes. The only adjust would be to use $dB$ and $dW$ as independent and reconstruct the dependency in the $\mathbb{\sigma}$ matrix. Commented Feb 4, 2022 at 21:12