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.

  • 1
    $\begingroup$ This is just a multi-dimensional version of the Ornstein–Uhlenbeck process, the solution is on wikipedia. $\endgroup$
    – Freelunch
    Commented Feb 4, 2022 at 7:18
  • $\begingroup$ 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. $\endgroup$ Commented Feb 4, 2022 at 21:12


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