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I'm attempting to solve the following two-dimensional stochastic differential equation:

$$ dX_1(t) = \big(\alpha X_2(t) - \theta_1 X_1(t)\big)dt + \sigma_1 dW_1(t), $$ $$ dX_2(t) = - \theta_2 X_2(t)dt + \sigma_2 dW_2(t), $$

where $\alpha, \theta_1, \theta_2, \sigma_1,$ and $\sigma_2$ are constants. I've already figured out that the solution to $X_2(t)$ using Ito's formula with $f(t,x_t) = x_t e^{\theta_2 t}$, which gives the solution

$$ X_2(t) = X_2(0) e^{-\theta_2 t} + \sigma_2 e^{\theta_2 t} \int_0^t e^{\theta_2 s} dW_2(s). $$

However I'm stuck on the next step for finding a solution for $X_1(t)$. I think I have to use the multidimensional form of Ito's formula, but I'm not sure how to approach the problem.

WilliamMorris
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1 Answers1

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Thanks to the suggestion by Thomas Kojar, I was able to use a special case of the linked answer in "Solution to General Linear SDE" to find a solution to this problem. From the general case they consider the stochastic differential equation

$$ dX_t = \big(a(t)X_t+ b(t)\big) dt + \big(g(t)X_t+ h(t)\big)dW_t. $$

In my case I have

\begin{align*} a(t) &= -\theta_1 = \text{constant}, \\ b(t) &= \alpha X_2(t), \\ g(t) &= 0, \\ h(t) &= \sigma_1 = \text{constant}, \end{align*}

and $X_t = X_1(t)$, and $W_t = W_1(t)$.

The integral solution to my original problem is then given by

$$ X_1(t) = X_1(0)e^{-\theta_1 t} + e^{-\theta_1 t}\Big[ \alpha\int_0^t e^{\theta_1 s} X_2(s)ds + \sigma_1\int_0^t e^{\theta_1 s} dW_1(s) \Big], $$

from their "case 2." I have validated this solution numerically against an Euler-Maruyama numerical solution, and the solutions appear to converge as $\Delta t \rightarrow 0$.

WilliamMorris
  • 404