Coupled linear stochastic differential equations

[Tyler_D]

Homework Statement

In the following system of coupled stochastic differential equations, solve for $x$ (note that $dW$ is a Wiener process)

Relevant Equations

$$dx = (\omega y - \gamma x)dt \\ dy = (-\omega x - \gamma y)dt + gdW \\$$

In order to solve for $x$, I need to re-write the equation for $dx$ so it is independent of $y$ and $dy$. However, I am having some issues with this. Can someone give me a push in the right direction?

 

[Delta2]

Solve the first equation for $y$ (will be a function of $x$ and $dx/dt$), then from this $y$ calculate $dy/dt$ (which will be a function of $dx/dt$ and $d^2x/dt^2$) and then replace in the second equation the $y$ and the $dy/dt$ you found.

 

[Tyler_D]

Solve the first equation for $y$ (will be a function of $x$ and $dx/dt$), then from this $y$ calculate $dy/dt$ (which will be a function of $dx/dt$ and $d^2x/dt^2$) and then replace in the second equation the $y$ and the $dy/dt$ you found.

There's no $dy/dt$ in the second equation. I could of course divide by $dt$, but I don't believe we are allowed to as $dW$ is not differentiable due to it being Wiener noise (?)

 

[Delta2]

To tell you the truth it beats me how you can define a differential $dW$ without the function W being differentiable, but anyway I think if it isn't differentiable then you probably can't solve this system.
Anyway after you calculate $y'=dy/dt$ as I told you before , replace $dy$ in the second equation with $y'dt$ and see what you can get.

 

[Fred Wright]

Please see equ. 3.31 here https://users.aalto.fi/~ssarkka/course_s2012/pdf/sde_course_booklet_2012.pdf. Of course you will have to read everything leading up to that equation to make sense of it:-)

 

[Tyler_D]

Please see equ. 3.31 here https://users.aalto.fi/~ssarkka/course_s2012/pdf/sde_course_booklet_2012.pdf. Of course you will have to read everything leading up to that equation to make sense of it:-)

Thanks Fred. But this is just a generic solution to the Ornstein-Uhlenbeck equation as far as I can tell? The question is, how do I go from this coupled system of SDEs I have above to an SDE in a single variable that I can solve. Or am I misunderstanding you?

Hope you can say a bit more about what you are thinking :) thanks