Coupled linear stochastic differential equations

Tyler_D
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Homework Statement
In the following system of coupled stochastic differential equations, solve for ##x## (note that ##dW## is a Wiener process)
Relevant Equations
[tex]
dx = (\omega y - \gamma x)dt \\
dy = (-\omega x - \gamma y)dt + gdW \\
[/tex]
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?
 
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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.
 
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Delta2 said:
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 (?)
 
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 said:
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
 
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