# How to solve a SDE with an Ornstein Uhlenbeck process driving it?

Thanks for having a look. I have come across this stochastic differential equation (SDE) "in the wild" and I am just learning Ito calculus for a few months, so I would appreciate any help in this regard. The SDE is for the Stochastic process $$O_t$$ dependent upon another stochastic process $$X_t$$, given by: $$\frac{dO_t}{O_t} = (f(O_t)-X_t)dt,$$ where $$X_t$$ is the standard Ornstein-Uhlenbeck process with some mean $$m$$ (https://en.wikipedia.org/wiki/Ornstein-Uhlenbeck_process): $$dX_t = \gamma(m-X_t)dt + \sigma dW_t,$$ where $$dW_t$$ is the standard Weiner process. Any bright ideas how I can attack this problem and get a general solution of $$O_t$$ if at all possible? $$f(O_t)$$ is atmost a quadratic polynomial of $$O_t$$.

The above equation for $$O_t$$ is rather non standard and I am not sure if it is well defined at all and how to think about the mean, covariance or variance of the process $$O_t$$, further the cross quadratic variation of $$[dX_t dO_t]$$ is also unclear. Any insight about this is really welcome.

In particular I am trying to write down, using Ito's multidimensional lemma $$d\log(O_t)$$, is the quadratic variation $$[dO_t^2] =0$$ since there is no diffusion?

Further, can I write $$X_t dt$$ as some process $$X_t dt=dY_t= A(t)dt +A'(t)dW_t$$?

Finally, to consider $$X_tdt$$, I was using the general solution of $$X_t=\frac{\sigma}{\sqrt{2\gamma}}e^{-\gamma t}W_{e^{2\theta t}}$$. In this regard are the following manipulations possible:

1. Can one write $$W_{f(t)} = W_{t\frac{f(t)}{t}}=^{??} \sqrt{\frac{f(t)}{t}}W_t$$?
2. This one is probably wrong but I am not sure why. If we can write $$W_t=\int_0 ^t dW_s$$, can one write $$W_{f(t)} = \int^{f(t)}_0dW_s$$?

References, Ideas/comments/criticisms, everything is welcome. Thanks for any and all help.

Edit: There might be a hint in this paper if in the equations in the abstract the $$Z_t$$ process is just $$0$$, but it's hard for me to decipher the rest. https://research.sabanciuniv.edu/7102/1/soner2.pdf

• I wouldn't be very hopeful for a solution. Maybe study its Fokker-Planck equation? Commented Feb 18, 2022 at 22:23
• Thanks! I am having a look at the FP equations. Any further comments or insights regarding the other questions? Commented Feb 18, 2022 at 23:12

$$y'=y(f(y)+g(t)).$$
So the main issue here is dealing with the polynomial $$f(y)$$. We need more information on $$f$$ to solve this. Even for
$$y'=y^2+y+1+t$$