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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

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

1 Answer 1

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This actually just a first order ode

$$y'=y(f(y)+g(t)).$$

So the main issue here is dealing with the polynomial $f(y)$. We need more information on $f$ to solve this. Even for

$$y'=y^2+y+1+t$$

the solution can be quite complicated

enter image description here

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