I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$

where $W_{t}^{1}$ and $W_{t}^{1}$ are two independent brownian motions and $$A(x,y)=a\frac{\Phi(y)\Phi(-y)e^{0.5(y^2-x^2)}+\Phi(x)\Phi(-x)e^{0.5(x^2-y^2)}}{1+a(1-2\Phi(x))(1-2\Phi(y))}, \ \ \ \ \ \ \ \ B=\sqrt{1-A^2}$$

It seems unlikely, but I was wondering if there may be an explicit solution to this stochastic differential equation. Perhaps it can be reduced to a linear SDE with a suitable function?

Any help would be greatly appreciated.

  • $\begingroup$ This is a monster but you may have help with the fact $$\Phi(-x) = 1 - \Phi(x)$$ and $$\sqrt{2\pi}\Phi'(x) = \mathrm{e}^{-0.5x^2}$$. Good luck with your work! $\endgroup$ – Chinny84 Aug 25 '14 at 8:20
  • $\begingroup$ further to my comment $\left(dY_t\right)^2=dt$. $\endgroup$ – Chinny84 Aug 25 '14 at 8:30
  • $\begingroup$ Yeah, Ive already used all of that in the derivation of A. Any comments relating the derivation of an actual explicit solution? $\endgroup$ – Math Girl Aug 25 '14 at 10:52
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    $\begingroup$ @Math Girl : The last comment of Chinny84 is very strong because if a solution exists it tells you that is has to be a Brownian motion tank's to Lévy's theorem on continuous local martingale, which is quite explicit in my opinion. Best regards $\endgroup$ – TheBridge Aug 25 '14 at 15:26
  • $\begingroup$ Yes it is and I was already aware of this. My question is whether there exists an explicit solution in terms of $W_{t}^{1}$ and $W_{t}^{2}$. Sorry, I should have made my question more clear. $\endgroup$ – Math Girl Aug 26 '14 at 7:29

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