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 in terms of $W_{t}^{1}$ and $W_{t}^{2}$? Perhaps it can be reduced to a linear SDE with a suitable function?

Any help would be greatly appreciated.


1 Answer 1


When $W_1$ and $W_2$ are independent, $Y$ is equal in law to a Brownian Motion:

  1. It is obviously a local martingale under its natural filtration, and
  2. From Ito isometry, $$\mathbf{E}Y_t^2 = \mathbf{E} \int_0^t A(...)^2 + B(...)^2 ds = t.$$ Then, from Levy's theorem (http://almostsure.wordpress.com/2010/04/13/levys-characterization-of-brownian-motion/), these two conditions imply that $Y$ is a Brownian Motion.

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