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I'm trying to complete a course on SDEs and I need to solve two stochastic differential equations. They are supposed to be easy, but I'm still a beginner and to be honest I'm quite stuck.

The pair of equations are the following:

$dX_t = dt + dW_t^x , \quad X_0=0$

$dY_t = X_tdW_t^y, \quad Y_0=0$

Where $dW_x^t$ and $dW_t^y$ are uncorrelated Brownian motions.

The first equation can be solved directly to obtain $X_t = t + W_t^x$. And substituting in the second equation I obtain $dY_t = t dW_t^y + W_t^x dW_t^y$

The problem is that I don’t know how to integrate $W_t^x$ with respect to $dW_t^y$. This integral seems to be beyond my course level, but I cannot find any other way to solve these equations.

Any hint or comment will be greatly appreaciated! thanks!

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