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I have came across the following stochastic integrals: $$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$ which was claimed to be standard normally distributed ($W_x$ and $W_y$ are independent standard Wiener processes). I simulated it and it is clearly the case, but I cannot show it formally. Any suggestions?

Thanks, Tamas

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    $\begingroup$ are you sure you don't miss an expectation in the denominator? It should help to use the properties of the Itô integral $I_t = \int_0^t W^x_u dW^y_u$ (mean is zero and variance given by Itô isometry) and then specify to the case $t=1$. $\endgroup$ – Quantuple Apr 11 '18 at 15:27
  • $\begingroup$ Any background information for this question? $\endgroup$ – Gordon Apr 12 '18 at 19:58

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