Given a random variable X~N(0,1). What is the correlation coefficient between X and Y=X^2. The simple solution: the square of a Gaussian process is itself a Gaussian process. In this case Y has zero mean and unit variance Y~N(0,1) and thus the correlation coefficient between X and Y is 1. Is there a way of seeing this using Ito's lemma, as follows: dX = dW, dY = 2XdX+(dX)^2. Clearly, <dX>=<dW>=0, <(dX)^2>=dt, <dY>=2<X dX> + dt and <dX dY> = 2<X (dX)^2> + O(dt^1.5) ~ 2<X (dX)^2>. How does one prove that 2<X (dX)^2> = dt?
Last edited by AlanB
on January 31st, 2012, 11:00 pm, edited 1 time in total.