PSD from autocorrelation in MATLAB
There are two mistakes in your code/method. The first is the term $\sqrt{\Delta t}$ in your second formula; it should be replaced by $\Delta t$. The second is in the computation of the  power spectrum from the estimated auto-correlation. What you do is square the result of the FFT Y to obtain mY, but that's not correct. First of all, Y is complex-valued, and second, the Fourier transform of the auto-correlation is the power spectrum, so there's no need to square the outcome of the FFT, you just need to compute its magnitude.

In sum, replace the line

v*sqrt(sigma)*sqrt(dt).*normrnd(0,1);

by

v*sqrt(sigma)*dt.*normrnd(0,1);

and

mY = (Y).^2/N;

by

mY = abs(Y)/N;

This should give you the desired result, as shown in the figure below: