I am working with stochastic PDES:
If one takes white Gaussian noise and subtract off the mean of the discretized noise, does this impact the Ito change of variables formula? I know that this shift does not change the variance.
The way I have thought about this is suppose I take a SDE $$\mathrm{d}X_t(r) = A(X_t(r))\mathrm{d}t+\sigma(X_t(r))\mathrm{d}W_t(r)$$
$$\langle W_t(r) \rangle = 0$$
$$\langle W_t(r)W_{t'}(r') \rangle = 2\frac{\Delta t}{\Delta V}\delta(r-r')$$
Since the SDE is already defined with a mean of zero, any non-zero mean in an instantaneous generation of a noise vector could be interpreted as a discretization error. I am working with systems requiring mass conservation, so a few errors like these can lead to mass conservation issues.