Question: The space ? of Brownian trajectories remains unchanged under the change of scale, however, the same sets of trajectories acquire different probabilities under the Wiener measure corresponding to wα(s). What is the relationship between the two Wiener measures?

If the dependent variable in an Itô equation, x(t), is transformed to y(t) = f(x(t), t), the differential equation for y(t) is found from Itô's formula. We consider here two applications of this principle, first, to reducing equations to explicitly solvable form and second, to change coordinates. Equations of the form

If x(s) = x, we denote the solution xx,s(t). When the coefficients a(t) and b(t) are deterministic functions, the solution is a Gaussian process with conditional mean.