A stochastic differential equation (SDE) \(\displaystyle dX_t = f(t,X_t) \, dt + g(t,X_t) \, dW_t \) is, in fact, not a differential equation at all, but only a symbolic representation for the stochastic integral equation \(\displaystyle X_t = X_{t_0} + \int _{t_0}^t f(s,X_s) \, ds + \int _{t_0}^t g(s,X_s) \, dW_s, \) where the first integral is a deterministic Riemann integral for each sample path. The second integral is an Itô stochastic integral, which is defined as the mean-square limit of sums of products of the integrand g evaluated at the start of each discretization subinterval times the increment of the Wiener process W t (which is often called a Brownian motion). It is not possible to define this stochastic integral pathwise as a Riemann-Stieltjes integral, because the sample paths of a Wiener process, although continuous, are nowhere differentiable and not even of bounded variation on any bounded time interval.