We have seen in the previous chapter how to define a stochastic process using a sequence of Gaussian infinitesimal increments dW, and how to obtain new stochastic processes as the solutions to stochastic differential equations driven by this Gaussian noise. We have seen that a stochastic process is a random variable x(t) at each time t, and we have calculated its probability density, P(x, t), average 〈x(t)〉 and variance V[x(t)]. In this chapter we will discuss and calculate some further properties of a stochastic process, in particular its sample paths, two-time correlation function, and power spectral density (or power spectrum). We also discuss the fact that Wiener noise is white noise.

Sample paths

A sample path of the Wiener process is a particular choice (or realization) of each of the increments dW. Since each increment is infinitesimal, we cannot plot a sample path with infinite accuracy, but must choose some time discretization Δt, and plot W(t) at the points nΔt. Note that in doing so, even though we do not calculate W(t) for the points in-between the values nΔt, the points we plot do lie precisely on a valid sample path, because we know precisely the probability density for each increment ΔW on the intervals Δt. If we chose Δt small enough, then we cannot tell by eye that the resolution is limited.