Week 3 Continuous time Gaussian processes

This week we take the limit ∆t → 0. The limit is a process Xt that is defined for all t in some range, such as t ∈ [0, T ]. The process takes place in continuous time. This week, Xt is a continuous function of t. The process has continuous sample paths. It is natural to suppose that the limit of a Markov process is a continuous time Markov process. The limits we obtain this week will be either Brownian motion or the Ornstein Uhlenbeck process. Both of these are Gaussian. We will see how such processes arise as the limits of discrete time Gaussian processes (week 1) or discrete time random walks and urn processes (week 2). The scalings of random processes are different from the scalings of differentiable paths you see in ordinary ∆t→ 0 calculus. Consider a small but non-zero ∆t. The net change in X over that interval is ∆X = Xt+∆t−Xt. If a path has well defined velocity, Vt = dX/dt, then ∆X ≈ V∆t. Mathematicians say that ∆X is on the order of ∆t, because ∆X is approximately proportional to ∆t for small ∆t. In this linear scaling, reducing ∆t by a factor of 2 (say) reduces ∆X approximately by the same factor of 2. Brownian motion, and the Ornstein Uhlenbeck process, have more complicated scalings. There is one scaling for ∆X, and a different one for E[∆X]. The change itself, ∆X, is on the order of √ ∆t. If ∆t is small, this is larger than the ∆t scaling differentiable processes have. Brownian motion moves much further in a small amount of time than differentiable processes do. The change in expected value is smaller, on the order of ∆t. It is impossible for the expected value to change by order √ ∆t, because the total change in the expected value over a finite time interval would be infinite. The Brownian motion manages to have ∆X on the order of √ ∆t through cancellation. The sign of ∆X goes back and forth, so that the net change is far smaller than the sum of |∆X| over many small intervals of time. That is |∆X1 + ∆X2 + · · · | << |∆X1|+ |∆X2|+ · · · .