Stochastic Processes


Sequence Order

Let {x n} be a sequence of real numbers and let {a n} be a sequence of positive real numbers. The sequence x n is at most of order a n if there exists a constant B< such that |x n|/a nB for all n. In this case we write x n=O(a n).

The sequence x n is of order less than a n if {x n/a n} converges to zero as n. In this case we write x n=o(a n).

Source: Davidson (1994, p. 31).

Properties of Sequence Order

If x n=O(N α) and y n=O(n β), then:

Source: Davidson (1994, p. 32).

Absolute Stochastic Order

A sequence of random variables {X n} is bounded in probability if for any ε>0 there exists a number B ε< such that sup nPr(|X n|>B ε)<ε. In this case we write X n=O p(1).

A sequence of random variables {X n} converges to 0 in probability if for any ε>0, lim nPr(|X n|>0)=0. In this case we write X n=o p(1).

Source: Davidson (1994, p. 187).

Relative Stochastic Order

Let {Y n} be another sequence, stochastic or deterministic. If X n/Y n=O p(1), we write X n=O p(Y n) and say that X n is at most of order Y n in probability. If X n/Y n=o p(1), we write X n=o p(Y n) and say that X n is of order less than Y n in probability.

Source: Davidson (1994, p. 187).

Note: The use of O p(1) and o p(1) for stochastic order is due to Mann and Wald (1943) (Source: Davidson (1994, p. 187)).


A discrete-time martingale is a stochastic process X 1,X 2,X 3, with the following properties:

The first property states that the random variable X t is integrable while the second says that the conditional expectation of the next observation, given the complete history of realizations of the process up to time t, is simply equal to the previous value.

A submartingale is a sequence of integrable random variables such that E[X t+1|X 1,,X t]X t. Similarly, a supermartingale satisfies E[X t+1|X 1,,X t]X t.

A martingale is therefore both a submartingale and a supermartingale.

Brownian Motion

A Brownian motion, also called a Wiener process, is a continuous-time stochastic process W t, t0. It is named in honor of botanist Robert Brown who noted the seemingly random movements of particles suspended in fluid.


A Brownian motion is typically characterized in terms of its increments. Given two distinct points in time t and s with s<t, the increment is the change in the process between s and t, given by W tW s.

A Brownian motion has the following three properties:

Pr(W t iW t i1H i,i=0,1,,k)= i=0 kPr(W t iW t i1H i).

Note that characterizing the process in terms of increments is especially useful for empirical studies as the process can only feasibly be observed at a finite number of points in time and the observed intervals follow a well-known distribution.


Geometric Brownian Motion

A geometric Brownian motion with drift μ and volatility σ satisfies dX t=μX tdt+σX tdW t where W t is a Wiener process. μ is the percentage drift, the expected percentage change in X t per unit of time. σ is the percentage volatility, the expected standard deviation over one unit of time.


A Brownian motion is a continuous phenomenon that we can only sample at a finite number of points. We can then interpolate linearly between these sampled values to create a plot. One can achieve closer approximations by choosing successively smaller sampling intervals.

To simulate a standard Brownian motion on [0,1] with N intervals of length 1/N, draw a sequence of independent Normally distributed random variables with mean 0 and variance 1/N. Then the value of the Brownian motion at time t=i/N for i{1,,N} is the sum of the first i draws.

Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process can be represented as a stochastic differential equation dX(t)=θ(X(t)μ)dt+σdW(t) with r(0)=r 0 and where W(t) is a Brownian motion (see above). It is the continuous-time analog of a discrete AR(1) process.