Let be a sequence of real numbers and let be a sequence of positive real numbers. The sequence is at most of order if there exists a constant such that for all . In this case we write .
The sequence is of order less than if converges to zero as . In this case we write .
Source: Davidson (1994, p. 31).
Properties of Sequence Order
If and , then:
- , whenever is defined
Source: Davidson (1994, p. 32).
Absolute Stochastic Order
A sequence of random variables is bounded in probability if for any there exists a number such that . In this case we write .
A sequence of random variables converges to 0 in probability if for any , . In this case we write .
Source: Davidson (1994, p. 187).
Relative Stochastic Order
Let be another sequence, stochastic or deterministic. If , we write and say that is at most of order in probability. If , we write and say that is of order less than in probability.
Source: Davidson (1994, p. 187).
Note: The use of and for stochastic order is due to Mann and Wald (1943) (Source: Davidson (1994, p. 187)).
A discrete-time martingale is a stochastic process with the following properties:
The first property states that the random variable 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 , is simply equal to the previous value.
A submartingale is a sequence of integrable random variables such that . Similarly, a supermartingale satisfies .
A martingale is therefore both a submartingale and a supermartingale.
A Brownian motion, also called a Wiener process, is a continuous-time stochastic process , . 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 and with , the increment is the change in the process between and , given by .
A Brownian motion has the following three properties:
The initial condition is known: a.s.
The increments are independent. That is, if , then
- For and with , where denotes the normal distribution.
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 and volatility satisfies where is a Wiener process. is the percentage drift, the expected percentage change in per unit of time. 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 with intervals of length , draw a sequence of independent Normally distributed random variables with mean and variance . Then the value of the Brownian motion at time for is the sum of the first draws.
The Ornstein-Uhlenbeck process can be represented as a stochastic differential equation with and where is a Brownian motion (see above). It is the continuous-time analog of a discrete AR(1) process.
Billingsley, P. (1995). Probability and Measure.
Chung, K. L. (2001). A Course in Probability Theory. Academic Press, London.
Cox J. C., J. Ingersoll, and S. Ross (1985). A Theory of the Term Structure of Interest Rates. Econometrica 53, 385–407.
Davidson, J. (1994). Stochastic Limit Theory. Oxford University Press.
Mann, H.B. and A. Wald (1943). On stochastic limit and order relationships. Annals of Mathematical Statistics 14, 217–226.