Stochastic Processes

Definitions

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 $B<\mathrm{\infty }$ such that $\left|x_n\right|/a_n\le B$ 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 $n\to \mathrm{\infty }$. In this case we write $x_n=o(a_n)$.

Source: Davidson (1994, p. 31).

Properties of Sequence Order

If $x_n=O(N^{\alpha })$ and $y_n=O(n^{\beta })$, then:

• $x_n+y_n=O(n^{\max \{\alpha ,\beta \}})$,
• $x_n{y}_n=O(n^{\alpha +\beta })$,
• $x_n^{\beta }=O(n^{\alpha \beta })$, whenever $x_n^{\beta }$ is defined

Source: Davidson (1994, p. 32).

Absolute Stochastic Order

A sequence of random variables $\{X_n\}$ is bounded in probability if for any $\epsilon >0$ there exists a number $B_{\epsilon }<\mathrm{\infty }$ such that ${\sup }_n\mathrm{Pr}(\left|X_n\right|>B_{\epsilon })<\epsilon$. 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 $\epsilon >0$, ${\lim }_{n\to \mathrm{\infty }}\mathrm{Pr}(\left|X_n\right|>0)=0$. 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)).

Martingales

A discrete-time martingale is a stochastic process $X_1,X_2,X_3,\dots$ with the following properties:

• $E[|X_t|]<\mathrm{\infty }$,
• $E[X_{t+1}|X_1,\dots ,X_t]=X_t.$

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 $E[X_{t+1}|X_1,\dots ,X_t]\ge X_t$. Similarly, a supermartingale satisfies $E[X_{t+1}|X_1,\dots ,X_t]\le X_t$.

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 $W_t$, $t\ge 0$. It is named in honor of botanist Robert Brown who noted the seemingly random movements of particles suspended in fluid.

Characterization

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 $W_t-W_s$.

A Brownian motion has the following three properties:

• The initial condition is known: $W_0=0$ a.s.

• The increments are independent. That is, if $0\le t_0\le t_1\le \cdots \le t_k$, then

$$\mathrm{Pr}(W_{t_i}-W_{t_{i-1}}\in H_i,i=0,1,\dots ,k)=\prod _{i=0}^k\mathrm{Pr}(W_{t_i}-W_{t_{i-1}}\in H_i).$$

• For $s$ and $t$ with $0\le s<t$, $$W_t-W_s\sim N(0,t-s)$$ where $N$ 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.

Variants

Geometric Brownian Motion

A geometric Brownian motion with drift $\mu$ and volatility $\sigma$ satisfies $$d{X}_t=\mu X_t\mathrm{dt}+\sigma X_t{\mathrm{dW}}_t$$ where $W_t$ is a Wiener process. $\mu$ is the percentage drift, the expected percentage change in $X_t$ per unit of time. $\sigma$ is the percentage volatility, the expected standard deviation over one unit of time.

Simulation

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 $i\in \{1,\dots ,N\}$ is the sum of the first $i$ draws.

Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process can be represented as a stochastic differential equation $$dX(t)=-\theta (X(t)-\mu )\mathrm{dt}+\sigma \mathrm{dW}(t)$$ 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.

References

• 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.