# Stochastic Processes

## Definitions

### Sequence Order

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

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

Source: Davidson (1994, p. 31).

#### Properties of Sequence Order

If ${x}_{n}=O\left({N}^{\alpha }\right)$ and ${y}_{n}=O\left({n}^{\beta }\right)$, then:

• ${x}_{n}+{y}_{n}=O\left({n}^{\mathrm{max}\left\{\alpha ,\beta \right\}}\right)$,
• ${x}_{n}{y}_{n}=O\left({n}^{\alpha +\beta }\right)$,
• ${x}_{n}^{\beta }=O\left({n}^{\alpha \beta }\right)$, whenever ${x}_{n}^{\beta }$ is defined

Source: Davidson (1994, p. 32).

### Absolute Stochastic Order

A sequence of random variables $\left\{{X}_{n}\right\}$ is bounded in probability if for any $\epsilon >0$ there exists a number ${B}_{\epsilon }<\infty$ such that ${\mathrm{sup}}_{n}\mathrm{Pr}\left(|{X}_{n}|>{B}_{\epsilon }\right)<\epsilon$. In this case we write ${X}_{n}={O}_{p}\left(1\right)$.

A sequence of random variables $\left\{{X}_{n}\right\}$ converges to 0 in probability if for any $\epsilon >0$, ${\mathrm{lim}}_{n\to \infty }\mathrm{Pr}\left(|{X}_{n}|>0\right)=0$. In this case we write ${X}_{n}={o}_{p}\left(1\right)$.

Source: Davidson (1994, p. 187).

### Relative Stochastic Order

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

Source: Davidson (1994, p. 187).

Note: The use of ${O}_{p}\left(1\right)$ and ${o}_{p}\left(1\right)$ 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:

• $E\left[|{X}_{t}|\right]<\infty$,
• $E\left[{X}_{t+1}|{X}_{1},\dots ,{X}_{t}\right]={X}_{t}.$

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 $E\left[{X}_{t+1}|{X}_{1},\dots ,{X}_{t}\right]\ge {X}_{t}$. Similarly, a supermartingale satisfies $E\left[{X}_{t+1}|{X}_{1},\dots ,{X}_{t}\right]\le {X}_{t}$.

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, 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

$\mathrm{Pr}\left({W}_{{t}_{i}}-{W}_{{t}_{i-1}}\in {H}_{i},i=0,1,\dots ,k\right)=\prod _{i=0}^{k}\mathrm{Pr}\left({W}_{{t}_{i}}-{W}_{{t}_{i-1}}\in {H}_{i}\right).$

• For $s$ and $t$ with $0\le s, ${W}_{t}-{W}_{s}\sim N\left(0,t-s\right)$ 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 $d{X}_{t}=\mu {X}_{t}\phantom{\rule{thickmathspace}{0ex}}\mathrm{dt}+\sigma {X}_{t}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{dW}}_{t}$ 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 $\left[0,1\right]$ 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 $i\in \left\{1,\dots ,N\right\}$ is the sum of the first $i$ draws.

## Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process can be represented as a stochastic differential equation $dX\left(t\right)=-\theta \left(X\left(t\right)-\mu \right)\phantom{\rule{thickmathspace}{0ex}}\mathrm{dt}+\sigma \phantom{\rule{thickmathspace}{0ex}}\mathrm{dW}\left(t\right)$ with $r\left(0\right)={r}_{0}$ and where $W\left(t\right)$ is a Brownian motion (see above). It is the continuous-time analog of a discrete AR(1) process.