# 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:

- ${x}_{n}+{y}_{n}=O({n}^{\mathrm{max}\{\alpha ,\beta \}})$,
- ${x}_{n}{y}_{n}=O({n}^{\alpha +\beta})$,
- ${x}_{n}^{\beta}=O({n}^{\alpha \beta})$, 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 ${\mathrm{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 $\epsilon >0$, ${\mathrm{lim}}_{n\to \mathrm{\infty}}\mathrm{Pr}(\left|{X}_{n}\right|>0)=0$. 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 $$d{X}_{t}=\mu {X}_{t}\phantom{\rule{thickmathspace}{0ex}}\mathrm{dt}+\sigma {X}_{t}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{dW}}_{t}$$ 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 $[0,1]$ 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
$$dX(t)=-\theta (X(t)-\mu )\phantom{\rule{thickmathspace}{0ex}}\mathrm{dt}+\sigma \phantom{\rule{thickmathspace}{0ex}}\mathrm{dW}(t)$$
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.