Stochastic process

Introduction

When you have a function $f$ , interpreted as a table

$x$	$f (x)$
0	1
1	2
2	5
you can think of the "discrete derivative" $Δ f$ as the table

$x$	$f (x)$	$Δ f (x)$
0	1	-
1	2	1
2	5	3
The usual derivative is no a "real" thing: it emerges when you take increasingly smaller steps $Δ x$ for the $x$ column and you consider the ratio $Δ f (x) / Δ x$ .
The problem of recover $f$ from $Δ f$ , or even from $d f / d x$ is called integration. So far, so good. But, what if $Δ f (x) / Δ x$ does not correspond to a fixed list of values but, instead, to a random list?

Formal definition

A stochastic process is a collection of random variables ${X_{t}}_{t \in T}$ defined on a common probability space $(Ω, F, P)$ , where:

$Ω$ is the sample space (set of all possible outcomes),
$F$ is a $σ$ -algebra (set of possible events),
$P$ is a probability measure,
$T$ is an index set (often representing time, e.g., $T = N$ or $T = [0, \infty)$ ),
For each $t \in T$ , $X_{t} : Ω \to S$ is a random variable mapping outcomes to states in a measurable space $S$ (e.g., $S = R$ ).

A simple example

Let’s explore a basic stochastic process using a random walk.
You start at position $0$ . At each step $t = 1, 2, \dots$ , you flip a fair coin:

Heads (H): Move +1 (right).
Tails (T): Move −1 (left).
Your position at time $t$ is a random variable $P_{t}$ , forming a sequence $P_{0}, P_{1}, P_{2}, \dots$ .
The set ${P_{t}}$ , a simple symmetric random walk, is a fundamental stochastic process where each $P_{t}$ depends on past coin flips.

Sample Space $Ω$

$Ω$ contains all infinite sequences of H/T (e.g., $(H, T, H, \dots)$ ). For finite $t$ , we restrict to the first $t$ flips. I.e.,

Ω = {H, T}^{N} .

The Random Variable $P_{2}$

At each step, you move +1 (if Heads) or −1 (if Tails). Starting at $P_{0} = 0$ , after 2 steps, the possible positions are:

$(H, H)$ : $0 \to + 1 \to + 2$
$(H, T)$ : $0 \to + 1 \to 0$
$(T, H)$ : $0 \to - 1 \to 0$
$(T, T)$ : $0 \to - 1 \to - 2$

Thus, $P_{2}$ can take values in ${- 2, 0, + 2}$ .
Let $ω = (ω_{1}, ω_{2})$ be the outcomes of the first two flips (where $ω_{i} \in {H, T}$ ). Then:

P_{2} (ω) = {\begin{cases} + 2 & if ω = (H, H), \\ 0 & if ω = (H, T) or (T, H), \\ - 2 & if ω = (T, T) . \end{cases}

Another example

Now imagine shrinking the step size $Δ t \to 0$ and the step size in position $Δ P \to 0$ , in just the right way, so that the resulting process still has randomness but now varies continuously in time.

The result — in the limit — is the Brownian motion $B_{t}$ .

Important property

Any function of a stochastic process is itself a stochastic process.

More formally, if ${X (t)}_{t \geq 0}$ is a stochastic process and $g$ is a measurable function, then ${Y (t)}_{t \geq 0}$ defined by:

Y (t) = g (t, X (t))

is also a stochastic process.

This is because:

For each fixed time $t$ , $Y (t)$ is a random variable (since it's a function of the random variable $X (t)$ )
The collection ${Y (t)}$ indexed by time $t$ forms a new stochastic process