Stochastic process

See stochastic calculus.

Formal definition

A stochastic process is a collection of random variables $\{X_t{\}}_{t\in T}$ defined on a common probabilistic space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, where:

• $\mathrm{\Omega }$ is the sample space (set of all possible outcomes),
• $\mathcal{F}$ is a $\sigma$-algebra (set of possible events),
• $\mathbb{P}$ is a probability measure,
• $T$ is an index set (often representing time, e.g., $T=\mathbb{N}$ or $T=[0,\mathrm{\infty })$).

Interpretation: Curve-Valued Random Variables

• In classical probability, a random variable assigns a real number to each outcome $\omega \in \mathrm{\Omega }$.
• A stochastic process (or, loosely speaking, a curve-valued random variable) assigns an entire function (a curve $t\mapsto x(t)$) to each $\omega$. Formally:

A simple example: random walk

Let’s explore a basic stochastic process using a random walk (it is an example of Markov process and Markov chain).
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 random variable $P_t$ depends on past coin flips. Who is who in this example?

Sample Space $\mathrm{\Omega }$

$\mathrm{\Omega }$ 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.,

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 $\omega =({\omega }_1,{\omega }_2)$ be the outcomes of the first two flips (where ${\omega }_i\in \{H,T\}$ and $\omega \in \mathrm{\Omega }$). Then:

Another example

Now imagine shrinking the step size $\mathrm{\Delta }t\to 0$ and the step size in position $\mathrm{\Delta }P\to 0$, 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 $\{X(t){\}}_{t\ge 0}$ is a stochastic process and $g$ is a measurable function, then $\{Y(t){\}}_{t\ge 0}$ defined by:

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

Curve-valued random variables

Given a stochastic process $X_t$ on $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, it let us to identify each $\omega \in \mathrm{\Omega }$ with the curve

Therefore, we can consider $\mathrm{\Omega }$ as a subset of

We can endow this set with a $\sigma$-algebra and then $X_t$ induces a probability distribution by means of the pushforward measure.