Stochastic Processes — Mathematics Knowledge System

Overview

A stochastic process is a family of random variables {X_t}_{t ∈ T} indexed by a parameter set T (usually time) and defined on a common probability space. The subject provides the rigorous framework for modelling random evolution: from discrete Markov chains to continuous-time diffusions, from queuing systems to financial models. Key concepts include sample paths, filtrations, stopping times, and the distinction between finite-dimensional distributions and the full process.

Intuition

A stochastic process is a random function of time. Think of recording the temperature every second — you get a sequence of random variables. The full object — not just each individual measurement but their joint distribution and mutual dependence — is the stochastic process. Its realisation (a single observed path) is called a sample path or trajectory.

Formal Definition

Definition

Let (Ω, F, P) be a probability space and (S, S) a measurable state space.

X = \{X_t : t \in T\}, \quad X_t : \Omega \to S \text{ measurable for each } t

Stochastic process as a family of random variables

SP-1

X : \Omega \times T \to S, \quad (\omega, t) \mapsto X_t(\omega)

Jointly measurable view of the process

SP-2

\mathcal{F}_t = \sigma(X_s : s \leq t), \quad t \in T

Natural filtration generated by the process

SP-3

Notation

Notation	Meaning
$\{X_t\}_{t \in T}$	Stochastic process indexed by T
$\mathcal{F}_t$	σ-algebra of events observable up to time t
$\tau$	Stopping time: {τ ≤ t} ∈ F_t for all t

Properties

Modification

Y \text{ is a modification of } X \text{ if } P(X_t = Y_t) = 1 \text{ for each fixed } t

Condition: Does not imply a.s. equal sample paths

Indistinguishable

P(X_t = Y_t \; \forall t) = 1

Condition: Stronger than modification; requires separability or right-continuity

Markov property

P(X_{t+s} \in A \mid \mathcal{F}_t) = P(X_{t+s} \in A \mid X_t)

Condition: Future depends only on present state, not history

Stationarity

(X_{t_1+h}, \ldots, X_{t_k+h}) \overset{d}{=} (X_{t_1}, \ldots, X_{t_k})

Condition: Distribution invariant under time shifts

Theorems

Theorem 1: Kolmogorov Extension Theorem

\text{A consistent family of finite-dimensional distributions } \{\mu_{t_1,\ldots,t_n}\} \text{ determines a unique probability measure on } (S^T, \mathcal{S}^{\otimes T})

Theorem 2: Kolmogorov Continuity Criterion

\text{If } E[|X_t - X_s|^\alpha] \leq C|t-s|^{1+\beta} \text{ for some } \alpha, \beta, C > 0, \text{ then } X \text{ has a continuous modification}

Applications

Asset price models (geometric Brownian motion, Heston, SABR) are continuous-time stochastic processes.

Worked Examples

N_t counts the number of events in [0,t]. By definition N_0 = 0, increments N_t - N_s ~ Poisson(λ(t-s)) for s<t, and non-overlapping increments are independent.
$N_t - N_s \sim \text{Poisson}(\lambda(t-s))$
For Poisson(λ(t-s)): mean = variance = λ(t-s).
$E[N_t] = \lambda t, \quad \text{Var}(N_t) = \lambda t$
The compensated process M_t = N_t - λt is a martingale:
$E[M_t \mid \mathcal{F}_s] = M_s + E[N_t - N_s] - \lambda(t-s) = M_s$

Answer: E[N_t] = Var(N_t) = λt; the compensated Poisson process M_t = N_t - λt is a martingale.

Practice Problems

Difficulty 7/10

Prove that for a stochastic process with independent increments and X_0 = 0, the natural filtration makes it a martingale if and only if E[X_t] = 0 for all t.

Difficulty 8/10

Let X_n be i.i.d. with E[X_1] = 0 and E[X_1²] = σ². Define M_n = (Σ_{k=1}^n X_k)² - nσ². Show M_n is a martingale.

Difficulty 9/10

State the Kolmogorov extension theorem precisely and explain why it is needed (i.e. what goes wrong without it).

Common Mistakes

Common Mistake

A modification is the same as an indistinguishable process

Two processes can be modifications of each other (equal in distribution at each fixed t) while having very different sample paths (e.g. one right-continuous, one not). Indistinguishability requires P(∀t: X_t = Y_t) = 1.

Common Mistake

The natural filtration is always right-continuous

The natural filtration F_t = σ(X_s: s≤t) need not be right-continuous. One usually works with the augmented filtration F_t⁺ = ∩_{s>t} F_s made right-continuous and complete.

Historical Background

Bachelier (1900) modelled stock prices as stochastic processes in his Paris thesis, predating Einstein's 1905 paper on Brownian motion. Kolmogorov (1931) gave the first rigorous treatment via the Kolmogorov extension theorem, showing how finite-dimensional distributions determine a process on a product space. Doob (1953) placed the theory on a modern measure-theoretic foundation.

1900
Bachelier models stock prices as random walks
Louis Bachelier
1905
Einstein's theory of Brownian motion
Albert Einstein
1931
Kolmogorov extension theorem and diffusion equations
Andrei Kolmogorov
1953
Doob's Stochastic Processes provides the measure-theoretic foundation
Joseph Doob

Summary

A stochastic process is a family of random variables {X_t} indexed by time, on a common probability space.
The Kolmogorov extension theorem guarantees existence from consistent finite-dimensional distributions.
The natural filtration F_t encodes information up to time t; stopping times are adapted to it.
Kolmogorov's continuity criterion gives a verifiable condition for continuous modifications.
Important classes: Markov processes, martingales, processes with independent increments, stationary processes.

References

BookKaratzas, I. & Shreve, S. (1991). Brownian Motion and Stochastic Calculus (2nd ed.). Springer.
BookRevuz, D. & Yor, M. (1999). Continuous Martingales and Brownian Motion (3rd ed.). Springer.