Stochastic Processes — Brownian Motion and Martingales

⚡ Key Takeaways

Brownian motion is the canonical continuous-time stochastic process with independent, stationary, Gaussian increments and continuous paths; a martingale is a process whose conditional expected future value equals its present value. Key result: Brownian motion is the prototypical martingale, and the two notions are linked by Lévy's characterization (a continuous martingale with quadratic variation $t$ is Brownian motion) and by the Itô stochastic calculus. Applies to: mathematical finance, statistical physics, PDE theory, filtering, population genetics. Originated with Brown (1827), Einstein (1905), Bachelier (1900), formalized by Wiener (1923), Lévy, Doob, and Itô (1940s–50s).

💡 The Core Insight

Randomness accumulated continuously over time, when made to have independent increments of variance proportional to elapsed time, forces a Gaussian limit (via the CLT) and a path so rough it has no derivative but a perfectly predictable "energy" $\langle W \rangle_t = t$ . The martingale property—conditional fairness—is what allows this roughness to be tamed: stochastic integrals against martingales are again martingales, and quadratic variation replaces the chain rule, giving the Itô formula. Brownian motion is simultaneously a Markov process, a Gaussian process, and a continuous martingale; these three identities, when forced to coexist, pin it down uniquely.

DefinitionFiltered Probability Space

A filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\ge 0}, \mathbb{P})$ consists of a probability space together with a filtration: an increasing family of $\sigma$ -algebras $\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}$ for $s \le t$ . It satisfies the usual conditions if it is right-continuous ( $\mathcal{F}_t = \bigcap_{u>t}\mathcal{F}_u$ ) and $\mathcal{F}_0$ contains all $\mathbb{P}$ -null sets. A process $X$ is adapted if $X_t$ is $\mathcal{F}_t$ -measurable for all $t$ .

DefinitionBrownian Motion

A real-valued process $\{W_t\}_{t\ge 0}$ is a standard Brownian motion (Wiener process) if: 1. $W_0 = 0$ almost surely; 2. independent increments: for $0\le t_0 < t_1 < \cdots < t_n$ , the increments $W_{t_{i}}-W_{t_{i-1}}$ are independent; 3. stationary Gaussian increments: $W_t - W_s \sim \mathcal{N}(0, t-s)$ for $s<t$ ; 4. continuous paths: $t \mapsto W_t$ is continuous almost surely.

1

\quad \mathbb{P}(W_t - W_s \in dx) = \frac{1}{\sqrt{2\pi(t-s)}}\exp\!\left(-\frac{x^2}{2(t-s)}\right)dx

Derivation hint: this is the transition density; it solves the heat equation $\partial_t p = \tfrac12 \partial_{xx} p$ with $p(0,\cdot)=\delta_0$ , encoding the Gaussian increment law.

DefinitionMartingale, Submartingale, Supermartingale

An adapted, integrable ( $\mathbb{E}|M_t|<\infty$ ) process is a martingale if

2

\quad \mathbb{E}[M_t \mid \mathcal{F}_s] = M_s \quad \text{for all } s \le t.

Derivation hint: replace " $=$ " with " $\ge$ " for a submartingale and " $\le$ " for a supermartingale; the equality is the tower property applied to a fair-game increment with zero conditional mean. The defining relation says the best forecast of $M_t$ given information up to $s$ is the current value $M_s$ .

TheoremBrownian Motion is a Martingale; Associated Martingales

Let $W$ be a standard Brownian motion with its natural filtration. Then each of the following is a martingale:

3

\quad W_t,\qquad W_t^2 - t,\qquad \exp\!\left(\lambda W_t - \tfrac12 \lambda^2 t\right)\ (\lambda\in\mathbb{R}).

Derivation hint: for $W_t$ , use independent increments and zero mean; for $W_t^2-t$ , expand $W_t^2=(W_t-W_s)^2+2W_s(W_t-W_s)+W_s^2$ and take conditional expectation; for the exponential, use the Gaussian MGF $\mathbb{E}[e^{\lambda(W_t-W_s)}\mid\mathcal F_s]=e^{\lambda^2(t-s)/2}$ .

TheoremLévy's Characterization

Let $M$ be a continuous local martingale with $M_0=0$ and quadratic variation $\langle M\rangle_t = t$ . Then $M$ is a standard Brownian motion.

4

\quad \langle M\rangle_t = t \ \text{and } M \text{ cont. local mart.} \ \Longrightarrow\ M_t \stackrel{d}{=} W_t

Derivation hint: apply Itô's formula to $\exp(i\lambda M_t + \tfrac12\lambda^2\langle M\rangle_t)$ ; the stochastic integral term is a martingale, forcing the conditional characteristic function to be that of $\mathcal N(0,t-s)$ , the law of an increment.

ProofQuadratic Variation of Brownian Motion

We show that over $[0,t]$ with partition $\Pi_n = \{0=t_0<\cdots<t_n=t\}$ of mesh $\|\Pi_n\|\to 0$ , the sums of squared increments converge to $t$ .

Step 1. Define $Q_n = \sum_{i=1}^n (W_{t_i}-W_{t_{i-1}})^2$ . By stationary increments, $\mathbb{E}[(W_{t_i}-W_{t_{i-1}})^2] = t_i - t_{i-1}$ , so

5

\quad \mathbb{E}[Q_n] = \sum_{i=1}^n (t_i - t_{i-1}) = t.

Derivation hint: variance of a centered Gaussian increment equals its time span; sum telescopes.

Step 2. Increments are independent, so the variance of $Q_n$ adds:

6

\quad \operatorname{Var}(Q_n) = \sum_{i=1}^n \operatorname{Var}\big((W_{t_i}-W_{t_{i-1}})^2\big) = 2\sum_{i=1}^n (t_i-t_{i-1})^2.

Derivation hint: for $Z\sim\mathcal N(0,\sigma^2)$ , $\operatorname{Var}(Z^2)=2\sigma^4$ ; here $\sigma^2=t_i-t_{i-1}$ .

Step 3. Bound the sum by the mesh:

7

\quad \operatorname{Var}(Q_n) = 2\sum_{i=1}^n (t_i-t_{i-1})^2 \le 2\,\|\Pi_n\|\sum_{i=1}^n (t_i-t_{i-1}) = 2\,\|\Pi_n\|\, t \to 0.

Derivation hint: factor out the largest interval; the remaining sum telescopes to $t$ .

Step 4. Since $\mathbb{E}[Q_n]=t$ and $\operatorname{Var}(Q_n)\to 0$ , Chebyshev gives $Q_n \to t$ in $L^2$ (and in probability). For nested partitions or summable meshes, the convergence is almost sure. Hence $\langle W\rangle_t = t$ . $\blacksquare$

∎

Proof

W_t^2 - t

is a Martingale

Step 1. For $s<t$ write $W_t = W_s + (W_t - W_s)$ , so

8

\quad W_t^2 = W_s^2 + 2W_s(W_t - W_s) + (W_t - W_s)^2.

Derivation hint: binomial expansion of $(W_s + \Delta)^2$ .

Step 2. Take $\mathbb{E}[\cdot\mid\mathcal F_s]$ . Since $W_s$ is $\mathcal F_s$ -measurable and the increment is independent of $\mathcal F_s$ with mean $0$ and variance $t-s$ :

9

\quad \mathbb{E}[W_t^2\mid\mathcal F_s] = W_s^2 + 2W_s\cdot 0 + (t-s) = W_s^2 + (t-s).

Derivation hint: pull out $\mathcal F_s$ -measurable factors; use $\mathbb{E}[\Delta\mid\mathcal F_s]=0$ and $\mathbb{E}[\Delta^2\mid\mathcal F_s]=t-s$ .

Step 3. Subtract $t$ from both sides:

10

\quad \mathbb{E}[W_t^2 - t \mid \mathcal F_s] = W_s^2 + (t-s) - t = W_s^2 - s.

Derivation hint: rearrange; the result is the value at $s$ , confirming the martingale property. $\blacksquare$

∎

TheoremItô's Formula

Let $f\in C^2(\mathbb{R})$ and $W$ a Brownian motion. Then a.s.,

11

\quad f(W_t) = f(W_0) + \int_0^t f'(W_s)\,dW_s + \frac12\int_0^t f''(W_s)\,ds.

Derivation hint: Taylor-expand $f(W_{t_{i}})-f(W_{t_{i-1}})$ to second order; the first-order terms build the Itô integral, while $\sum f''(\cdot)(W_{t_i}-W_{t_{i-1}})^2 \to \int f''\,d\langle W\rangle = \int f''\,ds$ because $(\,dW\,)^2 = dt$ .

Figure

Brownian sample path with its variance envelope

\pm\sqrt{t}

mathematician · Norbert Wiener | 1894–1964

Wiener gave the first rigorous mathematical construction of Brownian motion (1923), proving that a probability measure—the Wiener measure—exists on the space $C[0,\infty)$ of continuous functions realizing Einstein's physical predictions. He established the continuity and nowhere-differentiability of almost every path, and developed the abstract machinery (the Wiener integral) that anticipated Itô's calculus. Beyond probability, Wiener founded cybernetics, contributed to harmonic analysis (the Paley–Wiener theorems), and shaped wartime work on prediction and filtering. The Wiener process bears his name precisely because he transformed a physical phenomenon into a well-defined object of pure mathematics.

🧠 Intuition

Picture a drunkard taking $n$ steps of size $\pm 1/\sqrt n$ per unit time. After time $t$ , position is a sum of $\sim nt$ tiny independent steps; the CLT makes this Gaussian with variance $t$ . Send $n\to\infty$ : the discrete walk becomes Brownian motion. The variance growing linearly in time, not quadratically, is the signature of diffusion—random cancellation means displacement scales like $\sqrt t$ , not $t$ . The martingale property is "no information advantage": knowing the entire past gives no edge in predicting the next increment's direction, because $\mathbb{E}[W_t - W_s \mid \mathcal F_s] = 0$ . The roughness ( $(\,dW\,)^2 = dt$ ) is what distinguishes stochastic from ordinary calculus: a Brownian path wiggles so much that second-order effects survive in the limit, producing the extra $\tfrac12 f''$ term in Itô's formula. Think of it as a deterministic "drift" generated purely by the path's roughness.

AE

Albert Einstein

1879–1955

In his 1905 annus mirabilis paper on Brownian motion, Einstein derived the diffusion equation for suspended particles and predicted that the mean-squared displacement grows linearly in time: $\langle x^2\rangle = 2Dt$ , with $D = k_B T/(6\pi\eta r)$ relating the diffusion constant to temperature, viscosity, and particle radius. This gave a route to measuring Avogadro's number, providing decisive evidence for the atomic hypothesis (confirmed experimentally by Perrin, Nobel 1926). Einstein's physical reasoning—random molecular bombardment producing a Gaussian spreading—is exactly mirrored by the mathematical transition density $[1]$ .

ExampleHitting Time and Optional Stopping

Given: Standard Brownian motion $W$ starting at $0$ . Let $a>0$ , $b>0$ , and $\tau = \inf\{t: W_t = a \text{ or } W_t = -b\}$ be the first exit time from $(-b, a)$ . Find $\mathbb{P}(W_\tau = a)$ .

Steps: 1. $W_t$ is a martingale; $\tau$ is a stopping time with $\mathbb{E}[\tau]<\infty$ (since the gap is bounded). By the Optional Stopping Theorem, $\mathbb{E}[W_\tau] = W_0 = 0$ . 2. $W_\tau$ takes value $a$ with probability $p$ and $-b$ with probability $1-p$ . So

12

\quad 0 = \mathbb{E}[W_\tau] = a\,p + (-b)(1-p) = p(a+b) - b.

Derivation hint: equate the martingale's expected stopped value to its start, then solve the two-point distribution. 3. Solve: $p = \dfrac{b}{a+b}$ .

Boxed answer: $\displaystyle \boxed{\mathbb{P}(W_\tau = a) = \frac{b}{a+b}}$ . With $a=2,\ b=3$ : probability $= 3/5 = 0.6$ (dimensionless).

ExampleExpected Exit Time via

W_t^2 - t

Given: Same setup; compute $\mathbb{E}[\tau]$ , the expected time to exit $(-b,a)$ .

Steps: 1. The process $W_t^2 - t$ is a martingale (Eq. $[10]$ ). Optional stopping gives $\mathbb{E}[W_\tau^2 - \tau] = 0$ , hence $\mathbb{E}[\tau] = \mathbb{E}[W_\tau^2]$ . 2. Using $p = b/(a+b)$ from the previous example:

13

\quad \mathbb{E}[W_\tau^2] = a^2\cdot\frac{b}{a+b} + b^2\cdot\frac{a}{a+b} = \frac{ab(a+b)}{a+b} = ab.

Derivation hint: weight the squared exit values by their probabilities; factor the common $ab$ . 3. Therefore $\mathbb{E}[\tau] = ab$ .

Boxed answer: $\displaystyle \boxed{\mathbb{E}[\tau] = ab}$ . With $a=2,\ b=3$ : $\mathbb{E}[\tau] = 6$ (time units).

⚠️ Common mistake

Brownian motion is not differentiable. It is a frequent error to write " $dW_t/dt$ " as if it were an ordinary velocity. Almost every path is nowhere differentiable, has infinite total variation on every interval, and "white noise" $\dot W$ exists only as a distribution (generalized process). Consequently the Itô integral $\int H\,dW$ is not a Riemann–Stieltjes integral; it must be defined as an $L^2$ limit of left-endpoint Riemann sums, and the choice of evaluation point matters—the Stratonovich integral (midpoint) obeys the ordinary chain rule but is not a martingale.

⚠️ Common mistake

Optional stopping requires hypotheses. The identity $\mathbb{E}[M_\tau]=\mathbb{E}[M_0]$ can fail without integrability/uniform-integrability conditions or a bounded stopping time. A classic trap: $\tau_a = \inf\{t: W_t = a\}$ for $a>0$ has $\mathbb{P}(\tau_a<\infty)=1$ yet $\mathbb{E}[\tau_a]=\infty$ , and $\mathbb{E}[W_{\tau_a}] = a \neq 0 = W_0$ —optional stopping fails because $\{W_{t\wedge\tau_a}\}$ is not uniformly integrable.

Counterexample

A martingale need not converge. One might guess every martingale settles down. But $W_t$ itself is a martingale that oscillates forever: by the law of the iterated logarithm, $\limsup_{t\to\infty} W_t/\sqrt{2t\log\log t} = 1$ and $\liminf = -1$ a.s., so $W_t$ does not converge—it visits $+\infty$ and $-\infty$ in the limsup sense. Martingale convergence requires an $L^1$ -bound: $\sup_t \mathbb{E}|M_t|<\infty$ . Brownian motion has $\mathbb{E}|W_t| = \sqrt{2t/\pi}\to\infty$ , so the hypothesis fails.

connection

Brownian motion and martingales sit at the crossroads of several theories. Via the Feynman–Kac formula, expectations of functionals of $W$ solve parabolic PDEs, linking Partial Differential Equations to probability: $u(t,x) = \mathbb{E}[\phi(W_t)\mid W_0=x]$ solves the heat equation. The Itô integral and Itô's formula generate the theory of Stochastic Differential Equations, whose solutions are diffusions. The Martingale Representation Theorem states every $L^2$ Brownian martingale is a stochastic integral $\int H\,dW$ , underpinning Mathematical Finance and market completeness. Lévy's characterization $[4]$ shows Brownian motion is the only continuous martingale with quadratic variation $t$ , and the Dambis–Dubins–Schwarz theorem reveals every continuous martingale is a time-changed Brownian motion. The increment structure makes $W$ a Markov process and a Gaussian process simultaneously, connecting to Markov Processes and Gaussian field theory.

🔬 Where this appears

⏳ History

- 1827 — Robert Brown observes erratic motion of pollen grains suspended in water under a microscope, describing the phenomenon now bearing his name.

- 1900 — Louis Bachelier, in Théorie de la spéculation, models stock prices with Brownian motion—five years before Einstein—introducing the heat-equation connection.

- 1905 — Albert Einstein derives the diffusion law

\langle x^2\rangle = 2Dt

, linking microscopic molecular motion to macroscopic diffusion and Avogadro's number.

- 1906 — Marian Smoluchowski independently develops a kinetic theory of Brownian motion.

- 1908 — Paul Langevin introduces the stochastic differential equation

m\ddot x = -\gamma\dot x + \xi(t)

, the first SDE in physics.

- 1923 — Norbert Wiener constructs Brownian motion rigorously as a measure on continuous-function space, proving path continuity and nowhere-differentiability.

- 1933–1939 — Paul Lévy develops fine path properties (law of iterated logarithm, local time, characterization theorem) and the modulus of continuity.

- 1940s — Joseph Doob systematizes martingale theory: the convergence theorem, optional stopping, and maximal inequalities, providing the abstract backbone.

- 1944–1951 — Kiyosi Itô defines the stochastic integral and proves Itô's formula, founding stochastic calculus and the theory of diffusion processes.

open-problem

While Brownian motion itself is fully understood, surrounding questions remain deep and active. The construction and uniqueness of solutions to SDEs with irregular (merely measurable or distributional) coefficients—regularization by noise—is an ongoing program (Davie, Krylov–Röckner, and the rough-paths/regularity-structures approaches of Lyons and Hairer). The multidimensional and infinite-dimensional analogues—stochastic PDEs driven by space-time white noise—require renormalization, as in Hairer's Fields-medal-winning theory of regularity structures resolving the KPZ and $\Phi^4_3$ equations. Fine geometric properties of the Brownian zero set, the exact Hausdorff measure of paths, and intersection local times in critical dimensions continue to be studied. In finance, rough volatility models (Gatheral–Jaisson–Rosenbaum) replace Brownian volatility with fractional Brownian motion of Hurst index $H\approx 0.1$ , raising new questions about non-Markovian, non-semimartingale calculus where Itô's formula no longer directly applies.

Stochastic Processes — Brownian Motion and Martingales

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also