Brownian Motion Martingales¶

Brownian motion is the concrete testing ground for everything developed so far — Martingales, Martingale Convergence, Uniform Integrability, and the Doob-Meyer structure. The abstract theory becomes calculable here: \(W_t\) itself is a martingale, simple polynomial corrections produce more, and a single exponential object generates them all.

Throughout, \((W_t)_{t\ge 0}\) is a standard Brownian motion on \((\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})\) with the usual conditions. Recall (see § Brownian Motion): \(W_0=0\), paths are continuous, and increments \(W_t-W_s\sim\mathcal{N}(0,t-s)\) are independent of \(\mathcal{F}_s\).

The Basic Martingale: W_t¶

Recall (see § Conditional Expectation): \(\mathbb{E}[W_t \mid \mathcal{F}_s] = W_s\) for \(s \le t\). The martingale property is the formal statement that Brownian motion has no drift: the best prediction of the future is the present.

Polynomial Martingales¶

Powers of \(W_t\) are not themselves martingales, but the deterministic growth in their moments can be compensated.

Compensated square and cube

The processes

\[ W_t^2 - t \quad\text{and}\quad W_t^3 - 3tW_t \]

are martingales.

Idea. Write \(W_t = W_s + \Delta\) with \(\Delta = W_t - W_s \sim N(0, t-s)\) independent of \(\mathcal{F}_s\). Expanding the power and taking conditional expectations replaces \(\Delta^k\) by its moment — \(\mathbb{E}\Delta = 0\), \(\mathbb{E}\Delta^2 = t-s\), \(\mathbb{E}\Delta^3 = 0\). The expansion of \(W_t^2\) gives \(\mathbb{E}[W_t^2 \mid \mathcal{F}_s] = W_s^2 + (t-s)\), so the compensation by \(t\) yields a martingale. Similarly \(\mathbb{E}[W_t^3 \mid \mathcal{F}_s] = W_s^3 + 3W_s(t-s)\), matched by the compensator \(3tW_t\).

The compensator of \(W_t^2\) is \([W]_t = t\) (see § Quadratic Variation of Brownian Motion) — a foreshadowing of the Doob-Meyer decomposition (see Doob-Meyer Decomposition).

Hermite Structure¶

The polynomial martingales are not ad hoc: they are Hermite polynomials of \((W_t, t)\). Define the probabilist's Hermite polynomials \(H_n\) by the generating function

\[ \exp\!\left(\theta x - \tfrac{\theta^2}{2}\right) = \sum_{n=0}^\infty \frac{\theta^n}{n!} H_n(x), \]

and the "time-scaled" version \(H_n(W_t, t) = t^{n/2} H_n(W_t/\sqrt t)\). The first four:

\[ H_0 = 1,\quad H_1 = W_t,\quad H_2 = W_t^2 - t,\quad H_3 = W_t^3 - 3tW_t,\quad H_4 = W_t^4 - 6tW_t^2 + 3t^2. \]

Each \(H_n(W_t, t)\) is a martingale. The next section explains why.

The Exponential Martingale¶

The central object of the theory is

\[ Z_t^\theta = \exp\!\left(\theta W_t - \tfrac{\theta^2 t}{2}\right), \qquad \theta \in \mathbb{R}. \]

Exponential martingale

For every \(\theta \in \mathbb{R}\), \(Z_t^\theta\) is a strictly positive martingale with \(\mathbb{E}[Z_t^\theta] = 1\).

Proof. Factor \(Z_t^\theta = Z_s^\theta \cdot Y\) where \(Y = \exp(\theta(W_t - W_s) - \tfrac{\theta^2(t-s)}{2})\). Since \(W_t - W_s \sim N(0, t-s)\) is independent of \(\mathcal{F}_s\), the Gaussian mgf \(\mathbb{E}[e^{\theta Z}] = e^{\theta^2(t-s)/2}\) for \(Z \sim N(0,t-s)\) gives \(\mathbb{E}[Y] = 1\), so \(\mathbb{E}[Z_t^\theta \mid \mathcal{F}_s] = Z_s^\theta\). \(\square\)

The quadratic term \(-\theta^2 t/2\) is exactly what the exponential growth of \(e^{\theta W_t}\) demands. This is the prototype of the general fact that \(\exp(M_t - \tfrac{1}{2}[M]_t)\) is a local martingale for every continuous local martingale \(M\).

Generating all polynomial martingales

Expanding the exponential in powers of \(\theta\),

\[ Z_t^\theta = \sum_{n=0}^\infty \frac{\theta^n}{n!} H_n(W_t, t), \]

and each coefficient is a martingale. Indeed, \(\mathbb{E}[Z_t^\theta \mid \mathcal{F}_s] = Z_s^\theta\) holds for all \(\theta\); dominated convergence justifies interchanging sum and conditional expectation (using the Gaussian moment bound \(\sum |\theta|^n |H_n(W_t,t)|/n! \le \exp(|\theta||W_t| + \theta^2 t/2) \in L^1\)). Matching coefficients gives \(\mathbb{E}[H_n(W_t, t) \mid \mathcal{F}_s] = H_n(W_s, s)\).

So a single exponential object encodes all moment-compensation martingales at once.

Applications¶

Girsanov

Defining \(d\mathbb{Q}/d\mathbb{P}\big|_{\mathcal{F}_t} = Z_t^\theta\) turns \(\widetilde W_t = W_t - \theta t\) into a \(\mathbb{Q}\)-Brownian motion. The exponential martingale is the density process for drift changes — the mechanism behind risk-neutral pricing.

Moment generating functions and hitting times

When optional sampling applies at a stopping time \(\tau\), \(\mathbb{E}[\exp(\theta W_\tau - \theta^2\tau/2)] = 1\), yielding the joint Laplace transform of \((W_\tau, \tau)\) — the standard route to hitting time distributions.

Gaussian tail bound

Applying Markov to \(Z_t^\theta\) and optimizing in \(\theta\) gives \(\mathbb{P}(W_t \ge a) \le e^{-a^2/2t}\) for \(a > 0\).

The Stochastic Exponential¶

Recall (see § Local martingales and stochastic exponential): for a continuous local martingale \(M\) with \(M_0 = 0\), the Doléans-Dade exponential \(\mathcal{E}(M)_t = \exp(M_t - \tfrac{1}{2}[M]_t)\) is a local martingale solving \(dZ_t = Z_t\,dM_t\), \(Z_0 = 1\). Taking \(M = \theta W\) with \([M]_t = \theta^2 t\) recovers \(Z_t^\theta\).

Summary¶

Martingale	Role
\(W_t\)	The driftless process itself
\(W_t^2 - t\)	Variance compensation; \([W]_t = t\)
\(W_t^3 - 3tW_t\)	Third-moment compensation
\(H_n(W_t, t)\)	\(n\)-th Hermite martingale
\(\exp(\theta W_t - \theta^2 t/2)\)	Generating function; density process for Girsanov

The exponential martingale is the organizing principle: polynomial martingales are its Taylor coefficients, Girsanov its change-of-measure content, and large deviations its Markov-inequality consequence.

Exercises¶

Exercise 1. Prove \(\mathbb{E}[Z_t^\theta] = 1\) and \(\operatorname{Var}(Z_t^\theta) = e^{\theta^2 t} - 1\), and show \(Z_t^\theta \to 0\) a.s. for \(\theta \ne 0\).

Solution to Exercise 1

\(\mathbb{E}[Z_t^\theta] = \mathbb{E}[Z_0^\theta] = 1\) by the martingale property. For the variance,

\[ \mathbb{E}[(Z_t^\theta)^2] = e^{-\theta^2 t}\,\mathbb{E}[e^{2\theta W_t}] = e^{-\theta^2 t}\,e^{2\theta^2 t} = e^{\theta^2 t}, \]

so \(\operatorname{Var}(Z_t^\theta) = e^{\theta^2 t} - 1\).

For \(\theta \ne 0\): \(\log Z_t^\theta / t = \theta W_t/t - \theta^2/2\). By the law of the iterated logarithm, \(W_t/t \to 0\) a.s., so \(\log Z_t^\theta / t \to -\theta^2/2 < 0\) and \(Z_t^\theta \to 0\) a.s. (This also shows the family is not UI; see Uniform Integrability.) \(\square\)

Exercise 2. Expand \(\exp(\theta W_t - \theta^2 t/2)\) as a power series in \(\theta\) through order \(4\) and identify the Hermite martingales \(H_0, \ldots, H_4\).

Solution to Exercise 2

Multiplying \(\sum (\theta W_t)^k/k!\) by \(\sum (-\theta^2 t/2)^j/j!\) and collecting terms:

\(\theta^0\): \(1\)
\(\theta^1\): \(W_t\)
\(\theta^2\): \(\tfrac{1}{2}(W_t^2 - t)\)
\(\theta^3\): \(\tfrac{1}{6}(W_t^3 - 3tW_t)\)
\(\theta^4\): \(\tfrac{1}{24}(W_t^4 - 6tW_t^2 + 3t^2)\)

The coefficients of \(\theta^n/n!\) are \(H_n(W_t, t)\); each is a martingale. \(\square\)

Exercise 3. Show that \(\cosh(\theta W_t) e^{-\theta^2 t/2}\) and \(\sinh(\theta W_t) e^{-\theta^2 t/2}\) are martingales.

Solution to Exercise 3

Using \(\cosh x = (e^x + e^{-x})/2\) and \(\sinh x = (e^x - e^{-x})/2\),

\[ \cosh(\theta W_t) e^{-\theta^2 t/2} = \tfrac{1}{2}(Z_t^\theta + Z_t^{-\theta}),\qquad \sinh(\theta W_t) e^{-\theta^2 t/2} = \tfrac{1}{2}(Z_t^\theta - Z_t^{-\theta}). \]

Linear combinations of martingales are martingales. \(\square\)

Exercise 4. Prove Lévy's characterization: a continuous martingale \(M\) with \(M_0 = 0\) and \([M]_t = t\) is a standard Brownian motion.

Solution to Exercise 4

By the stochastic exponential principle, \(\mathcal{E}(\theta M)_t = \exp(\theta M_t - \tfrac{\theta^2}{2}[M]_t) = \exp(\theta M_t - \theta^2 t/2)\) is a local martingale, and it is a true martingale because \([M]_t = t\) is deterministic (Novikov: \(\mathbb{E}[\exp(\tfrac{1}{2}\theta^2 t)] < \infty\)).

The martingale identity \(\mathbb{E}[\exp(\theta M_t - \theta^2 t/2)\mid \mathcal{F}_s] = \exp(\theta M_s - \theta^2 s/2)\) becomes, on setting \(\theta = i\alpha\),

\[ \mathbb{E}[e^{i\alpha(M_t - M_s)}\mid \mathcal{F}_s] = e^{-\alpha^2(t-s)/2}. \]

The right side is deterministic, so \(M_t - M_s\) is independent of \(\mathcal{F}_s\) with characteristic function of \(N(0, t-s)\). Thus \(M\) has continuous paths, \(M_0 = 0\), and stationary independent Gaussian increments: it is a standard Brownian motion. \(\square\)

Exercise 5. State the martingale problem for Brownian motion and explain its connection to the heat equation.

Solution to Exercise 5

For \(f \in C^2(\mathbb{R})\), Itô's formula gives

\[ f(W_t) - \tfrac{1}{2}\int_0^t f''(W_s)\,ds = f(W_0) + \int_0^t f'(W_s)\,dW_s, \]

which is a local martingale (and a true martingale when \(f'\) has suitable growth). This is the martingale problem for Brownian motion: the generator is \(\tfrac{1}{2}\partial_{xx}\).

If \(u\) solves the heat equation \(\partial_t u = \tfrac{1}{2}\partial_{xx} u\), then Itô's formula applied to \(u(W_t, T-t)\) yields

\[ du(W_t, T-t) = \partial_x u\,dW_t + \bigl(-\partial_t u + \tfrac{1}{2}\partial_{xx} u\bigr)\,dt = \partial_x u\,dW_t, \]

so \(u(W_t, T-t)\) is a local martingale. This is the Feynman-Kac connection: heat-equation solutions are expectations of Brownian functionals. \(\square\)