Brownian motion and martingales and their relations

Brownian motion

Brownian motion is defined as a stochastic process $\{X_t\}_{t \geq 0}$ which is Gaussian, i.e. such that for any $0 \leq t_0 < t_1 < \cdots < t_n$ , $X_{t_j}$ for $j = 0, 1, \cdots, n$ have a jointly Gaussian (normal distribution), $\mathbb{E}X_t = 0$ for all $t$ , and the covariance is $\mathbb{E}X_sX_t = \min(s,t)$ for all $s, t \geq 0$ . A normal distribution is uniquely determined by its mean vector $\mu$ and covariance matrix $C$ . Each $X_t$ has a normal distribution $\mathcal{N}(0, t)$ .

Brownian motion – multiple sample paths from t = 0 to t = 5.

Density at t = 2.5.

It follows from the definition that Brownian motion has independent increments, in other words whenever $0 \leq s < t \leq u < v$ , $X_t - X_s$ and $X_v - X_u$ are independent.

A consequence of the definition is that for any $c > 0$ , the process $X_{c^2t}/ c$ is also a Brownian motion.

By a therorem of Norbert Wiener, Brownian motion can be chosen such that with probability 1 (or for all $w$ ), $X_t(w)$ is continuous as a function of $t$ . a Brownian motion with this property is called the standard Brownian motion.

A Markov time $\tau$ for Brownian motion is defined as a random variable $\tau \geq 0$ such that for each $t > 0$ , the event $\{ \tau < t\}$ is a function of $X_s$ for $0 \leq s \eq t$ .

The strong Markov property of Brownian motion says that if $\tau$ is a Markov time such that $\math{P}(\tau < \infty) > 0$ , then conditional on $\tau < \infty$ , the process $V_{t} \def X_{\tau + t} - X_{\tau}$ for $t \geq 0$ is itself a Brownian motion and is independent of $X_s$ for $s \leq \tau$ .

Using the strong Markov property one can prove the reflection principle for Brownian motion: for any $b > 0$ and $t > 0$ , the probability that $X_s \geq b$ for some $s$ with $0 < s \leq t$ equals $2\mathbb{P}\left(X_t \geq b\right)$ .

What may be called “Wald’s identity“ for Brownian motion says that if $\tau$ is a Markov time with $\mathbb{E}\left[\tau\right] < \infty$ , and so $\tau < \infty$ with probability 1, $\mathbb{E}\left[X_{\tau}\right] = 0$ and $\mathbb{E}\left[X_{\tau}^2\right] = \mathbb{E}\left[\tau\right]$ .

The Brownian Bridge is a Gaussian process $Y_t$ defined for $0 \leq t \leq 1$ with $\mathbb{E}\left[Y_t\right] \equiv 0$ and covariance $\mathbb{E}\left[Y_SY_t\right] = s\left(1 - t\right)$ for $0 \leq s \leq t \leq 1$ .

The path starts and ends at 0 (a.s.).
The dashed curves plot the $\pm 2\sqrt{t(1-t)}$ band (two standard deviations), and the dotted line marks the zero mean.
Note that $Y_0=Y_1=0$ a.s., and the fluctuations are largest near $t=\tfrac12$ .

Relations between the Brownian motion and Brownian bridge:

If $\{X_t\}_{t \geq 0}$ is a Brownian motion then $Y_t = X_t - tX_1$ for $0 \leq t \leq 1$ is a Brownian bridge, which is independent of $X_1$ . This shows that $Y_t$ also can (and will) be taken to be continuity as a function of $t$ .
Conversely, if $\{Y_t\}_{0 \leq t \leq 1}$ is a Brownian bridge and $Z$ is a $\mathcal{N}\left(0, 1\right)$ variable independent of $\{Y_t\}_{0 \leq t \leq 1}$ then $Y_t + t Z$ has the distribution of Brownian motion for $0 \leq t \leq 1$ .
The conditional distribution of $\{X_t\}_{0 \leq t \leq 1}$ given that $|X_1| < \epsilon$ converges to that of $\{Y_t\}_{0 \leq t \leq 1$ as $\epsilon \downarrow 0$ .
If $\{Y_t\}_{0\leqt\leq1}$ is a Brownian bridge and $X_t := (1 + t) Y_{t/ (1 + t)}$ then $\{ X_t \}_{t \geq 0}$ is a Brownian motion.

Brownian motion (BM) — where it shows up

Physics & PDEs: Diffusion of particles; probabilistic solution of the heat equation; harmonic measure & potential theory.
Finance: Geometric BM for asset prices; correlated BMs in stochastic volatility (e.g., Heston); first-passage ideas for default/credit models; Monte Carlo pricing.
Statistics & sequential analysis: Likelihood-ratio tests/SPRT; drift-diffusion models for decision making; estimation of diffusion parameters; functional CLT (random walk ⇒ BM).
Machine learning & MCMC: Langevin Monte Carlo; score-based/diffusion generative models (continuous-time SDE viewpoint).
Biology & neuroscience: Wright–Fisher diffusion (genetic drift); membrane-potential and evidence-accumulation models (first-passage times).
Queues, control, and signals: Heavy-traffic limits (Reflected BM); Kalman–Bucy filtering (continuous time); integrated white noise as a model for 1/f21/f^21/f2-like signals.

Brownian bridge (BB) — why it’s special

A BB is BM on [0,1]conditioned to be 0 at t=1: Yt=Bt−tB1

Goodness-of-fit tests: The exact null distributions of Kolmogorov–Smirnov, Cramér–von Mises, and Anderson–Darling statistics are functionals of a BB.
Empirical process theory: Donsker’s theorem ⇒ centered empirical CDF processes converge to a BB; yields asymptotic CIs/bands for distribution functions and QQ-plots.
Conditioned path sampling: “Diffusion bridges” in Bayesian inference, particle filters, and SDE calibration when you know endpoints (e.g., smoothing between observations).
Finance (variance reduction): Brownian-bridge corrections in Monte Carlo for barrier options: between time steps, condition on endpoints to estimate barrier crossings and cut bias/variance.
Boundary crossing & change-point: Many suprema/infima of centered cumulative sums have BB limits, giving p-values and thresholds for change-point and scan statistics.
Random geometry/combinatorics: BB/bridges underpin Brownian excursions, which connect to random trees (Aldous’ CRT) and scaling limits in random maps.

Let $\{Y_t\}_{0 \leq t \leq1}$ be a Brownian bridge and $b > 0$ . Then,

(a) the probability that $Y_t \geq b$ for some $t$ , which is the same that probability that $Y_t = b$ for some $t$ by sample continuity and the intermediate value theorem, equals $\exp(-2b^2)$ .

(b) the probability that $|Y_t| \geq b$ for some $t$ , which is the same as the probability that $|Y_t| = b$ for some $t$ , equals $2\sum_{j = 1}^{\infty} (-1)^{j - 1}exp(-2j^2b^2)$ . The series converges quite fast except for small b.

For a Brownian motion $\{X_t\}_{t \geq 0}$ and two real numbers $a$ and $b$ , let $\tau$ be the least time $t$ such that $X_t = at + b$ , if such a $t$ exists, or $+\infty$ otherwise. Then $\tau$ is a Markov time.

(a) If $b = 0$ then $\tau \equiv 0$ .

(b) If $a = 0 \neq b$ then $\mathbb{P}(\tau < \infty ) = 1$ but $\mathbb{E}\left[\tau\right] = +\infty$ , otherwise we’d get a contradiction to the Wald identity $\mathbb{E}\left[X_{\tau}\right] = 0$ .

(d) If $ab > 0$ then $0 < \mathbb{P}(\tau < \infty) = e^{-2ab} < 1$ .

Martingales

Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables. Let $\mathcal{F}_n$ be a collection of random variables such that $\mathcal{F}_m \subset \mathcal{F}_n$ for $m \leq n$ and $X_n \in \mathcal{F}_n$ for each $n$ . Then, $\{X_n\}$ is called a martingale sequence if whenever $m < n$ , $X_m = \mathbb{E}(X_n | \mathcal{F}_m)$ , or a submartingale sequence if $X_m \leq \mathbb{E}(X_n | \mathcal{F}_m)$ , or a supermartingale sequence if $X_m \geq \mathbb{E}(X_n | \mathcal{F}_m)$ .

To make the collection $\mathcal{F}_n$ as small as possible one can and often will take $\mathcal{F}_n = \{X_1, X_2, \dots, X_n\}$ .

A stopping time for a martingale (or sub- or supermartingale) sequence is a positive integer-valued random variable $\tau$ such that for each $n = 1, 2, \dots,$ the event $\{\tau \leq n\}$ is a function of the variables in $\mathcal{F}_{n}$ .

Optional stopping. Let $\{X_n\}_{n \geq 1}$ be a martingale sequence for $\mathcal{F}_{n} = \{X_1, \dots, X_n \}$ . Let $N < \infty$ and let $\sigma$ and $\tau$ be two stopping times for $\{\mathcal{F}_n\}_{n \geq 1}$ such that $\sigma \leq \tau \leq N$ with probability 1. Then $\mathbb{E}(X_{\tau} | X_j , j \leq \sigma) = X_{\sigma}$ . For a submartingale or supermartingale, the equality is replaced by $\geq$ or $\leq$ respectively.

Doob’s maximal inequality. Let $\{X_j\}_{j \geq 1}$ be a submartingale sequence, let $M > 0$ , and let $A(M, n)$ be the event $\{max_{1 \leq j \leq n} X_j \geq M \}$ . Then

$MP(A(M, n)) \leq \mathbb{E}(X_n \mathds{1}_{A(M, n)}) \leq \mathbb{E}(\max(X_n, 0))$

Let $X_n \geq 0$ for all $n$ and suppose $\{X_n \}$ form a supermartingale with respect to $\mathcal{F}_n = \{X_1, \dots, X_n\}$ . Let $\sigma \leq \tau \leq +\infty$ be stopping times for $\{X_n \}$ . Then