Brownian Motion

Contents (hide)

1. Definition and existence
2. Construction of Brownian motion
3. Brownian motion is a martingale
4. Brownian motion is Gaussian, Markov and continuous
5. Brownian motion is a diffusion
6. Brownian motion is wild

1. Definition and existence

Consider the (one-dimensional for example) Gaussian transition density

$\fs4 p(x,y;t) = \frac{1}{\sqrt{2\pi t}} \exp \left\{ -\frac{(x-y)^2}{2 t} \right\}$

Define the measures, for all $\fs4 t_1,\ldots,t_k \in T$

$\fs4 \mu_{t_1 \ldots t_k}(F_1 \times \cdots \times F_k) = \int_{F_1 \times \cdots \times F_k} p(x,x_1;t_1)p(x_1,x_2;t_2-t_1)\cdots p(x_{k-1},x_k;t_{k-1}-t_k) d x_1 \cdots d x_k$

Kolmogorov's extension theorem will provide us with the existence of a process with Gaussian increments $\fs4 \{B_t\}_{t\geq 0}$ , which we define as the Brownian motion (started at $\fs4 B_0 = x$ )

2. Construction of Brownian motion

Assume an infinite collection of standard normal random variables $\fs4 B_k^n$ , $\fs4 n\in \mathbb{N}$ , $\fs4 k$ odd, $\fs4 k \leq 2^n$ ; define the functions (called Haar functions)

$\fs4 g_1^0 = 1$
$\fs4 g_k^n(u) = \left\{\begin{array}{rl} 2^{(n-1)/2} & \text{for }2^{-n}(k-1) < u \leq 2^{-n}k \\ -2^{(n-1)/2} & \text{for }2^{-n}k < u \leq 2^{-n}(k+1) \\ 0 & \text{elsewhere} \end{array} \right.$
$\fs4 f_k^n(t) = \int_0^t g_k^n(u) d u$

The latter is a "tent shaped" function that we use to define an approximation on $\fs4 t\in [0,1]$ of the BM

$\fs4 B_n(t) = \sum_{\text{appr }k,n} f_k^n(t) B_k^n$

The figure below illustrates this method of construction of the Brownian motion.

3. Brownian motion is a martingale

Now assume that $\fs4 \{B_t\}_{t \geq 0}$ is a Brownian motion and $\fs4 \mathcal{F}_t$ is the generated filtration. Then $\fs4 B_t$ is a martingale with respect to the probability measure $\fs4 P$

$\fs4 E [B_t - B_s | \mathcal{F}_s] = 0 \text{, or } E [B_t | \mathcal{F}_s] = B_s$

Also $\fs4 B^2_t - t$ is a martingale. Levy's theorem states the converse: if $\fs4 \{X_t\}_{t \geq 0}$ is a continuous martingale, and also $\fs4 X^2_t - t$ is a martingale, then $\fs4 X_t$ is a BM. A consequence is that every continuous martingale is a time-changed Brownian motion.

The process $\fs4 \exp \left( \theta B_t - \frac{1}{2} \theta^2 t \right)$ , for all $\fs4 \theta \in \mathbb{R}$ , is the exponential martingale.

4. Brownian motion is Gaussian, Markov and continuous

Brownian motion is Gaussian by definition, that is for all $\fs4 t_1,\ldots,t_k \in T$ the random variable $\fs4 B = (B_1,\ldots,B_k)$ has a multi-normal distribution

Also BM is Markov, as $\fs4 E [f(B_{s+t})|\mathcal{F}_s] = P_t f(B_s)$ for the transition semigroup

$\fs4 P_t f(x) = \int_{\mathbb{R}} p(x, y; t) f(y) d y$

Kolmogorov's continuity theorem states that if for all $\fs4 t\in T$ we can find constants $\fs4 \alpha, \beta, \gamma > 0$ such that

$\fs4 E |X_{t_1}-X_{t_2}|^\alpha \leq \gamma |t_1 - t_2|^{1+\beta}, \text{ for all } 0\leq t_1,t_2 \leq t$

then $\fs4 X_t$ has continuous paths (or at least a version). For the Brownian motion

$\fs4 E |B_{t_1}-B_{t_2}|^4 = 3 |t_1 - t_2|^2$

and therefore $\fs4 B_t$ has continuous sample paths

5. Brownian motion is a diffusion

A diffusion $\fs4 \{X_t\}_{t\geq 0}$ is a continuous, time homogeneous, Markov process that is "characterized" by its local drift $\fs4 \mu$ and volatility $\fs4 \sigma$ . For "small" $\fs4 h$

$\fs4 E[X_{t+h}-X_t|\mathcal{F}_t] = h \cdot \mu(X_t) + o(h)$

$\fs4 E[(X_{t+h}-X_t-h\mu(X_t))^2|\mathcal{F}_t] = h \cdot \sigma^2(X_t) + o(h)$

If the drift and volatility is constant, the process $\fs4 X_t = \mu t + \sigma B_t$ for a BM $\fs4 \{B_t\}_{t\geq 0}$ will also be a diffusion. More generally the instantaneous drift and volatility do not have to be constant, but can depend on the location $\fs4 X_t$ and the time $\fs4 t$

6. Brownian motion is wild

Brownian motion as a function of time $\fs4 t \rightarrow B(t,\omega)$ is a lot wilder than most "normal" functions. In particular, although a Brownian motion is everywhere continuous, it is nowhere differentiable. Also, the total variation of the BM

$\fs4 \sum |B_{t_{k+1}}-B_{t_k}| \rightarrow \infty$

and the quadratic variation

$\fs4 \sum |B_{t_{k+1}}-B_{t_k}| \rightarrow t$

(for $\fs4 t_{k+1}-t_k \rightarrow 0$ , $\fs4 t_k\leq t$ )

For "normal" functions the total variation would be the length of the curve: to draw a Brownian motion on a finite interval we need infinite ink. The quadratic variation of "normal" functions is zero.

Finally, it is impossible to find a monotonic interval, therefore we cannot split a Brownian motion in two parts with a (non-vertical) line. These features are apparent if we zoom into the sample paths of a Brownian motion, as the figure below illustrates

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