---
title: "Wiener process"
firstLetter: "W"
publishDate: 2021-10-29
categories:
- Physics
- Mathematics

date: 2021-10-21T19:40:02+02:00
draft: false
markup: pandoc
---

# Wiener process

The **Wiener process** is a stochastic process that provides
a pure mathematical definition of the physical phenomenon of **Brownian motion**,
and hence is also called *Brownian motion*.

A Wiener process $B_t$ is defined as any
time-indexed [random variable](/know/concept/random-variable/)
$\{B_t: t \ge 0\}$ (i.e. stochastic process)
that has the following properties:

1.  Initial condition $B_0 = 0$.
2.  Each **increment** of $B_t$ is independent of the past:
    given $0 \le s < t \le u < v$, then
    $B_t \!-\! B_s$ and $B_v \!-\! B_u$ are independent random variables.
3.  The increments of $B_t$ are Gaussian with mean $0$
    and variance $h$, where $h$ is the time step,
    such that $B_{t+h} \!-\! B_t \sim \mathcal{N}(0, h)$.
4.  $B_t$ is a continuous function of $t$.

There exist stochastic processes that satisfy these requirements,
infinitely many in fact.
In other words, Brownian motion exists,
and can be constructed in various ways.

Since the variance of an increment is expressed in units of time $t$,
the physical unit of the Wiener process is the square root of time $\sqrt{t}$.

Brownian motion is **self-similar**:
if we define a rescaled $W_t = \sqrt{\alpha} B_{t/\alpha}$ for some $\alpha$,
then $W_t$ is also a valid Wiener process,
meaning that there are no fundemental scales.
A consequence of this is that:
$\mathbf{E}|B_t|^p = \mathbf{E}|\sqrt{t} B_1|^p = t^{p/2} \mathbf{E}|B_1|^p$.
Another consequence is invariance under "time inversion",
by defining $\sqrt{\alpha} = t$, such that $W_t = t B_{1/t}$.

Despite being continuous by definition,
the **total variation** $V(B)$ of $B_t$ is infinite
(informally, the curve is infinitely long).
For $t_i \in [0, 1]$ in $n$ steps of maximum size $\Delta t$:

$$\begin{aligned}
    V_t
    = \lim_{\Delta t \to 0} \sup \sum_{i = 1}^n \big|B_{t_i} - B_{t_{i-1}}\big|
    = \infty
\end{aligned}$$

However, curiously, the **quadratic variation**, written as $[B]_t$,
turns out to be deterministically finite and equal to $t$,
while a differentiable function $f$ would have $[f]_t = 0$:

$$\begin{aligned}
    \:[B]_t
    = \lim_{\Delta t \to 0} \sum_{i = 1}^n \big|B_{t_i} - B_{t_{i - 1}}\big|^2
    = t
\end{aligned}$$

Therefore, despite being continuous by definition,
the Wiener process is not differentiable,
not even in the mean square, because:

$$\begin{aligned}
    \frac{B_{t+h} - B_t}{h}
    \sim \frac{1}{h} \mathcal{N}(0, h)
    \sim \mathcal{N}\Big(0, \frac{1}{h}\Big)
    \qquad \quad
    \lim_{h \to 0} \mathbf{E} \bigg|\mathcal{N}\Big(0, \frac{1}{h}\Big) \bigg|^2
    = \infty
\end{aligned}$$



## References
1.  U.F. Thygesen,
    *Lecture notes on diffusions and stochastic differential equations*,
    2021, Polyteknisk Kompendie.