From f9f062d4382a5f501420ffbe4f19902fe94cf480 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 31 Oct 2021 13:54:31 +0100
Subject: Expand knowledge base

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+---
+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.
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