From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/wiener-process/index.md | 94 ++++++++++++++---------------
 1 file changed, 47 insertions(+), 47 deletions(-)

(limited to 'source/know/concept/wiener-process')

diff --git a/source/know/concept/wiener-process/index.md b/source/know/concept/wiener-process/index.md
index 629da20..c6fcf8e 100644
--- a/source/know/concept/wiener-process/index.md
+++ b/source/know/concept/wiener-process/index.md
@@ -14,34 +14,34 @@ 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
-stochastic process $\{B_t: t \ge 0\}$ that satisfies:
-
-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 $t$.
+A Wiener process $$B_t$$ is defined as any
+stochastic process $$\{B_t: t \ge 0\}$$ that satisfies:
+
+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 $\sqrt{t}$.
+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,
+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:
-$\mathbf{E}|B_t|^p = \mathbf{E}|\sqrt{t} B_1|^p = t^{p/2} \mathbf{E}|B_1|^p$.
+$$\mathbf{E}|B_t|^p = \mathbf{E}|\sqrt{t} B_1|^p = t^{p/2} \mathbf{E}|B_1|^p$$.
 Another consequence is invariance under "time inversion",
-by defining $\sqrt{\alpha} = t$, such that $W_t = t B_{1/t}$.
+by defining $$\sqrt{\alpha} = t$$, such that $$W_t = t B_{1/t}$$.
 
 Despite being continuous by definition,
 the Wiener process is not differentiable in general,
@@ -68,13 +68,13 @@ and all increments are independent (so it is a Markov process).
 An important question about the Wiener process
 is whether it is **recurrent** or **transient**:
 given a hypersphere (interval in 1D, circle in 2D, sphere in 3D)
-away from the origin, will $B_t$ visit it after a finite time $\tau\!<\!\infty$?
-It is *recurrent* if yes, i.e. $P(\tau \!<\! \infty) = 1$, or *transient* otherwise.
+away from the origin, will $$B_t$$ visit it after a finite time $$\tau\!<\!\infty$$?
+It is *recurrent* if yes, i.e. $$P(\tau \!<\! \infty) = 1$$, or *transient* otherwise.
 The answer to this question turns out to depend on the number of dimenions.
 
-To demonstrate this, we model the $d$-dimensional Wiener process
-as an [Itō diffusion](/know/concept/ito-calculus/) $X_t$,
-which also allows us to shift the initial condition $X_0$
+To demonstrate this, we model the $$d$$-dimensional Wiener process
+as an [Itō diffusion](/know/concept/ito-calculus/) $$X_t$$,
+which also allows us to shift the initial condition $$X_0$$
 (or resume a "paused" process):
 
 $$\begin{aligned}
@@ -82,10 +82,10 @@ $$\begin{aligned}
     = X_0 + \int_0^t \dd{B_s}
 \end{aligned}$$
 
-Consider two hyperspheres, the inner with radius $R_i$,
-and the outer with $R_o > R_i$.
-Let the initial condition $|X_0| \in \: ]R_i, R_o[$,
-then we define the stopping times $\tau_i$, $\tau_o$ and $\tau$ like so:
+Consider two hyperspheres, the inner with radius $$R_i$$,
+and the outer with $$R_o > R_i$$.
+Let the initial condition $$|X_0| \in \: ]R_i, R_o[$$,
+then we define the stopping times $$\tau_i$$, $$\tau_o$$ and $$\tau$$ like so:
 
 $$\begin{aligned}
     \tau_i
@@ -98,15 +98,15 @@ $$\begin{aligned}
     \equiv \min\{\tau_i, \tau_o\}
 \end{aligned}$$
 
-We stop when the inner or outer hypersphere is touched by $X_t$,
+We stop when the inner or outer hypersphere is touched by $$X_t$$,
 whichever happens first.
 
 [Dynkin's formula](/know/concept/dynkins-formula/)
-is applicable to this situation, if we define $h(x)$ as follows,
-where the *terminal reward* $\Gamma$ equals $1$ for $|X_\tau| = R_i$,
-and $0$ for $|X_\tau| = R_o$,
-such that $h(X_0)$ equals the probability
-that we touch $R_i$ before $R_o$ for a given $X_0$:
+is applicable to this situation, if we define $$h(x)$$ as follows,
+where the *terminal reward* $$\Gamma$$ equals $$1$$ for $$|X_\tau| = R_i$$,
+and $$0$$ for $$|X_\tau| = R_o$$,
+such that $$h(X_0)$$ equals the probability
+that we touch $$R_i$$ before $$R_o$$ for a given $$X_0$$:
 
 $$\begin{aligned}
     h(X_0)
@@ -114,8 +114,8 @@ $$\begin{aligned}
     = P\Big[|X_\tau| \!=\! R_i \:\Big|\: X_0\Big]
 \end{aligned}$$
 
-Dynkin's formula then tells us that $h(x)$ is given by the following equation,
-with the boundary conditions $h(R_i) = 1$ and $h(R_o) = 0$:
+Dynkin's formula then tells us that $$h(x)$$ is given by the following equation,
+with the boundary conditions $$h(R_i) = 1$$ and $$h(R_o) = 0$$:
 
 $$\begin{aligned}
     0
@@ -124,9 +124,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Thanks to this problem's spherical symmetry,
-$h$ only depends on the radial coodinate $r$,
-so the Laplacian $\nabla^2$ can be written as follows
-in $d$-dimensional [spherical coordinates](/know/concept/spherical-coordinates/):
+$$h$$ only depends on the radial coodinate $$r$$,
+so the Laplacian $$\nabla^2$$ can be written as follows
+in $$d$$-dimensional [spherical coordinates](/know/concept/spherical-coordinates/):
 
 $$\begin{aligned}
     0
@@ -134,8 +134,8 @@ $$\begin{aligned}
     = \pdvn{2}{h}{r} + \frac{d - 1}{r} \pdv{h}{r}
 \end{aligned}$$
 
-For $d = 1$, the solution $h_1(r)$ is as follows,
-of which we take the limit for $R_o \to \infty$:
+For $$d = 1$$, the solution $$h_1(r)$$ is as follows,
+of which we take the limit for $$R_o \to \infty$$:
 
 $$\begin{aligned}
     h_1(r)
@@ -144,14 +144,14 @@ $$\begin{aligned}
     1
 \end{aligned}$$
 
-The outer hypersphere becomes harder to reach for larger $R_o$,
-and for $R_o \to \infty$ we are left with
-the probability of hitting $R_i$ only.
-This turns out to be $1$, so in 1D the Wiener process is recurrent:
+The outer hypersphere becomes harder to reach for larger $$R_o$$,
+and for $$R_o \to \infty$$ we are left with
+the probability of hitting $$R_i$$ only.
+This turns out to be $$1$$, so in 1D the Wiener process is recurrent:
 it always comes close to the origin in finite time.
 
-For $d = 2$, the solution $h_2(r)$ is as follows,
-whose limit turns out to be $1$,
+For $$d = 2$$, the solution $$h_2(r)$$ is as follows,
+whose limit turns out to be $$1$$,
 so the Wiener process is also recurrent in 2D:
 
 $$\begin{aligned}
@@ -161,8 +161,8 @@ $$\begin{aligned}
     1
 \end{aligned}$$
 
-However, for $d \ge 3$, the solution $h_d(r)$
-does not converge to $1$ for $R_o \to \infty$,
+However, for $$d \ge 3$$, the solution $$h_d(r)$$
+does not converge to $$1$$ for $$R_o \to \infty$$,
 meaning the Wiener process is transient in 3D or higher:
 
 $$\begin{aligned}
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