Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 68 insertions, 66 deletions

diff --git a/source/know/concept/ito-process/index.md b/source/know/concept/ito-process/index.md
index b82835d..f192e28 100644
--- a/source/know/concept/ito-process/index.md
+++ b/source/know/concept/ito-process/index.md
@@ -9,8 +9,8 @@ layout: "concept"
 ---
 
 Given two [stochastic processes](/know/concept/stochastic-process/)
-$F_t$ and $G_t$, consider the following random variable $X_t$,
-where $B_t$ is the [Wiener process](/know/concept/wiener-process/),
+$$F_t$$ and $$G_t$$, consider the following random variable $$X_t$$,
+where $$B_t$$ is the [Wiener process](/know/concept/wiener-process/),
 i.e. Brownian motion:
 
 $$\begin{aligned}
@@ -19,25 +19,25 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where the latter is an [Itō integral](/know/concept/ito-integral/),
-assuming $G_t$ is Itō-integrable.
-We call $X_t$ an **Itō process** if $F_t$ is locally integrable,
-and the initial condition $X_0$ is known,
-i.e. $X_0$ is $\mathcal{F}_0$-measurable,
-where $\mathcal{F}_t$ is the filtration
-to which $F_t$, $G_t$ and $B_t$ are adapted.
-The above definition of $X_t$ is often abbreviated as follows,
-where $X_0$ is implicit:
+assuming $$G_t$$ is Itō-integrable.
+We call $$X_t$$ an **Itō process** if $$F_t$$ is locally integrable,
+and the initial condition $$X_0$$ is known,
+i.e. $$X_0$$ is $$\mathcal{F}_0$$-measurable,
+where $$\mathcal{F}_t$$ is the filtration
+to which $$F_t$$, $$G_t$$ and $$B_t$$ are adapted.
+The above definition of $$X_t$$ is often abbreviated as follows,
+where $$X_0$$ is implicit:
 
 $$\begin{aligned}
     \dd{X_t}
     = F_t \dd{t} + G_t \dd{B_t}
 \end{aligned}$$
 
-Typically, $F_t$ is referred to as the **drift** of $X_t$,
-and $G_t$ as its **intensity**.
-Because the Itō integral of $G_t$ is a
+Typically, $$F_t$$ is referred to as the **drift** of $$X_t$$,
+and $$G_t$$ as its **intensity**.
+Because the Itō integral of $$G_t$$ is a
 [martingale](/know/concept/martingale/),
-it does not contribute to the mean of $X_t$:
+it does not contribute to the mean of $$X_t$$:
 
 $$\begin{aligned}
     \mathbf{E}[X_t]
@@ -45,25 +45,25 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Now, consider the following **Itō stochastic differential equation** (SDE),
-where $\xi_t = \idv{B_t}{t}$ is white noise,
-informally treated as the $t$-derivative of $B_t$:
+where $$\xi_t = \idv{B_t}{t}$$ is white noise,
+informally treated as the $$t$$-derivative of $$B_t$$:
 
 $$\begin{aligned}
     \dv{X_t}{t}
     = f(X_t, t) + g(X_t, t) \: \xi_t
 \end{aligned}$$
 
-An Itō process $X_t$ is said to satisfy this equation
-if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$,
-in which case $X_t$ is also called an **Itō diffusion**.
+An Itō process $$X_t$$ is said to satisfy this equation
+if $$f(X_t, t) = F_t$$ and $$g(X_t, t) = G_t$$,
+in which case $$X_t$$ is also called an **Itō diffusion**.
 All Itō diffusions are [Markov processes](/know/concept/markov-process/),
-since only the current value of $X_t$ determines the future,
-and $B_t$ is also a Markov process.
+since only the current value of $$X_t$$ determines the future,
+and $$B_t$$ is also a Markov process.
 
 
 ## Itō's lemma
 
-Classically, given $y \equiv h(x(t), t)$,
+Classically, given $$y \equiv h(x(t), t)$$,
 the chain rule of differentiation states that:
 
 $$\begin{aligned}
@@ -71,8 +71,8 @@ $$\begin{aligned}
     = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{x}
 \end{aligned}$$
 
-However, for a stochastic process $Y_t \equiv h(X_t, t)$,
-where $X_t$ is an Itō process,
+However, for a stochastic process $$Y_t \equiv h(X_t, t)$$,
+where $$X_t$$ is an Itō process,
 the chain rule is modified to the following,
 known as **Itō's lemma**:
 
@@ -89,7 +89,7 @@ $$\begin{aligned}
 <div class="hidden" markdown="1">
 <label for="proof-lemma">Proof.</label>
 We start by applying the classical chain rule,
-but we go to second order in $x$.
+but we go to second order in $$x$$.
 This is also valid classically,
 but there we would neglect all higher-order infinitesimals:
 
@@ -98,7 +98,7 @@ $$\begin{aligned}
     = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{X_t} + \frac{1}{2} \pdvn{2}{h}{x} \dd{X_t}^2
 \end{aligned}$$
 
-But here we cannot neglect $\dd{X_t}^2$.
+But here we cannot neglect $$\dd{X_t}^2$$.
 We insert the definition of an Itō process:
 
 $$\begin{aligned}
@@ -109,8 +109,8 @@ $$\begin{aligned}
     + \frac{1}{2} \pdvn{2}{h}{x} \Big( F_t^2 \dd{t}^2 + 2 F_t G_t \dd{t} \dd{B_t} + G_t^2 \dd{B_t}^2 \Big)
 \end{aligned}$$
 
-In the limit of small $\dd{t}$, we can neglect $\dd{t}^2$,
-and as it turns out, $\dd{t} \dd{B_t}$ too:
+In the limit of small $$\dd{t}$$, we can neglect $$\dd{t}^2$$,
+and as it turns out, $$\dd{t} \dd{B_t}$$ too:
 
 $$\begin{aligned}
     \dd{t} \dd{B_t}
@@ -120,8 +120,8 @@ $$\begin{aligned}
     \longrightarrow 0
 \end{aligned}$$
 
-However, due to the scaling property of $B_t$,
-we cannot ignore $\dd{B_t}^2$, which has order $\dd{t}$:
+However, due to the scaling property of $$B_t$$,
+we cannot ignore $$\dd{B_t}^2$$, which has order $$\dd{t}$$:
 
 $$\begin{aligned}
     \dd{B_t}^2
@@ -131,8 +131,8 @@ $$\begin{aligned}
     \longrightarrow \dd{t}
 \end{aligned}$$
 
-Where $\chi_1^2(\dd{t})$ is the generalized chi-squared distribution
-with one term of variance $\dd{t}$.
+Where $$\chi_1^2(\dd{t})$$ is the generalized chi-squared distribution
+with one term of variance $$\dd{t}$$.
 </div>
 </div>
 
@@ -144,9 +144,9 @@ to make the solution of a given Itō SDE easier.
 ## Coordinate transformations
 
 The simplest coordinate transformation is a scaling of the time axis.
-Defining $s \equiv \alpha t$, the goal is to keep the Itō process.
-We know how to scale $B_t$, be setting $W_s \equiv \sqrt{\alpha} B_{s / \alpha}$.
-Let $Y_s \equiv X_t$ be the new variable on the rescaled axis, then:
+Defining $$s \equiv \alpha t$$, the goal is to keep the Itō process.
+We know how to scale $$B_t$$, be setting $$W_s \equiv \sqrt{\alpha} B_{s / \alpha}$$.
+Let $$Y_s \equiv X_t$$ be the new variable on the rescaled axis, then:
 
 $$\begin{aligned}
     \dd{Y_s}
@@ -156,14 +156,14 @@ $$\begin{aligned}
     &= \frac{1}{\alpha} f(Y_s) \dd{s} + \frac{1}{\sqrt{\alpha}} g(Y_s) \dd{W_s}
 \end{aligned}$$
 
-$W_s$ is a valid Wiener process,
+$$W_s$$ is a valid Wiener process,
 and the other changes are small,
 so this is still an Itō process.
 
 To solve SDEs analytically, it is usually best
-to have additive noise, i.e. $g = 1$.
+to have additive noise, i.e. $$g = 1$$.
 This can be achieved using the **Lamperti transform**:
-define $Y_t \equiv h(X_t)$, where $h$ is given by:
+define $$Y_t \equiv h(X_t)$$, where $$h$$ is given by:
 
 $$\begin{aligned}
     \boxed{
@@ -173,8 +173,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Then, using Itō's lemma, it is straightforward
-to show that the intensity becomes $1$.
-Note that the lower integration limit $x_0$ does not enter:
+to show that the intensity becomes $$1$$.
+Note that the lower integration limit $$x_0$$ does not enter:
 
 $$\begin{aligned}
     \dd{Y_t}
@@ -185,9 +185,9 @@ $$\begin{aligned}
     &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g'(X_t) \bigg) \dd{t} + \dd{B_t}
 \end{aligned}$$
 
-Similarly, we can eliminate the drift $f = 0$,
+Similarly, we can eliminate the drift $$f = 0$$,
 thereby making the Itō process a martingale.
-This is done by defining $Y_t \equiv h(X_t)$, with $h(x)$ given by:
+This is done by defining $$Y_t \equiv h(X_t)$$, with $$h(x)$$ given by:
 
 $$\begin{aligned}
     \boxed{
@@ -197,8 +197,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The goal is to make the parenthesized first term (see above)
-of Itō's lemma disappear, which this $h(x)$ does indeed do.
-Note that $x_0$ and $x_1$ do not enter:
+of Itō's lemma disappear, which this $$h(x)$$ does indeed do.
+Note that $$x_0$$ and $$x_1$$ do not enter:
 
 $$\begin{aligned}
     0
@@ -212,8 +212,8 @@ $$\begin{aligned}
 
 It is worth knowing under what condition a solution to a given SDE exists,
 in the sense that it is finite on the entire time axis.
-Suppose the drift $f$ and intensity $g$ satisfy these inequalities,
-for some known constant $K$ and for all $x$:
+Suppose the drift $$f$$ and intensity $$g$$ satisfy these inequalities,
+for some known constant $$K$$ and for all $$x$$:
 
 $$\begin{aligned}
     x f(x) \le K (1 + x^2)
@@ -222,8 +222,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 When this is satisfied, we can find the following upper bound
-on an Itō process $X_t$,
-which clearly implies that $X_t$ is finite for all $t$:
+on an Itō process $$X_t$$,
+which clearly implies that $$X_t$$ is finite for all $$t$$:
 
 $$\begin{aligned}
     \boxed{
@@ -237,7 +237,7 @@ $$\begin{aligned}
 <label for="proof-existence">Proof</label>
 <div class="hidden" markdown="1">
 <label for="proof-existence">Proof.</label>
-If we define $Y_t \equiv X_t^2$,
+If we define $$Y_t \equiv X_t^2$$,
 then Itō's lemma tells us that the following holds:
 
 $$\begin{aligned}
@@ -253,7 +253,7 @@ $$\begin{aligned}
     = Y_0 + \mathbf{E}\! \int_0^t 2 X_s f(X_s) + g^2(X_s) \dd{s}
 \end{aligned}$$
 
-Given that $K (1 \!+\! x^2)$ is an upper bound of $x f(x)$ and $g^2(x)$,
+Given that $$K (1 \!+\! x^2)$$ is an upper bound of $$x f(x)$$ and $$g^2(x)$$,
 we get an inequality:
 
 $$\begin{aligned}
@@ -267,7 +267,7 @@ $$\begin{aligned}
 
 We then apply the
 [Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/),
-noting that $(Y_0 \!+\! 3 K t)$ does not decrease with time, leading us to:
+noting that $$(Y_0 \!+\! 3 K t)$$ does not decrease with time, leading us to:
 
 $$\begin{aligned}
     \mathbf{E}[Y_t]
@@ -275,12 +275,13 @@ $$\begin{aligned}
     \\
     &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big)
 \end{aligned}$$
+
 </div>
 </div>
 
 If a solution exists, it is also worth knowing whether it is unique.
-Suppose that $f$ and $g$ satisfy the following inequalities,
-for some constant $K$ and for all $x$ and $y$:
+Suppose that $$f$$ and $$g$$ satisfy the following inequalities,
+for some constant $$K$$ and for all $$x$$ and $$y$$:
 
 $$\begin{aligned}
     \big| f(x) - f(y) \big| \le K \big| x - y \big|
@@ -288,10 +289,10 @@ $$\begin{aligned}
     \big| g(x) - g(y) \big| \le K \big| x - y \big|
 \end{aligned}$$
 
-Let $X_t$ and $Y_t$ both be solutions to a given SDE,
+Let $$X_t$$ and $$Y_t$$ both be solutions to a given SDE,
 but the initial conditions need not be the same,
-such that the difference is initially $X_0 \!-\! Y_0$.
-Then the difference $X_t \!-\! Y_t$ is bounded by:
+such that the difference is initially $$X_0 \!-\! Y_0$$.
+Then the difference $$X_t \!-\! Y_t$$ is bounded by:
 
 $$\begin{aligned}
     \boxed{
@@ -305,8 +306,8 @@ $$\begin{aligned}
 <label for="proof-uniqueness">Proof</label>
 <div class="hidden" markdown="1">
 <label for="proof-uniqueness">Proof.</label>
-We define $D_t \equiv X_t \!-\! Y_t$ and $Z_t \equiv D_t^2 \ge 0$,
-together with $F_t \equiv f(X_t) \!-\! f(Y_t)$ and $G_t \equiv g(X_t) \!-\! g(Y_t)$,
+We define $$D_t \equiv X_t \!-\! Y_t$$ and $$Z_t \equiv D_t^2 \ge 0$$,
+together with $$F_t \equiv f(X_t) \!-\! f(Y_t)$$ and $$G_t \equiv g(X_t) \!-\! g(Y_t)$$,
 such that Itō's lemma states:
 
 $$\begin{aligned}
@@ -322,9 +323,9 @@ $$\begin{aligned}
     = Z_0 + \mathbf{E}\! \int_0^t 2 D_s F_s + G_s^2 \dd{s}
 \end{aligned}$$
 
-The *Cauchy-Schwarz inequality* states that $|D_s F_s| \le |D_s| |F_s|$,
-and then the given fact that $F_s$ and $G_s$ satisfy
-$|F_s| \le K |D_s|$ and $|G_s| \le K |D_s|$ gives:
+The *Cauchy-Schwarz inequality* states that $$|D_s F_s| \le |D_s| |F_s|$$,
+and then the given fact that $$F_s$$ and $$G_s$$ satisfy
+$$|F_s| \le K |D_s|$$ and $$|G_s| \le K |D_s|$$ gives:
 
 $$\begin{aligned}
     \mathbf{E}[Z_t]
@@ -333,12 +334,12 @@ $$\begin{aligned}
     &\le Z_0 + \int_0^t (2 K \!+\! K^2) \: \mathbf{E}[Z_s] \dd{s}
 \end{aligned}$$
 
-Where we have implicitly used that $D_s F_s = |D_s F_s|$
-because $Z_t$ is positive for all $G_s^2$,
-and that $|D_s|^2 = D_s^2$ because $D_s$ is real.
+Where we have implicitly used that $$D_s F_s = |D_s F_s|$$
+because $$Z_t$$ is positive for all $$G_s^2$$,
+and that $$|D_s|^2 = D_s^2$$ because $$D_s$$ is real.
 We then apply the
 [Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/),
-recognizing that $Z_0$ does not decrease with time (since it is constant):
+recognizing that $$Z_0$$ does not decrease with time (since it is constant):
 
 $$\begin{aligned}
     \mathbf{E}[Z_t]
@@ -346,13 +347,14 @@ $$\begin{aligned}
     \\
     &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big)
 \end{aligned}$$
+
 </div>
 </div>
 
 Using these properties, it can then be shown
 that if all of the above conditions are satisfied,
 then the SDE has a unique solution,
-which is $\mathcal{F}_t$-adapted, continuous, and exists for all times.
+which is $$\mathcal{F}_t$$-adapted, continuous, and exists for all times.