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+---
+title: "Itō calculus"
+firstLetter: "I"
+publishDate: 2021-11-06
+categories:
+- Mathematics
+
+date: 2021-11-06T14:34:00+01:00
+draft: false
+markup: pandoc
+---
+
+# Itō calculus
+
+Given two time-indexed [random variables](/know/concept/random-variable/)
+(i.e. stochastic processes) $F_t$ and $G_t$,
+then consider the following random variable $X_t$,
+where $B_t$ is the [Wiener process](/know/concept/wiener-process/):
+
+$$\begin{aligned}
+    X_t
+    = X_0 + \int_0^t F_s \dd{s} + \int_0^t G_s \dd{B_s}
+\end{aligned}$$
+
+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](/know/concept/sigma-algebra/)
+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**.
+Now, consider the following **Itō stochastic differential equation** (SDE),
+where $\xi_t = \dv*{B_t}{t}$ is white noise:
+
+$$\begin{aligned}
+    \dv{X_t}{t}
+    = f(X_t, t) + g(X_t, t) \: \xi_t
+\end{aligned}$$
+
+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**.
+
+Because the Itō integral of $G_t$ is a
+[martingale](/know/concept/martingale/),
+it does not contribute to the mean of $X_t$:
+
+$$\begin{aligned}
+    \mathbf{E}[X_t]
+    = \int_0^t \mathbf{E}[F_s] \dd{s}
+\end{aligned}$$
+
+
+## Itō's lemma
+
+Classically, given $y \equiv h(x(t), t)$,
+the chain rule of differentiation states that:
+
+$$\begin{aligned}
+    \dd{y}
+    = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{x}
+\end{aligned}$$
+
+However, for a stochastic process $Y_t \equiv h(X_t, t)$,
+where $X_t$ is an Itō process,
+the chain rule is modified to the following,
+known as **Itō's lemma**:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{Y_t}
+        = \pdv{h}{t} \dd{t} + \bigg( \pdv{h}{x} F_t + \frac{1}{2} G_t^2 \pdv[2]{h}{x} \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t}
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-lemma"/>
+<label for="proof-lemma">Proof</label>
+<div class="hidden">
+<label for="proof-lemma">Proof.</label>
+We start by applying the classical chain rule,
+but we go to second order in $x$.
+This is also valid classically,
+but there we would neglect all higher-order infinitesimals:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{X_t} + \frac{1}{2} \pdv[2]{h}{x} \dd{X_t}^2
+\end{aligned}$$
+
+But here we cannot neglect $\dd{X_t}^2$.
+We insert the definition of an Itō process:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    &= \pdv{h}{t} \dd{t} + \pdv{h}{x} \Big( F_t \dd{t} + G_t \dd{B_t} \Big) + \frac{1}{2} \pdv[2]{h}{x} \Big( F_t \dd{t} + G_t \dd{B_t} \Big)^2
+    \\
+    &= \pdv{h}{t} \dd{t} + \pdv{h}{x} \Big( F_t \dd{t} + G_t \dd{B_t} \Big)
+    + \frac{1}{2} \pdv[2]{h}{x} \Big( F_t^2 \dd{t}^2 + 2 F_t G_t \dd{t} \dd{B_t} + G_t^2 \dd{B_t}^2 \Big)
+\end{aligned}$$
+
+In the limit of small $\dd{t}$, we can neglect $\dd{t}^2$,
+and as it turns out, $\dd{t} \dd{B_t}$ too:
+
+$$\begin{aligned}
+    \dd{t} \dd{B_t}
+    &= (B_{t + \dd{t}} - B_t) \dd{t}
+    \sim \dd{t} \mathcal{N}(0, \dd{t})
+    \sim \mathcal{N}(0, \dd{t}^3)
+    \longrightarrow 0
+\end{aligned}$$
+
+However, due to the scaling property of $B_t$,
+we cannot ignore $\dd{B_t}^2$, which has order $\dd{t}$:
+
+$$\begin{aligned}
+    \dd{B_t}^2
+    &= (B_{t + \dd{t}} - B_t)^2
+    \sim \big( \mathcal{N}(0, \dd{t}) \big)^2
+    \sim \chi^2_1(\dd{t})
+    \longrightarrow \dd{t}
+\end{aligned}$$
+
+Where $\chi_1^2(\dd{t})$ is the generalized chi-squared distribution
+with one term of variance $\dd{t}$.
+</div>
+</div>
+
+The most important application of Itō's lemma
+is to perform coordinate transformations,
+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:
+
+$$\begin{aligned}
+    \dd{Y_s}
+    = \dd{X_t}
+    &= f(X_t) \dd{t} + g(X_t) \dd{B_t}
+    \\
+    &= \frac{1}{\alpha} f(Y_s) \dd{s} + \frac{1}{\sqrt{\alpha}} g(Y_s) \dd{W_s}
+\end{aligned}$$
+
+$W_s$ is a valid Wiener process,
+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. $g = 1$.
+This can be achieved using the **Lamperti transform**:
+define $Y_t \equiv h(X_t)$, where $h$ is given by:
+
+$$\begin{aligned}
+    \boxed{
+        h(x)
+        = \int_{x_0}^x \frac{1}{g(y)} \dd{y}
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    &= \bigg( f(X_t) \: h'(X_t) + \frac{1}{2} g^2(X_t) \: h''(X_t) \bigg) \dd{t} + g(X_t) \: h'(X_t) \dd{B_t}
+    \\
+    &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g^2(X_t) \frac{g'(X_t)}{g^2(X_t)} \bigg) \dd{t} + \frac{g(X_t)}{g(X_t)} \dd{B_t}
+    \\
+    &= \bigg( \frac{f(X_t)}{g(X_t)} - \frac{1}{2} g'(X_t) \bigg) \dd{t} + \dd{B_t}
+\end{aligned}$$
+
+Similarly, we can eliminate the drift $f = 0$,
+thereby making the Itō process a martingale.
+This is done by defining $Y_t \equiv h(X_t)$, with $h(x)$ given by:
+
+$$\begin{aligned}
+    \boxed{
+        h(x)
+        = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg)
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    0
+    &= f(x) \: h'(x) + \frac{1}{2} g^2(x) \: h''(x)
+    \\
+    &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg)
+\end{aligned}$$
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.