From f091bf0922c26238d16bf175a8ea916a16d11fba Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 6 Nov 2021 21:47:08 +0100 Subject: Expand knowledge base --- content/know/concept/ito-calculus/index.pdc | 215 ++++++++++++++++++++++++++++ 1 file changed, 215 insertions(+) create mode 100644 content/know/concept/ito-calculus/index.pdc (limited to 'content/know/concept/ito-calculus') diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-calculus/index.pdc new file mode 100644 index 0000000..576e09a --- /dev/null +++ b/content/know/concept/ito-calculus/index.pdc @@ -0,0 +1,215 @@ +--- +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**: + +$$\begin{aligned} + \boxed{ + \dd{Y_t} + = \pdv{h}{t} \dd{t} + \bigg( \pdv{h}{x} F_t + \frac{1}{2} G_t^2 \pdv[2]{h}{x} \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t} + } +\end{aligned}$$ + +
+ + + +
+ +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. -- cgit v1.2.3