From 62759ea3f910fae2617d033bf8f878d7574f4edd Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 7 Nov 2021 19:34:18 +0100
Subject: Expand knowledge base, reorganize measure theory, update gitignore
---
content/know/concept/ito-calculus/index.pdc | 180 +++++++++++++++++++++++++---
1 file changed, 164 insertions(+), 16 deletions(-)
(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
index 576e09a..3527b1d 100644
--- a/content/know/concept/ito-calculus/index.pdc
+++ b/content/know/concept/ito-calculus/index.pdc
@@ -12,10 +12,10 @@ 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/):
+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/),
+i.e. Brownian motion:
$$\begin{aligned}
X_t
@@ -27,7 +27,7 @@ assuming $G_t$ is Itō-integrable.
We call $X_t$ an **Itō process** if $F_t$ is locally integrable,
and the initial condition $X_0$ is known,
i.e. $X_0$ is $\mathcal{F}_0$-measurable,
-where $\mathcal{F}_t$ is the [filtration](/know/concept/sigma-algebra/)
+where $\mathcal{F}_t$ is the filtration
to which $F_t$, $G_t$ and $B_t$ are adapted.
The above definition of $X_t$ is often abbreviated as follows,
where $X_0$ is implicit:
@@ -39,8 +39,18 @@ $$\begin{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
+[martingale](/know/concept/martingale/),
+it does not contribute to the mean of $X_t$:
+
+$$\begin{aligned}
+ \mathbf{E}[X_t]
+ = \int_0^t \mathbf{E}[F_s] \dd{s}
+\end{aligned}$$
+
Now, consider the following **Itō stochastic differential equation** (SDE),
-where $\xi_t = \dv*{B_t}{t}$ is white noise:
+where $\xi_t = \dv*{B_t}{t}$ is white noise,
+informally treated as the $t$-derivative of $B_t$:
$$\begin{aligned}
\dv{X_t}{t}
@@ -51,15 +61,6 @@ An Itō process $X_t$ is said to satisfy this equation
if $f(X_t, t) = F_t$ and $g(X_t, t) = G_t$,
in which case $X_t$ is also called an **Itō diffusion**.
-Because the Itō integral of $G_t$ is a
-[martingale](/know/concept/martingale/),
-it does not contribute to the mean of $X_t$:
-
-$$\begin{aligned}
- \mathbf{E}[X_t]
- = \int_0^t \mathbf{E}[F_s] \dd{s}
-\end{aligned}$$
-
## Itō's lemma
@@ -204,9 +205,156 @@ $$\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)
+ &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g^2(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg)
+\end{aligned}$$
+
+
+## Existence and uniqueness
+
+It is worth knowing under what condition a solution to a given SDE exists,
+in the sense that it is finite on the entire time axis.
+Suppose the drift $f$ and intensity $g$ satisfy these inequalities,
+for some known constant $K$ and for all $x$:
+
+$$\begin{aligned}
+ x f(x) \le K (1 + x^2)
+ \qquad \quad
+ g^2(x) \le K (1 + x^2)
+\end{aligned}$$
+
+When this is satisfied, we can find the following upper bound
+on an Itō process $X_t$,
+which clearly implies that $X_t$ is finite for all $t$:
+
+$$\begin{aligned}
+ \boxed{
+ \mathbf{E}[X_t^2]
+ \le \big(X_0^2 + 3 K t\big) \exp\!\big(3 K t\big)
+ }
+\end{aligned}$$
+
+
+
+
+
+
+If we define $Y_t \equiv X_t^2$,
+then Itō's lemma tells us that the following holds:
+
+$$\begin{aligned}
+ \dd{Y_t}
+ = \big( 2 X_t \: f(X_t) + g^2(X_t) \big) \dd{t} + 2 X_t \: g(X_t) \dd{B_t}
\end{aligned}$$
+Integrating and taking the expectation value
+removes the Wiener term, leaving:
+
+$$\begin{aligned}
+ \mathbf{E}[Y_t]
+ = Y_0 + \mathbf{E}\! \int_0^t 2 X_s f(X_s) + g^2(X_s) \dd{s}
+\end{aligned}$$
+
+Given that $K (1 \!+\! x^2)$ is an upper bound of $x f(x)$ and $g^2(x)$,
+we get an inequality:
+
+$$\begin{aligned}
+ \mathbf{E}[Y_t]
+ &\le Y_0 + \mathbf{E}\! \int_0^t 2 K (1 \!+\! X_s^2) + K (1 \!+\! X_s^2) \dd{s}
+ \\
+ &\le Y_0 + \int_0^t 3 K (1 + \mathbf{E}[Y_s]) \dd{s}
+ \\
+ &\le Y_0 + 3 K t + \int_0^t 3 K \big( \mathbf{E}[Y_s] \big) \dd{s}
+\end{aligned}$$
+
+We then apply the
+[Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/),
+noting that $(Y_0 \!+\! 3 K t)$ does not decrease with time, leading us to:
+
+$$\begin{aligned}
+ \mathbf{E}[Y_t]
+ &\le (Y_0 + 3 K t) \exp\!\bigg( \int_0^t 3 K \dd{s} \bigg)
+ \\
+ &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big)
+\end{aligned}$$
+
+
+
+If a solution exists, it is also worth knowing whether it is unique.
+Suppose that $f$ and $g$ satisfy the following inequalities,
+for some constant $K$ and for all $x$ and $y$:
+
+$$\begin{aligned}
+ \big| f(x) - f(y) \big| \le K \big| x - y \big|
+ \qquad \quad
+ \big| g(x) - g(y) \big| \le K \big| x - y \big|
+\end{aligned}$$
+
+Let $X_t$ and $Y_t$ both be solutions to a given SDE,
+but the initial conditions need not be the same,
+such that the difference is initially $X_0 \!-\! Y_0$.
+Then the difference $X_t \!-\! Y_t$ is bounded by:
+
+$$\begin{aligned}
+ \boxed{
+ \mathbf{E}\big[ (X_t - Y_t)^2 \big]
+ \le (X_0 - Y_0)^2 \exp\!\Big( \big(2 K \!+\! K^2 \big) t \Big)
+ }
+\end{aligned}$$
+
+
+
+
+
+
+We define $D_t \equiv X_t \!-\! Y_t$ and $Z_t \equiv D_t^2 \ge 0$,
+together with $F_t \equiv f(X_t) \!-\! f(Y_t)$ and $G_t \equiv g(X_t) \!-\! g(Y_t)$,
+such that Itō's lemma states:
+
+$$\begin{aligned}
+ \dd{Z_t}
+ = \big( 2 D_t F_t + G_t^2 \big) \dd{t} + 2 D_t G_t \dd{B_t}
+\end{aligned}$$
+
+Integrating and taking the expectation value
+removes the Wiener term, leaving:
+
+$$\begin{aligned}
+ \mathbf{E}[Z_t]
+ = Z_0 + \mathbf{E}\! \int_0^t 2 D_s F_s + G_s^2 \dd{s}
+\end{aligned}$$
+
+The *Cauchy-Schwarz inequality* states that $|D_s F_s| \le |D_s| |F_s|$,
+and then the given fact that $F_s$ and $G_s$ satisfy
+$|F_s| \le K |D_s|$ and $|G_s| \le K |D_s|$ gives:
+
+$$\begin{aligned}
+ \mathbf{E}[Z_t]
+ &\le Z_0 + \mathbf{E}\! \int_0^t 2 K D_s^2 + K^2 D_s^2 \dd{s}
+ \\
+ &\le Z_0 + \int_0^t (2 K \!+\! K^2) \: \mathbf{E}[Z_s] \dd{s}
+\end{aligned}$$
+
+Where we have implicitly used that $D_s F_s = |D_s F_s|$
+because $Z_t$ is positive for all $G_s^2$,
+and that $|D_s|^2 = D_s^2$ because $D_s$ is real.
+We then apply the
+[Grönwall-Bellman inequality](/know/concept/gronwall-bellman-inequality/),
+recognizing that $Z_0$ does not decrease with time (since it is constant):
+
+$$\begin{aligned}
+ \mathbf{E}[Z_t]
+ &\le Z_0 \exp\!\bigg( \int_0^t 2 K \!+\! K^2 \dd{s} \bigg)
+ \\
+ &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big)
+\end{aligned}$$
+
+
+
+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.
+
## References
--
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