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+---
+title: "Itō process"
+date: 2021-11-06
+categories:
+- Mathematics
+- Stochastic analysis
+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/),
+i.e. Brownian motion:
+
+$$\begin{aligned}
+    X_t
+    = X_0 + \int_0^t F_s \dd{s} + \int_0^t G_s \dd{B_s}
+\end{aligned}$$
+
+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
+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
+[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 $\xi_t = \idv{B_t}{t}$ 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**.
+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.
+
+
+## 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**:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{Y_t}
+        = \bigg( \pdv{h}{t} + \pdv{h}{x} F_t + \frac{1}{2} \pdvn{2}{h}{x} G_t^2 \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t}
+    }
+\end{aligned}$$
+
+<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:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{X_t} + \frac{1}{2} \pdvn{2}{h}{x} \dd{X_t}^2
+\end{aligned}$$
+
+But here we cannot neglect $\dd{X_t}^2$.
+We insert the definition of an Itō process:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    &= \pdv{h}{t} \dd{t} + \pdv{h}{x} \Big( F_t \dd{t} + G_t \dd{B_t} \Big) + \frac{1}{2} \pdvn{2}{h}{x} \Big( F_t \dd{t} + G_t \dd{B_t} \Big)^2
+    \\
+    &= \pdv{h}{t} \dd{t} + \pdv{h}{x} \Big( F_t \dd{t} + G_t \dd{B_t} \Big)
+    + \frac{1}{2} \pdvn{2}{h}{x} \Big( F_t^2 \dd{t}^2 + 2 F_t G_t \dd{t} \dd{B_t} + G_t^2 \dd{B_t}^2 \Big)
+\end{aligned}$$
+
+In the limit of small $\dd{t}$, we can neglect $\dd{t}^2$,
+and as it turns out, $\dd{t} \dd{B_t}$ too:
+
+$$\begin{aligned}
+    \dd{t} \dd{B_t}
+    &= (B_{t + \dd{t}} - B_t) \dd{t}
+    \sim \dd{t} \mathcal{N}(0, \dd{t})
+    \sim \mathcal{N}(0, \dd{t}^3)
+    \longrightarrow 0
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \boxed{
+        h(x)
+        = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^y \frac{2 f(z)}{g^2(z)} \dd{z} \bigg) \dd{y}
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    0
+    &= f(x) \: h'(x) + \frac{1}{2} g^2(x) \: h''(x)
+    \\
+    &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g^2(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg)
+\end{aligned}$$
+
+
+## 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 $t$:
+
+$$\begin{aligned}
+    \boxed{
+        \mathbf{E}[X_t^2]
+        \le \big(X_0^2 + 3 K t\big) \exp\!\big(3 K t\big)
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-existence"/>
+<label for="proof-existence">Proof</label>
+<div class="hidden" markdown="1">
+<label for="proof-existence">Proof.</label>
+If we define $Y_t \equiv X_t^2$,
+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:
+
+$$\begin{aligned}
+    \mathbf{E}[Y_t]
+    &\le (Y_0 + 3 K t) \exp\!\bigg( \int_0^t 3 K \dd{s} \bigg)
+    \\
+    &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big)
+\end{aligned}$$
+</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 $y$:
+
+$$\begin{aligned}
+    \big| f(x) - f(y) \big| \le K \big| x - y \big|
+    \qquad \quad
+    \big| g(x) - g(y) \big| \le K \big| x - y \big|
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \boxed{
+        \mathbf{E}\big[ (X_t - Y_t)^2 \big]
+        \le (X_0 - Y_0)^2 \exp\!\Big( \big(2 K \!+\!  K^2 \big) t \Big)
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-uniqueness"/>
+<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 $G_t \equiv g(X_t) \!-\! g(Y_t)$,
+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):
+
+$$\begin{aligned}
+    \mathbf{E}[Z_t]
+    &\le Z_0 \exp\!\bigg( \int_0^t 2 K \!+\! K^2 \dd{s} \bigg)
+    \\
+    &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big)
+\end{aligned}$$
+</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.
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.