path: root/source/know/concept/ito-process/index.md

---
title: "Itō process"
sort_title: "Ito process" # sic
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**:

$$\begin{aligned}
    \boxed{
        \dd{Y_t}
        = \bigg( \pdv{h}{t} + \pdv{h}{x} F_t + \frac{1}{2} \pdvn{2}{h}{x} G_t^2 \bigg) \dd{t} + \pdv{h}{x} G_t \dd{B_t}
    }
\end{aligned}$$


{% include proof/start.html id="proof-lemma" -%}
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 $$\dd{t}$$.
{% include proof/end.html id="proof-lemma" %}


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 $$t$$:

$$\begin{aligned}
    \boxed{
        \mathbf{E}[X_t^2]
        \le \big(X_0^2 + 3 K t\big) \exp\!\big(3 K t\big)
    }
\end{aligned}$$


{% include proof/start.html id="proof-existence" -%}
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:

$$\begin{aligned}
    \mathbf{E}[Y_t]
    &\le (Y_0 + 3 K t) \exp\!\bigg( \int_0^t 3 K \dd{s} \bigg)
    \\
    &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big)
\end{aligned}$$
{% include proof/end.html id="proof-existence" %}


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:

$$\begin{aligned}
    \boxed{
        \mathbf{E}\big[ (X_t - Y_t)^2 \big]
        \le (X_0 - Y_0)^2 \exp\!\Big( \big(2 K \!+\!  K^2 \big) t \Big)
    }
\end{aligned}$$


{% include proof/start.html id="proof-uniqueness" -%}
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):

$$\begin{aligned}
    \mathbf{E}[Z_t]
    &\le Z_0 \exp\!\bigg( \int_0^t 2 K \!+\! K^2 \dd{s} \bigg)
    \\
    &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big)
\end{aligned}$$
{% include proof/end.html id="proof-uniqueness" %}


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.