Categories: Mathematics, Stochastic analysis.

Itō process

Given two stochastic processes \(F_t\) and \(G_t\), consider the following random variable \(X_t\), where \(B_t\) is the 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, 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, 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, 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, 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:

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

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:

\[\begin{aligned} \boxed{ h(x) = \int_{x_0}^x \exp\!\bigg( \!-\!\! \int_{x_1}^y \frac{2 f(z)}{g^2(z)} \dd{z} \bigg) \dd{y} } \end{aligned}\]

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 \(t\):

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

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:

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

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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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