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{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.

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.