Categories: Mathematics, Stochastic analysis.

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.

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**:

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

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:

\[\begin{aligned} \dd{Y_t} = \pdv{h}{t} \dd{t} + \pdv{h}{x} \dd{X_t} + \frac{1}{2} \pdv[2]{h}{x} \dd{X_t}^2 \end{aligned}\]

But here we cannot neglect \(\dd{X_t}^2\). We insert the definition of an Itō process:

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

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 \(\dd{t}\).

The most important application of Itō’s lemma is to perform coordinate transformations, to make the solution of a given Itō SDE easier.

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:

\[\begin{aligned} 0 &= f(x) \: h'(x) + \frac{1}{2} g^2(x) \: h''(x) \\ &= \Big( f(x) - \frac{1}{2} g^2(x) \frac{2 f(x)}{g^2(x)} \Big) \exp\!\bigg( \!-\!\! \int_{x_1}^x \frac{2 f(y)}{g^2(y)} \dd{y} \bigg) \end{aligned}\]

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, 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, 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.

- U.H. Thygesen,
*Lecture notes on diffusions and stochastic differential equations*, 2021, Polyteknisk Kompendie.

© Marcus R.A. Newman, a.k.a. "Prefetch".
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