11. Linear Stochastic Differential Equations

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11. Linear Stochastic Differential Equations#

We now consider a class of higher order stochastic differential equations that are formed by driving a linear, constant coefficient \(n^{th}\) order differential equation with a white noise. Letting \(D^j\) be the \(j^{th}\) mean square derivative operator, we consider the equation

(58)#\[\theta(D)\, x(t) = w(t)\]

or

(59)#\[(D^n + \theta_{n-1}\, D^{n-1} + \ldots + \theta_0)\, x(t) = w(t)\]

where

\[ E w(t)\, w(t-\tau) = \delta(\tau) \]

and

\[\begin{split} \begin{aligned} \theta(D) &= (D^n + \theta_{n-1}\, D^{n-1} + \ldots + \theta_0) \\ &= (D - \lambda_1)\ (D - \lambda_2) \ldots (D - \lambda_n) \end{aligned} \end{split}\]

We assume that the real parts of \(\lambda_j\), which we denote \(re(\lambda_j)\), are less than zero for \(j = 1, \ldots, n\). We assume that the \(\lambda_j\)’s, which are the zeroes of \(\theta(s)\), are distinct.

By writing (58) we intend to denote that \(x(t)\) is the stochastic process given by

(60)#\[x(t) = \int^\infty_0 p(\tau)\, w(t - \tau)\, d\tau = \tilde P(D)\, w(t)\]

where

\[ \tilde P(D) \equiv \theta(D)^{-1} =\ \frac{1}{(D - \lambda_1)\, \cdots\, (D - \lambda_n)}. \]

Note

The white noise \(w(t)\) is not an ordinary function. Chapters 4 and 6 construct it as a generalized process with the \(\delta\)-function autocovariance \(E\,w(t)\,w(t-\tau)=\delta(\tau)\). Its instantaneous variance is infinite, and it means something only underneath an integral sign. The moving average (60) is such an integral. Read it as a mean square (Wiener) stochastic integral against the increments of the Wiener process \(W\) of Chapter 6. Write \(dW(t) = w(t)\,dt\). Then

\[ x(t) = \int_0^\infty p(\tau)\, w(t-\tau)\, d\tau = \int_0^\infty p(\tau)\, dW(t-\tau), \]

The right side is the \(L^2\) limit of the approximating sums \(\sum_i p(\tau_i)\,[\,W(t-\tau_i) - W(t-\tau_{i+1})\,]\) built from the independent Gaussian increments of \(W\). The limit exists when \(\int_0^\infty p(\tau)^2\,d\tau < \infty\). The Itô isometry then gives \(E\,x(t)^2 = \int_0^\infty p(\tau)^2\,d\tau\) and \(R(s) = \int_0^\infty p(\tau)\,p(\tau+s)\,d\tau\). Integration converts a generalized noise into an ordinary process with finite variance, covariance stationary and mean square continuous. The kernel is a causal linear filter that weights the recent past of the innovations by \(p\). Square integrability holds here because \(p(\tau)=\sum_j g_j e^{\lambda_j\tau}\) decays exponentially. Each factor \(\frac{1}{D-\lambda_j}\,w(t) = \int_0^\infty e^{\lambda_j u}\,w(t-u)\,du\) is a one-sided exponential smoother, the Ornstein–Uhlenbeck building block of Chapter 7. The stationary solution sums these smoothers.

The kernel \(p(\tau)\) is deterministic. It does not depend on the path of \(W\), so how we form the stochastic integral does not matter. The Itô and Stratonovich integrals differ by a quadratic variation correction that appears when the integrand depends on the Brownian motion against which it is integrated, as the \(g(x)\,dW\) terms of Chapter 7 do. A deterministic integrand kills that correction. Here the Itô integral, the Stratonovich integral, and the mean square integral coincide.

A partial fraction representation of \(\tilde P(D)\) is

\[ \tilde P(D) = \sum_{j=1}^{n}\ \frac{g_j}{D - \lambda_j} \]

where

\[ g_j = \lim_{s\to \lambda_j}\ \tilde P(s)\ (s - \lambda_j). \]

Now \(1/(s-\lambda_j)\) for \(re(\lambda_j) < 0\) is the Laplace transform of the function \([e^{\lambda_j t},\, t \geq 0\); 0 for \(t < 0]\). We express this transform pair relationship as

\[ \frac{1}{s - \lambda_j}\ \leftrightarrow\ e^{\lambda_j\, t}. \]

This relationship justifies the operational calculus

\[ \frac{1}{D - \lambda_j}\ w(t) = \int^\infty_0 e^{\lambda_j u}\ w(t-u)\, du \]

for \(re(\lambda_j) < 0\).

Thus, we can express (60) as

\[ x(t) = \sum_{j=1}^{n}\ \frac{g_j}{D - \lambda_j}\ w(t) \]

or

\[ x(t) = \sum_{j=1}^{n} g_j \int^\infty_0 e^{\lambda_j u}\ w(t - u)\, du. \]

This equation is said to represent the stationary solution of the stochastic differential equation. These rational-spectral-density processes are precisely the ones that admit a finite-dimensional state-space realization; 15. State-Space Models, the Kalman Filter, and Spectral Factorization recasts their Wold representation and spectral factorization as the time-domain output of the Kalman–Bucy filter.

We now show that the solution is mean square differentiable \(n - 1\) times, but not \(n\) times, by applying the initial value theorem of 9. Characterizations of Mean Square Differentiability and Mean Square Continuity, \(p(0) = \lim_{s\to\infty} s\tilde P(s)\). The Laplace transform of \(p(\tau)\) is \(\tilde P(s) = 1/[(s - \lambda_1) \ldots (s - \lambda_n)]\), that of \(p'(\tau)\) is \(s \tilde P(s) = s/[(s - \lambda_1) \ldots (s - \lambda_n)]\), and so on. For \(x(t)\) to be mean square differentiable, we require that

\[ p(0) = \lim_{s\to \infty}\ s \tilde P(s) = \lim_{s\to \infty}\ \frac{s}{(s-\lambda_1) \ldots (s - \lambda_n)} = 0, \]

which obtains so long as \(n \geq 2\). For \(x(t)\) to be \(j\) times mean square differentiable, we require that

\[ p^{(j - 1)}(0) = \lim_{s \to \infty}\ s^j\, \tilde P(s) = \lim_{s \to \infty}\ \frac{s^j}{(s - \lambda_1) \ldots (s - \lambda_n)} = 0 \]

which holds for \(j < n\), but is violated for \(j \geq n\).

Therefore, where \(x(t)\) is governed by the solution of the \(n^{th}\) order stochastic differential equation, \(x(t)\) is mean square differentiable up to orders \((n-1)\), but is not differentiable \(n\) times. (The reader can verify that in the present case of a rational \(p(\tau)\) function, \(\int^\infty_0 |D^j p(\tau)|^2\, d\tau < + \infty\) for all \(j\), so that the second condition for mean square differentiability is satisfied.)

Next, we consider a broader class of linear stochastic differential equations generated by driving an \(n^{th}\) order constant coefficient linear differential equation with a linear combination of a white noise and \(m \leq n - 1\) generalized derivatives of this same white noise. The differential equation is

\[ \theta(D)\, x(t) = \Psi(D)\, w(t) \]

where

\[\begin{split} \begin{aligned} \theta(D) &= (D^n + \theta_{n-1}\, D^{n-1}+\ldots + \theta_0) \\ &= (D - \lambda_1)\ (D - \lambda_2) \ldots (D - \lambda_n),\ re(\lambda_j)< 0,\ j = 1,\ldots, n \\ \Psi(D) &= \Psi_0 + \Psi_1 D + \ldots + \Psi_m D^m,\ \Psi_0 = 1,\ m \leq n-1. \\ E w(t) &\, w(t - \tau) = \delta(\tau). \end{aligned} \end{split}\]

In this equation, \(w(t)\) is a fundamental white noise for \(x(t)\) if and only if the zeroes of \(\Psi(s)\) lie in the left hand of the complex plane. By the stationary solution of this equation we mean

\[ x(t) = \tilde P(D)\, w(t) = \int^\infty_0 p(\tau)\, w(t-\tau)\, d\tau \]

where

\[ \tilde P(s) = \theta(s)^{-1}\, \Psi(s). \]

A partial fraction representation of \(\tilde P(s)\) is

\[ \tilde P(s) = \sum_{j=1}^{n}\ \frac{g_j}{s - \lambda_j} \]

where \(g_j = \lim_{s \to \lambda_j}\ \tilde P(s)\ (s-\lambda_j)\). Therefore \(p(\tau) = \sum_{j=1}^{n}\ g_j\ e^{\lambda_j \tau}\). Proceeding as above, once again we see that the solution has representation

\[ x(t) = \sum_{j=1}^{n}\ g_j \int^\infty_0 e^{\lambda_j u}\, w(t-u)\, du. \]

Evidently, \(p(\tau) = \sum_{j=1}^{n} g_j\, e^{\lambda_j \tau}\) is continuous in \(\tau\). Therefore \(x_t\) is mean square continuous. To test for mean square differentiability, we must evaluate

\[ p(0) = \lim_{s\to \infty}\, s \tilde P(s) = \lim_{s\to \infty}\ \frac{\Psi_0 s + \Psi_1 s^2 + \ldots + \Psi_m s^{m+1}}{\theta_0 + \theta_1 s + \ldots + s^n} \]

which equals zero only if \(n > m + 1\). Proceeding in this way, it can be verified that \(x(t)\) is mean square differentiable \(n - 1 - m\) times, but no more.

Note

The kernels \(p(\tau) = \sum_j g_j\, e^{\lambda_j \tau}\) that rational linear stochastic differential equations deliver are analytic. The processes of this chapter are therefore mean square continuous, and their differentiability depends only on how many of the leading values \(p(0), p'(0), \ldots\) vanish. That is the count \(n - 1 - m\) just obtained, an application of the criteria of 9. Characterizations of Mean Square Differentiability and Mean Square Continuity. 23. Temporal Aggregation of Economic Time Series leaves this smooth class. Marcet posits one-sided kernels that are discontinuous, the leading case being a jump at the origin, \(p(0) \neq 0\). Such a process remains mean square continuous, because Theorem 14 of 9. Characterizations of Mean Square Differentiability and Mean Square Continuity asks only that the kernel be square integrable, not that it be continuous. It is not mean square differentiable, so it is locally unpredictable in the sense of 13. Locally Unpredictable Stochastic Processes. Those discontinuities, and the jumps they force on the sampled discrete-time kernel at every integer, drive the aliasing and the manufactured or destroyed Granger causality that Chapter 23 analyzes.

The following theorem pulls together a number of our earlier results.

Theorem 15

Let \(x(t)\) be a covariance stationary, zero mean, linearly indeterministic stochastic process, with Wold representation

\[ x(t) = \int^\infty_0 p(\tau)\, w(t - \tau)\, d\tau, \]

and covariogram \(E x(t)\, x(t-\tau) = R(\tau)\). Suppose that \(R(\tau)\) and the Wold kernel \(p(\tau)\) are analytic functions, \(R\) having a convergent Taylor series representation

(61)#\[R(\tau) = \sum_{n=0}^{\infty} R^{(n)}(0)\ \frac{\tau^n}{n!}\]

where \(R^{(n)}(\tau) \equiv \frac{d^n}{d\tau^n}\ R(\tau)\). Then it follows that \(x(t)\) is mean square differentiable \(k\) times, for every \(k \geq 1\) (the letter \(m\) is reserved in this chapter for the degree of \(\Psi\)). Indeed, it follows from (61) that

\[ R^{(2k)}(\tau) = \sum_{n=2k}^{\infty}\, R^{(n)}(0)\ \frac{\tau^{(n-2k)}}{(n - 2k)!}\, , \]

so that \(R^{(2k)}(\tau)\) exists and itself has a convergent Taylor series representation. Furthermore, \(p(\tau)\) is by hypothesis analytic and so has a convergent Taylor series representation

\[ p(\tau) = \sum_{n=0}^{\infty} p^{(n)}(0)\ \frac{\tau^n}{n!}\, . \]

Since \(x(t)\) is mean square differentiable an infinite number of times, it follows that \(p^{(n)}(0) = 0\) for all \(n \geq 0\), and therefore that \(p(\tau) = 0\) for all \(\tau \geq 0\). Therefore \(x(t) = 0\) for all \(t\).

The theorem states that the only covariance stationary linearly indeterministic process with analytic autocovariance function \(R(\tau)\) and analytic Wold integrating kernel \(p(\tau)\) is the trivial process \(x(t) = 0\).

Note

A real-valued function \(R\) of a real variable is analytic on an interval if around every point \(\tau_0\) of that interval it agrees with a convergent power series

\[ R(\tau) = \sum_{n=0}^{\infty} \frac{R^{(n)}(\tau_0)}{n!}\,(\tau - \tau_0)^n, \]

having a strictly positive radius of convergence. Equivalently, \(R\) is the restriction to the real axis of a function holomorphic on an open neighborhood of that interval in the complex plane \(\mathbb{C}\). A theorem of complex analysis says that a function complex-differentiable once on an open set is infinitely complex-differentiable there and equals its own Taylor series. For complex functions, “differentiable,” “infinitely differentiable,” and “equal to a convergent power series” say the same thing. For real functions they do not.

Analytic functions are rigid. The identity theorem says that an analytic function on a connected domain is determined by its values on any subset having a limit point. If all its derivatives vanish at a single point, or if it vanishes on an interval however short, then it vanishes identically.

That rigidity drives Theorem 15, and it asks more than smoothness. A function can be infinitely differentiable (\(C^\infty\)) without being analytic. Take \(f(\tau) = e^{-1/\tau^2}\) with \(f(0)=0\). Every \(f^{(n)}(0)\) vanishes, yet \(f\) is not identically zero, so \(f\) is not analytic at the origin. Smoothness of \(p(\tau)\) would therefore permit a nonzero kernel all of whose derivatives vanish at \(\tau = 0\). Analyticity of \(R\), inherited by \(p\), upgrades \(p^{(n)}(0) = 0\) for all \(n\) into \(p \equiv 0\), and so into \(x(t) \equiv 0\).

Exercises#

import numpy as np

Exercise 13

The differentiability count. The text shows that \(\theta(D)x = \psi(D)w\) delivers a process that is mean square differentiable exactly \(n - 1 - m\) times, where \(n = \deg\theta\) and \(m = \deg\psi\). Verify this two ways for the third-order system

\[ (D^3 + .6D^2 + .4D + .2)\, x(t) = w(t), \qquad n = 3,\ m = 0, \]

which is the numerical example of 18. Time Aggregation, Table 1.

(a) Compute the roots \(\lambda_j\) of \(\theta\), form the residues \(g_j = \lim_{s\to\lambda_j}(s-\lambda_j)\tilde P(s)\), and build \(p(\tau) = \sum_j g_j e^{\lambda_j\tau}\). Confirm numerically that \(p(0) = p'(0) = 0\) while \(p''(0) \neq 0\), so that the process is twice but not three times mean square differentiable, i.e. \(n-1-m = 2\).

(b) Confirm the same count with the initial value theorem of 9. Characterizations of Mean Square Differentiability and Mean Square Continuity, by evaluating \(s^j \tilde P(s)\) at large real \(s\) for \(j = 1, 2, 3\) and seeing which limits vanish.

(c) Now put a numerator of degree \(m = 2\) on the same denominator, say \(\psi(s) = 1 + s + s^2\). Show that \(p(0) \neq 0\): the process is differentiable \(n - 1 - m = 0\) times, and is therefore locally unpredictable in the sense of 13. Locally Unpredictable Stochastic Processes.

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