# Economic and financial modeling techniques in the frequency domain

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## Abstract

I provide some results on continuous-time frequency domain methods that can be used in dynamic models of noisy information and strategic behavior, including Fourier transform methods, spectral factorization, and some notes on numerical implementation.

## Keywords

Continuous time stochastic control Frequency domain Fourier transform methods Spectral factorization## JEL Classification

C61 C73## 1 Introduction

In this report, I describe some frequency domain modeling techniques using a continuous time approach, with a highly abbreviated version of the model in Taub (2018) to motivate the technical elements. The main tools are the Laplace and Fourier transforms and the continuous-time analogue of the Wiener–Hopf equation. These tools are described in sections 6.A (pp. 216–220), 7.1–7.2 (pp. 221–228), and 7.A (262–264) of Kailath et al. (2000). An additional reference is Hansen et al. (1991).

## 2 White noise and serially correlated processes in continuous time

*n*(

*t*); we would like to consider this as the continuous-time analogue of discrete-time white noise, that is, Gaussian, zero-mean, and serially uncorrelated. We are interested in processes that are convolutions of

*n*(

*t*):

Doob and also Hansen and Sargent draw a similar conclusion.The conclusion we arrive at from the above discussion is that we cannot represent mathematically white noise itself, but if it appears in integrated form then Brownian motion is an appropriate model (Davis 1977 p. 82).

*u*(

*t*) can be viewed as white noise in the limit as \(\eta \) goes to infinity. Given that white noise is itself ill-defined, it is mathematically more sound to view the Laplace transform of

*u*(

*t*) as occurring before taking that limit, that is, it is the transform of the integrated process:

As Kailath, Sayed, and Hassibi, and also Hansen and Sargent note, the Laplace transform is a special case of the Fourier transform, and care must be taken to ensure that the integration inherent in the transforms converges. Kailath, Sayed, and Hassibi refer to exponential boundedness of the processes; this is a specialization of the more general requirement that the functions in question have no poles in the domain of interest. Specifically, as elaborated by Hansen and Sargent, I assume that functions are analytic in a strip along the imaginary line, and the imaginary line itself is in this strip because of discounting. Moreover the endogenous processes that will later be generated by optimization and equilibrium will automatically satisfy this criterion.

*u*(

*t*) is an Ornstein–Uhlenbeck processes, that is, the analogue of an autoregressive process in discrete time settings. The weighting filter is then in exponential form:

Using the continuous time transform as described in Kailath et al. (2000, p. 217), and also in Sect. 3, the *s*-transforms of the filters for the input processes of a typical economic model are Ornstein–Uhlenbeck processes—again, the analogue of first-order autoregressive process—is \(\varPhi (s) = \frac{1}{s+\rho }\), (see Davis (1977, p. 80); similarly the *s*-transform of a white noise process is the identity matrix *I* (again, see Davis 1977 p. 80).

## 3 Fourier transforms of continuous-time processes

*z*(which corresponds to white noise) is the \(\delta \) function. (A reference is Igloi and Terdik 1999, p. 4.) The spectral density is:

*s*-domain is, therefore, also rational functions, except that causality is associated with poles in the left half plane instead of the unit circle. An additional reference is Hansen et al. (1991).

Observe that \(a=1\) yields the Fourier transform of a standard Brownian motion.

## 4 Optimizing in the frequency domain

Whiteman (1985) constructed a discrete time model and then converted the objective itself into *z*-transform form. The optimization was then over linear operators or filters that were found via a variational derivative of the transformed objective.^{1} This was achieved by imposing the constraint that the controls must be a linear filter of the information, and taking the expectation of the objective prior to optimizing over those filters. However, it is essential to reduce the covariance function of the fundamental processes—the white noise fundamentals—to a scalar covariance matrix. In continuous time, the equivalent operation is to make the fundamental covariance function \(R_x(t)\) a Dirac \(\delta \)-function.^{2}

*f*and

*g*that are elements of \(L_2\)

*F*and

*G*are elements of the Hardy space \(H^2\), the set of square integrable functions on the right half plane with inner product

*G*, \(G^*(r-s^*)\).

^{3}The \(r-s^*\) term captures discounting, and where the integration is along a strip parallel to the imaginary axis in which \(\hbox {Re}(s) = a\), where the functions

*F*and

*G*are analytic in the right half plane—that is,

*F*and

*G*have no poles or singularities in the region \(\hbox {Re}(s) > - r\), and with

*a*small enough to avoid poles and thus yield convergence, that is, \(a<r\).

^{4}There are two parts to the integrand: the causal part

*F*(

*s*) and the anti-causal part \(G^*(r-s^*)\), reflecting the inner product that is expressed in the objective.

### 4.1 An objective in the frequency domain

*s*-transform of an objective; as an example, a much-simplified version of the objective in Taub (2018) is:

*s*-transform of an exogenous driving process, and \(\varLambda \) and \(\varGamma \) are elements of \(H^2\). The causal and anti-causal parts reflect the inner product that is expressed in the objective, and

*R*is the covariance matrix function of the Dirac-\(\delta \) fundamentals

*e*(

*t*) and

*u*(

*t*).

^{5}The covariance function

*R*is block diagonal:

*B*.

#### 4.1.1 The variational condition and solution

*s*-transformed objectives can be stated. First, the notation

*s*-domain that is anti-causal, that is, \(A^*(s)=0\), for

*s*in the right half-plane.

*B*in the large shareholder’s objective (4), can be expressed as:

The equation could be solved if we could simply invert the coefficient of B. However, this inverse would involve values of *s* in the right half plane that correspond to putting weights on future values of the underlying stochastic processes, which cannot be predicted. This asymmetry in future versus past values of the underlying processes prevents solution via such direct inversion. The solution, therefore, requires an indirect method.

There are three elements of the solution method: factorization, inversion, and projection. In the factorization step, the coefficient matrix is factored into two factors, one that is a function whose zeroes are only in the left half plane, and one whose zeroes are only in the right half plane. There are two inversion steps. In the first inversion step the factor with zeroes in the right half plane is inverted: as a result of the inversion the right-half-plane zeroes then become poles. The projection step is then undertaken: the projection or annihilator operator is applied to eliminate terms that have poles in the right half plane, while preserving elements with poles in the left half plane. Finally, the factor that has zeroes in the left half plane is inverted, yielding a solution that only has poles in the left half plane, corresponding to functions that operate only on the current and past history of the underlying stochastic processes.

*G*can be chosen to be analytic and invertible. Then, the solution is:

## 5 Practical details of spectral factorization and annihilator operations in continuous time

*K*(

*s*) is the Laplace transform (

*s*-transform) of the unknown filter that is to be found;

*G*(

*s*) is the Laplace transform of a function

*g*(

*t*) that is a purely anticausal function, that is, a function that is analytic on the left half plane only and zero in the right half plane, but which is otherwise arbitrary, corresponding to the principal part function \(\sum _{-\infty }^{-1}\) in the discrete time setting: \(g(t)=0\), \(t>0\); \(S_{sy}(s)\) and \(S_y(s)\) are the Laplace transforms of variance and covariance functions:

*s*, which is completely different. Thus, \(S_y\) is the Laplace transform of the observed process, and \(\mathbf{s}(t)\) is the signal process that the observer wants to extract; \(R_{ys}\) is then the covariance function between the observed and signal processes.

*R*is a positive constant, and

*L*(

*s*) is causal, that is, both

*L*and \(L^{-1}\) are analytic on the right half plane.

*L*is analytic in the right half plane because its pole, \(-\alpha \), is in the left half plane, and the inverse is analytic in the right half plane because the zero, \(- \sqrt{\alpha ^2 + 2\alpha }\), is in the left half plane.

### 5.1 A small lemma about the annihilator

*t*, that is, in the right half plane. The following small lemma holds, which is a variation of Whittle’s theorem.

### Lemma 1

*F*be analytic in the right half plane, and let \(a>0\). Then

### Proof

*F*, namely \(F(s)=\frac{1}{s+b}\), \(b>0\)—namely when

*F*is also the filter for an Ornstein–Uhlenbeck process. In that case, the inner integral of (9) is:

*f*is analytic, then it can be represented in power series form:

*s*-transform of this function is:

*k*. \(\square \)

This result is stated and proved in greater generality for matrix systems in Seiler and Taub (2008), Lemma C.18, using state space methods. When general compound expressions of the sort \(\left\{ F G^*\right\} _+\), where both *F* and *G* are analytic, that is, their poles are in the left half plane, are viewed from a state space perspective, it is clear that the product will be a function with poles in the left half place inherited from the poles of *F* and poles in the right half plane inherited from *G*. The annihilator removes the latter poles, while the poles of *F* survive.

## 6 Conclusion

Frequency domain methods are a useful tool for modeling economic situations in which there are dynamics, noise, and strategic behavior. After mapping a typical model to an appropriate function space, the essential features of optimization by agents, and the establishment and characterization of equilibria, can be tractably carried out via variational methods and fixed point methods respectively using operations that are essentially algebraic in nature. Furthermore, numerical simulations of the resulting models are a straightforward application of complementary methods that have been developed by the applied engineering literature.

## Footnotes

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