E. Elton and M. Gruber, Modern Portfolio Theory and Investment Analysis (1984, p. 273)

What role have theoretical methods initially developed in mathematics and physics played in the progress of financial economics? What is the relationship between financial economics and econophysics? What is the relevance of the “classical ergodicity hypothesis” to modern portfolio theory? This paper addresses these questions by reviewing the etymology and history of the classical ergodicity hypothesis in 19th century statistical mechanics. An explanation of classical ergodicity is provided that establishes a connection to the fundamental empirical problem of using nonexperimental data to verify theoretical propositions in modern portfolio theory. The role of the ergodicity assumption in the

At least since Markowitz [

Physical theory has evolved considerably from the constrained optimization, static equilibrium approach of rational mechanics which underpins MPT. In detailing historical developments in physics since the 19th century, it is conventional to jump from the determinism of rational mechanics to quantum mechanics to recent developments in chaos theory, overlooking the relevance of the initial steps toward modeling stochastic behavior of physical phenomena by Ludwig Boltzmann (1844–1906), James Maxwell (1831–1879), and Josiah Gibbs (1839–1903). As such, there is a point of demarcation between the intellectual prehistories of MPT and econophysics that can, arguably, be traced to the debate over energistics around the end of the 19th century. While the evolution of physics after energistics involved the introduction and subsequent stochastic generalization of ergodic concepts, fueled by the emergence of MPT following Markowitz, financial economics incorporated ergodicity into empirical methods aimed at generalizing and testing the capital asset pricing model and other elements of MPT.^{1}

In contrast, from the early ergodic models of Boltzmann to the fractals and chaos theory of Mandlebrot, physics has employed a wider variety of ergodic and nonergodic stochastic models aimed at capturing key empirical characteristics of various physical problems at hand. Such models typically have a mathematical structure that varies from the constrained optimization techniques underpinning MPT, restricting the straightforward application of many physical models. Yet, the demarcation between the use of ergodic notions in physics and financial economics was blurred substantively by the introduction of diffusion process techniques to solve contingent claims valuation problems. Following contributions by Sprenkle [

Schinckus [

To this end, this paper provides an etymology and history of the “classical ergodicity hypothesis” in 19th century statistical mechanics. Subsequent use of ergodicity in financial economics, in general, and MPT, in particular, is then examined. A modern interpretation of classical ergodicity is provided that uses Sturm-Liouville theory, a mathematical method central to classical statistical mechanics, to decompose the transition probability density of a one-dimensional diffusion process subject to regular upper and lower reflecting barriers. This “classical” decomposition divides the transition density of an ergodic process into a possibly multimodal limiting stationary density which is independent of time and initial condition and a power series of time and boundary dependent transient terms. In contrast, empirical theory aimed at estimating relationships from MPT typically ignores the implications of the initial and boundary conditions that generate transient terms and focuses on properties of a particular class of unimodal limiting stationary densities with finite parameters. To illustrate the implications of the expanded class of ergodic processes available to econophysics, properties of the bimodal quartic exponential stationary density are considered and used to assess the ability of the classical ergodicity hypothesis to explain certain “stylized facts” associated with the

The

From the perspective of modern mathematics, statistical physics, or systems theory, Birkhoff [

Mirowski [^{2}

It was during the transition from rational to statistical mechanics during the last third of the 19th century that Boltzmann made contributions leading to the transformation of theoretical physics from the microscopic mechanistic models of Rudolf Clausius (1835–1882) and James Maxwell to the macroscopic probabilistic theories of Josiah Gibbs and Albert Einstein (1879–1955).^{3}

Having descended from the deterministic rational mechanics of mid-19th century physics, defining works of MPT do not capture the probabilistic approach to modeling systems initially introduced by Boltzmann and further clarified by Gibbs.^{4}^{5}

The Maxwell distribution is defined over the velocity of gas molecules and provides the probability for the relative number of molecules with velocities in a certain range. Using a mechanical model that involved molecular collision, Maxwell [

At least since Samuelson [

Initial empirical estimation for the deterministic models of neoclassical economics proceeded with the addition of a stationary, usually Gaussian, error term to produce a discrete time general linear model (GLM) leading to estimation using ordinary least squares or maximum likelihood techniques. In the history of MPT, such early estimations were associated with tests of the capital asset pricing model such as the “market model,” for example, Elton and Gruber [

The emergence of ARCH, GARCH, and related empirical models was part of a general trend toward the use of inductive methods in economics, often employing discrete, linear time series methods to model transformed economic variables, for example, Hendry [^{6}

The conventional view of ergodicity in economics, in general, and financial economics, in particular, is reflected by Hendry [^{7}^{8}

Even though the formal solutions proposed were inadequate by standards of modern mathematics, the thermodynamic model introduced by Boltzmann to explain the dynamic properties of the Maxwell distribution is a pedagogically useful starting point to develop the implications of ergodicity for MPT. To be sure, von Neumann [^{9}

Halmos [

The mean ergodic theorem of von Neumann [^{10}^{11}

Employing conventional econometrics in empirical studies, MPT requires that the real world distribution for

In physics, phenomenology lies at the intersection of theory and experiment. Theoretical relationships between empirical observations are modeled without deriving the theory directly from first principles, for example, Newton’s laws of motion. Predictions based on these theoretical relationships are obtained and compared to further experimental data designed to test the predictions. In this fashion, new theories that can be derived from first principles are motivated. Confronted with nonexperimental data for important financial variables, such as common stock prices, interest rates, and the like, financial economics has developed some theoretical models that aim to fit the “stylized facts” of those variables. In contrast, the MPT is initially derived directly from the “first principles” of constrained expected utility maximizing behavior by individuals and firms. Given the difficulties in economics of testing model predictions with “new” experimental data, physics and econophysics have the potential to provide a rich variety of mathematical techniques that can be adapted to determining mathematical relationships among financial variables that explain the “stylized facts” of observed nonexperimental data.^{12}

The evolution of financial economics from the deterministic models of neoclassical economics to more modern stochastic models has been incremental and disjointed. The preference for linear models of static equilibrium relationships has restricted the application of theoretical frameworks that capture more complex nonlinear dynamics, for example, chaos theory; truncated Levy processes. Yet, important financial variables have relatively innocuous sample paths compared to some types of variables encountered in physics. There is an impressive range of mathematical and statistical models that, seemingly, could be applied to almost any physical or financial situation. If the process can be verbalized, then a model can be specified. This begs the following questions: are there transformations, ergodic or otherwise, that capture the basic “stylized facts” of observed financial data? Is the random instability in the observed sample paths identified in, say, stock price time series consistent with the

Boltzmann was concerned with demonstrating that the Maxwell distribution emerged in the limit as

Because the particle movements in a kinetic gas model are contained within an enclosed system, for example, a vertical glass tube, classical Sturm-Liouville (S-L) methods can be applied to obtain solutions for the transition densities. These classical results for the distributional implications of imposing regular reflecting boundaries on diffusion processes are representative of the modern phenomenological approach to random systems theory which “studies qualitative changes of the densites^{13}

The distributional implications of boundary restrictions, derived by modeling the random variable as a diffusion process subject to reflecting barriers, have been studied for many years, for example, Feller [

The use of the diffusion model to represent the nonlinear dynamics of stochastic processes is found in a wide range of subjects. Physical restrictions such as the rate of observed genetic mutation in biology or character of heat diffusion in engineering or physics often determine the specific formalization of the diffusion model. Because physical interactions can be complex, mathematical results for diffusion models are pitched at a level of generality sufficient to cover such cases.^{14}^{15}

If the diffusion process is subject to upper and lower reflecting boundaries that are regular and fixed ^{16}^{17}

In order to more accurately capture the^{18}^{19}

In general, solving the forward equation (

The solution procedure employed by Wong [

Because the methods for solving the S-L problem are ODE-based, some method of eliminating the time derivative in (

The classical S-L problem of solving (

The regular, self-adjoint Sturm-Liouville problem has an infinite sequence of real eigenvalues,

This proposition provides the general solution to the regular, self-adjoint S-L problem of deriving

The theoretical advantage obtained by imposing regular reflecting barriers on the diffusion state space for the forward equation is that an ergodic decomposition of the transition density is assured. The relevance of bounding the state space and imposing regular reflecting boundaries can be illustrated by considering the well known solution (e.g., [

One possible method of obtaining a stationary distribution without imposing both upper and lower boundaries is to impose only a lower (upper) reflecting barrier and construct the stochastic process such that positive (negative) infinity is nonattracting, for example, Linetsky [

Following Linetsky [

The roots of bifurcation theory can be found in the early solutions to certain deterministic ordinary differential equations. Consider the deterministic dynamics described by the pitchfork bifurcation ODE:

It is well known that the introduction of randomness to the pitchfork ODE changes the properties of the equilibrium solution, for example, [

Models in MPT are married to the transition probability densities associated with unimodal stationary distributions, especially the class of Gaussian-related distributions. Yet, it is well known that more flexibility in the shape of the stationary distribution can be achieved using a higher order exponential density, for example, Fisher [

To this end, assume that the stationary distribution is a fourth degree or “general quartic” exponential:^{21}

As illustrated in Figure ^{22}

Family of stationary densities for

Following Chiarella et al. [

The classical ergodicity hypothesis provides a point of demarcation in the prehistories of MPT and econophysics. To deal with the problem of making statistical inferences from “nonexperimental” data, theories in MPT typically employ stationary densities that are time reversible, are unimodal, and allow no short or long term impact from initial and boundary conditions. The possibility of bimodal processes or

Due to the widespread application in a wide range of subjects, textbook presentations of the Sturm-Liouville problem possess subtle differences that require some clarification to be applicable to the formulation used in this paper. In particular, to derive the canonical form (

Another specification of the forward equation that is of importance is found in Wong [

(a)

(b) For

(c) For some

(d) Consider

It follows from part (b) that

(e) Consider

(f) Obtaining the solution for

The authors are Professor of Finance and Associate Professor of Finance at Simon Fraser University.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Helpful comments from Chris Veld, Yulia Veld, Emmanuel Havens, Franck Jovanovic, Marcel Boumans, and Christoph Schinckus are gratefully acknowledged.

For example, Black and Scholes [

In rational mechanics, once the initial positions of the particles of interest, for example, molecules, are known, the mechanical model fully determines the future evolution of the system. This scientific and philosophical approach is often referred to as Laplacian determinism.

Boltzmann and Max Planck were vociferous opponents of energetics. The debate over energetics was part of a larger intellectual debate concerning determinism and reversibility. Jevons [

As such, Boltzmann was part of the larger: “Second Scientific Revolution, associated with the theories of Darwin, Maxwell, Planck, Einstein, Heisenberg and Schrödinger, (which) substituted a world of process and chance whose ultimate philosophical meaning still remains obscure” [

There are many interesting sources on these points which provide citations for the historical papers that are being discussed. Cercignani [

Kapetanios and Shin [

The second law of thermodynamics is the universal law of increasing entropy, a measure of the randomness of molecular motion and the loss of energy to do work. First recognized in the early 19th century, the second law maintains that the entropy of an isolated system, not in equilibrium, will necessarily tend to increase over time. Entropy approaches a maximum value at thermal equilibrium. A number of attempts have been made to apply the entropy of information to problems in economics, with mixed success. In addition to the second law, physics now recognizes the zeroth law of thermodynamics that “any system approaches an equilibrium state” [

In this process, the ergodicity hypothesis is required to permit the one observed sample path to be used to estimate the parameters for the

Heterodox critiques are associated with views considered to originate from within economics. Such critiques are seen to be made by “economists,” for example, Post-Keynesian economists, institutional economists, radical political economists, and so on. Because such critiques take motivation from the theories of mainstream economics, these critiques are distinct from econophysics. Following Schinckus [

Dhrymes [

Critiques of mainstream economics that are rooted in the insights of

In this context though not in all contexts, econophysics provides a “macroscopic” approach. In turn, ergodicity is an assumption that permits the time average from a single observed sample path to (phenomenologically) model the ensemble of sample paths. Given this, econophysics does contain a substantively richer toolkit that encompasses both ergodic and nonergodic processes. Many works in econophysics implicitly assume ergodicity and develop models based on that assumption.

The distinction between invariant and ergodic measures is fundamental. Recognizing a number of distinct definitions of ergodicity are available, following Medio [

The phenomenological approach is not without difficulties. For example, the restriction to Markov processes ignores the possibility of invariant measures that are not Markov. In addition, an important analytical construct in bifurcation theory, the Lyapunov exponent, can encounter difficulties with certain invariant Markov measures. Primary concern with the properties of the stationary distribution is not well suited to analysis of the dynamic paths around a bifurcation point. And so it goes.

A diffusion process is “regular” if starting from any point in the state space

The classification of boundary conditions is typically an important issue in the study of solutions to the forward equation. Important types of boundaries include regular, exit, entrance, and natural. Also the following are important in boundary classification: the properties of attainable and unattainable, whether the boundary is attracting or non-attracting, and whether the boundary is reflecting or absorbing. In the present context, regular, attainable, reflecting boundaries are usually being considered, with a few specific extensions to other types of boundaries. In general, the specification of boundary conditions is essential in determining whether a given PDE is self-adjoint.

Heuristically, if the ergodic process runs long enough, then the stationary distribution can be used to estimate the constant mean value. This definition of ergodic is appropriate for the one-dimensional diffusion cases considered in this paper. Other combinations of transformation, space, and function will produce different requirements. Various theoretical results are available for the case at hand. For example, the existence of an invariant Markov measure and exponential decay of the autocorrelation function are both assured.

For ease of notation it is assumed that

The mathematics at this point are heuristic. It would be more appropriate to observe that

A more detailed mathematical treatment can be found in de Jong [

In what follows, except where otherwise stated, it is assumed that

A number of simplifications were used to produce the 3D image in Figure