Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2015, Article ID 737905, 17 pages http://dx.doi.org/10.1155/2015/737905

Research Article Classical Ergodicity and Modern Portfolio Theory

Geoffrey Poitras and John Heaney

Beedie School of Business, Simon Fraser University, Vancouver, BC, Canada V5A lS6

Correspondence should be addressed to Geoffrey Poitras; [email protected]

Received 6 February 2015; Accepted 5 April 2015

Academic Editor: Wing K. Wong

Copyright © 2015 G. Poitras and J. Heaney. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ex post/ex ante quandary confronting modern portfolio theory is also examined.

“As the physicist builds models of the movement of matter in a frictionless environment, the economist builds models where there are no institutional frictions to the movement of stock prices.” E. Elton and M. Gruber, Modern Portfolio Theory and Investment Analysis (1984, p. 273)

1. Introduction attention in financial economics. In contrast, econophysi- cists generally consider financial economics to be primarily At least since Markowitz [1] initiated modern portfolio concerned with a core theory that is inconsistent with the theory (MPT), it has often been maintained that the tradeoff empirical orientation of physical theory. between systematic risk and expected return is the most Physical theory has evolved considerably from the con- important theoretical element of financial economics, for strained optimization, static equilibrium approach of rational example, Campbell [2]. Extending Mirowski [3], the static mechanics which underpins MPT. In detailing historical equilibrium methods used to develop propositions in MPT developments in physics since the 19th century, it is conven- such as the capital asset pricing model can be traced to tional to jump from the determinism of rational mechanics mathematical concepts developed from the deterministic to quantum mechanics to recent developments in chaos “rational mechanics” approach to 19th century physics. In theory, overlooking the relevance of the initial steps toward theyearssinceMarkowitz[1], financial economics has also modeling stochastic behavior of physical phenomena by adopted alternative mathematical methods from more recent Ludwig Boltzmann (1844–1906), James Maxwell (1831–1879), contributions to physics, especially the diffusion processes and Josiah Gibbs (1839–1903). As such, there is a point of employed by Black and Scholes [4] to determine option demarcation between the intellectual prehistories of MPT prices. The emergence of econophysics during the last decade and econophysics that can, arguably, be traced to the debate of the twentieth century, for example, Roehner [5]and over energistics around the end of the 19th century. While the Jovanovic and Schinckus [6], has provided a variety of evolution of physics after energistics involved the introduc- theoretical and empirical methods adapted from physics, tion and subsequent stochastic generalization of ergodic con- ranging from statistical mechanics to chaos theory, to analyze cepts, fueled by the emergence of MPT following Markowitz, financial phenomena. Yet, despite considerable overlap in financial economics incorporated ergodicity into empirical method, contributions to econophysics have gained limited methods aimed at generalizing and testing the capital asset 2 Chinese Journal of Mathematics pricing model and other elements of MPT.1 Significantly, ergodicity is provided that uses Sturm-Liouville theory, a stochastic generalization of the static equilibrium approach mathematical method central to classical statistical mechan- of MPT required the adoption of a restricted class of ergodic ics, to decompose the transition probability density of a processes, that is, “time reversible” probabilistic models, one-dimensional diffusion process subject to regular upper especially the unimodal likelihood functions associated with and lower reflecting barriers. This “classical” decomposition certain stationary distributions. divides the transition density of an ergodic process into In contrast, from the early ergodic models of Boltzmann a possibly multimodal limiting stationary density which to the fractals and chaos theory of Mandlebrot, physics is independent of time and initial condition and a power has employed a wider variety of ergodic and nonergodic series of time and boundary dependent transient terms. In stochastic models aimed at capturing key empirical charac- contrast, empirical theory aimed at estimating relationships teristics of various physical problems at hand. Such models from MPT typically ignores the implications of the initial typically have a mathematical structure that varies from and boundary conditions that generate transient terms and the constrained optimization techniques underpinning MPT, focuses on properties of a particular class of unimodal lim- restricting the straightforward application of many physical iting stationary densities with finite parameters. To illustrate models. Yet, the demarcation between the use of ergodic the implications of the expanded class of ergodic processes notions in physics and financial economics was blurred sub- available to econophysics, properties of the bimodal quartic stantively by the introduction of diffusion process techniques exponential stationary density are considered and used to to solve contingent claims valuation problems. Following assess the ability of the classical ergodicity hypothesis to contributions by Sprenkle [7]andSamuelson[8], Black explain certain “stylized facts” associated with the ex post/ex and Scholes [4]andMerton[9] provided an empirically ante quandary confronting MPT. viable method of using diffusion methods to determine, using Ito’s lemma, a partial differential equation that can be solved for an option price. Use of Ito’s lemma to solve 2. A Brief History of Classical Ergodicity stochastic optimization problems is now commonplace in financial economics, for example, Brennan and Schwartz10 [ ], The Encyclopedia of Mathematics [16] defines ergodic theory including the continuous time generalizations of MPT, for as the “metric theory of dynamical systems, the branch of example, Epstein and Ji [11]. In spite of the considerable the theory of dynamical systems that studies systems with progression of certain mathematical techniques employed in an invariant measure and related problems.” This modern physics into financial economics, overcoming the difficulties definition implicitly identifies the birth of ergodic theory of applying the wide range of models developed for physical with proofs of the mean ergodic theorem by von Neumann situations to fit the empirical properties of financial data is [17] and the pointwise ergodic theorem by Birkhoff [18]. still a central problem confronting econophysics. These early proofs have had significant impact in a wide Schinckus [12, p. 3816] accurately recognizes that the range of modern subjects. For example, the notions of positivist philosophical foundation of econophysics depends invariant measure and metric transitivity used in the proofs fundamentally on empirical observation: “The empiricist arefundamentaltothemeasuretheoreticfoundationof dimension is probably the first positivist feature of econo- modern probability theory [19, 20]. Building on a seminal physics.” Following McCauley [13]andothers,thiscon- contribution to probability theory by Kolmogorov [21], in cern with empiricism often focuses on the identification of the years immediately following it was recognized that the macrolevel statistical regularities that are characterized by ergodic theorems generalize the strong law of large numbers. the scaling laws identified by Mandelbrot14 [ ]andMan- Similarly, the equality of ensemble and time averages—the dlebrot and Hudson [15] for financial data. Unfortunately, essence of the mean ergodic theorem—is necessary to the this empirically driven ideal is often confounded by the concept of a strictly stationary stochastic process. Ergodic “nonrepeatable” experiment that characterizes most observed theory is the basis for the modern study of random dynamical economic and financial data. There is quandary posed by systems, for example, Arnold [22]. In mathematics, ergodic having only a single observed ex post time path to estimate theory connects measure theory with the theory of transfor- the distributional parameters for the ensemble of ex ante mation groups. This connection is important in motivating time paths needed to make decisions involving future values the generalization of harmonic analysis from the real line to of financial variables. In contrast to natural sciences, such locally compact groups. as physics, in the human sciences there is no assurance From the perspective of modern mathematics, statistical that ex post statistical regularity translates into ex ante fore- physics, or systems theory, Birkhoff [18]andvonNeumann casting accuracy. Resolution of this quandary highlights the [17] are excellent starting points for a modern history of usefulness of employing a “phenomenological” approach to ergodic theory. Building on the modern ergodic theorems, modeling stochastic properties of financial variables relevant subsequent developments in these and related fields have to MPT. been dramatic. These contributions mark the solution to a To this end, this paper provides an etymology and problem in statistical mechanics and thermodynamics that history of the “classical ergodicity hypothesis” in 19th cen- was recognized sixty years earlier when Ludwig Boltzmann tury statistical mechanics. Subsequent use of ergodicity in introduced the classical ergodic hypothesis to permit the the- financial economics, in general, and MPT, in particular, oretical phase space average to be interchanged with the mea- is then examined. A modern interpretation of classical surable time average. For the purpose of contrasting methods Chinese Journal of Mathematics 3 from physics and econophysics with those used in MPT, the Having descended from the deterministic rational selection of the less formally correct and rigorous classical mechanics of mid-19th century physics, defining works of ergodic hypothesis of Boltzmann is a more auspicious begin- MPT do not capture the probabilistic approach to modeling ning. Problems of interest in mathematics are generated by a systems initially introduced by Boltzmann and further range of subjects, such as physics, chemistry, engineering, and clarified by Gibbs.4 Mathematical problems raised by biology. The formulation and solution of physical problems Boltzmann were subsequently solved using tools introduced in, say, statistical mechanics or particle physics will have in a string of later contributions by the likes of the Ehrenfests mathematical features which are inapplicable or unnecessary andCantorinsettheory,GibbsandEinsteininphysics, for MPT. For example, in statistical mechanics, points in the Lebesgue in measure theory, Kolmogorov in probability phase space are often multidimensional functions represent- theory,andWeinerandLevyinstochasticprocesses. ing the mechanical state of the system; hence the desirability Boltzmann was primarily concerned with problems in the of a group-theoretic interpretation of the ergodic hypothesis. kinetic theory of gases, formulating dynamic properties of From the perspective of MPT, such complications are largely the stationary Maxwell distribution, the velocity distribution irrelevant. The history of classical ergodic theory captures of gas molecules in thermal equilibrium. Starting in 1871, the etymology and basic physical interpretation providing a Boltzmann took this analysis one step further to determine more revealing prehistory of the relevant MPT mathematics. the evolution equation for the distribution function. The This arguably more revealing prehistory begins with the mathematical implications of this classical analysis still formulation of theoretical problems that von Neumann and resonate in many subjects of the modern era. The etymology Birkhoff were later able to solve. for “ergodic” begins with an 1884 paper by Boltzmann, Mirowski [3][23, esp. ch. 5] establishes the importance of though the initial insight to use probabilities to describe a gas 19th century physics in the development of the neoclassical system can be found as early as 1857 in a paper by Clausius economic system advanced by W. Stanley Jevons (1835– and in the famous 1860 and 1867 papers by Maxwell.5 1882) and Leon Walras (1834–1910) during the marginalist The Maxwell distribution is defined over the velocity of revolution of the 1870s. Being derived using principles from gas molecules and provides the probability for the relative neoclassical economic theory, MPT also inherited essential number of molecules with velocities in a certain range. features of mid-19th century physics: deterministic rational Using a mechanical model that involved molecular collision, mechanics; conservation of energy; and the nonatomistic Maxwell [25] was able to demonstrate that, in thermal continuum view of matter that inspired the energetics equilibrium, this distribution of molecular velocities was 2 movement later in the 19th century. More precisely, from a “stationary” distribution that would not change shape neoclassical economics MPT inherited a variety of static equi- due to ongoing molecular collision. Boltzmann aimed to librium techniques and tools such as mean-variance utility determine whether the Maxwell distribution would emerge functions and constrained optimization. As such, failings in the limit, whatever the initial state of the gas. In order to of neoclassical economics identified by econophysicists also study the dynamics of the equilibrium distribution over time, apply to central propositions of MPT. Included in the failings Boltzmann introduced the probability distribution of the is an overemphasis on theoretical results at the expense of relative time a gas molecule has a velocity in a certain range identifying models that have greater empirical validity, for while still retaining the notion of probability for velocities example, Roehner [5] and Schinckus [12]. of a relative number of gas molecules. Under the classical It was during the transition from rational to statistical ergodic hypothesis, the average behavior of the macroscopic mechanics during the last third of the 19th century that gas system, which can objectively be measured over time, can Boltzmann made contributions leading to the transformation be interchanged with the average value calculated from the of theoretical physics from the microscopic mechanistic ensemble of unobservable and highly complex microscopic models of Rudolf Clausius (1835–1882) and James Maxwell molecular motions at a given point in time. In the words to the macroscopic probabilistic theories of Josiah Gibbs and of Weiner [26,p.1]“BothintheolderMaxwelltheoryand 3 Albert Einstein (1879–1955). Coming largely after the start in the later theory of Gibbs, it is necessary to make some of the marginalist revolution in economics, this fundamental sort of logical transition between the average behavior of all transformation in theoretical physics had little impact on dynamical systems of a given family or ensemble, and the the progression of financial economics until the appearance historical average of a single system.” of diffusion equations in contributions on continuous time finance that started in the 1960s and culminated in Black and Scholes [4]. The deterministic mechanics of the energistic 3. Use of the Ergodic Hypothesis in approach was well suited to the axiomatic formalization of Financial Economics neoclassical economic theory which culminated in the fol- lowing: the von Neumann and Morgenstern expected utility At least since Samuelson [27], it has been recognized that approach to modeling uncertainty; the Bourbaki inspired empirical theory and estimation in economics, in general, Arrow-Debreu general equilibrium theory, for example, and financial economics, in particular, relies heavily on the Weintraub [24], and, ultimately, MPT. In turn, empirical use of specific unimodal stationary distributions associated estimation and the subsequent extension of static equilibrium with a particular class of ergodic processes. As reflected in MPT results to continuous time were facilitated by the the evolution of the concept in economics, the specification adoption of a narrow class of ergodic processes. andimplicationsofergodicityhaveonlydevelopedgradually. 4 Chinese Journal of Mathematics

The early presentation of ergodicity by Samuelson [27] processeshavealsobeenemployedwherethetimeseriesis involves the addition of a discrete Markov error term into the only subject to fractional differencing. Significantly, recent deterministic cobweb model to demonstrate that estimated contributions on Markov switching processes and exponen- forecasts of future values, such as prices, “should be less tial smooth transition autoregressive processes have demon- variable than the actual data.”Considerable opaqueness about strated the “possibility that nonlinear ergodic processes can the definition of ergodicity is reflected in the statement that be misinterpreted as unit root nonstationary processes” [37, a “‘stable’ stochastic process...eventually forgets its past and p. 620]. Bonomo et al. [38] illustrates the recent application therefore in the far future can be expected to approach an of Markov switching processes in estimating the asset pricing ergodic probability distribution” [27,p.2].Theconnection models of MPT. between ergodic processes and nonlinear dynamics that char- The conventional view of ergodicity in economics, in acterizes present efforts in economics goes unrecognized, general, and financial economics, in particular, is reflected by for example, [27,p.1,5].Whilesomeexplicitapplications Hendry [35, p. 100]: “Whether economic reality is an ergodic of ergodic processes to theoretical modeling in economics process after suitable transformation is a deep issue” which have emerged since Samuelson [27], for example, Horst and is difficult to analyze rigorously. As a consequence, inthe Wenzelburger [28] and Dixit and Pindyck [29], financial limited number of instances where ergodicity is examined econometrics has produced the bulk of the contributions. in economics a variety of different interpretations appear. Initial empirical estimation for the deterministic models In contrast, the ergodic hypothesis in classical statistical of neoclassical economics proceeded with the addition of a mechanics is associated with the more physically transparent stationary, usually Gaussian, error term to produce a discrete kinetic gas model than the often technical and targeted time general linear model (GLM) leading to estimation using concepts of ergodicity encountered in modern economics, ordinary least squares or maximum likelihood techniques. In in general, and MPT, in particular. For Boltzmann, the the history of MPT, such early estimations were associated classical ergodic hypothesis permitted the unobserved com- with tests of the capital asset pricing model such as the “mar- plex microscopic interactions of individual gas molecules ket model,” for example, Elton and Gruber [30]. Iterations to obey the second law of thermodynamics, a concept that 7 and extensions of the GLM to deal with complications arising has limited application in economics. Despite differences in in empirical estimates dominated early work in economet- physical interpretation, there is similarity to the problem of rics, for example, Dhrymes [31] and Theil [32], leading to modeling “macroscopic” financial variables, such as common application of generalized least squares estimation techniques stock prices, foreign exchange rates, “asset” prices, or interest that encompassed autocorrelated and heteroskedastic error rates. By construction, when it is not possible to derive a terms. Employing 𝐿2 vector space methods with stationary theory for describing and predicting empirical observations Gaussian-based error term distributions ensured these early from known first principles about the (microscopic) rational stochastic models implicitly assumed ergodicity. The general- behavior of individuals and firms. By construction, this 8 ization of this discrete time estimation approach to the class of involves a phenomenological approach to modeling. ARCH and GARCH error term models by Engle and Granger Even though the formal solutions proposed were inad- was of such significance that a Nobel prize in economics equate by standards of modern mathematics, the thermo- was awarded for this contribution, for example, Engle and dynamic model introduced by Boltzmann to explain the Granger [33]. By modeling the evolution of the volatility, this dynamic properties of the Maxwell distribution is a peda- approach permitted a limited degree of nonlinearity to be gogically useful starting point to develop the implications modeled providing a substantively better fit of MPT models of ergodicity for MPT. To be sure, von Neumann [17]and to observed financial time series, for example, Beaulieu et al. Birkhoff [18] correctly specify ergodicity using Lebesgue inte- [34]. gration, an essential analytical tool unavailable to Boltzmann, The emergence of ARCH, GARCH, and related empirical but the analysis is too complex to be of much value to all models was part of a general trend toward the use of inductive but the most mathematically specialized economists. The physical intuition of the kinetic gas model is lost in the methods in economics, often employing discrete, linear time generality of the results. Using Boltzmann as a starting point, series methods to model transformed economic variables, for the large number of mechanical and complex molecular example, Hendry [35]. At least since Dickey and Fuller [36], collisions could correspond to the large number of micro- it has been recognized that estimates of univariate time series scopic, atomistic liquidity providers, and traders interacting models for many financial times series reveals evidence of to determine the macroscopic financial market price. In this “non-stationarity.” A number of approaches have emerged context, it is variables such as the asset price or the interest 6 to deal with this apparent empirical quandary. In particu- rate or the exchange rate, or some combination, that is lar, transformation techniques for time series models have being measured over time and ergodicity would be associated received considerable attention. Extension of the Box-Jenkins with the properties of the transition density generating the methodology led to the concept of economic time series macroscopic variables. Ergodicity can fail for a number of being I(0)—stationary in the level—and I(1)—nonstationary reasons and there is value in determining the source of the in the level but stationary after first differencing. Two(1) I failure. In this vein, there are two fundamental difficulties economic variables could be cointegrated if differencing the associated with the classical ergodicity hypothesis in Boltz- two series produced an I(0) process, for example, Hendry mann’s statistical mechanics, reversibility and recurrence, [35]. Extending early work on distributed lags, long memory that are largely unrecognized in financial economics.9 Chinese Journal of Mathematics 5

Halmos [39, p. 1017] is a helpful starting point to sort There is a persistent belief that increasing the length or outthedifferingnotionsofergodicitythatareofrelevance sampling frequency of a financial time series will improve to the issues at hand: “The ergodic theorem is a statement the precision of a statistical estimate, for example, Dimson about a space, a function and a transformation.” In mathe- et al. [41]. Similarly, focus on the tradeoff between “risk and matical terms, ergodicity or “metric transitivity” is a property return” requires the use of unimodal stationary densities for of “indecomposable,” measure preserving transformations. transformed financial variables such as the rate of return. Because the transformation acts on points in the space, there The impact of initial and boundary conditions on financial is a fundamental connection to the method of measuring decision making is generally ignored. The inconsistency of relationships such as distance or volume in the space. In von reversible processes with key empirical facts, such as the Neumann [17]andBirkhoff[18], this is accomplished using asymmetric tendency for downdrafts in prices to be more the notion of Lebesgue measure: the admissible functions severe than upswings, is ignored in favor of adhering to are either integrable (Birkhoff) or square integrable (von “reversible” theoretical models that can be derived from first Neumann). In contrast to, say, statistical mechanics where principles associated with constrained optimization tech- spaces and functions account for the complex physical niques, for example, Constantinides [42]. interaction of large numbers of particles, economic theories such as MPT usually specify the space in a mathematically convenient fashion. For example, in the case where there is a 4. A Phenomenological Interpretation of single random variable, then the space is “superfluous” [20, Classical Ergodicity p. 182] as the random variable is completely described by the distribution. Multiple random variables can be handled In physics, phenomenology lies at the intersection of theory by assuming the random variables are discrete with finite and experiment. Theoretical relationships between empiri- state spaces. In effect, conditions for an “invariant measure” cal observations are modeled without deriving the theory areassumedinMPTinordertofocusattentionon“finding directly from first principles, for example, Newton’s laws of and studying the invariant measures” [22,p.22],where motion.Predictionsbasedonthesetheoreticalrelationships in the terminology of financial econometrics, the invariant are obtained and compared to further experimental data measure usually corresponds to the stationary distribution or designed to test the predictions. In this fashion, new theories likelihood function. that can be derived from first principles are motivated. Con- The mean ergodic theorem of von Neumann [17]pro- fronted with nonexperimental data for important financial vides an essential connection to the ergodicity hypothesis variables, such as common stock prices, interest rates, and in financial econometrics. It is well known that, in the the like, financial economics has developed some theoretical Hilbert and Banach spaces common to econometric work, models that aim to fit the “stylized facts” of those variables. the mean ergodic theorem corresponds to the strong law In contrast, the MPT is initially derived directly from the of large numbers. In statistical applications where strictly “first principles” of constrained expected utility maximizing stationary distributions are assumed, the relevant ergodic behavior by individuals and firms. Given the difficulties in ∗ ∗ transformation, 𝐿 ,istheunitshiftoperator:𝐿 Ψ[𝑥(𝑡)] = economics of testing model predictions with “new” experi- ∗ ∗ 𝑘 Ψ[𝐿 𝑥(𝑡)] = Ψ[𝑥(𝑡 +1)]; [(𝐿 ) ]Ψ[𝑥(𝑡)] = Ψ[𝑥(𝑡 + mental data, physics and econophysics have the potential to ∗ −𝑘 𝑘)];and{(𝐿 ) }Ψ[𝑥(𝑡)] = Ψ[𝑥(𝑡 −𝑘)] with 𝑘 being an provide a rich variety of mathematical techniques that can integer and Ψ[𝑥] the strictly stationary distribution for 𝑥 be adapted to determining mathematical relationships among 𝑡 10 financial variables that explain the “stylized facts” of observed that in the strictly stationary case is replicated at each . 12 Significantly, this reversible transformation is independent nonexperimental data. of initial time and state. Because this transformation can The evolution of financial economics from the deter- ∗ ministic models of neoclassical economics to more modern be achieved by imposing strict stationarity on Ψ[𝑥], 𝐿 will stochastic models has been incremental and disjointed. The only work for certain ergodic processes. In effect, the ergodic preference for linear models of static equilibrium relation- requirement that the transformation is measure preserving is ships has restricted the application of theoretical frameworks weaker than the strict stationarity of the stochastic process 𝐿∗ that capture more complex nonlinear dynamics, for exam- sufficient to achieve . The practical implications of the ple, chaos theory; truncated Levy processes. Yet, important 𝐿∗ reversible ergodic transformation are described by David- financial variables have relatively innocuous sample paths son [40,p.331]:“Inaneconomicworldgovernedentirelyby compared to some types of variables encountered in physics. ... [time reversible] ergodic processes economic relationships There is an impressive range of mathematical and statistical among variables are timeless, or ahistoric in the sense that the models that, seemingly, could be applied to almost any physi- 11 future is merely a statistical reflection of the past [sic].” calorfinancialsituation.Iftheprocesscanbeverbalized,then Employing conventional econometrics in empirical stud- a model can be specified. This begs the following questions: ies, MPT requires that the real world distribution for 𝑥(𝑡),for are there transformations, ergodic or otherwise, that capture example, the asset return, is sufficiently similar to those for thebasic“stylizedfacts”ofobservedfinancialdata?Isthe both 𝑥(𝑡 + 𝑘) and 𝑥(𝑡 ;− 𝑘) that is, the ergodic transformation random instability in the observed sample paths identified ∗ 𝐿 is reversible. The reversibility assumption is systemic in in, say, stock price time series consistent with the ex ante MPT appearing in the use of long estimation periods to deter- stochastic bifurcation of an ergodic process, for example, mine important variables such as the “equity risk premium.” Chiarella et al. [43]? In the bifurcation case, the associated 6 Chinese Journal of Mathematics ex ante stationary densities are bimodal and irreversible, a provides sufficient generality to resolve certain empirical situationwherethemeancalculatedfrompastvaluesofa difficulties arising from key stylized facts observed in the single, nonexperimental ex post realization of the process is nonexperimental time series from financial economics. More not necessarily informative about the mean for future values. generally, the framework suggests a method of allowing Boltzmann was concerned with demonstrating that the MPT to encompass the nonlinear dynamics of diffusion pro- Maxwell distribution emerged in the limit as 𝑡→∞for cesses. In other words, within the more formal mathematical systems with large numbers of particles. The limiting process framework of classical statistical mechanics, it is possible for 𝑡 requires that the system run long enough that the to reformulate the classical ergodicity hypothesis to permit initial conditions do not impact the stationary distribution. a useful stochastic generalization of theories in financial At the time, two fundamental criticisms were aimed at this economics such as MPT. general approach: reversibility and recurrence. In the context The use of the diffusion model to represent the nonlinear of financial time series, reversibility relates to the use of past dynamics of stochastic processes is found in a wide range values of the process to forecast future values. Recurrence of subjects. Physical restrictions such as the rate of observed relates to the properties of the long run average which genetic mutation in biology or character of heat diffusion in involves the ability and length of time for an ergodic process engineering or physics often determine the specific formal- to return to its stationary state. For Boltzmann, both these ization of the diffusion model. Because physical interactions criticisms have roots in the difficulty of reconciling the second can be complex, mathematical results for diffusion models are law of thermodynamics with the ergodicity hypothesis. Using pitched at a level of generality sufficient to cover such cases.14 Sturm-Liouvillemethods,itcanbeshownthatclassical Such generality is usually not required in financial economics. ergodicity requires the transition density of the process to In this vein, it is possible to exploit mathematical properties be decomposable into the sum of a stationary density and of bounded state spaces and one-dimensional diffusions a mean zero transient term that captures the impact of the to overcome certain analytical problems that can confront initial condition of the system on the individual sample paths; continuous time Markov solutions. The key construct in irreversibility relates to properties of the stationary density the S-L method is the ergodic transition probability density and nonrecurrence to the behavior of the transient term. function 𝑈 which is associated with the random (financial) Because the particle movements in a kinetic gas model are variable 𝑥 at time 𝑡(𝑈=𝑈[𝑥,𝑡|𝑥0]) that follows a regular, contained within an enclosed system, for example, a vertical time homogeneous diffusion process. While it is possible to glass tube, classical Sturm-Liouville (S-L) methods can be allow the state space to be an infinite open interval 𝐼𝑜 =(𝑎,𝑏: applied to obtain solutions for the transition densities. These ∞≤𝑎<𝑏≤∞),afiniteclosedinterval𝐼𝑐 =[𝑎,𝑏: classical results for the distributional implications of impos- −∞<𝑎<𝑏<+∞]or the specific interval 𝐼𝑠 =[0= ing regular reflecting boundaries on diffusion processes are 𝑎<𝑏<∞)are applicable to financial variables.15 Assuming representative of the modern phenomenological approach to that 𝑈 is twice continuously differentiable in 𝑥 and once in 𝑡 random systems theory which “studies qualitative changes of and vanishes outside the relevant interval, then 𝑈 obeys the the densites [sic] of invariant measures of the Markov semi- forwardequation(e.g.,[47,p.102–4]): group generated by random dynamical systems induced by stochastic differential equations” [44,p.27].13 Because the ini- 𝜕2 𝜕 𝜕𝑈 {𝐵 [𝑥] 𝑈} − {𝐴 [𝑥] 𝑈} = , (1) tial condition of the system is explicitly recognized, ergodicity 𝜕𝑥2 𝜕𝑥 𝜕𝑡 in these models takes a different form than that associated 2 with the unit shift transformation of unimodal stationary where 𝐵[𝑥] (=(1/2)𝜎 [𝑥] > 0) is the one half the infinitesimal densities typically adopted in financial economics, in general, variance and 𝐴[𝑥] the infinitesimal drift of the process. 𝐵[𝑥] and MPT, in particular. The ergodic transition densities are is assumed to be twice and 𝐴[𝑥] once continuously differ- derived as solutions to the forward differential equation entiable in 𝑥. Being time homogeneous, this formulation associated with one-dimensional diffusions. The transition permits state, but not time, variation in the drift and variance densities contain a transient term that is dependent on the parameters. initial condition of the system and boundaries imposed on If the diffusion process is subject to upper and lower the state space. Path dependence, that is, irreversibility, can reflecting boundaries that are regular and fixed (−∞<𝑎< be introduced by employing multimodal stationary densities. 𝑏<∞),theclassical“Sturm-Liouvilleproblem”involves The distributional implications of boundary restrictions, solving (1) subject to the separated boundary conditions:16 derived by modeling the random variable as a diffusion 󵄨 process subject to reflecting barriers, have been studied for 𝜕 󵄨 {𝐵 [𝑥] 𝑈 [𝑥,] 𝑡 }󵄨 −𝐴[𝑎] 𝑈 [𝑎,] 𝑡 = 0, (2) many years, for example, Feller [45]. The diffusion process 𝜕𝑥 󵄨𝑥=𝑎 framework is useful because it imposes a functional structure 󵄨 𝜕 󵄨 that is sufficient for known partial differential equation {𝐵 [𝑥] 𝑈 [𝑥,] 𝑡 }󵄨 −𝐴[𝑏] 𝑈 [𝑏,] 𝑡 = 0. (3) (PDE)solutionprocedurestobeusedtoderivetherelevant 𝜕𝑥 󵄨𝑥=𝑏 transition probability densities. Wong [46] demonstrated that with appropriate specification of parameters in the PDE, the And the initial condition is as follows: transition densities for popular stationary distributions such 𝑏 as the exponential, uniform, and normal distributions can 𝑈 [𝑥, 0] =𝑓[𝑥0], where: ∫ 𝑓[𝑥0]=1 (4) be derived using classical S-L methods. The S-L framework 𝑎 Chinese Journal of Mathematics 7

and 𝑓[𝑥0] is the continuous density function associated with to permit closed form solutions for the transition density 𝑥0 where 𝑎≤𝑥0 ≤𝑏. When the initial starting value, 𝑥0, to be obtained. Wong [46] provides an illustration of this is known with certainty, the initial condition becomes the approach. The PDE (1) is reduced to an ODE by only Dirac delta function, 𝑈[𝑥, 0] = 𝛿[𝑥0 −𝑥 ],andtheresulting considering the strictly stationary distributions arising from solution for 𝑈 is referred to as the “principal solution.”Within the Pearson system. Restrictions on the associated Ψ[𝑥] are the framework of the S-L method, a stochastic process has constructed by imposing the fundamental ODE condition for the property of classical ergodicity when the transition density the unimodal Pearson system of distributions: 17 satisfies the following: 𝑑Ψ [𝑥] 𝑒 𝑥+𝑒 = 1 0 Ψ [𝑥] . 2 (6) 𝑏 𝑑𝑥 𝑑2𝑥 +𝑑1𝑥+𝑑0 lim 𝑈 [𝑥,0 𝑡|𝑥 ]=∫ 𝑓[𝑥0] 𝑈 [𝑥,0 𝑡|𝑥 ]𝑑𝑥0 𝑡→∞ 𝑎 (5) The transition probability density 𝑈 for the ergodic process =Ψ[𝑥] . can then be reconstructed by working back from a specific closed form for the stationary distribution using known Important special cases occur for the principal solution results for the solution of specific forms of the forward equation. In this procedure, the 𝑑0, 𝑑1, 𝑑2, 𝑒0,and𝑒1 in the (𝑓[𝑥0]=𝛿[𝑥−𝑥0])andwhen𝑓[𝑥0] is from a specific PearsonODEareusedtospecifytherelevantparameters class such as the Pearson distributions. To be ergodic, the 𝑈 time invariant stationary density Ψ[𝑥] is not permitted to in (1).The for important stationary distributions that fall within the Pearson system, such as the normal, beta, central “decompose” the sample space with a finite number of 𝑡 indecomposable subdensities, each of which is time invariant. , and exponential, can be derived by this method. Such irreversible processes are not ergodic, even though each The solution procedure employed by Wong [46] depends of the subdensities could be restricted to obey the ergodic crucially on restricting the PDE problem sufficiently to apply classical S-L techniques. Using S-L methods, various studies theorem. To achieve ergodicity, a multimodal stationary 𝑈 density can be used instead of decomposing the sample space have generalized the set of solutions for to cases where the stationary distribution is not a member of the Pearson using subdensities with different means. In turn, multimodal 𝑈 irreversible ergodic processes have the property that the mean system or is otherwise unknown, for example, Linetsky calculated from past values of the process are not necessarily [49]. In order to employ the separation of variables technique informative enough about the modes of the ex ante densities used in solving S-L problems, (1) has to be transformed into to provide accurate predictions. the canonical form of the forward equation. To do this, the In order to more accurately capture the ex ante properties following function associated with the invariant measure is of financial time series, there are some potentially restrictive introduced: features in the classical S-L framework that can be identified. 𝑥 𝐴 [𝑠] 𝑟 [𝑥] =𝐵[𝑥] exp [− ∫ 𝑑𝑠] . (7) For example, time homogeneity of the process eliminates 𝑎 𝐵 [𝑠] 18 the need to explicitly consider the location of 𝑡0. Time homogeneity is a property of 𝑈 and, as such, is consistent with Using this function, the forward equation can be rewritten in ∗ “ahistorical” MPT. In the subclass of {𝑈} denoted as {𝑈 }, the form a time homogeneous and reversible stationary distribution 𝜕 𝜕𝑈 𝜕𝑈 1 {𝑝 [𝑥] }+𝑞[𝑥] 𝑈= , governs the dynamics of 𝑥(𝑡).Significantly,while𝑈 is time 𝑟 [𝑥] 𝜕𝑥 𝜕𝑥 𝜕𝑡 (8) homogeneous, there are some 𝑈 consistent with irreversible processes. A relevant issue for econophysicists is to deter- where mine which concept—time homogeneity or reversibility— 𝑝 [𝑥] =𝐵[𝑥] 𝑟 [𝑥] , is inconsistent with economic processes that capture col- lapsing pricing conventions in asset markets; liquidity traps 𝜕2𝐵 𝜕𝐴 (9) 𝑞 [𝑥] = − . in money markets; and collapse induced structural shifts 𝜕𝑥2 𝜕𝑥 in stock markets. In the S-L framework, the initial state of the system (𝑥0) is known and the ergodic transition density Equation (8) is the canonical form of (1). The S-L problem provides information about how a given point 𝑥0 shifts 𝑡 nowinvolvessolving(8) subject to appropriate initial and units along a trajectory.19 In contrast, applications in financial boundary conditions. ∗ econometrics employ the strictly stationary 𝑈 ,wherethe Because the methods for solving the S-L problem are location of 𝑥0 is irrelevant while 𝑈 incorporates 𝑥0 as an ODE-based, some method of eliminating the time deriva- initial condition associated with the solution of a partial tive in (1) is required. Exploiting the assumption of time differential equation. homogeneity, the eigenfunction expansion approach applies separation of variables, permitting (8) to be specified as 20 −𝜆𝑡 5. Density Decomposition Results 𝑈 [𝑥,] 𝑡 =𝑒 𝜑 [𝑥] , (10) In general, solving the forward equation (1) for 𝑈 subject where 𝜑[𝑥] is only required to satisfy the easier-to-solve ODE: to (2), (3) and some admissible form of (4) is difficult, 1 𝑑 𝑑𝜑 󸀠 for example, Feller [45]andRisken[48]. In such circum- [𝑝 [𝑥] ] + [𝑞 [𝑥] +𝜆] 𝜑 [𝑥] = 0.(1 ) stances, it is expedient to restrict the problem specification 𝑟 [𝑥] 𝑑𝑥 𝑑𝑥 8 Chinese Journal of Mathematics

Transforming the boundary conditions involves substitution a power series of transient terms associated with the remaining of (10) into (2) and (3) and solving to get eigenvalues, that are boundary and initial condition dependent: 󵄨 𝑈 [𝑥, 𝑡 |𝑥 ] =Ψ 𝑥 +𝑇[𝑥, 𝑡 |𝑥 ] , 𝑑 󵄨 0 [ ] 0 (15) {𝐵 [𝑥] 𝜑 [𝑥]}󵄨 −𝐴[𝑎] 𝜑 [𝑎] = , ( 󸀠) 󵄨 0 2 𝑑𝑥 󵄨𝑥=𝑎 where 󵄨 −1 𝑑 󵄨 𝑟 [𝑥] {𝐵 [𝑥] 𝜑 [𝑥]}󵄨 −𝐴[𝑏] 𝜑 [𝑏] = . ( 󸀠) Ψ [𝑥] = . 󵄨 0 3 𝑏 (16) 𝑑𝑥 󵄨𝑥=𝑏 ∫ 𝑟 [𝑥]−1 𝑑𝑥 𝑎

Significant analytical advantages are obtained by making the Using the specifications of 𝜆𝑛, 𝑐𝑛,and𝜓𝑛,thepropertiesof S-L problem “regular” which involves assuming that [𝑎, 𝑏] is a 𝑇[𝑥, 𝑡] are defined as 𝑟[𝑥] 𝑝[𝑥] 𝑞[𝑥] closed interval with , ,and beingrealvaluedand ∞ 𝑝[𝑥] [𝑎, 𝑏] 𝑟[𝑥] >0 −𝜆 𝑡 having a continuous derivative on and , 𝑇 [𝑥, 𝑡|𝑥 ]=∑ 𝑐 𝑒 𝑛 𝜓 [𝑥] 𝑝[𝑥] >0 [𝑎, 𝑏] 0 𝑛 𝑛 at every point in . “Singular” S-L problems arise 𝑛=1 wheretheseconditionsareviolateddueto,say,aninfinite (17) [𝑎, 𝑏] ∞ state space or a vanishing coefficient in the interval . 1 −𝜆𝑛𝑡 = ∑ 𝑒 𝜓𝑛 [𝑥] 𝜓𝑛 [𝑥0] The separated boundary conditions (2) and (3) ensure the 𝑟 [𝑥] 𝑛=1 problem is self-adjoint [50,p.91]. The classical S-L problem of solving (8) subject to the with initial and boundary conditions admits a solution only for 𝑏 certain critical values of 𝜆, the eigenvalues. Further, since ∫ 𝑇 [𝑥,0 𝑡|𝑥 ]𝑑𝑥=0, (1) is linear in 𝑈, the general solution for (10) is given by a 𝑎 (18) linear combination of solutions in the form of eigenfunction lim 𝑇 [𝑥,0 𝑡|𝑥 ]=0. expansions. Details of these results can be found in Hille 𝑡→∞ [51,ch.8],BirkhoffandRota[52, ch. 10], and Karlin and This proposition provides the general solution to the Taylor [53]. When the S-L problem is self-adjoint and regular regular, self-adjoint S-L problem of deriving 𝑈 when the the solutions for the transition probability density can be process is subject to regular reflecting barriers. Taking the summarized in the following (see the appendix for proof). limit as 𝑡→∞in (15),itfollowsfrom(16) and (17) that the transition density of the stochastic process satisfies Proposition 1 (ergodic transition density decomposition). the classical ergodic property. Considerable effort has been The regular, self-adjoint Sturm-Liouville problem has an infi- given to determining the convergence behavior of different 0=𝜆 <𝜆 <𝜆 <⋅⋅⋅ nite sequence of real eigenvalues, 0 1 2 processes. The distributional impact of the initial conditions with and boundary restrictions enter through 𝑇[𝑥, 𝑡0 |𝑥 ].From the restrictions on 𝑇[𝑥, 𝑡0 |𝑥 ] in (17),thetotalmassof lim 𝜆𝑛 =∞. 𝑛→∞ (11) the transient term is zero so the mean ergodic theorem still applies. The transient only acts to redistribute the mass of To each eigenvalue there corresponds a unique eigenfunction the stationary distribution, thereby causing a change in shape 𝜑 ≡𝜑[𝑥] 𝑛 𝑛 . Normalization of the eigenfunctions produces which can impact the ex ante calculation of the expected value. The specific degree and type of alteration depends on 𝑏 −1/2 2 the relevant assumptions made about the parameters and ini- 𝜓𝑛 [𝑥] =[∫ 𝑟 [𝑥] 𝜑𝑛 𝑑𝑥] 𝜑𝑛. (12) 𝑎 tial functional forms. Significantly, stochastic generalization of static and deterministic MPT almost always ignores the The 𝜓𝑛[𝑥] eigenfunctions form a complete orthonormal system impact of transients by only employing the limiting stationary in 𝐿2[𝑎, 𝑏].Theuniquesolutionin𝐿2[𝑎, 𝑏] to (1),subjectto distribution component. the boundary conditions (2)-(3) and initial condition (4) is, in The theoretical advantage obtained by imposing regular general form, reflecting barriers on the diffusion state space for the forward equation is that an ergodic decomposition of the transition ∞ −𝜆 𝑡 density is assured. The relevance of bounding the state space 𝑈 [𝑥,] 𝑡 = ∑ 𝑐 𝜓 [𝑥] 𝑒 𝑛 , 𝑛 𝑛 (13) and imposing regular reflecting boundaries can be illustrated 𝑛=0 by considering the well known solution (e.g., [54,p.209])for 𝑈 involving a constant coefficient standard normal variate where 𝑌(𝑡) = ({𝑥0 −𝑥 − 𝜇𝑡}/𝜎) overtheunboundedstatespace 𝑏 𝐼𝑜 =(∞≤𝑥≤∞).Inthiscasetheforwardequation 2 2 𝑐𝑛 = ∫ 𝑟 [𝑥] 𝑓[𝑥0]𝜓𝑛 [𝑥] 𝑑𝑥. (14) (1) reduces to (1/2){𝜕 𝑈/𝜕𝑌 }=𝜕𝑈/𝜕𝑡.Byevaluatingthese 𝑎 derivatives, it can be verified that the principal solution for 𝑈 Giventhis,thetransitionprobabilitydensityfunctionfor𝑥 is at time 𝑡 can be reexpressed as the sum of a stationary 2 1 (𝑥−𝑥0 −𝜇𝑡) limiting equilibrium distribution associated with the 𝜆0 =0 𝑈 [𝑥,0 𝑡|𝑥 ]= exp [− ] (19) 𝜎√(2𝜋𝑡) 𝜎2𝑡 eigenvalue, that is linearly independent of the boundaries and 2 Chinese Journal of Mathematics 9 and as 𝑡→−∞or 𝑡→+∞then 𝑈→0and boundary conditions and then solving for Ψ[𝑥] and 𝑇[𝑥, 𝑡| the stochastic process is nonergodic because it does not 𝑥0].Wong[46] uses a different approach, initially selecting possess a nontrivial stationary distribution. The mean ergodic a stationary distribution and then solving for 𝑈 using the theorem fails: if the process runs long enough, then 𝑈 will restrictions of the Pearson system to specify the forward evolve to where there is no discernible probability associated equation. In this approach, the functional form of the desired with starting from 𝑥0 and reaching the neighborhood of a stationary distribution determines the appropriate boundary given point 𝑥. The absence of a stationary distribution raises conditions. While application of this approach has been a number of questions, for example, whether the process limited to the restricted class of distributions associated with has unit roots. Imposing regular reflecting boundaries is a the Pearson system, it is expedient when a known stationary certain method of obtaining a stationary distribution and a distribution, such as the standard normal distribution, is of discrete spectrum [55,p.13].Alternativemethods,suchas interest. More precisely, let specifying the process to admit natural boundaries where the 2 parameters of the diffusion are zero within the state space, 1 𝑥 Ψ [𝑥] = exp [− ], cangiverisetocontinuousspectrumandraisesignificant √ 2𝜋 2 (22) analytical complexities. At least since Feller [45], the search for useful solutions, including those for singular diffusion 𝐼𝑜 = (−∞<𝑥<∞) . problems, has produced a number of specific cases of interest. However, without the analytical certainty of the classical S-L In this case, the boundaries of the state space are nonat- framework, analysis proceeds on a case by case basis. tracting and not regular. Solving the Pearson equation gives Onepossiblemethodofobtainingastationarydistribu- 𝑑Ψ[𝑥]/𝑑𝑥 = −𝑥Ψ[𝑥] and a forward equation of the OU form: tion without imposing both upper and lower boundaries is to 𝜕2𝑈 𝜕 𝜕𝑈 impose only a lower (upper) reflecting barrier and construct + 𝑥𝑈 = . (23) the stochastic process such that positive (negative) infinity 𝜕𝑥2 𝜕𝑥 𝜕𝑡 is nonattracting, for example, Linetsky [49]andA¨ıt-Sahalia FollowingWong[46,p.268]Mehler’sformulacanbeusedto [56]. This can be achieved by using a mean-reverting drift 𝑈 term. In contrast, Cox and Miller [54, p. 223–5] use the express the solution for as Brownian motion, constant coefficient forward equation with 2 𝑈 [𝑥,0 𝑡|𝑥 ] 𝑥0 >0, 𝐴[𝑥] = 𝜇,and <0 𝐵[𝑥] = (1/2)𝜎 subjecttothelower 𝑥=0 2 reflecting barrier at given in (2) to solve for both the −(𝑥−𝑥 𝑒−𝑡) (24) 𝑈 and the stationary density. The principal solution is solved = 1 [ 0 ]. exp ( −𝑒−2𝑡) using the “method of images” to obtain √2𝜋(1 −𝑒−2𝑡) 2 1 2 (𝑥 − 𝑥 −𝜇𝑡) 𝑡→−∞ 𝑈→0 𝑡→+∞ 𝑈 [𝑥, 𝑡|𝑥 ]= 1 { −( 0 ) Given this, as then and as then 0 exp 2 𝜎√2𝜋𝑡 2𝜎 𝑡 𝑈 achieves the stationary standard normal distribution.

2 4𝑥0𝜇𝑡−(𝑥−𝑥0 −𝜇𝑡) + exp −( )} 6. The Quartic Exponential Distribution 2𝜎2𝑡 (20) The roots of bifurcation theory can be found in the early 𝜇 + 1 {2 solutions to certain deterministic ordinary differential equa- 2 𝜎√2𝜋𝑡 𝜎 tions. Consider the deterministic dynamics described by the pitchfork bifurcation ODE: 2𝜇𝑥 𝑥+𝑥0 +𝜇𝑡 ⋅ exp ( )(1 −𝑁[ ])}, 𝜎2 √ 𝑑𝑥 3 𝜎 𝑡 =−𝑥 +𝜌 𝑥+𝜌 , (25) 𝑑𝑡 1 0 where 𝑁[𝑥] is the cumulative standard normal distribution 𝜌 𝜌 function. Observing that 𝐴[𝑥]=𝜇>0again produces 𝑈→ where 0 and 1 are the “normal” and “splitting” control 𝜌 0 as 𝑡→+∞, the stationary density for 𝐴[𝑥] = 𝜇 <0 has variables, respectively (e.g., [58, 59]). While 0 has significant the Maxwell form information in a stochastic context, this is not usually the case 𝜌 =0 󵄨 󵄨 󵄨 󵄨 in the deterministic problem so 0 is assumed. Given 󵄨𝜇󵄨 󵄨𝜇󵄨 𝑥 2 󵄨 󵄨 2 󵄨 󵄨 this, for 𝜌1 ≤0,thereisonerealequilibrium({𝑑𝑥/𝑑𝑡} =0) Ψ [𝑥] = exp −( ). (21) 𝜎2 𝜎2 solution to this ODE at 𝑥=0where “all initial conditions converge to the same final point exponentially fast with time” Though 𝑥0 does not enter the solution, combined with the [60,p.260].For𝜌1 >0, the solution bifurcates into three location of the boundary at 𝑥=0, it does implicitly impose equilibrium solutions 𝑥={0,±√𝜌1},oneunstableandtwo the restriction 𝑥>0.FromtheProposition,𝑇[𝑥, 𝑡0 |𝑥 ] can stable. In this case, the state space is split into two physically be determined as 𝑈[𝑥, 𝑡0 |𝑥 ]−Ψ[𝑥]. distinct regions (at 𝑥=0) with the degree of splitting Following Linetsky [49], Veerstraeten [57], and others, controlled by the size of 𝜌1. Even for initial conditions that the analytical procedure used to determine 𝑈 involves are “close,” the equilibrium achieved will depend on the specifying the parameters of the forward equation and the sign of the initial condition. Stochastic bifurcation theory 10 Chinese Journal of Mathematics extends this model to incorporate Markovian randomness. where 𝐾 is a constant determined such that the density 21 In this theory, “invariant measures are the random analogues integrates to one; and 𝛽4 >0. Following Fisher [61], of deterministic fixed points”22 [ ,p.469].Significantly, the class of distributions associated with the general quartic ergodicity now requires that the component densities that exponential admits both unimodal and bimodal densities and bifurcate out of the stationary density at the bifurcation point nests the standard normal as a limiting case where 𝛽4 =𝛽3 = √ be invariant measures, for example, Crauel et al. [44,sec.3]. 𝛽1 =0and 𝛽2 =1/2with 𝐾=1/( 2𝜋).Thestationary As such, the ergodic bifurcating process is irreversible in the distribution of the bifurcating double well process is a special sensethatpastsamplepaths(priortothebifurcation)cannot case of the symmetric quartic exponential distribution: reliably be used to generate statistics for the future values of 2 4 thevariable(afterthebifurcation). Ψ[𝑦]=𝐾𝑆 exp [− {𝛽2 (𝑥−𝜇) +𝛽4 (𝑥 − 𝜇) }] , It is well known that the introduction of randomness to (28) the pitchfork ODE changes the properties of the equilibrium where 𝛽4 ≥ 0, solution, for example, [22,sec.9.2].Itisnolongernecessary 𝜇 that the state space for the principal solution be determined where is the population mean and the symmetry restriction 𝛽 =𝛽 =0 by the location of the initial condition relative to the bifur- requires 1 3 . Such multimodal stationary densities cation point. The possibility for randomness to cause some have received scant attention in financial economics, in paths to cross over the bifurcation point depends on the size general,andinMPT,inparticular.Toseewhythecondition 𝛽 𝑋=𝑌−{𝛽/4𝛽 } of volatility of the process, 𝜎, which measures the nonlinear on 1 is needed, consider change of origin 3 4 to signaltowhitenoiseratio.Ofthedifferentapproaches remove the cubic term from the general quartic exponential to introducing randomness (e.g., multiplicative noise), the [65,p.480]: simplest approach to converting from a deterministic to a 2 Ψ[𝑦]=𝐾 [− {𝜅 (𝑦 −𝜇 )+𝛼(𝑦−𝜇 ) stochastic context is to add a Weiner process (𝑑𝑊(𝑡)) to the 𝑄 exp 𝑦 𝑦 𝜎 (29) ODE. Augmenting the diffusion equation to allow for to 4 control the relative impact of nonlinear drift versus random +𝛾(𝑦−𝜇𝑦) }] , where 𝛾≥0. noise produces the “pitchfork bifurcation with additive noise” [22,p.475]whichinsymmetricformis The substitution of 𝑦 for 𝑥 indicates the change of origin 3 which produces the following relations between coefficients 𝑑𝑋 (𝑡) =(𝜌1𝑋 (𝑡) −𝑋(𝑡) )𝑑𝑡+𝜎𝑑𝑊(𝑡) . (26) for the general and specific cases:

Applications in financial economics, for example,¨ Aıt-Sahalia 𝛽 𝛽 2 − 𝛽 𝛽 𝛽 +𝛽 3 𝜅=8 1 4 4 2 3 4 3 , [56], refer to this diffusion process as the double well process. 2 8𝛽4 While consistent with the common use of diffusion equations in financial economics, the dynamics of the pitchfork process 𝛽 𝛽 − 𝛽 2 (30) 𝛼=8 2 4 3 3 , captured by 𝑇[𝑥, 𝑡0 |𝑥 ] have been “forgotten” [22,p.473]. 8𝛽4 Models in MPT are married to the transition probability densities associated with unimodal stationary distributions, 𝛾=𝛽4. especially the class of Gaussian-related distributions. Yet, it is well known that more flexibility in the shape of the stationary The symmetry restriction 𝜅=0can only be satisfied if both distribution can be achieved using a higher order exponential 𝛽3 and 𝛽1 =0. Given the symmetry restriction, the double density, for example, Fisher [61], Cobb et al. [62], and Crauel well process further requires −𝛼=𝛾=𝜎=1.Solvingfor and Flandoli [60]. Increasing the degree of the polynomial the modes of Ψ[𝑦] gives ±√{|𝛼|/(2𝛾)} which reduces to ±1 in the exponential comes at the expense of introducing addi- for the double well process, as in A¨ıt-Sahalia [56,Figure6B, tional parameters resulting in a substantial increase in the p. 1385]. analytical complexity, typically defying a closed form solution As illustrated in Figure 1, the selection of 𝑎𝑖 in the station- 4 2 for the transition densities. However, following Elliott [63], it ary density Ψ𝑖[𝑥] =𝑄 𝐾 exp{−(0.25𝑥 − 0.5𝑥 −𝑎𝑖𝑥)} defines hasbeenrecognizedthatthesolutionoftheassociatedregular a family of general quartic exponential densities, where 𝑎𝑖 S-L problem will still have a discrete spectrum, even if the is the selected value of 𝜅 for that specific density.22 The specific form of the eigenfunctions and eigenvalues in 𝑇[𝑥, 𝑡| coefficient restrictions on the parameters 𝛼 and 𝛾 dictate that 𝑥0] is not precisely determined [64,sec.6.7].Inferencesabout these values cannot be determined arbitrarily. For example, transient stochastic behavior can be obtained by examining given that 𝛽4 issetat0.25,then,for𝑎𝑖 =0,itfollowsthat the solution of the deterministic nonlinear dynamics. In this 𝛼=𝛽2 = 0.5. “Slicing across” the surface in Figure 1 at process, attention initially focuses on the properties of the 𝑎𝑖 =0reveals a stationary distribution that is equal to the higher order exponential distributions. double well density. Continuing to slice across as 𝑎𝑖 increases To this end, assume that the stationary distribution is a in size, the bimodal density becomes progressively more fourth degree or “general quartic” exponential: asymmetrically concentrated in positive 𝑥 values. Though the location of the modes does not change, the amount of density Ψ [𝑥] =𝐾 [−Φ [𝑥]] exp between the modes and around the negative mode decreases. (27) 𝑎 =𝐾 [− (𝛽 𝑥4 +𝛽 𝑥3 +𝛽 𝑥2 +𝛽 𝑥)] , Similarly, as 𝑖 decreases in size the bimodal density becomes exp 4 3 2 1 more asymmetrically concentrated in positive 𝑥 values. Chinese Journal of Mathematics 11

sample paths that provide a better approximation to the sample paths of observed financial data.

0.4 7. Conclusion 0.3 ] 1 x

[ 0.2 i The classical ergodicity hypothesis provides a point of demar- Ψ 0.1 0.5 0 cation in the prehistories of MPT and econophysics. To 0 a deal with the problem of making statistical inferences from −2 “nonexperimental” data, theories in MPT typically employ 0 −0.5 stationary densities that are time reversible, are unimodal, x and allow no short or long term impact from initial and 2 −1 boundary conditions. The possibility of bimodal processes or ex ante impact from initial and boundary conditions is not Ψ [𝑥] = Figure 1: Family of stationary densities for 𝑖 recognized or, it seems, intended. Significantly, as illustrated 𝐾 {−(0.25𝑥4 − 0.5𝑥2 −𝑎𝑥)} 𝑄 exp 𝑖 . Each of the continuous values for in the need to select an 𝑎𝑖 in Figure 1 in order to specify the 𝑎 𝑎=0 signifies a different stationary density. For example, at the “real world” ex ante stationary density, a semantic connection density is the double well density which symmetric about zero and can be established between the subjective uncertainty about with modes at ±1. encountering a future bifurcation point and, say, the possible collapse of an asset price bubble impacting future market valuations. Examining the quartic exponential stationary distribution associated with a bifurcating ergodic process, it is While the stationary density is bimodal over 𝑎𝑖𝜀{−1, 1},for apparent that this distribution nests the Gaussian distribution |𝑎𝑖| large enough the density becomes so asymmetric that as a special case. In this sense, results from classical statistical only a unimodal density appears. For the general quartic, mechanics can be employed to produce a stochastic general- asymmetry arises as the amount of the density surrounding ization of the unimodal, time reversible processes employed each mode (the subdensity) changes with 𝑎𝑖.Inthis,the in modern portfolio theory. individual stationary subdensities have a symmetric shape. To introduce asymmetry in the subdensities, the reflecting boundaries at 𝑎 and 𝑏 that bound the state space for the Appendix regular S-L problem can be used to introduce positive Preliminaries on Solving the asymmetry in the lower subdensity and negative asymmetry in the upper subdensity. Forward Equation Following Chiarella et al. [43], the stochastic bifurcation Due to the widespread application in a wide range of sub- process has a number of features which are consistent with jects, textbook presentations of the Sturm-Liouville problem the ex ante behavior of a securities market driven by a possess subtle differences that require some clarification combination of chartists and fundamentalists. Placed in to be applicable to the formulation used in this paper. In the context of the classical S-L framework, because the particular, to derive the canonical form (8) of the Fokker- stationary distributions are bimodal and depend on forward Planck equation (1), observe that evaluating the derivatives parameters, such as 𝜅, 𝛼, 𝛾,and𝑎𝑖 in Figure 1,thatare in (1) gives notknownonthedecisiondate,therisk-returntradeoff models employed in MPT are uninformative. What use is 𝜕2𝑈 𝜕𝐵 𝜕𝑈 𝜕2𝐵 𝜕𝐴 𝐵 [𝑥] +[ −𝐴[𝑥]] +[ − ]𝑈 theforecastprovidedby𝐸[𝑥(𝑇)] when it is known that 2 2 2 𝑥(𝑇) 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 there are other modes for values that are more likely (A.1) to occur? A mean estimate that is close to a bifurcation 𝜕𝑈 = . point would even be unstable. The associated difficulty of 𝜕𝑡 calculating an expected return forecast or other statistical estimate such as volatility from past data can be compounded This can be rewritten as by the presence of transients that originate from boundaries 1 𝜕 𝜕𝑈 𝜕𝑈 and initial conditions. For example, the presence of a recent [𝑃 [𝑥] ] +𝑄[𝑥] 𝑈= , (A.2) structural break, for example, the collapse in global stock 𝑟 [𝑥] 𝜕𝑥 𝜕𝑥 𝜕𝑡 prices from late 2008 to early 2009, can be accounted for where by appropriate selection of 𝑥0, the fundamental dependence of investment decisions on the relationship between 𝑥0 𝑃 [𝑥] =𝐵[𝑥] 𝑟 [𝑥] , and future (not past) performance is not captured by the reversible ergodic processes employed in MPT that ignore 1 𝜕𝑃 𝜕𝐵 = 2 −𝐴[𝑥] , transients arising from initial conditions. Mathematical tools 𝑟 [𝑥] 𝜕𝑥 𝜕𝑥 (A.3) such as classical S-L methods are able to demonstrate this 𝜕2𝐵 𝜕𝐴 fundamental dependence by exploiting properties of ex ante 𝑄 [𝑥] = − . bifurcating ergodic processes to generate theoretical ex post 𝜕𝑥2 𝜕𝑥 12 Chinese Journal of Mathematics

It follows that 𝐵[𝑥] and 𝐴[𝑥] from the Pearson system. The associated boundary condition follows from observing the 𝜌[𝑥] will be 𝜕𝐵 1 𝜕𝑃 1 𝜕𝑟 = − 𝑃 [𝑥] theergodicdensityandmakingappropriatesubstitutionsinto 𝜕𝑥 𝑟 [𝑥] 𝜕𝑥 𝑟2 𝜕𝑥 the boundary condition: (A.4) 𝜕𝐵 𝐵 [𝑥] 𝜕𝑟 = −𝐴[𝑥] − . 𝜕 2 𝜕𝑥 𝑟 [𝑥] 𝜕𝑥 {𝐵 [𝑥] 𝑓 [𝑡] 𝜌 [𝑥] 𝜃 [𝑥]}−𝐴[𝑥] 𝑓 [𝑡] 𝜌 [𝑥] 𝜃 [𝑥] 𝜕𝑥 This provides the solution for the key function 𝑟[𝑥]: = 0 (A.10) 1 𝜕𝑟 1 𝜕𝐵 𝐴 [𝑥] 𝑑 − =− 󳨀→ [𝐵𝜌𝜃] − 𝐴𝜌𝜃= . 𝑟 [𝑥] 𝜕𝑥 𝐵 [𝑥] 𝜕𝑥 𝐵 [𝑥] 𝑑𝑥 0 󳨀→ [𝑟] − [𝑘] ln ln Evaluating the derivative and taking values at the lower (or 𝑥 𝐴 [𝑠] (A.5) upper) boundary give =−∫ 𝑑𝑠, 𝐵 [𝑠] 𝑑𝜃 [𝑎] 𝑑𝐵𝜌 𝐵 [𝑎] 𝜌 [𝑎] +𝜃[𝑎] −𝐴[𝑎] 𝜌 [𝑎] 𝜃 [𝑎] 𝑥 𝐴 [𝑠] 𝑑𝑥 𝑑𝑥 𝑟 [𝑥] =𝐵[𝑥] [− ∫ 𝑑𝑠] . exp 𝐵 [𝑠] = 0 This 𝑟[𝑥] function is used to construct the scale and speed 𝑑𝜃 [𝑎] (A.11) =𝐵[𝑎] 𝜌 [𝑎] densities commonly found in presentations of solutions to 𝑑𝑥 the forward equation, for example, Karlin and Taylor [53]and 𝑑𝐵 [𝑎] 𝜌 [𝑎] Linetsky [49]. +𝜃[𝑎] [ −𝐴[𝑎] 𝜌 [𝑎]]. Another specification of the forward equation that is of 𝑑𝑥 importance is found in Wong [46,eq.6-7]: Observing that the expression in the last bracket is the 𝑑 𝑑𝜃 𝑈 [𝐵 [𝑥] 𝜌 [𝑥] ]+𝜆𝜌[𝑥] 𝜃 [𝑥] = 0 original condition with the ergodic density serving as gives 𝑑𝑥 𝑑𝑥 the boundary condition stated in Wong [46,eq.7]. (A.6) 𝑑𝜃 𝜓 𝑛 [𝑎, 𝑏] with b.c. 𝐵 [𝑥] 𝜌 [𝑥] = 0. Proof of Proposition 1. (a) 𝑛 has exactly zeroes in : 𝑑𝑥 Hille [51,p.398,Theorem8.3.3] and Birkhoff and Rota [52,p. 320, Theorem 5] show that the eigenfunctions of the Sturm- This formulation occurs after separating variables, say with ( 󸀠) ( 󸀠) ( 󸀠) 𝑛 𝑈[𝑥] = 𝑔[𝑥]ℎ[𝑡].Substitutingthisresultinto(1) gives Liouville system 1 with 2 , 3 ,and(4) have exactly zeroes in the interval (𝑎, 𝑏).Moreprecisely,sinceitisassumed 2 𝜕 𝜕 𝜕ℎ that 𝑟>0,theeigenfunction𝜓𝑛 corresponding to the 𝑛th [𝐵𝑔ℎ] − [𝐴𝑔ℎ] =𝑔[𝑥] . 2 (A.7) eigenvalue has exactly 𝑛 zeroes in (𝑎, 𝑏). 𝜕𝑥 𝜕𝑥 𝜕𝑡 𝑏 (b) For 𝜓𝑛 =0, ∫ 𝜓𝑛[𝑥]𝑑𝑥 = 0. Using the separation of variables substitution (1/ℎ){𝜕ℎ/𝜕𝑡} = 𝑎 −𝜆 and redefining 𝑔[𝑥] = 𝜌𝜃 gives Proof.For𝜓𝑛 =0the following applies: 𝑑 𝑑 [ 𝐵𝑔−𝐴𝑔] 1 𝑑 𝑑 𝑑𝑥 𝑑𝑥 𝜓𝑛 = { [𝐵 [𝑥] 𝜓𝑛]−𝐴[𝑥] 𝜓𝑛}, 𝜆𝑛 𝑑𝑥 𝑑𝑥 =−𝜆𝑔 󵄨 𝑏 𝑑 󵄨 (A.8) ∴ ∫ 𝜓 [𝑥] 𝑑𝑥 = 1 { [𝐵 [𝑥] 𝜓 ]󵄨 𝑛 𝑛 󵄨 𝑑 𝑑 𝑎 𝜆𝑛 𝑑𝑥 󵄨𝑥=𝑏 = [ 𝐵 [𝑥] 𝜌 [𝑥] 𝜃 [𝑥] −𝐴[𝑥] 𝜌 [𝑥] 𝜃 [𝑥]] (A.12) 𝑑𝑥 𝑑𝑥 󵄨 𝑑 󵄨 −𝐴[𝑏] 𝜓𝑛 [𝑏] − [𝐵 [𝑥] 𝜓𝑛]󵄨 =−𝜆𝜌𝜃. 𝑑𝑥 󵄨𝑥=𝑎

Evaluating the derivative inside the bracket and using the +𝐵[𝑎] 𝜓𝑛 [𝑎]}=0. condition {𝑑/𝑑𝑥}[𝐵𝜌] −𝐴𝜌=0 to specify admissible 𝜌 give 𝑑 𝑑 𝑑 𝜓 [𝑥] [𝜃 (𝐵𝜌) + 𝐵𝜌 𝜃 − 𝐴𝜌𝜃] Since each 𝑛 satisfies the boundary conditions. 𝑑𝑥 𝑑𝑥 𝑑𝑥 (c) For some 𝑘, 𝜆𝑘 =0. (A.9) 𝑑 𝑑 = [𝐵𝜌 𝜃]=−𝜆𝜌[𝑥] 𝜃 [𝑥] Proof.From(13) 𝑑𝑥 𝑑𝑥 ∞ −𝜆𝑘𝑡 which is equation (6) in Wong [46]. The condition used 𝑈 [𝑥,] 𝑡 = ∑ 𝑐𝑘𝑒 𝜓𝑘 [𝑥] . (A.13) to define 𝜌 is then used to identify the specification of 𝑘=0 Chinese Journal of Mathematics 13

𝑏 ∫ 𝑈[𝑥, 𝑡]𝑑𝑥 = Since 𝑎 1then Conflict of Interests ∞ 𝑏 −𝜆 𝑡 The authors declare that there is no conflict of interests = ∑ 𝑐 𝑒 𝑘 ∫ 𝜓 𝑥 𝑑𝑥. 1 𝑘 𝑘 [ ] (A.14) regarding the publication of this paper. 𝑘=0 𝑎 But from part (b) this will = 0 (which is a contradiction) Acknowledgments unless 𝜆𝑘 =0for some 𝑘. (d) Consider 𝜆0 =0. Helpful comments from Chris Veld, Yulia Veld, Emmanuel Havens, Franck Jovanovic, Marcel Boumans, and Christoph Proof. From part (a), 𝜓0[𝑥] hasnozeroesin(𝑎, 𝑏). Therefore, 𝑏 𝑏 Schinckus are gratefully acknowledged. ∫ 𝜓 [𝑥]𝑑𝑥 > ∫ 𝜓 [𝑥]𝑑𝑥 < either 𝑎 0 0or 𝑎 0 0. 𝜆 Itfollowsfrompart(b)that 0 =0. Endnotes (e) Consider 𝜆𝑛 >0for 𝑛 =0̸ .Thisfollowsfrompart(d) and the strict inequality conditions provided in part (a). 1. For example, Black and Scholes [4]discussthecon- (f) Obtaining the solution for 𝑇[𝑥] in the Proposition, sistency of the option pricing formula with the capi- from part (d) it follows that tal asset pricing model. More generally, financial eco- 𝑑 nomics employs primarily Gaussian-based finite param- {[𝐵 [𝑥] 𝜓 [𝑥,] 𝑡 ] −𝐴[𝑥] 𝜓 [𝑥,] 𝑡 }= . (A.15) 𝑑𝑥 0 𝑥 0 0 eter models that agree with the “tradeoff between risk and return.” Integrating this equation from 𝑎 to 𝑥 and using the boundary condition give 2. In rational mechanics, once the initial positions of the particles of interest, for example, molecules, are known, [𝐵 [𝑥] 𝜓0 [𝑥,] 𝑡 ]𝑥 −𝐴[𝑥] 𝜓0 [𝑥,] 𝑡 = 0. (A.16) the mechanical model fully determines the future evo- 𝜓 lution of the system. This scientific and philosophical This equation can be solved for 0 to get approach is often referred to as Laplacian determinism. 𝑥 𝐴 [𝑠] 𝜓 =𝐴[𝐵 [𝑥]]−1 [∫ 𝑑𝑠]=𝐶[𝑟 [𝑥]]−1 , 3. Boltzmann and Max Planck were vociferous opponents 0 exp 𝐵 [𝑠] 𝑎 (A.17) of energetics. The debate over energetics was part ofa where: 𝐶=constant. larger intellectual debate concerning determinism and reversibility. Jevons [66,p.738-9]reflectstheentrenched Therefore, determinist position of the marginalists: “We may safely 𝑏 −1/2 accept as a satisfactory scientific hypothesis the doctrine 2 −2 −1 so grandly put forth by Laplace, who asserted that 𝜓0 [𝑥] =[∫ 𝑟 [𝑥] 𝐶 [𝑟 [𝑥]] 𝑑𝑥] 𝐶 [𝑟 [𝑥]] 𝑎 a perfect knowledge of the universe, as it existed at (A.18) any given moment, would give a perfect knowledge [𝑟 [𝑥]]−1 = . of what was to happen thenceforth and for ever after. 𝑏 1/2 [∫ 𝑟 [𝑥]−1 𝑑𝑥] Scientific inference is impossible, unless we may regard 𝑎 the present as the outcome of what is past, and the cause Using this definition and observing that the integral of 𝑓[𝑥] of what is to come. To the view of perfect intelligence over the state space is one it follows nothing is uncertain.” What Boltzmann, Planck, and othershadobservedinstatisticalphysicswasthateven though the behavior of one or two molecules can be 𝑏 { 𝑟 [𝑥]−1 } 𝑐0 = ∫ 𝑓 [𝑥] 𝑟 [𝑥] 𝑑𝑥 completely determined, it is not possible to generalize { 𝑏 1/2 } 𝑎 [∫ 𝑟 [𝑥] 𝑑𝑥] these mechanics to the describe the macroscopic motion { 𝑎 } of molecules in large, complex systems, for example, Brush [67, esp. ch. II]. = 1 , 𝑏 1/2 (A.19) [∫ 𝑟 [𝑥] 𝑑𝑥] 4.Assuch,Boltzmannwaspartofthelarger:“Second 𝑎 Scientific Revolution, associated with the theories of 𝑟 [𝑥]−1 Darwin, Maxwell, Planck, Einstein, Heisenberg and ∴𝑐𝜓 [𝑥] = . 0 𝑏 Schrodinger,¨ (which) substituted a world of process [∫ [𝑟 [𝑥]]−1 𝑑𝑥] 𝑎 andchancewhoseultimatephilosophicalmeaningstill remains obscure” [67,p.79].Thisrevolutionsuperceded (g) The Proof of the Proposition now follows from parts (f), the following: “First Scientific Revolution, dominated by (e), and (b). the physical astronomy of Copernicus, Kepler, Galileo, and Newton,... in which all changes are cyclic and all Disclosure motions are in principle determined by causal laws.” The irreversibility and indeterminism of the Second The authors are Professor of Finance and Associate Professor Scientific Revolution replaces the reversibility and deter- of Finance at Simon Fraser University. minism of the first. 14 Chinese Journal of Mathematics

5. There are many interesting sources on these points political economists, and so on. Because such critiques which provide citations for the historical papers that are take motivation from the theories of mainstream eco- being discussed. Cercignani [68, p. 146–50] discusses nomics, these critiques are distinct from econophysics. theroleofMaxwellandBoltzmanninthedevelopment Following Schinckus [12,p.3818],“Econophysicistshave of the ergodic hypothesis. Maxwell [25] is identified as then allies within economics with whom they should “perhaps the strongest statement in favour of the ergodic become acquainted.” hypothesis.” Brush [69]hasadetailedaccountofthe development of the ergodic hypothesis. Gallavotti [70] 10. Dhrymes [31,p.1–29]discussesthealgebraofthelag traces the etymology of “ergodic” to the “ergode” in an operator. 1884 paper by Boltzmann. More precisely, an ergode 11. Critiques of mainstream economics that are rooted in is shorthand for “ergomonode” which is a “monode the insights of The General Theory recognize the dis- with given energy” where a “monode” can be either tinction between fundamental uncertainty and objective a single stationary distribution taken as an ensemble probability. As a consequence, the definition of ergodic or a collection of such stationary distributions with theory in heterodox criticisms of mainstream economics some defined parameterization. The specific use is clear lacks formal precision, for example, the short term from the context. Boltzmann proved that an ergode dependence of ergodic processes on initial conditions is is an equilibrium ensemble and, as such, provides a not usually recognized. Ergodic theory is implicitly seen mechanical model consistent with the second law of as another piece of the mathematical formalism inspired thermodynamics. It is generally recognized that the byHilbertandBourbakiandcapturedintheArrow- modern usage of “the ergodic hypothesis” originates Debreu general equilibrium model of mainstream eco- with Ehrenfest [71]. nomics. 6. Kapetanios and Shin [37,p.620]capturetheessenceof 12. In this context though not in all contexts, econophysics this quandary: “Interest in the interface of nonstation- provides a “macroscopic” approach. In turn, ergodicity arity and nonlinearity has been increasing in the econo- is an assumption that permits the time average from a metric literature. The motivation for this development single observed sample path to (phenomenologically) maybetracedtotheperceivedpossibilitythatnonlinear model the ensemble of sample paths. Given this, econo- ergodic processes can be misinterpreted as unit root physicsdoescontainasubstantivelyrichertoolkitthat nonstationary processes. Furthermore, the inability of encompasses both ergodic and nonergodic processes. standard unit root tests to reject the null hypothesis of Manyworksineconophysicsimplicitlyassumeergod- unit root for a large number of macroeconomic vari- icity and develop models based on that assumption. ables, which are supposed to be stationary according to economic theory, is another reason behind the increased 13. The distinction between invariant and ergodic measures interest.” is fundamental. Recognizing a number of distinct defi- nitions of ergodicity are available, following Medio [74, 7. The second law of thermodynamics is the universal law p. 70], the Birkhoff-Khinchin ergodic (BK) theorem for of increasing entropy, a measure of the randomness of invariant measures can be used to demonstrate that molecular motion and the loss of energy to do work. ergodic measures are a class of invariant measures. First recognized in the early 19th century, the second More precisely, the BK theorem permits the limit of law maintains that the entropy of an isolated system, the time average to depend on initial conditions. In not in equilibrium, will necessarily tend to increase effect, the invariant measure is permitted to decompose over time. Entropy approaches a maximum value at into invariant “submeasures.”The physical interpretation thermal equilibrium. A number of attempts have been of this restriction is that sample paths starting from a made to apply the entropy of information to problems particular initial condition may only be able to access a in economics, with mixed success. In addition to the part of the sample space, no matter how long the process second law, physics now recognizes the zeroth law of is allowed to run. For an ergodic process, sample paths thermodynamics that “any system approaches an equi- starting from any admissible initial condition will be able librium state” [72, p. 54]. This implication of the second to “fill the sample space”; that is, if the process is allowed law for theories in economics was initially explored by to run long enough, the time average will not depend on Georgescu-Roegen [73]. the initial condition. Medio [74,p.73]providesauseful 8. In this process, the ergodicity hypothesis is required example of an invariant measure that is not ergodic. to permit the one observed sample path to be used to 14. The phenomenological approach is not without difficul- estimate the parameters for the ex ante distribution of ties. For example, the restriction to Markov processes the ensemble paths. In turn, these parameters are used ignores the possibility of invariant measures that are to predict future values of the economic variable. not Markov. In addition, an important analytical con- 9. Heterodox critiques are associated with views consid- struct in bifurcation theory, the Lyapunov exponent, ered to originate from within economics. Such critiques can encounter difficulties with certain invariant Markov areseentobemadeby“economists,”forexample,Post- measures. Primary concern with the properties of the Keynesian economists, institutional economists, radical stationary distribution is not well suited to analysis of Chinese Journal of Mathematics 15

the dynamic paths around a bifurcation point. And so 22. A number of simplifications were used to produce the 3D it goes. image in Figure 1: 𝑥 has been centered about 𝜇 and 𝜎= 𝐾 =1 15. A diffusion process is “regular” if starting from any 𝑄 . Changing these values will impact the specific 𝑥 point in the state space 𝐼,anyotherpointin𝐼 can be size of the parameter values for a given but will not reached with positive probability (Karlin and Taylor [53, change the general appearance of the density plots. p. 158]. This condition is distinct from other definitions ofregularthatwillbeintroduced:“regularboundary References conditions” and “regular S-L problem.” [1] H. Markowitz, “Portfolio selection,” The Journal of Finance,vol. 16. The classification of boundary conditions is typically an 7,pp.77–91,1952. important issue in the study of solutions to the forward [2]J.Y.Campbell,“Understandingriskandreturn,”Journal of equation. 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