Econophysics: Financial White Noise Switch. Thought of the Day 115.0

What is the cause of large market fluctuation? Some economists blame irrationality behind the fat-tail distribution. Some economists observed that social psychology might create market fad and panic, which can be modeled by collective behavior in statistical mechanics. For example, the bi-modular distribution was discovered from empirical data in option prices. One possible mechanism of polarized behavior is collective action studied in physics and social psychology. Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when field parameter changes across some threshold. The Ising model in equilibrium statistical mechanics was borrowed to study social psychology. Its phase transition from uni-modular to bi-modular distribution describes statistical features when a stable society turns into a divided society. The problem of the Ising model is that its key parameter, the social temperature, has no operational definition in social system. A better alternative parameter is the intensity of social interaction in collective action.

A difficult issue in business cycle theory is how to explain the recurrent feature of business cycles that is widely observed from macro and financial indexes. The problem is: business cycles are not strictly periodic and not truly random. Their correlations are not short like random walk and have multiple frequencies that changing over time. Therefore, all kinds of math models are tried in business cycle theory, including deterministic, stochastic, linear and nonlinear models. We outline economic models in terms of their base function, including white noise with short correlations, persistent cycles with long correlations, and color chaos model with erratic amplitude and narrow frequency band like biological clock.

 

The steady state of probability distribution function in the Ising Model of Collective Behavior with h = 0 (without central propaganda field). a. Uni-modular distribution with low social stress (k = 0). Moderate stable behavior with weak interaction and high social temperature. b. Marginal distribution at the phase transition with medium social stress (k = 2). Behavioral phase transition occurs between stable and unstable society induced by collective behavior. c. Bi-modular distribution with high social stress (k = 2.5). The society splits into two opposing groups under low social temperature and strong social interactions in unstable society. 

Deterministic models are used by Keynesian economists for endogenous mechanism of business cycles, such as the case of the accelerator-multiplier model. The stochastic models are used by the Frisch model of noise-driven cycles that attributes external shocks as the driving force of business fluctuations. Since 1980s, the discovery of economic chaos and the application of statistical mechanics provide more advanced models for describing business cycles. Graphically,

The steady state of probability distribution function in socio-psychological model of collective choice. Here, “a” is the independent parameter; “b” is the interaction parameter. a Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. b Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment. 

Numerical 1 autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. The broken line is a sine wave with period P = 1. 

Linear harmonic cycles with unique frequency are introduced in business cycle theory. The auto-correlations from harmonic cycle and white noise are shown in the above figure. Auto-correlation function from harmonic cycles is a cosine wave. The amplitude of cosine wave is slightly decayed because of limited data points in numerical experiment. Auto-correlations from a random series are an erratic series with rapid decade from one to residual fluctuations in numerical calculation. The auto-regressive (AR) model in discrete time is a combination of white noise term for simulating short-term auto-correlations from empirical data.

The deterministic model of chaos can be classified into white chaos and color chaos. White chaos is generated by nonlinear difference equation in discrete-time, such as one-dimensional logistic map and two-dimensional Henon map. Its autocorrelations and power spectra look like white noise. Its correlation dimension can be less than one. White noise model is simple in mathematical analysis but rarely used in empirical analysis, since it needs intrinsic time unit.

Color chaos is generated by nonlinear differential equations in continuous-time, such as three-dimensional Lorenz model and one-dimensional model with delay-differential model in biology and economics. Its autocorrelations looks like a decayed cosine wave, and its power spectra seem a combination of harmonic cycles and white noise. The correlation dimension is between one and two for 3D differential equations, and varying for delay-differential equation.

History shows the remarkable resilience of a market that experienced a series of wars and crises. The related issue is why the economy can recover from severe damage and out of equilibrium? Mathematically speaking, we may exam the regime stability under parameter change. One major weakness of the linear oscillator model is that the regime of periodic cycle is fragile or marginally stable under changing parameter. Only nonlinear oscillator model is capable of generating resilient cycles within a finite area under changing parameters. The typical example of linear models is the Samuelson model of multiplier-accelerator. Linear stochastic models have similar problem like linear deterministic models. For example, the so-called unit root solution occurs only at the borderline of the unit root. If a small parameter change leads to cross the unit circle, the stochastic solution will fall into damped (inside the unit circle) or explosive (outside the unit circle) solution.

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Superstrings as Grand Unifier. Thought of the Day 86.0

The first step of deriving General Relativity and particle physics from a common fundamental source may lie within the quantization of the classical string action. At a given momentum, quantized strings exist only at discrete energy levels, each level containing a finite number of string states, or particle types. There are huge energy gaps between each level, which means that the directly observable particles belong to a small subset of string vibrations. In principle, a string has harmonic frequency modes ad infinitum. However, the masses of the corresponding particles get larger, and decay to lighter particles all the quicker.

Most importantly, the ground energy state of the string contains a massless, spin-two particle. There are no higher spin particles, which is fortunate since their presence would ruin the consistency of the theory. The presence of a massless spin-two particle is undesirable if string theory has the limited goal of explaining hadronic interactions. This had been the initial intention. However, attempts at a quantum field theoretic description of gravity had shown that the force-carrier of gravity, known as the graviton, had to be a massless spin-two particle. Thus, in string theory’s comeback as a potential “theory of everything,” a curse turns into a blessing.

Once again, as with the case of supersymmetry and supergravity, we have the astonishing result that quantum considerations require the existence of gravity! From this vantage point, right from the start the quantum divergences of gravity are swept away by the extended string. Rather than being mutually exclusive, as it seems at first sight, quantum physics and gravitation have a symbiotic relationship. This reinforces the idea that quantum gravity may be a mandatory step towards the unification of all forces.

Unfortunately, the ground state energy level also includes negative-mass particles, known as tachyons. Such particles have light speed as their limiting minimum speed, thus violating causality. Tachyonic particles generally suggest an instability, or possibly even an inconsistency, in a theory. Since tachyons have negative mass, an interaction involving finite input energy could result in particles of arbitrarily high energies together with arbitrarily many tachyons. There is no limit to the number of such processes, thus preventing a perturbative understanding of the theory.

An additional problem is that the string states only include bosonic particles. However, it is known that nature certainly contains fermions, such as electrons and quarks. Since supersymmetry is the invariance of a theory under the interchange of bosons and fermions, it may come as no surprise, post priori, that this is the key to resolving the second issue. As it turns out, the bosonic sector of the theory corresponds to the spacetime coordinates of a string, from the point of view of the conformal field theory living on the string worldvolume. This means that the additional fields are fermionic, so that the particle spectrum can potentially include all observable particles. In addition, the lowest energy level of a supersymmetric string is naturally massless, which eliminates the unwanted tachyons from the theory.

The inclusion of supersymmetry has some additional bonuses. Firstly, supersymmetry enforces the cancellation of zero-point energies between the bosonic and fermionic sectors. Since gravity couples to all energy, if these zero-point energies were not canceled, as in the case of non-supersymmetric particle physics, then they would have an enormous contribution to the cosmological constant. This would disagree with the observed cosmological constant being very close to zero, on the positive side, relative to the energy scales of particle physics.

Also, the weak, strong and electromagnetic couplings of the Standard Model differ by several orders of magnitude at low energies. However, at high energies, the couplings take on almost the same value, almost but not quite. It turns out that a supersymmetric extension of the Standard Model appears to render the values of the couplings identical at approximately 10¹⁶ GeV. This may be the manifestation of the fundamental unity of forces. It would appear that the “bottom-up” approach to unification is winning. That is, gravitation arises from the quantization of strings. To put it another way, supergravity is the low-energy limit of string theory, and has General Relativity as its own low-energy limit.

Fortune of the Individuals Restricted to Integers: Random Economic Exchange Between Populations of Traders.

Consider a population of traders, each of which possesses a certain amount of capital which is assumed to be quantized in units of minimal capital. Taking this latter quantity as the basic unit, the fortune of an individual is restricted to the integers. The wealth of the population evolves by the repeated interaction of random pairs of traders. In each interaction, one unit of capital is transferred between the trading partners. To complete the description, we specify that if a poorest individual (with one unit of capital) loses all remaining capital by virtue of a “loss”, the bankrupt individual is considered to be economically dead and no longer participates in economic activity.

In the following, we consider a specific realization of additive capital exchange, the “random” exchange, where the direction of the capital exchange is independent of the relative capital of the traders. While this rule has little economic basis, the model is completely soluble and thus provides a helpful pedagogical point.

In a random exchange, one unit of capital is exchanged between trading partners as represented by the reaction scheme (j, k) → (j ± 1, k ∓ 1). Let c_k(t) be the density of individuals with capital k. within a mean-field description, c_k(t) evolves according to

dc_k(t)/dt = N(t) [c_k+1(t) + c_k-1(t) – 2c_k(t)] —– (1)

with N(t) ≡ M₀(t) = ∑_k=1^∞ c_k(t), the population density. The first two terms account for gain in c_k(t) due to the interactions (j, k + 1) → (j + 1, k) and (j, k − 1) → (j−1, k), respectively, while the last term accounts for the loss in c_k(t) due to the interactions (j, k) → (j±1, k∓1).

By defining a modified time variable,

T = ∫₀^t dt’N(t’) —– (2)

equation (1) is reduced to the discrete diffusion equation

dc_k(T)/dT = c_k+1(T) + c_k-1(T) – 2c_k(T) —– (3)

The rate equation for the poorest density has the slightly different form, dc₁/dT = c₂ − 2c₁, but may be written in the same form as equation (3) if we impose the boundary condition c₀(T) = 0.

For illustrative purposes, let us assume that initially all individuals have one unit of capital, c_k(0) = δ_k1. The solution to equation (3) subject to these initial and boundary conditions is

c_k(T) = e^−2T [I_k−1(2T) − I_k+1(2T)] —– (4)

where I_n denotes the modified Bessel function of order n. consequently, the total density N(t) is

N(T) = e^−2T [I₀(2T) + I₁(2T)] —– (5)

To re-express this exact solution in terms of the physical time t, we first invert equation (2) to obtain t(T) = ∫₀^T dT′/N(T′), and then eliminate T in favor of t in the solution for c_k(T). For simplicity and concreteness, let us consider the long-time limit. From equation (4),

c_k(T) ≅ k/√(4πT³) exp (-k²/4T) —– (6)

and from equation (5),

N(T) ≅ (πT)^−1/2 —– (7)

Equation (7) also implies t ≅ 2/3 √(πT³) which gives

N(T) ≅ (2/3πt)^1/3 —– (8)

and

c_k(t) ≅ k/3t exp [-(π/144)^1/3 k²/t^2/3] —– (9)

Note that this latter expression may be written in the scaling form c_k(t) ∝ N²xe^−x2, with the scaling variable x ∝ kN. One can also confirm that the scaling solution represents the basin of attraction for almost all exact solutions. Indeed, for any initial condition with c_k(0) decaying faster than k⁻², the system reaches the scaling limit c_k(t) ∝ N²xe^−x2. On the other hand, if c_k(0) ∼ k^−1−α, with 0 < α < 1, such an initial state converges to an alternative scaling limit which depends on α. These solutions exhibit a slower decay of the total density, N ∼ t^−α/(1+α), while the scaling form of the wealth distribution is

c_k(t) ∼ N^2/αC_α(x), x ∝ kN^1/α —– (10)

with the scaling function

C_α(x) = e^−x2 ∫₀^∞ du e^−u2 sinh(2ux)/u^1+α —– (11)

Evaluating the integral by the Laplace method gives an asymptotic distribution which exhibits the same x^−1−α as the initial distribution. This anomalous scaling in the solution to the diffusion equation is a direct consequence of the extended initial condition. This latter case is not physically relevant, however, since the extended initial distribution leads to a divergent initial wealth density.