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

circle24

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.

 

Untitled

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,

Untitled

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. 

Untitled

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.

Untitled

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.

Finsler Space as a Locally Minkowskian Space: Caught Between Curvature and Torsion Tensors.

main

The extension of Riemannian “point”-space {xi} into a “line-space” {xi, dxi} make things clearer but not easier: how do you explain to a physicist a geometry supporting at least 3 curvature tensors and five torsion tensors? Not to speak of its usefulness for physics! Fortunately, the “impenetrable forest” by now has become a real, enjoyable park: through the application of the concepts of fibre bundle and non-linear connection. The different curvatures and torsion tensors result from vertical and horizontal parts of geometric objects in the tangent bundle, or in the Finsler bundle of the underlying manifold.

In essence, Finsler geometry is analogous to Riemannian geometry: there, the tangent space in a point p is euclidean space; here, the tangent space is just a normed space, i.e., Minkowski Space. Put differently: A Finsler metric for a differentiable manifold M is a map that assigns to each point x ∈ M a norm on the tangent space TxM. When referred to the almost exclusive use of methods from Riemannian geometry it means that this norm is demanded to derive from the length of a smooth path γ : [a, b] → M defined by ∫ab ∥ dγ(t)/dt ∥ dt. Then Finsler space becomes an example for the class of length spaces.

Starting from the length of the curve,

dγ(p, q):= ∫pq Lx(t), dx(t)/dt dt

the variational principle δdγ(p, q) = 0 leads to the Euler-Lagrange equation

d/dt(∂L/∂x ̇i) – ∂L/∂xi = 0

which may be rewritten into

d2xi/dt2 + 2Gi(xl, x ̇m) = 0

with Gi(xl, x ̇m) = 1/4gkl(-∂L/∂xl + ∂2L/∂xl∂x ̇m), and 2gik = ∂2L/∂x ̇l∂x ̇m, gilgjl = δij. The theory then is developed from the Lagrangian defined in this way. This involves an important object Nil := ∂Gi/∂yl, the geometrical meaning of which is a non-linear connection.

In general, a Finsler structure L(x, y) with y := dx(t))/dt = x ̇ and homogeneous degree 1 in y is introduced, from which the Finsler metric follows as:

fij = fji = ∂(1/2L2)/∂yi∂yj, fijyiyj = L2, yl∂L/∂yl = L, fijyj = L∂L/∂yi

A further totally symmetric tensor Cijk ensues:

Cijk := ∂(1/2L2)/∂yi∂yj∂yk

which will be interpreted as a torsion tensor. As an example of a Finsler metric is the Randers metric.

L(x.y) = bi(x)yi + √(aij(x)yiyj)

The Finsler metric following is

fik = bibk + aik + 2b(iak)lyˆl − aillakmm(bnn)

with yˆk := yk(alm(x)ylym)−1/2. Setting aij = ηij, yk = x ̇k, and identifying bi with the electromagnetic 4-potential eAi leads back to the Lagrangian for the motion of a charged particle.

In this context, a Finsler space thus is called a locally Minkowskian space if there exists a coordinate system, in which the Finsler structure is a function of yi alone. The use of the “element of support” (xi, dxk ≡ yk) essentially amounts to a step towards working in the tangent bundle TM of the manifold M.

Beginning of Matter, Start to Existence Itself

__beginning_of_matter___by_ooookatioooo

When the inequality

μ+3p/c2 >0 ⇔ w > −1/3

is satisfied, one obtains directly from the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ +3p/c2) + Λ

the Friedmann-Lemaître (FL) Universe Singularity Theorem, which states that:

In a FL universe with Λ ≤ 0 and μ + 3p/c2 > 0 at all times, at any instant t0 when H0 ≡ (S ̇/S)0 > 0 there is a finite time t: t0 − (1/H0) < t < t0, such that S(t) → 0 as t → t; the universe starts at a space-time singularity there, with μ → ∞ and T → ∞ if μ + p/c2 > 0.

This is not merely a start to matter – it is a start to space, to time, to physics itself. It is the most dramatic event in the history of the universe: it is the start of existence of everything. The underlying physical feature is the non-linear nature of the Einstein’s Field Equations (EFE): going back into the past, the more the universe contracts, the higher the active gravitational density, causing it to contract even more. The pressure p that one might have hoped would help stave off the collapse makes it even worse because (consequent on the form of the EFE) p enters algebraically into the Raychaudhuri equation with the same sign as the energy density μ. Note that the Hubble constant gives an estimate of the age of the universe: the time τ0 = t0 − t since the start of the universe is less than 1/H0.

This conclusion can in principle be avoided by a cosmological constant, but in practice this cannot work because we know the universe has expanded by at least a ratio of 11, as we have seen objects at a redshift 6 of 10, the cosmological constant would have to have an effective magnitude at least 113 = 1331 times the present matter density to dominate and cause a turn-around then or at any earlier time, and so would be much bigger than its observed present upper limit (of the same order as the present matter density). Accordingly, no turnaround is possible while classical physics holds. However energy-violating matter components such as a scalar field can avoid this conclusion, if they dominate at early enough times; but this can only be when quantum fields are significant, when the universe was at least 1012 smaller than at present.

Because Trad ∝ S−1, a major conclusion is that a Hot Big Bang must have occurred; densities and temperatures must have risen at least to high enough energies that quantum fields were significant, at something like the GUT energy. The universe must have reached those extreme temperatures and energies at which classical theory breaks down.

Hyperbolic Brownian Sheet, Parabolic and Elliptic Financials. (Didactic 3)

Fig-3-Realizations-of-the-fractional-Brownian-sheet-on-the-plane-with-graph-dimensions

Financial and economic time series are often described to a first degree of approximation as random walks, following the precursory work of Bachelier and Samuelson. A random walk is the mathematical translation of the trajectory followed by a particle subjected to random velocity variations. The analogous physical system described by SPDE’s is a stochastic string. The length along the string is the time-to-maturity and the string configuration (its transverse deformation) gives the value of the forward rate f(t,x) at a given time for each time-to-maturity x. The set of admissible dynamics of the configuration of the string as a function of time depends on the structure of the SPDE. Let us for the time being restrict our attention to SPDE’s in which the highest derivative is second order. This second order derivative has a simple physical interpretation : the string is subjected to a tension, like a piano chord, that tends to bring it back to zero transverse deformation. This tension forces the “coupling” among different times-to-maturity so that the forward rate curve is at least continuous. In principle, the most general formulation would consider SPDE’s with terms of arbitrary derivative orders. However, it is easy to show that the tension term is the dominating restoring force, when present, for deformations of the string (forward rate curve) at long “wavelengths”, i.e. for slow variations along the time-to-maturity axis. Second order SPDE’s are thus generic in the sense of a systematic expansion.

In the framework of second order SPDE’s, we consider hyperbolic, parabolic and elliptic SPDE’s, to characterize the dynamics of the string along two directions : inertia or mass, and viscosity or subjection to drag forces. A string that has “inertia” or, equivalently, “mass” per unit length, along with the tension that keeps it continuous, is characterized by the class of hyperbolic SPDE’s. For these SPDE’s, the highest order derivative in time has the same order as the highest order derivative in distance along the string (time-to-maturity). As a consequence, hyperbolic SPDE’s present wave-like solutions, that can propagate as pulses with a “velocity”. In this class, we find the so-called “Brownian sheet” which is the direct generalization of Brownian motion to higher dimensions, that preserves continuity in time-to-maturity. The Brownian sheet is the surface spanned by the string configurations as time goes on. The Brownian sheet is however non-homogeneous in time-to-maturity.

If the string has no inertia, its dynamics are characterized by parabolic SPDE’s. These stochastic processes lead to smoother diffusion of shocks through time, along time-to-maturity. Finally, the third class of SPDE’s of second-order, namely elliptic partial differential equations. Elliptic SPDE’s give processes that are differentiable both in x and t. Therefore, in the strict limit of continuous trading, these stochastic processes correspond to locally riskless interest rates.

The general form of SPDE’s reads

A(t,x) ∂2f(t,x)/∂t2 + 2B(t,x) ∂2f(t,x)/∂t∂x + C(t,x) ∂2f(t,x)/∂x2 = F(t,x,f(t,x), ∂f(t,x)/∂t, ∂f(t,x)/∂x, S) —– (1)

where f (t, x) is the forward rate curve. S(t, x) is the “source” term that will be generally taken to be Gaussian white noise η(t, x) characterized by the covariance

Cov η(t, x), η(t′, x′) = δ(t − t′) δ(x − x′) —– (2)

where δ denotes the Dirac distribution. Equation (1) is the most general second-order SPDE in two variables. For arbitrary non-linear terms in F, the existence of solutions is not warranted and a case by case study must be performed. For the cases where F is linear, the solution f(t,x) exists and its uniqueness is warranted once “boundary” conditions are given, such as, for instance, the initial value of the function f(0,x) as well as any constraints on the particular form of equation (1).

Equation (1) is defined by its characteristics, which are curves in the (t, x) plane that come in two families of equation :

Adt = (B + √(B2 − AC))dx —– (3)

Adt = (B − √(B2 − AC))dx —– (4)

These characteristics are the geometrical loci of the propagation of the boundary conditions.

Three cases must be considered.

• When B2 > AC, the characteristics are real curves and the corresponding SPDE’s are called “hyperbolic”. For such hyperbolic SPDE’s, the natural coordinate system is formed from the two families of characteristics. Expressing (1) in terms of these two natural coordinates λ and μ, we get the “normal form” of hyperbolic SPDE’s :

2f/∂λ∂μ = P (λ,μ) ∂f/∂λ +Q (λ,μ) ∂f/∂μ + R (λ,μ)f + S(λ,μ) —– (5)

The special case P = Q = R = 0 with S(λ,μ) = η(λ,μ) corresponds to the so-called Brownian sheet, well studied in the mathematical literature as the 2D continuous generalization of the Brownian motion.

• When B2 = AC, there is only one family of characteristics, of equation

Adt = Bdx —– (6)

Expressing (1) in terms of the natural characteristic coordinate λ and keeping x, we get the “normal form” of parabolic SPDE’s :

2f/∂x2 = K (λ,μ)∂f/∂λ +L (λ,μ)∂f/∂x +M (λ,μ)f + S(λ,μ) —– (7)

The diffusion equation, well-known to be associated to the Black-Scholes option pricing model, is of this type. The main difference with the hyperbolic equations is that it is no more invariant with respect to time-reversal t → −t. Intuitively, this is due to the fact that the diffusion equation is not conservative, the information content (negentropy) continually decreases as time goes on.

• When B2 < AC, the characteristics are not real curves and the corresponding SPDE’s are called “elliptic”. The equations for the characteristics are complex conjugates of each other and we can get the “normal form” of elliptic SPDE’s by using the real and imaginary parts of these complex coordinates z = u ± iv :

2f/∂u2 + ∂2f/∂v2 = T ∂f/∂u + U ∂f/∂v + V f + S —– (8)

There is a deep connection between the solution of elliptic SPDE’s and analytic functions of complex variables.

Hyperbolic and parabolic SPDE’s provide processes reducing locally to standard Brownian motion at fixed time-to-maturity, while elliptic SPDE’s give locally riskless time evolutions. Basically, this stems from the fact that the “normal forms” of second-order hyperbolic and parabolic SPDE’s involve a first-order derivative in time, thus ensuring that the stochastic processes are locally Brownian in time. In contrast, the “normal form” of second-order elliptic SPDE’s involve a second- order derivative with respect to time, which is the cause for the differentiability of the process with respect to time. Any higher order SPDE will be Brownian-like in time if it remains of order one in its time derivatives (and higher-order in the derivatives with respect to x).