Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime

In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X _α,H (t)=X _H (S _α (t)), 0<α,H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equation and Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X _α,H (t) and the corresponding Black-Scholes formula for the fair prices of European option.     

Black-Scholes Formula in Subdiffusive Regime

In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained results with the classical ones.     

A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes

Anomalous diffusion processes are often classified by their mean square displacement. If the mean square displacement grows linearly in time, the process is considered classical. If it grows like t ^β with β<1 or β>1, the process is considered subdiffusive or superdiffusive, respectively. Processes with infinite mean square displacement are considered superdiffusive. We begin by examining the ways in which power-law mean square displacements can arise; namely via non-zero drift, nonstationary increments, and correlated increments. Subsequently, we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present. Finally, we introduce an alternative classification scheme based on renormalization groups. This scheme classifies processes with stationary increments such as Brownian motion and fractional Brownian motion in the same groups as the mean square displacement scheme, but does a better job of classifying processes with nonstationary increments and/or processes with infinite second moments such as α-stable Lévy motion. A numerical approach to analyzing data based on the renormalization group classification is also presented.     

Two competing species in super-diffusive dynamical regimes

The dynamics of two competing species within the framework of the generalized Lotka-Volterra equations, in the presence of multiplicative α-stable Lévy noise sources and a random time dependent interaction parameter, is studied. The species dynamics is characterized by two different dynamical regimes, exclusion of one species and coexistence of both, depending on the values of the interaction parameter, which obeys a Langevin equation with a periodically fluctuating bistable potential and an additive α-stable Lévy noise. The stochastic resonance phenomenon is analyzed for noise sources asymmetrically distributed. Finally, the effects of statistical dependence between multiplicative noise and additive noise on the dynamics of the two species are studied.     

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

Continuous time Black-Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(S_α(t)), 0<α<1, here dX(τ)=μX(τ)(dτ)^2H+σX(τ)dB_H(τ), as a model of asset prices, which captures the subdiffusive characteristic of financial markets. We find the corresponding subdiffusive Black-Scholes equation and the Black-Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs.     

Crossover from Rouse dynamics to the α-relaxation in poly (vinyl ethylene)

By means of neutron spin echo (NSE) we have explored the dynamics of poly(vinyl ethylene) on length scales covering Rouse dynamics and below. The results establish the simultaneous existence of a generic sublinear diffusion regime which underlies the α-process in addition to the Rouse process. Both regimes are separated by a well-defined dynamic crossover. From that the size of the Gaussian blobs making up the Rouse model is determined directly. The glassy dynamics may thus be identified with subdiffusive motions occurring within these Gaussian blobs.     

A precursor of market crashes: Empirical laws of Japan's internet bubble

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as X(t) = S(t)/S(0), where S(0), is the stock price for an onset time of the bubble. We selected approximately 3200 stocks traded on the Japanese Stock Exchange, and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which internet Bubble formed and crashed in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, P(S>x) ∼x^-α , with an exponent that moves over time. Furthermore we found that as the power-law exponents α approached two, the bubble burst. It is reasonable to suppose that it indicates that internet bubble is about to burst.     

Brownian particles in random and quasicrystalline potentials: How they approach the equilibrium

We study the Brownian motion of an ensemble of single colloidal particles in a random square and a quasicrystalline potential when they start from non-equlibrium. For both potentials, Brownian dynamics simulations reveal a widespread subdiffusive regime before the diffusive long-time limit is reached in thermal equilibrium. We develop a random trap model based on a distribution for the depths of trapping sites that reproduces the results of the simulations in detail. Especially, it gives analytic formulas for the long-time diffusion constant and the relaxation time into the diffusive regime. Aside from detailed differences, our work demonstrates that quasicrystalline potentials can be used to mimic aspects of random potentials.     

Anomalous Pulsation

Subordinating regular diffusion – namely, Brownian motion – to random time flows generated by Lévy noises may result in anomalous diffusion. Motivated by this phenomena, and by the recent interest in the phenomena of blinking in various physical systems, we explore the subordination of regular stochastic pulsation – namely, Poisson process – to random time flows generated by Lévy noises. We show that such subordination may yield, analogous to the case of diffusion, anomalous pulsation. Anomalous pulsation displays the following anomalous behaviors, which are impossible in the case of regular pulsation: (i) simultaneous emission of multiple pulses; (ii) non-linear local pulsation rates; (iii) clustering of pulsation epochs.     

The Hopfield Model on a Sparse Erdös-Renyi Graph

We analyze the storage capacity of the Hopfield model on a sparse G(N,p) random graph. We show that it is proportional to αpN in the entire regime where the corresponding random graph is asymptotically connected and for all value of α≤α _c =0.03.     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

A multidimensional subdiffusion model： An arbitrage-free market

<正>To capture the subdiffusive characteristics of financial markets,the subordinated process,directed by the inverse Q-stale subordinator S_α(t) for 0 <α< 1,has been employed as the model of asset prices.In this article,we introduce a multidimensional subdiffusion model that has a bond and K correlated stocks.The stock price process is a multidimensional subdiffusion process directed by the inverse Q-stable subordinator.This model describes the period of stagnation for each stock and the behavior of the dependency between multiple stocks.Moreover,we derive the multidimensional fractional backward Kolmogorov equation for the subordinated process using the Laplace transform technique.Finally, using a martingale approach,we prove that the multidimensional subdiffusion model is arbitrage-free,and also gives an arbitrage-free pricing rule for contingent claims associated with the martingale measure.     

Low-frequency fluctuations in stochastic processes with a 1/<Emphasis Type="Italic">f</Emphasis><Superscript>α</Superscript> spectrum

The results of numerical analysis of the Brownian movement of a particle in the force field of the potential corresponding to interacting subcritical and supercritical phase transitions are considered. If the white noise intensity corresponds to the critical intensity of the noise-induced transition, the system of stochastic differential equations describes random steady-state processes with fluctuation power spectra inversely proportional to frequency f, S(f) ∼ 1/f ^α, where exponent α varies in the interval 0.8 ≤ α ≤ 1.8. Exponent β of distribution function P(τ) ∼ τ^−β for the duration of low-frequency extremal fluctuations, which are analogous to avalanches considered in the models of self-organized criticality in many respects, varies between the same limits. It is shown that exponents α and β are connected through the relation α + β = 2.     

Time-changed geometric fractional Brownian motion and option pricing with transaction costs

This paper deals with the problem of discrete time option pricing by a fractional subdiffusive Black–Scholes model. The price of the underlying stock follows a time-changed geometric fractional Brownian motion. By a mean self-financing delta-hedging argument, the pricing formula for the European call option in discrete time setting is obtained.     

On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data

Consider an inviscid Burgers equation whose initial data is a Lévy α-stable process Z with α>1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a partially negative answer to a Conjecture of Janicki and Woyczynski (J. Stat. Phys. 86(1–2):277–299, 1997). Along the way, we contradict a recent Conjecture of Z. Shi () about the lower tails of integrated stable processes.     

Correlation Inequalities for Interacting Particle Systems with Duality

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other—a process which we call here the symmetric inclusion process (SIP)—or repel each other—a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction,—the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.     

Ratchet transport with subdiffusion

We introduce a model which combines the subdiffusive dynamics and the ratchet effect. Using a subordination ideology, we show that the resulting directed transport is sublinear, 〈 x(t)〉≃Jt^β, β< 1. The proposed model may be relevant to a phenomenon of saltatory microbiological motility.