In this Letter we introduce a generalization of the Lorenz dynamical system using fractional derivatives. Thus, the system can have an effective noninteger dimension Sigma defined as a sum of the orders of all involved derivatives. We found that the system with Sigma<3 can exhibit chaotic behavior. A striking finding is that there is a critical value of the effective dimension Sigma(cr), under which the system undergoes a transition from chaotic dynamics to regular one.

Chaotic dynamics of the fractional Lorenz system.

The Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodinger equation. The resulting Hamiltonian is found to be non-Hermitian and nonlocal in time. The resulting wave functions are thus not invariant under time reversal. The time fractional Schrodinger equation is solved for a free particle and for a potential well. Probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative. New identities for the Mittag–Leffler function are also found and presented in an Appendix.

Time fractional Schrödinger equation

This study examines the two most attractive characteristics, memory and chaos, in simulations of financial systems. A fractional-order financial system is proposed in this study. It is a generalization of a dynamic financial model recently reported in the literature. The fractional-order financial system displays many interesting dynamic behaviors, such as fixed points, periodic motions, and chaotic motions. It has been found that chaos exists in fractional-order financial systems with orders less than 3. In this study, the lowest order at which this system yielded chaos was 2.35. Period doubling and intermittency routes to chaos in the fractional-order financial system were found.

Nonlinear dynamics and chaos in a fractional-order financial system

In this paper, the first integral method is used to construct exact solutions of the time-fractional Wu---Zhang system. Fractional derivatives are described by conformable fractional derivative. This method is based on the ring theory of commutative algebra. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.

The first integral method for Wu---Zhang system with conformable time-fractional derivative

In this paper we numerically investigate the chaotic behaviors of the fractional-order Chen system. A striking finding is that the lowest order for this system to have chaos is 0.3, which is the lowest-order chaotic system among all the found chaotic systems to date.

A note on the fractional-order Chen system

A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.

Fractional Quantum Mechanics

A new extension of a fractality concept in nancial mathematics has been developed. We have introduced a new fractional Langevin-type stochastic dierential equation that diers from the standard Langevin equation: (i) by replacing the rst-order derivative with respect to time by the fractional derivative of order ; and (ii) by replacing \white noise" Gaussian stochastic force by the generalized \shot noise", each pulse of which has a random amplitude with the -stable Levy distribution. As an application of the developed fractional non-Gaussian dynamical approach the expression for the probability distribution function (pdf) of the returns has been established. It is shown that the obtained fractional pdf ts well the central part and the tails of the empirical distribution of S&P 500 returns. c 2000 Elsevier Science B.V. All rights reserved.

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Fractional market dynamics

A new exactly solvable option pricing model has been introduced and elaborated. It is assumed that a stock price follows a Geometric shot noise process. An arbitrage-free integro-differential option pricing equation has been obtained and solved. The new Greeks have been analytically calculated. It has been shown that in diffusion approximation the developed option pricing model incorporates the well-known Black–Scholes equation and its solution. The stochastic dynamic origin of the Black–Scholes volatility has been uncovered. To model the observed market stock price patterns consisting of high frequency small magnitude and low frequency large magnitude jumps, the superposition of two Geometric shot noises has been implemented. A new generalized option pricing equation has been obtained and its exact solution was found. Merton’s jump–diffusion formula for option price was recovered in diffusion approximation. Despite the non-Gaussian nature of probability distributions involved, the new option pricing model has the same degree of analytical tractability as the Black–Scholes model and the Merton jump–diffusion model. This attractive feature allows one to derive exact formulas to value options and option related instruments in the market with jump-like price patterns.

Valuing options in shot noise market

A unified analytical pricing framework with involvement of the shot noise random process has been introduced and elaborated. Two exactly solvable new models have been developed. The first model has been designed to value options. It is assumed that asset price stochastic dynamics follows a Geometric Shot Noise motion. A new arbitrage-free integro-differential option pricing equation has been found and solved. The put-call parity has been proved and the Greeks have been calculated. Three additional new Greeks associated with market model parameters have been introduced and evaluated. It has been shown that in diffusion approximation the developed option pricing model incorporates the well-known Black-Scholes equation and its solution. The stochastic dynamic origin of the Black-Scholes volatility has been uncovered. The new option pricing model has been generalized based on asset price dynamics modeled by the superposition of Geometric Brownian motion and Geometric Shot Noise. To model stochastic dynamics of a short term interest rate, the second model has been introduced and developed based on Langevin type equation with shot noise. A new bond pricing formula has been obtained. It has been shown that in diffusion approximation the developed bond pricing formula goes into the well-known Vasicek solution. The stochastic dynamic origin of the long-term mean and instantaneous volatility of the Vasicek model has been uncovered. A generalized bond pricing model has been introduced and developed based on short term interest rate stochastic dynamics modeled by superposition of a standard Wiener process and shot noise. Despite the non-Gaussianity of probability distributions involved, all newly elaborated models have the same degree of analytical tractability as the Black-Scholes model and the Vasicek model.

Nick Laskin

Papers

Fractional Quantum Mechanics

Fractional market dynamics

Fractional Quantum Mechanics

Valuing options in shot noise market

New Pricing Framework: Options and Bonds