Clustering in a model with repulsive long-range interactions

A striking clustering phenomenon in the antiferromagnetic Hamiltonian Mean-Field model has been previously reported. The numerically observed bicluster formation and stabilization is here fully explained by a non linear analysis of the Vlasov equation.Comment: 8 pages, 5 Fig

Local and nonlocal parallel heat transport in general magnetic fields

A novel approach that enables the study of parallel transport in magnetized plasmas is presented. The method applies to general magnetic fields with local or nonlocal parallel closures. Temperature flattening in magnetic islands is accurately computed. For a wave number $k$, the fattening time scales as $\chi_{\parallel} \tau \sim k^{-\alpha}$ where $\chi$ is the parallel diffusivity, and $\alpha=1$ ($\alpha=2$) for non-local (local) transport. The fractal structure of the devil staircase temperature radial profile in weakly chaotic fields is resolved. In fully chaotic fields, the temperature exhibits self-similar evolution of the form $T=(\chi_{\parallel} t)^{-\gamma/2} L \left[ (\chi_{\parallel} t)^{-\gamma/2} \delta \psi \right]$, where $\delta \psi$ is a radial coordinate. In the local case, $f$ is Gaussian and the scaling is sub-diffusive, $\gamma=1/2$. In the non-local case, $f$ decays algebraically, $L (\eta) \sim \eta^{-3}$, and the scaling is diffusive, $\gamma=1$

Fluid limit of the continuous-time random walk with general Lévy jump distribution functions.

The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions ψ(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times, and algebraic decaying jump distributions, corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order β in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integro-differential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as τc ∼ λ −α/β where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Green’s function) of the truncated fractional equation exhibits a transition from algebraic decay for t > τc

Fractional diffusion models of option prices in markets with jumps.

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derivedFractional-Black-Scholes; Lévy-Stable processes; FMLS; KoBoL; CGMY; Fractional calculus; Riemann-Liouville fractional derivative; Barrier options; Down-and-out; Up-and-out; Double knock-out;