52 research outputs found
Fluid limit of the continuous-time random walk with general LĂ©vy 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 Ï(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 LĂ©vy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order ÎČ in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general LĂ©vy stochastic processes in the LĂ©vy-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 LĂ©vy 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 Ïc ⌠λ âα/ÎČ where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Greenâs function) of the truncated fractional equation exhibits a transition from algebraic decay for t > Ïc
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 LĂ©vy 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 LĂ©vy 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-BlackâScholes; LĂ©vy-stable processes; FMLS; KoBoL; CGMY; Fractional calculus; RiemannâLiouville fractional derivative; Barrier options; Down-and-out; Up-and-out; Double knock-out;
On the fluid limit of the continuous-time random walk with general LĂ©vy 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 Ï(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 LĂ©vy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order ÎČ in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general LĂ©vy stochastic processes in the LĂ©vy-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 LĂ©vy 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 Ïc ⌠λ âα/ÎČ where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Greenâs function) of the truncated fractional equation exhibits a transition from algebraic decay for t << Ïc to stretched Gaussian decay for t >> Ïc
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 LĂ©vy 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 LĂ©vy 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; LĂ©vy-Stable processes; FMLS; KoBoL; CGMY; Fractional calculus; Riemann-Liouville fractional derivative; Barrier options; Down-and-out; Up-and-out; Double knock-out;
Multiscale statistical analysis of coronal solar activity
Multi-filter images from the solar corona are used to obtain temperature maps
which are analyzed using techniques based on proper orthogonal decomposition
(POD) in order to extract dynamical and structural information at various
scales. Exploring active regions before and after a solar flare and comparing
them with quiet regions we show that the multiscale behavior presents distinct
statistical properties for each case that can be used to characterize the level
of activity in a region. Information about the nature of heat transport is also
be extracted from the analysis.Comment: 24 pages, 18 figure
On the fluid limit of the continuous-time random walk with general LĂ©vy 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 Ï(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 LĂ©vy stable processes, the
fluid limit leads to the fractional diffusion equation of order α in space and order ÎČ in time. However,
these are two special cases of a wider class of models. Here we consider the CTRW for the most
general LĂ©vy stochastic processes in the LĂ©vy-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 LĂ©vy 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 Ïc ⌠λ âα/ÎČ where 1/λ is the truncation length scale.
The asymptotic behavior of the propagator (Greenâs function) of the truncated fractional equation
exhibits a transition from algebraic decay for t > Ï
Anomalous losses of energetic particles in fusion plasmas
The confinement of energetic particles (EP) in nuclear fusion devices is studied in the presence of an oscillating electrostatic potential and an axi-symmetric magnetic equilibrium. Despite the poloidal and toroidal symmetries, radial transport is observed. The transport leads to an algebraic decaying loss time, which is at odds with diffusive transport that predicts a faster exponential decay. A dynamical explanation of the observed anomalous loss time decay is presented. It is shown that transport is characterized by LĂ©vy flights that lead to super-diffusive poloidal transport and asym-metric non-Gaussian (LĂ©vy) probability distribution functions of displacements. The anomalous scaling exponents are shown to be consistent with the Continuous Time Random Walk (CTRW) theory. The results imply that EP might efficiently slowed down by the thermal population before leaving the system. Also, the asymmetric transport can potentially lead to a net torque
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