共查询到20条相似文献,搜索用时 546 毫秒
1.
<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. 相似文献
2.
To capture the subdiffusive characteristics of financial markets, the subordinated process, directed by the inverse α-stale subordinator Sα(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 α-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 by Laplace transform technique. Finally, using 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. 相似文献
3.
Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases
V. S. Kirchanov 《Russian Physics Journal》2009,52(4):343-353
Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with
0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic
fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is
simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been
obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps
over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations,
anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping. 相似文献
4.
B. Mazzi F. Okkels J.C. Vassilicos 《The European Physical Journal B - Condensed Matter and Complex Systems》2002,28(2):243-251
We present a shell-model of fractal induced turbulence which predicts that structure function scaling exponents decrease in
absolute value as the fractal dimension of the turbulence-inducing fractal object increases. This qualitative prediction is
in agreement with laboratory measurements. Finer details of the fractal induced turbulence statistics and dynamics depend
on the fractal force's phases, i.e. on the detailed construction of the fractal stirrer. In a case of deterministic forcing phases, a critical fractal dimension
exists below which the average rate of inter-scale energy transfer <T
n> is a decreasing function of the wavenumber kn and the structure function scaling exponents take close to Kolmogorov values. Above this critical fractal dimension, <T
n> is an increasing function of kn and the structure function scaling exponents deviate significantly from Kolmogorov values.
Received 25 June 2001 / Received in final form 5 April 2002 Published online 19 July 2002 相似文献
5.
Chen W 《Chaos (Woodbury, N.Y.)》2006,16(2):023126
This study makes the first attempt to use the 23-order fractional Laplacian modeling of Kolmogorov -53 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. Nonlinear inertial interactions and molecular Brownian diffusivity are considered to be the bifractal mechanism behind multifractal scaling of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 23-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we conjecture a fractional Reynolds equation, underlying fractal spacetime structures of Levy 23 stable distribution and the Kolmogorov scaling at inertial scales. The new perspective of this study is that the fractional calculus is an effective approach to modeling the chaotic fractal phenomena induced by nonlinear interactions. 相似文献
6.
Advection and dispersion in time and space 总被引:2,自引:0,他引:2
Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly. 相似文献
7.
8.
Jin-Rong Liang Jun Wang Long-Jin Lǔ Hui Gu Wei-Yuan Qiu Fu-Yao Ren 《Journal of statistical physics》2012,146(1):205-216
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. 相似文献
9.
This paper is devoted to the study of a fractional version of non-linear
\mathfrakMn(t)\mathfrak{M}^{\nu}(t), t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential
equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo.
We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions
and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions
of their classical counterparts with the random time process T
2ν(t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion
equation. 相似文献
10.
This paper presents finite-velocity random motions driven by fractional Klein–Gordon equations of order $\alpha \in (0,1]$ . A key tool in the analysis is played by the McBride’s theory which converts fractional hyper-Bessel operators into Erdélyi–Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein–Gordon equation. The distribution of the fractional telegraph process for $\alpha = 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein–Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,\alpha )$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,\alpha )$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of direction. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$ -dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler–Poisson–Darboux equation to which our theory applies. 相似文献
11.
《Physica A》1988,153(1):57-66
Persistent diffusion in one dimension, in which the velocity of the diffusing particle is a dichotomic Markov process, is considered. The flow is non-Markovian, but the position and the velocity together constitute a Markovian diffusion process. We solve the coupled forward Kolmogorov equations and the coupled backward Kolmogorov equations with appropriate initial conditions, to establish a generalized (matrix) form of the renewal equation connecting the probability densities and first passage time distributions for persistent diffusion. 相似文献
12.
The blinking of quantum dots under the action of laser radiation is described based on a model of a binary (two-state) renewal
process with on (fluorescent) and off (non fluorescent) states. The T
on and T
off sojourn times in the on and off states are random and power-law distributed with exponents 0 < α < 1 and 0 < β < 1; the averages
of the on and off times are infinite. As a consequence of this, the Gaussian statistics is inapplicable and the process is
described using a more general statistics. An equation for the density of distribution p(t
on|t) of the total on time during the observation time t is derived that contains derivatives of fractional orders α and β. A solution to this equation is found in terms of fractional
stable distributions. The Poisson transform of the density p(t
on|t) leads to the photon counting distribution and determines the fluorescence statistics. It is demonstrated that, if a blinking
process with exponents α < β is implemented, then, at fairly long times, the on time will considerably prevail over the off
time, i.e., blinking will be suppressed. This behavior is evidenced by the types of distributions of the total fluorescence
time, the decay of relative fluctuations, and the Monte Carlo simulated trajectories of the process. 相似文献
13.
R. T. Sibatov 《JETP Letters》2011,93(9):503-507
Distributions of the conductance G of a long quantum wire with the fractal distribution of barriers have been obtained in the successive incoherent tunneling
regime. The asymptotic behavior (in the limit L → ∞) of moments 〈G
k
(L)〉, average power of the shot noise 〈S(L)〉, and Fano factor agree with the results of the work [C. W. J. Beenakker et al., Phys. Rev. B 79, 024204 (2009)], and the distributions themselves describe well the Monte Carlo simulation results. The equation that has
been obtained for the distributions of the resistance and conductance agrees with the recent fractional differential generalization
of the Dorokhov-Mello-Pereyra-Kumar equation for the quasi-one-dimensional multichannel disordered semiconductors with a self-similar
distribution of scatterers. 相似文献
14.
SACHIN B BHALEKAR 《Pramana》2013,81(2):215-224
In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = a f\left(y(t-\tau)\right) - b y(t)$ , where D α is a Caputo fractional derivative of order 0?<?α?≤?1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay. 相似文献
15.
I. P. Guk 《Technical Physics》1998,43(4):353-357
Analogs of the Lagrange equation for particles evolving in a space of fractal dimension are obtained. Two cases are considered:
1) when the space is formed by a set of material points (a so-called fractal continuum), and 2) when the space is a true fractal.
In the latter case the fractional integrodifferential formalism is utilized, and a new principle for devising a fractal theory,
viz., a generalized principle of least action, is proposed and used to obtain the corresponding Lagrange equation. The Lagrangians
for a free particle and a closed system of interacting particles moving in a fractal continuum are derived.
Zh. Tekh. Fiz. 68, 7–11 (February 1990) 相似文献
16.
The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function \(f_j(r,t)\) of a particle moving on a substrate with time delays \(\tau _j\). Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability \(P_j\) is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, \(P_j \propto f_j^{\nu - 1}\), the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, \(P_j \propto \tau _j^{-1-\alpha }\) (with \(0< \alpha < 2\)), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents. 相似文献
17.
V. V. Uchaikin 《Technical Physics》2004,49(7):929-932
The one-dimensional walk of a particle executing instantaneous jumps between the randomly distributed “atoms” at which it
resides for a random time is considered. The random distances between the neighboring atoms and the time intervals between
jumps are mutually independent. The asymptotic (t → ∞) behavior of this process is studied in connection with the problem of interpretation of the generalized fractional diffusion
equation (FDE). It is shown that the interpretation of the FDE as the equation describing the walk (diffusion) in a fractal
medium is incorrect in the model problem considered. The reason is that the FDE implies that the consecutive jumps (fractal
walk) are independent, whereas they are correlated in the case under consideration: a particle leaving an atom in the direction
opposite to the preceding direction traverses the same path until arriving at the atom. 相似文献
18.
I. Montvay 《Zeitschrift fur Physik C Particles and Fields》1980,4(1):45-52
A unified picture of both soft and hard hadronic collisions is suggested. The basis idea is to use Regge trajectories of the type $$\alpha (t) = \alpha (0) - \gamma ln (1 + \beta \sqrt {t_0 - t} )$$ in dual models with Mandelstam analyticity. The idea is applied to elastic proton-proton scattering to derive kinematical boundaries of the asymptotic (Regge-t R and scaling-S sc.,t sc.) regimes, to fix the angular dependence in the scaling-and thet-dependence in the Regge domain of the scattering amplitude and to interpolate between the two asymptotic domains. 相似文献
19.
20.
We consider the solution of the equation r(t) = W(r(t)), r(0) = r
0 > 0 where W(⋅) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α∈ (0,1). We show that for almost all realizations of
W(⋅) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to
the left of r
0, depending on whether W(r
0) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying
the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast
to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property
of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m.,
see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys.
205 (1999) 97–111. 相似文献