اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLevy stable two-sided distributions: exact and explicit densities for asymmetric case
282
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study the one-dimensional Levy stable density distributions g(alpha, beta; x) for -infty < x < infty, for rational values of index alpha and the asymmetry parameter beta: alpha = l/k and beta = (l - 2r)/k, where l, k and r are positive integers such that 0 < l/k < 1 for 0 <= r <= l and 1 < l/k <= 2 for l-k <= r <= k. We treat both symmetric (beta = 0) and asymmetric (beta neq 0) cases. We furnish exact and explicit forms of g(alpha, beta; x) in terms of known functions for any admissible values of alpha and beta specified by a triple of integers k, l and r. We reproduce all the previously known exact results and we study analytically and graphically many new examples. We point out instances of experimental and statistical data that could be described by our solutions.
قيم البحث
اقرأ أيضاً
Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can
be modelled with Levy stable distributions. We review comprehensively the derivation of an analytical expression for the spectra of covariance matrices approximated by free Levy stable random variables and validate it by Monte Carlo simulation.
The functional method to derive the fractional Fokker-Planck equation for probability distribution from the Langevin equation with Levy stable noise is proposed. For the Cauchy stable noise we obtain the exact stationary probability density function
of Levy flights in different smooth potential profiles. We find confinement of the particle in the superdiffusion motion with a bimodal stationary distribution for all the anharmonic symmetric monostable potentials investigated. The stationary probability density functions show power-law tails, which ensure finiteness of the variance. By reviewing recent results on these statistical characteristics, the peculiarities of Levy flights in comparison with ordinary Brownian motion are discussed.
A version of the saddle point method is developed, which allows one to describe exactly the asymptotic behavior of distribution densities of Levy driven stochastic integrals with deterministic kernels. Exact asymptotic behavior is established for (a)
the transition probability density of a real-valued Levy process; (b) the transition probability density and the invariant distribution density of a Levy driven Ornstein-Uhlenbeck process; (c) the distribution density of the fractional Levy motion.
Levy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the jump lengths---are drawn from an $alpha$-stable jump length distribution with long-tailed, power-law asymptote. As a result, the var
iance of Levy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Levy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Levy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index $alpha$ and the skewness (asymmetry) parameter $beta$. The other approach is based on the stochastic Langevin equation with $alpha$-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.
We consider a stochastic volatility model with Levy jumps for a log-return process $Z=(Z_{t})_{tgeq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{tgeq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{tgeq 0}$ is an independent Levy proce
ss with absolutely continuous Levy measure $ u$. Small-time expansions, of arbitrary polynomial order, in time-$t$, are obtained for the tails $bbp(Z_{t}geq z)$, $z>0$, and for the call-option prices $bbe(e^{z+Z_{t}}-1)_{+}$, $z eq 0$, assuming smoothness conditions on the {PaleGrey density of $ u$} away from the origin and a small-time large deviation principle on $U$. Our approach allows for a unified treatment of general payoff functions of the form $phi(x){bf 1}_{xgeq{}z}$ for smooth functions $phi$ and $z>0$. As a consequence of our tail expansions, the polynomial expansions in $t$ of the transition densities $f_{t}$ are also {Green obtained} under mild conditions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
