Paweł Sztonyk
Estimates of densities of convolution semigroups of probability measures are given under specific assumptions on the corresponding Lévy measure and the Lévy--Khinchin exponent. The assumptions are satisfied, e.g., by tempered stable semigroups of J. Rosi{ń}ski.
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In this article, the infinite series form of the probability densities of tempered stable and inverse tempered stable subordinators are obtained using Mellin transform. Further, the densities of the products and quotients of stable and inverse stable subordinators are worked out. The asymptotic behaviours of these densities are obtained as $x \rightarrow 0^+$. Similar results for tempered and inverse tempered stable subordinators are discussed. Our results provide alternative methods to find the densities of these subordinators and complement the results available in literature.
Multivariate tempered stable random measures (ISRMs) are constructed and their corresponding space of integrable functions is characterized in terms of a quasi-norm utilizing the so-called Rosinski measure of a tempered stable law. In the special case of exponential tempered ISRMs operator-fractional tempered stable random fields are presented by a moving-average and a harmonizable representation, respectively.
In this article, we study the potential theory of normal tempered stable process which is obtained by time-changing the Brownian motion with a tempered stable subordinator. Precisely, we study the asymptotic behavior of potential density and Levy density associated with tempered stable subordinator and the Green function and the Levy density associated with the normal tempered stable process. We also provide the corresponding results for normal inverse Gaussian process which is a well studied process in literature.
Stable laws can be tempered by modifying the Lévy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random walk models can provide a basic physical model for the underlying physical phenomena.
This work defines two classes of processes, that we term {\it tempered fractional multistable motion} and {\it tempered multifractional stable motion}. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise Hölder exponent, Hölder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These proc…
Approximations Related to Tempered Stable Distributions
本文利用Stein方法推导了Tempered Stable分布分别与复合泊松分布、alpha稳定分布在Kolmogorov距离下的误差界，及两Tempered Stable分布在光滑Wasserstein距离下的误差界，还讨论了其对正态和方差伽马分布的近似并得到相应极限定理。
In this paper we define the closure under weak convergence of the class of p-tempered α-stable distributions. We give necessary and sufficient conditions for convergence of sequences in this class. Moreover, we show that any element in this class can be approximated by the distribution of a linear combination of elementary p-tempered α-stable random variables.
We introduce a large and flexible class of discrete tempered stable distributions, and analyze the domains of attraction for both this class and the related class of positive tempered stable distributions. Our results suggest that these are natural models for sums of independent and identically distributed random variables with tempered heavy tails tails, i.e. tails that appear to be heavy up to a point, but ultimately decay faster.
We give an explicit representation for the transition law of a tempered stable Ornstein-Uhlenbeck process and use it to develop a rejection sampling algorithm for exact simulation of increments from this process. Our results apply to general classes of both univariate and multivariate tempered stable distributions and contain a number of previously studied results as special cases.
In this article, we introduce mixtures of tempered stable subordinators (TSS). These mixtures define a class of subordinators which generalize tempered stable subordinators. The main properties like probability density function (pdf), Levy density, moments, governing Fokker-Planck-Kolmogorov (FPK) type equations, asymptotic form of potential density and asymptotic form of the renewal function for the corresponding inverse subordinator are discussed. We also generalize these results to n-th order mixtures of TSS.
Meerschaert and Sabzikar [12], [13] introduced tempered fractional Brownian/stable motion (TFBM/TFSM) by including an exponential tempering factor in the moving average representation of FBM/FSM. The present paper discusses another tempered version of FBM/FSM, termed tempered fractional Brownian/stable motion of second kind (TFBM II/TFSM II).We prove that TFBM/TFSM and TFBM II/TFSM II are different processes. Particularly, large time properties of TFBM II/TFSM II are similar to those of FBM/FSM and are in deep contrast to large time properties of TFBM/TFSM.
A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, with fractional calculus. Finally, some analytical approximations for the probability density function of tempered infinite divisible distribution, which elliptical tempered stable distributions are a subclass of them, are considered.
Exact Simulation from Tempered Stable Distributions with Infinite Variation ($α\ge1$)
本文提出了一种精确且计算可行的方法，用于模拟无限变分的 tempered稳定分布，α∈[1,2)。
On the convolution equivalence of tempered stable distributions on the real line
本文证明了单变量调和稳定分布的卷积等价性，从而严格化了其概率密度与Lévy密度渐近相似的经典启发式论证，并讨论了文献中的具体例子。
Rapidly decreasing tempered stable distributions are useful models for financial applications. However, there has been no exact method for simulation available in the literature. We remedy this by introducing an exact simulation method in the finite variation case. Our methodology works for the wider class of $p$-RDTS distributions.
Tempered stable distributions and processes
本文研究了 tempered稳定分布及其过程，探讨了极限分布、参数估计、密度变换和 p-变分指数，以及由这些过程驱动的指数股票模型。
We extend the class of tempered stable distributions first introduced in Rosinski 2007. Our new class allows for more structure and more variety of tail behaviors. We discuss various subclasses and the relation between them. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.
We derive an explicit representation for the transition law of a $p$-tempered $α$-stable process of Ornstein-Uhlenbeck-type and use it to develop a methodology for simulation. Our results apply in both the univariate and multivariate cases. Special attention is given to the case where $p\leα$, which is more complicated and requires additional care.
We study a class of stochastic differential equations driven by a possibly tempered L{é}vy process, under mild conditions on the coefficients. We prove the well-posedness of the associated martingale problem as well as the existence of the density of the solution. Two sided heat kernel estimates are given as well. Our approach is based on the Parametrix series expansion
In this article, we develop Stein characterization for two-sided tempered stable distribution. Stein characterizations for normal, gamma, Laplace, and variance-gamma distributions already known in the literature follow easily. One can also derive Stein characterizations for more difficult distributions such as the distribution of product of two normal random variables, a difference between two gamma random variables. Using the semigroup approach, we obtain estimates of the solution to Stein equation. Finally, we apply these estimates to obtain error bounds in the Wasserstein-type distance for …
We study the probability of ruin before time $t$ for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae are found to indicate some potential drawbacks to the use of the inverse Gaussian process as a …
We discuss the properties of three upsilon transforms, which are related to the class of $p$-tempered $α$-stable ($TS^p_α$) distributions. In particular, we characterize their domains and show how they can be represented as compositions of each other. Further, we show that if $-\infty<β<α<2$ and $0<q<p<\infty$ then they can be used to transform the Lévy measures of $TS^p_β$ distributions into those of $TS^q_α$.
On Fractional Tempered Stable Motion
本文定义并研究了分数阶缓和稳定运动（fTSm），揭示了其介于高斯与稳定分布之间的尾部特性及短长期近似行为，并推导其级数表示用于模拟和样本路径性质研究。
We consider the first-hitting time of a tempered $β$-stable subordinator, also called inverse tempered stable (ITS) subordinator. The density function of the ITS subordinator is obtained, for the index of stability $β\in (0,1)$. The series representation of the ITS density is also obtained, which could be helpful for computational purposes. The asymptotic behaviors of the $q$-th order moments of the ITS subordinator are investigated. In particular, the limiting behaviors of the mean of the ITS subordinator is given. The limiting form of the ITS density, as the space variable $x\rightarrow 0$, …
Parametric Estimation of Tempered Stable Laws
本文比较最大似然与广义矩等参数估计方法，证明缓增稳定分布估计量大样本性质，并通过模拟和金融数据验证其效果。
Tempered geometric stable distributions and processes
本文提出基于指数阻尼Mittag-Leffler函数的几何调和方法，研究单变量调和几何稳定分布，计算其特征指数、累积量和谱密度，并分析相关Lévy过程的标度极限。
We develop efficient methods for simulating processes of Ornstein-Uhlenbeck type related to the class of $p$-tempered $α$-stable ($\ts$) distributions. Our results hold for both the univariate and multivariate cases and we consider both the case where the $\ts$ distribution is the stationary law and where it is the distribution of the background driving Lévy process (BDLP). In the latter case, we also derive an explicit representation for the transition law as this was previous known only in certain special cases and only for $p=1$ and $α\in[0,1)$. Simulation results suggest that our methods w…
We study a class of multivariate tempered stable distributions and introduce the associated class of tempered stable Sato subordinators. These Sato subordinators are used to build additive inhomogeneous processes by subordination of a multiparameter Brownian motion. The resulting process is additive and time inhomogeneous. Furthermore, these processes are associated with the distribution at unit time of a class of Lévy process with good fit properties on fifinancial data. The main feature of the Sato subordinated Brownian motion is that it has time dependent correlation, whereas the Lévy count…
Since the turn of the century, there has been increased interest in the application of heavy-tailed distributions, particularly stable distributions, to problems in physics and finance. Although, the tails of stable distributions provide a better fit to real-world data, they are too fat to describe empirical distributions. To remedy this drawback of stable distributions, so-called tempered variants of stable distributions have been proposed. In this paper, we argue that the tempering should be connected to the model leading to heavy tailed distribution and propose several tempering procedures …
Simulating Continuous-Time Autoregressive Moving Average Processes Driven By p-Tempered α-Stable Lévy Processes
本文研究了由 p-tempered α-stable Lévy 过程驱动的 CARMA 过程的模拟方案，通过截断级数表示近似驱动噪声，推导了级数表示结果，证明了误差界限，并通过 Monte Carlo 实验验证了方法的有效性。
Stable and tempered stable distributions and processes: an overview toward trajectory simulation
本文综述稳定分布、α-稳定过程及经典缓变稳定（CTS）过程的核心性质，给出对应采样与增量模拟算法并附多尺度数值示例，为解决这类无限跳Lévy过程的轨迹模拟计算难题提供实践路线图。
Self-Decomposable Laws Associated with General Tempered Stable (GTS) Distribution and their Simulation Applications
本文研究广义调和稳定（GTS）分布相关的自分解分布及其背景驱动Lévy过程，并用于Ornstein-Uhlenbeck型过程的平稳分布，提出基于随机积分表示的模拟方法以模拟标普500指数和比特币日累积收益。
Information geometry of tempered stable processes
本文研究了 tempered稳定过程的信息几何，推导了 α-分歧、Fisher 信息矩阵和 α-连接，并展示了其在统计应用中的示例。
Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, \cite{MAI}, \cite{STAW} and \cite{GAR}). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index $ρ\in (0,1)$); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the $n$-times Laplace transform of its density) which is indexed by the …
A limit theorem for generalized tempered stable processes and their quadratic variations with stable index tending to two
本文研究稳定指数趋于二时广义调和稳定过程的极限，证明其二次协变差收敛到独立的一维稳定过程，并应用于金融建模，构造逼近Black-Scholes的纯跳跃模型，分析隐含波动率偏斜。
The paper investigates the rich class of Generalized Tempered Stable distribution, an alternative to Normal distribution and the $α$-Stable distribution for modelling asset return and many physical and economic systems. Firstly, we explore some important properties of the Generalized Tempered Stable (GTS) distribution. The theoretical tools developed are used to perform empirical analysis. The GTS distribution is fitted using S&P 500, SPY ETF and Bitcoin BTC. The Fractional Fourier Transform (FRFT) technique evaluates the probability density function and its derivatives in the maximum likeliho…
In this paper the running average of a subordinator with a tempered stable distribution is considered. We investigate a family of previously unexplored infinite-activity subordinators induced by the probability distribution of the running average process and determine their jump intensity measures. Special cases including gamma processes and inverse Gaussian processes are discussed. Then we derive easily implementable formulas for the distribution functions, cumulants, and moments, as well as provide explicit estimates for their asymptotic behaviors. Numerical experiments are conducted for ill…
We use the mean exit time to quantify macroscopic dynamical behaviors of stochastic dynamical systems driven by tempered Lévy fluctuations, which are solutions of nonlocal elliptic equations. Firstly, we construct a new numerical scheme to compute and solve the mean exit time associated with the one dimensional stochastic system. Secondly, we extend the analytical and numerical results to two dimensional case: horizontal-vertical and isotropic case. Finally, we verify the effectiveness of the presented schemes with numerical experiments in several examples.
This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $α$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{τ_D}\right|$ and the first exit time $τ_D$ are firstly obtained. It is found that the probability density function of $\left|X_{τ_D}\right|$ or $τ_D$ exponentially decays with the increase of $\left|X_{τ_D}\right|$ or $τ_D$, and $\mathrm{E}\left[τ_D\right]\sim \left|\mathrm{E}\left[X_{τ_D}\right]\right|$,\ $\mathrm{E}\left[τ_D\right]\sim\mathrm{E}\left[\left|X_{τ_D}-\mathrm{E}\left[X_{τ_D}\right]\right|^2\right] $…
Constructing \Levy-driven Ornstein-Uhlenbeck processes is a task closely related to the notion of self-decomposability. In particular, their transition laws are linked to the properties of what will be hereafter called the \emph{a-reminder} of their self-decomposable stationary laws. In the present study we fully characterize the Lévy triplet of these a-reminder s and we provide a general framework to deduce the transition laws of the finite variation Ornstein-Uhlenbeck processes associated with tempered stable distributions. We focus finally on the subclass of the exponentially-modulated temp…
Exponential stock models driven by tempered stable processes
本文研究了由 tempered 稳定过程驱动的指数股票模型，分析了等价鞅测度的存在性，并给出了欧式期权的定价公式。
Fast and explicit European option pricing under tempered stable processes
该研究针对修正稳定过程驱动的指数莱维模型，提出欧式期权定价的级数展开方法，无需精细选择超参数，数值表现可与先进方法媲美。
On Approximations of Subordinators in $L^p$ and the Simulation of Tempered Stable Distributions
本文研究正半直线上的从属子近似问题，证明缩放泊松混合可近似从属子并推导$L^p$收敛速率，提出近似模拟方法，在一般混合框架下重点针对tempered stable分布类展开分析。
Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence," and is kept fixed in between those times. This model interpolates between that of a homogenous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the correspond…
Optimal Harvesting of a Stochastic Logistic Model Driven by One-Sided Tempered Stable Process
本文研究由单侧 tempered 稳定过程驱动的随机逻辑捕捞模型，推导了种群灭绝与持久的阈值条件，证明了模型的分布稳定性，并得出最优捕捞努力和最大可持续产量的显式解。分析了白噪声强度和莱维跳跃强度对最优捕捞策略的影响，提出四种针对性干预策略。
Limit Theorems for Tempered Linear Processes with Innovations in the Domain of Attraction of a Stable Law
该研究针对新息属于稳定律吸引域的回火线性过程，推导了其部分和过程的极限定理，明确了不同回火强度下的归一化关系与极限性质，推广了未回火的对数边界定理。
In this study we consider the pricing of energy derivatives when the evolution of spot prices is modeled with a normal tempered stable driven Ornstein-Uhlenbeck process. Such processes are the generalization of normal inverse Gaussian processes that are widely used in energy finance applications. We first specify their statistical properties calculating their characteristic function in closed form. This result is instrumental for the derivation of non-arbitrage conditions such that the spot dynamics is consistent with the forward curve without relying on numerical approximations or on numerica…
In this paper we investigate two variants of $α$-stable processes, namely tempered stable subordinators and modified tempered stable process as well as their renormalization. We study the weak convergence in the Skorohod space and prove that they satisfy the uniform tightness condition. Finally, applications to the $α$-dependence of the solutions of SDEs driven by these processes are discussed.
There are given sufficient conditions under which mixtures of dilations of Lévy spectral measures, on a Hilbert space, are Lévy measures again. We introduce some random integrals with respect to infinite dimensional Lévy processes, which in turn give some integral mappings. New classes (convolution semigroups) are introduced. One of them gives an unexpected relation between the free (Voiculescu) and the classical Lévy-Khintchine formulae while the second one coincides with tempered stable measures (Mantegna nad Stanley) arisen in statistical physics.}
Monte Carlo algorithm for the extrema of tempered stable processes
本文提出一种基于未调和过程增量的蒙特卡洛算法，用于高效计算调和稳定过程的极值向量，并证明其最优复杂度，为障碍期权定价和风险度量提供置信区间。
Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data
针对GTS分布概率密度无显式表达式导致MLE参数估计失效的问题，本文基于FRFT提出七参数GTS分布金融数据拟合方法，在四类资产收益率数据上验证其拟合效果显著优于多种同类分布。
European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis
本文将GTS分布用于S&P 500指数收益并结合Esscher变换、扩展BS公式及两种数值算法进行欧式期权定价，发现GTS能修正BS在NTM和ITM期权上的低估。
Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function
本文提出一种快速精确算法，用于模拟调和稳定从属过程穿越任意非增绝对连续函数的首达时间及上下冲量，并证明其运行时间具有有限指数矩，期望运行时间在稳定参数上至多三次增长、在调和参数和函数初始值上线性增长，数值实验验证了理论界。
In this paper, we define a new and broad family of vector-valued random fields called tempered operator fractional operator-stable random fields (TRF, for short). TRF is typically non-Gaussian and generalizes tempered fractional stable stochastic processes. TRF comprises moving average and harmonizable-type subclasses that are constructed by tempering (matrix-) homogeneous, matrix-valued kernels in time- and Fourier-domain stochastic integrals with respect to vector-valued, strictly operator-stable random measures. We establish the existence and fundamental properties of TRF. Assuming both Gau…
The mean first exit (passage) time characterizes the average time of a stochastic process never leaving a fixed region in the state space, while the escape probability describes the likelihood of a transition from one region to another for a stochastic system driven by discontinuous (with jumps) Lévy motion. This paper discusses the two deterministic quantities, mean first exit time and escape probability, for the anomalous processes having the tempered Lévy stable waiting times with the tempering index $λ>0$ and the stability index $0<α\le 1$; as for the distribution of jump lengths or the ty…
Mixing rates for linear operators under infinitely divisible measures on Banach spaces
本文在 Banach 空间线性动力学中，利用余差泛函与控制测度研究无限可分测度下算子的混合速率，并给出了复合 Poisson、α-稳定及调和 α-稳定测度下加权移位算子的显式收敛速率。
We show that positive $α$-stable densities are hyperbolically completely monotone if and only if $α\le 1/2$. This gives a positive answer to a question raised by L. Bondesson in 1977.
On some extensions of generalized counting processes
本文研究了泊松过程和广义计数过程的分数扩展，通过引入由稳定分布和温度稳定子ordinator求和的逆变换时间变化，推导了概率分布和生成函数的控制方程，并给出了几种模型的概率分布和生成函数的闭式表达。
We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Levy process
We investigate the problem raised by L. Bondesson, about the hyperbolic complete monotonicity of $α$-stable densities. We prove that densitites of subordinators of order $α$ are HCM for $α\in ]0,1/4] \cup [1/3,1/2]$.
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