Friedrich Hubalek, Alexey Kuznetsov
We study the density of the supremum of a strictly stable Lévy process. We prove that for almost all values of the index $α$ -- except for a dense set of Lebesgue measure zero -- the asymptotic series which were obtained in A. Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the supremum.
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We study the density of the supremum of a strictly stable Lévy process. As was proved recently in F. Hubalek and A. Kuznetsov "A convergent series representation for the density of the supremum of a stable process" (Elect. Comm. in Probab., 16, 84-95, 2011), for almost all irrational values of the stability parameter $α$ this density can be represented by an absolutely convergent series. We show that this result is not valid for all irrational values of $α$: we construct a dense uncountable set of irrational numbers $α$ for which the series does not converge absolutely. Our second goal is to i…
We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $α$ is not rational. In the case when parameters $α$ and $ρ=\p(X_1>0)$ satisfy $ρ+k=l/α$ for some integers $k,l \ge 1$ we prove that these asymptotic expansions are in fact convergent series representations of the density of supremum.
We study the asymptotics, for small and large values, of the supremum of a product of symmetric stable processes. We show in particular that the persistence exponent remains the same as for only one process, up to some logarithmic terms.
An absolutely convergent double series representation for the density of the supremum of $α$-stable Levy process is given in [3, Theorem 2] for almost all irrational $α$. This result cannot be made stronger in the following sense: the series does not converge absolutely when $α$ belongs to a certain subset of irrational numbers of Lebesgue measure zero (see [6, Theorem 2]). Our main result in this note shows that for every irrational $α$ there is a way to rearrange the terms of the double series, so that it converges to the density of the supremum. We show how one can establish this stronger r…
We consider a process $Z$ on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$ and the time $T$ at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted Lévy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy proces…
In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative Lévy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $\Prob(\sup_{s\leq t}Z_α(s)\geq u)=α\Prob(Z_α(t)\geq u)$ for $u\geq 0$ where $Z_α$ is a spectrally negative Lévy process with $1<α\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive Lévy process w…
Joint density of a stable process and its supremum: regularity and upper bounds
本文利用模拟中的三个思想，结合凹极大子表示和Chambers-Mallows-Stuck表示，通过多层蒙特卡罗插值加速，建立了稳定过程及其上确界联合密度的无限可微性和近乎最优多项式上界。
If $X$ is a spectrally positive stable process of index $α\in(1,2)$ whose Lévy measure has density $cx^{-α-1}$ on $(0,\infty),$ and $S_1=\sup_{0<t\leq1}X_t,$ it is known that $P(S_1>x)\backsim cα^{-1}x^{-α}$ as $x\to\infty.$ It is also known that $S_1$has a continuous density, $s$ say. The point of this note is to show that $s(x)\backsim cx^{-(α+1)}$ as $x\to\infty.$
We study the distribution and various properties of exponential functionals of hypergeometric Levy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained by Kuznetsov "On extrema of stable processes" (2010) and Hubalek and Kuznetsov "A convergent series representation for the density of the supremum of a stable process"…
Let $X=(X_t)_{t\ge0}$ be a stable Lévy process of index $α\in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann--Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for $f_t$. Recalling the familiar relation between $S_t$ and the first entry t…
For a spectrally positive and strictly stable process with index in (1,2), a series representation is obtained for the joint distribution of the "first passage triple" that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including 1) the joint law of the first passage triple and the pre-passage running supremum, and 2) at a fixed time point, the joint law of the process' value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions …
We study the Wiener--Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener--Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener--Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum fu…
If $X$ is a stable process of index $α\in(0,2)$ whose Lévy measure has density $cx^{-α-1}$ on $(0,\infty)$, and $S_1=\sup_{0<t\leq1}X_t$, it is known that $P(S_1>x)\backsim Aα^{-1}x^{-α}$ as $x\to\infty$ and $P(S_1\leq x)\backsim Bα^{-1}ρ^{-1}x^{αρ}$ as $x\downarrow0$. [Here $ρ=P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known that $S_1$ has a continuous density, $m$ say. The main point of this note is to show that $m(x)\backsim Ax^{-(α+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{αρ-1}$ as $x\downarrow0$. Similar results are obtained for related densities.
We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct non-asymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We pro…
Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary max-stable process with Fréchet marginals $Φ_α, α>0$ and set $M_X(T)=\sup_{t \in [0,T]} X(t),T>0$. In the light of the seminal articles [1,2], it follows that $A_T=M_X(T)/T^{1/α}$ converges in distribution as $T\to \infty$ to $\mathcal{H}_Z^{1/α} X(1)$, where $\mathcal{H}_Z$ is the Pickands constant corresponding to the spectral process $Z$ of $X$. In this contribution we derive explicit formulas for $\mathcal{H}_Z$ in terms of $Z$ and show necessary and sufficient conditions for its positivity. From our analysis it follows t…
We prove that when a sequence of Lévy processes $X^{(n)}$ or a normed sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L^{(n)}$ of local times at the supremum of $X^{(n)}$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S^{(n)}$ conditioned to stay…
Given a stable Lévy process $X=(X_t)_{0\le t\le T}$ of index $α\in(1,2)$ with no negative jumps, and letting $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t\in [0,T]$, we consider the optimal prediction problem \[V=\inf_{0\leτ\le T}\mathsf{E}(S_T-X_τ)^p,\] where the infimum is taken over all stopping times $τ$ of $X$, and the error parameter $p\in(1,α)$ is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann--Liouville type, and finding an explicit solution to the latter, we show that there exists $α_*\in(1,2)$ (equal to 1.57 …
This paper is concerned with finite sample approximations to the supremum of a non-degenerate $U$-process of a general order indexed by a function class. We are primarily interested in situations where the function class as well as the underlying distribution change with the sample size, and the $U$-process itself is not weakly convergent as a process. Such situations arise in a variety of modern statistical problems. We first consider Gaussian approximations, namely, approximate the $U$-process supremum by the supremum of a Gaussian process, and derive coupling and Kolmogorov distance bounds.…
The normalized spectral representation of a max-stable process on a compact set is the unique representation where all spectral functions share the same supremum. Among the class of equivalent spectral representations of a process, the normalized spectral representation plays a distinctive role as a solution of two optimization problems in the context of an efficient simulation of max-stable processes. Our approach has the potential of considerably reducing the simulation time of max-stable processes.
We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).
We consider a one-dimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height $\log t$, $(L_X(t,\mathfrak m_{\log t}+x)/t,x\in \R)$, converges in law to a functional of two independent Lévy processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a two-sided Lévy process conditioned to stay positive. We also obtain the limit law of the supremum of the normalized local time. This result has been obtained by Andreoletti and Diel in the cas…
In this paper we introduce a ten-parameter family of Lévy processes for which we obtain Wiener-Hopf factors and distribution of the supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener-Hopf factors can be expressed as infinite products over roots of a certain transcendental equat…
In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions. As an example we consider gamma Lévy process, $α$-stable Lévy process and Brownian motion. Moreover we obtain identities for Laplace transform of the distribution for the supremum of Lévy processes with randomly broken drift and on random intervals.
An identity in law for the area of a spectrally positive Lévy stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse Beta random variable and the square of a positive stable random variable. This identity entails that the stopped area is distributed as the perpetuity of a spectrally negative Lévy process, and is hence self-decomposable. We also derive a convergent series representation for the density, whose behaviour at zero is shown to be Fréchet-like.
In this paper, we continue our understanding of the stable process from the perspective of the theory of self-similar Markov processes in the spirit of the recent papers of Kyprianou (2016) and Kyprianou et al. (2017). In particular, we turn our attention to the case of $d$-dimensional isotropic stable process, for $d\geq 2$. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develo…
We find an expression for the joint Laplace transform of the law of $(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a Lévy process $X$, where $T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$ is an $α$-stable Lévy process, with $1<α<2$, we show how to recover from this formula the law of $X_{T_{[x,+\infty[}}$; this result was already obtained by D. Ray, in the symmetric case and by N. Bingham, in the case when $X$ is non spectrally negative. Then, we study the behaviour of the time of first passage $T_{[x,+\infty[},$ conditioned to $\{X_{T_{[x,+\infty[}} -x \leq h\}$ wh…
We study the upper tail behaviors of the local times of the additive stable processes. Let $X_1(t),...,X_p(t)$ be independent, d-dimensional symmetric stable processes with stable index $0<α\le 2$ and consider the additive stable process $\bar{X}(t_1,...,t_p)=X_1(t_1)+... +X_p(t_p)$. Under the condition $d<αp$, we obtain a precise form of the large deviation principle for the local time \[η^x([0,t]^p)=\int_0^t...\int_0^tδ_x\bigl(X_1(s_1)+... +X_p(s_p)\bigr) ds_1... ds_p\] of the multiparameter process $\bar{X}(t_1,...,t_p)$, and for its supremum norm $\sup_{x\in\mathbb{R}^d}η^x([0,t]^p)$. Our …
Integral representations for expectations of functions of a stable Lévy process $X$ and its supremum $\bar X$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T, \barX_T$, the joint cpdf of $X_T$ and $\barX_T$, and the expectation of $(\be X_T-\barX_T)_+$, $\be>1$, are considered, and efficient numerical procedures for cpdfs are developed. The most efficient numerical methods use the conformal acceleration technique and simplified trapezoid rule.
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。
The article is devoted to the problem of calculating the probability density of a strictly stable law at $x\to\infty$. To solve this problem, it was proposed to use the expansion of the probability density in a power series. A representation of the probability density in the form of a power series and an estimate for the remainder term was obtained. This power series is convergent in the case $0<α<1$ and asymptotic at $x\to\infty$ in the case $1<α<2$. The case $α=1$ was considered separately. It was shown that in the case $α=1$ the obtained power series was convergent for any $|x|>1$ at $N\to\…
The aim of this paper is to give a wavelet series representation of Linear Multifractional Stable Motion (LMSM in brief), which is more explicit than that introduced in (Ayache & Hamonier 2012). Instead of using Daubechies wavelet, which is not given by a closed form, we use the Haar wavelet. In order to obtain this new representation, we introduce a Haar expansion of the high and low frequency parts of the $\mathcal{S}α\mathcal{S}$ random field $X$ generating LMSM. Then, by using Abel transforms, we show that these series are convergent, almost surely, in the space of continuous functions. Fi…
Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely convergent in all the Hölder spaces $C^γ([-M,M]^N)$, where $M>0$ and $γ\in (0, \min\{H_1,\ldots, H_N\})$ are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let $X=\{X(t), t \in \R^N\}$ be an $\R^d$-valued harmonizable fractional stable sheet whose components …
The paper considers the problem of calculating the distribution function of a strictly stable law at $x\to\infty$. To solve this problem, an expansion of the distribution function in a power series was obtained, and an estimate of the remainder term was also obtained. It was shown that in the case $α<1$ this series was convergent for any $x$, in the case $α=1$ the series was convergent at $N\to\infty$ in the domain $|x|>1$, and in the case $α>1$ the series was asymptotic at $x\to\infty$. The case $α=1$ was considered separately and it was demonstrated that in that case the series converges to …
The problem of calculating the probability density and distribution function of a strictly stable law is considered at $x\to0$. The expansions of these values into power series were obtained to solve this problem. It was shown that in the case $α<1$ the obtained series were asymptotic at $x\to0$, in the case $α>1$ they were convergent and in the case $α=1$ in the domain $|x|<1$ these series converged to an asymmetric Cauchy distribution. It has been shown that at $x\to0$ the obtained expansions can be successfully used to calculate the probability density and distribution function of strictly …
Loop Vertex Representation for Cumulants, Part II: Weingarten Calculus
本文利用 Weingarten 微积分为稳定随机矩阵模型构造标量累积量，并在 N 趋于无穷时给出显式收敛展开。
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