The lower envelope of positive self-similar Markov processes

Positive recurrence and transience of a two-station network with server states

We study positive recurrence and transience of a two-station network in which the behavior of the server in each station is governed by a Markov chain with a finite number of server states; this service process can represent various service disciplines such as a non-preemptive priority service and K-limited service. Assuming that exogenous customers arrive according to independent Markovian arrival processes (MAPs), we represent the behavior of the whole network as a continuous-time Markov chain and, by the uniformization technique, obtain the corresponding discrete-time Markov chain, which is…

A Jump Type SDE Approach to Positive Self-Similar Markov Processes

We present a new approach to positive self-similar Markov processes (pssMps) by reformulating Lamperti's transformation via jump type SDEs. As applications, we give direct constructions of pssMps (re)started continuously at zero if the Lamperti transformed Levy process is spectrally negative. Our paper can be seen as a continuation of similar studies for continuous state branching processes but the approach seems to be more fruitful in the context of pssMps.

Exit problems for positive self-similar Markov processes with one-sided jumps

A systematic exposition of scale functions is given for positive self-similar Markov processes (pssMp) with one-sided jumps. The scale functions express as convolution series of the usual scale functions associated with spectrally one-sided Lévy processes that underly the pssMp through the Lamperti transform. This theory is then brought to bear on solving the spatio-temporal: (i) two-sided exit problem; (ii) joint first passage problem upwards for the the pssMp and its multiplicative drawdown (resp. drawup) in the spectrally negative (resp. positive) case.

Self-similar scaling limits of Markov chains on the positive integers

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $X_{n}$ to reach…

Correction. Perfect simulation for a class of positive recurrent Markov chains

Correction to Annals of Applied Probability 17 (2007) 781--808 [doi:10.1214/105051607000000032].

Limit theorems for affine Markov walks conditioned to stay positive

Consider the real Markov walk $S_n = X_1+ \dots+ X_n$ with increments $\left(X_n\right)_{n\geq 1}$ defined by a stochastic recursion starting at $X_0=x$. For a starting point $y>0$ denote by $τ_y$ the exit time of the process $\left( y+S_n \right)_{n\geq 1}$ from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event $τ_y \geq n$ and of the conditional law of $y+S_n$ given $τ_y \geq n$ as $n \to +\infty$.

On some transformations between positive self--similar Markov processes

A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given a pssMp $X,$ that never hits 0, we construct a pssMp $X^{\downarrow}$ that hits 0 in a finite time. The latter can be viewed as $X$ conditioned to hit 0 in a finite time and we prove that this conditioning is determined by the pre-minimum part of $X.$ Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously.

Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption

Consider a Markov chain $(X_n)_{n\geqslant 0}$ with values in the state space $\mathbb X$. Let $f$ be a real function on $\mathbb X$ and set $S_0=0,$ $S_n = f(X_1)+\cdots + f(X_n),$ $n\geqslant 1$. Let $\mathbb P_x$ be the probability measure generated by the Markov chain starting at $X_0=x$. For a starting point $y \in \mathbb R$ denote by $τ_y$ the first moment when the Markov walk $(y+S_n)_{n\geqslant 1}$ becomes non-positive. Under the condition that $S_n$ has zero drift, we find the asymptotics of the probability $\mathbb P_x ( τ_y >n )$ and of the conditional law $\mathbb P_x ( y+S_n\leq…

A temporal factorization at the maximum for spectrally negative positive self-similar Markov processes

For a spectrally negative positive self-similar Markov process with an a.s. finite overall supremum we provide, in tractable detail, a kind of conditional Wiener-Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of said absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existent…

Entrance laws for positive self-similar Markov processes

In this paper we propose an alternative construction of the self-similar entrance laws for positive self-similar Markov processes. The study of entrance laws has been carried out in previous papers using different techniques, depending on whether the process hits zero in a finite time almost surely or not. The technique here used allows to obtain the entrance laws in a unified way. Besides, we show that in the case where the process hits zero in a finite time, if there exists a self-similar entrance law, then there are infinitely many, but they can all be embedded into a single one. We propose…

Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

Using Lamperti's relationship between Lévy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law $\mathbb{P}_x$ of a pssMp starting at $x>0$, in the Skorohod space of càdlàg paths, when $x$ tends to 0. To do so, we first give conditions which allow us to construct a càdlàg Markov process $X^{(0)}$, starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws $\mathbb{P}_x$ to converge weakly to the law of $X^{(0)}$ as $x$ goes to 0. In particular, this answers a quest…

Perfect simulation for a class of positive recurrent Markov chains

This paper generalizes the work of Kendall [Electron. Comm. Probab. 9 (2004) 140--151], which showed that perfect simulation, in the form of dominated coupling from the past, is always possible (although not necessarily practical) for geometrically ergodic Markov chains. Here, we consider the more general situation of positive recurrent chains and explore when it is possible to produce such a simulation algorithm for these chains. We introduce a class of chains which we name tame, for which we show that perfect simulation is possible.

具有正跳的Markov过程的最优停止

Optimal stopping for Markov processes with positive jumps

本文研究实值Markov过程的贴现最优停止问题，基于Dynkin特征和Green核给出价值函数表示公式，并利用Fourier变换求解了电力市场中Levy驱动的Ornstein-Uhlenbeck过程的最优停止实例。

Some explicit identities associated with positive self-similar Markov processes

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type $π(dx)=e^{γx}ν(e^x-1) dx$, where $ν$ is the density of the stable Lévy measure and $γ$ is a positive parameter which depends on its characteristics. These processes were introduced in \cite{CC} as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitti…

通过复活获得的正自相似马尔可夫过程

Positive self-similar Markov processes obtained by resurrection

本文利用 Lamperti 变换构造通过在首次离开 (0,∞) 时复活严格 α-稳定过程而得到的正自相似马尔可夫过程，分析其长期行为、有限时间吸收条件及常返延拓的存在性，并给出跳跃核的双边精确估计。

On the asymptotic behaviour of increasing positive self-similar Markov processes

We are interested by the rate of growth of increasing positive self-similar Markov processes (ipssMp) such that the subordinator associated to it via Lamperti's transformation has infinite mean. We prove that the logarithm of an ipssMp normalized by the logarithm of the time converges weakly, as the time tends to infinity, if and only if the Laplace exponent of the underlying subordinator is regularly varying at zero. Moreover, we prove that the regular variation at zero of the Laplace exponent is essentially nasc for the existence of a function that normalizes the logarithm of an ipssMp. We o…

具有严格正密度的有理到达过程不一定马尔可夫

Rational arrival processes with strictly positive densities need not be Markovian

论文通过构造一个三阶反例，证明具有严格正联合密度和唯一主导实特征值的有理到达过程不一定是马尔可夫到达过程，且在强条件下猜想也失效。

A Markov chain on the symmetric group which is Schubert positive?

We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.

非可逆马尔可夫链谱间隙的界及其在时序差分学习中的应用

Bounds on Spectral Gaps for Non-Reversible Markov Chains with Applications to Temporal Difference Learning

本文为分析时序差分算法，通过Dirichlet谱间隙界定非可逆马尔可夫链的特征值，推广了稳定性充分条件，并给出离散时间线性高斯系统的显式界。

Transience and Recurrence of Markov Processes with Constrained Local Time

We study Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in [5] and [33], we study transience and recurrence for a broad class of Markov processes. In order to understand the distribution of the local time, we determine the distribution of a non-decreasing Lévy process (the inverse local time) conditioned to remain above a given level which varies in time. We study a time-dependent region, in contrast to previous works in which a process is conditioned to remain in a fixe…

马尔可夫过程正连续加法函数的广义Kac动量公式

Generalized Kac's moment formula for positive continuous additive functionals of Markov processes

本文推导了马尔可夫过程正连续加法函数的广义Kac动量公式，适用于具有绝对连续转移函数的标准过程及对称Hunt过程。

Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fréchet distribution

We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time $\mathbb{T}$ of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen time-changed, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the $\mathbb{T}$'s within a class have the same law which we identify in a simple form for all classes and reduces, in the spect…

Law of the absorption time of some positive self-similar Markov processes

Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten's constant appearing in the study …

On Markov--up processes and their recurrence properties

A simple model of the new notion of "Markov up" processes is proposed; its positive recurrence and ergodic properties are shown under the appropriate conditions.

拉普拉斯对偶中的正马尔可夫过程

Positive Markov processes in Laplace duality

本文建立了一个关于正马尔可夫过程拉普拉斯对偶的一般框架，证明了过程具有拉普拉斯对偶当且仅当满足某种完全单调性条件，并探讨了在边界0和∞处的弱连续性和吸收性性质。

Fluctuation theory and exit systems for positive self-similar Markov processes

For a positive self-similar Markov process, X, we construct a local time for the random set, $Θ$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $Θ$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder p…

Coarse Ricci curvature for continuous-time Markov processes

The coarse Ricci curvature for Markov chains is generalized for continuous time. We show that a positive coarse Ricci curvature implies a contraction of the Markov process for the Wasserstein distance between probability measures. This leads to a spectral gap estimate in the reversible case.

无限可分随机向量的极小径向次伽马包络

Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors

本文研究无限可分随机向量，定义径向极点C_X(t)并推导其变分公式，证明相关规则，明确其几何性质，还给出加性伽马射线模型下的有限多面体公式。

On laws of large numbers in $L^2$ for supercritical branching Markov processes beyond $λ$-positivity

We give necessary and sufficient conditions for laws of large numbers to hold in $L^2$ for the empirical measure of a large class of branching Markov processes, including $λ$-positive systems but also some $λ$-transient ones, such as the branching Brownian motion with drift and absorption at $0$. This is a significant improvement over previous results on this matter, which had only dealt so far with $λ$-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly p…

Conditioning subordinators embedded in Markov processes

The running infimum of a Levy process relative to its point of issue is know to have the same range that of the negative of a certain subordinator. Conditioning a Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned t…

Martin kernels for Markov processes with jumps

We prove existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular domains, in the context of general metric measure spaces. As a corollary, we prove uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R^d with positive continuous density of the Lévy measure; stable-like processes in R^d and in domains; …

Functional limit theorems for random walks perturbed by positive alpha-stable jumps

Let $ξ_1$, $ξ_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $η_1$, $η_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $α$-stable distribution, $α\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $ξ_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $η_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. T…

Order of magnitude time-reversible Markov chains and characterization of clustering processes

We introduce the notion of order of magnitude reversibility (OM-reversibility) in Markov chains that are parametrized by a positive parameter $\ep$. OM-reversibility is a weaker condition than reversibility, and requires only the knowledge of order of magnitude of the transition probabilities. For an irreducible, OM-reversible Markov chain on a finite state space, we prove that the stationary distribution satisfies order of magnitude detailed balance (analog of detailed balance in reversible Markov chains). The result characterizes the states with positive probability in the limit of the stati…

不可约马尔可夫链平稳分布存在性与唯一性的初等证明

An elementary proof of existence and uniqueness of stationary distributions for irreducible Markov chains

该研究针对满足行和为1、元素非负且不可约的n×n矩阵P，利用基础代数和Bolzano-Weierstrass定理，给出了其严格正且和为1的平稳行向量存在唯一性这一经典结果的初等证明。

Ladder epochs and ladder chain of a Markov random walk with discrete driving chain

Let $(M_{n},S_{n})_{n\ge 0}$ be a Markov random walk with positive recurrent driving chain $(M_{n})_{n\ge 0}$ having countable state space $\mathcal{S}$ and stationary distribution $π$. It is shown in this note that, if the dual sequence $({}^{\#}M_{n},{}^{\#}S_{n})_{n\ge 0}$ is positive divergent, i.e. ${}^{\#}S_{n}\to\infty$ a.s., then the strictly ascending ladder epochs $σ_{n}^{>}$ of $(M_{n},S_{n})_{n\ge 0}$ are a.s. finite and the ladder chain $(M_{σ_{n}^{>}})_{n\ge 0}$ is positive recurrent on some $\mathcal{S}^{>}\subset\mathcal{S}$. We also provide simple expressions for its stationar…

A Lyapunov view on positive Harris recurrence of multiclass queueing networks

This paper addresses the question when the underlying Markov process of a multiclass queueing network is positive Harris recurrent. It is well-known that stability of the fluid limit model is a sufficient condition for this. Hence, stability of fluid (limit) models is of vital interest. Recently, it has been shown that if the fluid model satisfies certain properties, it is stable if and only if there exists a Lyapunov function. In this paper a new method is provided to conclude that the underlying Markov process is positive Harris recurrent if the fluid model is stable by using explicitly the …

Markovian lifts of positive semidefinite affine Volterra type processes

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein Uhlenbeck processes whose state space are matrix valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multi…

A linear programming approach to Markov reward error bounds for queueing networks

In this paper, we present a numerical framework for constructing bounds on stationary performance measures of random walks in the positive orthant using the Markov reward approach. These bounds are established in terms of stationary performance measures of a perturbed random walk whose stationary distribution is known explicitly. We consider random walks in an arbitrary number of dimensions and with a transition probability structure that is defined on an arbitrary partition of the positive orthant. Within each component of this partition the transition probabilities are homogeneous. This enab…

谱正马尔可夫加性模型下的最优分红与资本注入

Optimal dividend and capital injection under spectrally positive Markov additive models

本文在谱正马尔可夫加性模型下研究带资本注入的最优分红问题，通过辅助奇异控制问题与递归不动点论证，证明了状态调制的双障碍策略最优并给出障碍的解析刻画。

马尔可夫链上的正转移测度空间

The space of positive transition measures on a Markov chain

本文针对马尔可夫链转移概率空间对偶平坦结构统计意义不明的问题，基于Amari正测度理论构造正转移测度空间并研究其对偶平坦结构，为马尔可夫链几何提供新见解。

正超模相依马尔可夫树分布的比较结果

Comparison results for positive supermodular dependent Markov tree distributions

本文针对树结构马尔可夫分布建立超模序比较结果，利用随机单调性和边二元 copula 逐点序，得到极值次序统计量与正相依随机变量和的一阶、二阶随机占优结果，并应用于扰动随机游走最大值的分布稳健性分析。

Structure recovery for partially observed discrete Markov random fields on graphs under not necessarily positive distributions

We propose a penalized pseudo-likelihood criterion to estimate the graph of conditional dependencies in a discrete Markov random field that can be partially observed. We prove the convergence of the estimator in the case of a finite or countable infinite set of nodes. In the finite case, the underlying graph can be recovered with probability one, while in the countable infinite case, we can recover any finite sub-graph with probability one by allowing the candidate neighborhoods to grow as a function o(log n), with n the sample size. Our method requires minimal assumptions on the probability d…

量子马尔可夫半群的遍历性质

Ergodic Properties of Quantum Markov Semigroups

本文研究了无限维量子马尔可夫半群的遍历性质，提出了确保指数收敛的充分条件，并通过示例展示了正递归子空间在全局渐近稳定性研究中的作用。

隐马尔可夫基本面下的均衡价格

Equilibrium prices under hidden Markov fundamentals

本研究在隐马尔可夫基本面下分析 Epstein-Zin 经济，证明估值因子恒常性，给出两状态定价方程解的性质，并推导期权定价与偏度展开。

带状马尔可夫过程的正双对角分解

Positive Bidiagonal Factorizations for Banded Markov Processes

该研究提出正双对角分解（PBF）理论，建立无可逆性等条件下带状马尔可夫过程的谱与概率框架，推导相关公式并刻画其结构，给出显式示例。