Green Measures for Time Changed Markov Processes

使用时间变更马尔可夫过程进行采样

Sampling with time-changed Markov processes

本文提出通过时间变更马尔可夫过程加速MCMC算法收敛，通过状态依赖函数调整基础过程时间速度，证明了其理论性质并在多峰分布和稀有事件模拟中实现性能提升。

$L^p$-Green-tight measures of $L^p$-Kato class for symmetric Markov processes

In this paper, we introduce the notion of $L^p$-Green-tight measures of $L^p$-Kato class in the framework of symmetric Markov processes. The class of $L^p$-Green-tight measures of $L^p$-Kato class is defined by the $p$-th power of resolvent kernels. We first prove that under the $L^p$-Green tightness of the measure $μ$, the embedding of extended Dirichlet space into $L^{2p}(E;μ)$ is compact under the absolute continuity condition for transient Markov processes, which is an extension of recent seminal work by Takeda. Secondly, we prove the coincidence between two classes of $L^p$-Green-tightnes…

时间变换的马尔可夫过程与时空耦合非局部方程

Time-changed Markov processes and space-time coupled non-local equations

研究耦合完全非局部方程，通过基于异常过程的随机表示建立解的存在唯一性，利用最大值原理推导唯一性，还得到布莱克 - 斯科尔斯方程的非局部类似物，解决特定条件下衍生证券定价问题。

Limit theorems for prices of options written on semi-Markov processes

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordina…

Ergodicity for Time Changed Symmetric Stable Processes

In this paper we study the ergodicity and the related semigroup property for a class of symmetric Markov jump processes associated with time changed symmetric $α$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric $α$-stable processes, yield sharp criteria for their various ergodic properties and corre…

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…

无正跳跃时空变换过程的尺度函数

Scale functions of space-time changed processes with no positive jumps

本文研究无正跳跃标准过程经时空变换后所得过程的尺度函数表示问题，利用原过程的尺度函数与Volterra积分方程给出新过程尺度函数的表达式，并将其应用于自相似Markov过程和连续状态分支过程。

马尔可夫可加过程格林函数的渐近性：基于二进积分分裂的方法

Asymptotics of Green functions for Markov-additive processes: an approach via dyadic splitting of integrals

本文通过二进积分分裂技术，为有限状态空间E上的离散马尔可夫可加过程推导格林函数的渐近等价，推广了Ney和Spitzer关于独立同分布随机变量和的经典定理到空间非齐次情形。

On discrete-time semi-Markov processes

In the last years, many authors studied a class of continuous time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop the discrete-time version of such a theory. We show that a class of discrete-time semi-Markov chains can be seen as time-changed Markov chains and we obtain governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling li…

Martin Capacity for Markov Chains

The probability that a transient Markov chain, or a Brownian path, will ever visit a given set Lambda, is classically estimated using the capacity of Lambda with respect to the Green kernel G(x,y). We show that replacing the Green kernel by the Martin kernel G(x,y)/G(0,y) yields improved estimates, which are exact up to a factor of 2. These estimates are applied to random walks on lattices, and also to explain a connection found by R. Lyons between capacity and percolation on trees.

Quasipotential and logarithmic asymptotics of the Green's measures

It is proved that the weak large deviation principle of the scaled processes $Z^\eps(t) = \eps Z(t/\eps)$ implies the weak large deviation principle for the scaled Green's measures of the Markov process $Z(t)$.

Stochastic Volterra equations with time-changed Lévy noise and maximum principles

Motivated by a problem of optimal harvesting of natural resources, we study a control problem for Volterra type dynamics driven by time-changed Lévy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative introduced in [15]. We prove both a sufficient and necessary stochastic maximum principle.

一类非马尔可夫过程的格林测度

Green Measures for a Class of non-Markov Processes

本文研究广义灰色布朗运动族 $B_{β,α}$ 的格林测度，证明当 $dα>2$ 且 $1<α\le2$ 时永久积分以概率1存在，从而统一推广了分数布朗运动、布朗运动及其他非高斯过程的格林测度。

Compound Poisson Processes: Potentials, Green Measures and Random Times

In this paper we study the existence of Green measures for Markov processes with a nonlocal jump generator. The jump generator has no second moment and satisfies a suitable condition on its Fourier transform. We also study the same problem for certain classes of random time changes Markov processes with jump generator.

自相似高斯马尔可夫过程

Self-similar Gaussian Markov processes

本文刻画了所有多维实值自相似高斯马尔可夫过程，发现其协方差矩阵函数可分为白噪声型、连续矩阵半群型及 Cauchy 方程非连续解型三类，并指出一维情形退化为双参数时间变换 Brownian motion 族，从而简化了非马尔可夫性的证明。

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。

On the exit time from open sets of some semi-Markov processes

In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large $t$) and of the distribution function (for small $t$) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leacky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be…

生灭过程是经过时间变换的 Feller 布朗运动

Birth-death processes are time-changed Feller's Brownian motions

本文通过特定时间变换建立 Feller 布朗运动与生灭过程的双向等价关系，并给出参数精确对应与构造性路径表示框架。

作为时间变换Walsh布朗运动的星图上的一般扩散过程

General diffusions on the star graph as time-changed Walsh Brownian motion

本研究将星图上的一般正则扩散表示为时间变换Walsh布朗运动，据此推导粘性粘合扩散的时间变换表示、占据时间公式，还拓展了Feller相关判据并证明星图Dirichlet问题解的存在唯一性及显式形式。

On the first-passage time of an integrated Gauss-Markov process

It is considered the integrated process $X(t)= x + \int _0^t Y(s) ds ,$ where $Y(t)$ is a Gauss-Markov process starting from $y.$ The first-passage time (FPT) of $X$ through a constant boundary and the first-exit time of $X$ from an interval $(a,b)$ are investigated, generalizing some results on FPT of integrated Brownian motion. An essential role is played by a useful representation of $X,$ in terms of Brownian motion which allows to reduces the FPT of $X$ to that of a time-changed Brownian motion. Some explicit examples are reported; when theoretical calculation is not available, the quantit…

多参数马尔可夫过程的加法次协调

Additive subordination of multiparameter Markov processes

本文研究了多参数马尔可夫过程在加法次协调下的性质，通过扩展菲利普斯定理证明其为费勒过程，并推导了生成元和符号的莱维-辛钦表示，应用于金融领域的奥本海姆-乌尔登贝克过程的因子型规范。

Time fractional equations and probabilistic representation

In this paper, we study the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts. An explicit relation between occupation measure for Markov processes time-changed by inverse subordinator in open sets and that of the original Markov process in the open set is also given.

具有长程依赖性的非马尔可夫链及其缩放极限

Non-Markovian chains with long-range dependence and their scaling limits

本文研究了具有长程依赖性的非马尔可夫链及其缩放极限，探讨了para-马尔可夫链和时间改变的马尔可夫链模型，揭示了其与分数阶方程和异常扩散的关系。

Red Light Green Light Method for Solving Large Markov Chains

Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. Without loss of generality, and drawing our motivation from applications to large networks, we interpret this problem as one of computing the stationary distribution of a random walk on a graph. We propose a new controlled, easily distributed algorithm for this task, briefly summarized as follows: at the beginning, each node receives a fixed amount of cash (posi…

加性增加与乘性减少的马尔可夫调制过程的退出问题

Exit problems for additive-increase and multiplicative-decrease Markov-modulated processes

研究有限状态马尔可夫调制加性增加乘性减少过程的一侧、两侧及反射退出问题，通过首次跳跃分解、拉普拉斯变换和递归方法推导退出矩阵，并构造相关杜布鞅。

时间变换反射扩散：亚扩散、超扩散与爆炸现象

Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion

本文研究广义抛物域内受独立逆次ordinator时间变换驱动的斜反射扩散，强化了相关增长律，得到二阶矩渐近，分析了不同参数下的扩散 regime 与爆炸条件，确立了强存在性等性质。

Estimates of Dirichlet heat kernel for symmetric Markov processes

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.

On a class of Time-fractional Continuous-state Branching Processes

We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton--Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general mo…

On the exact asymptotics of exit time from a cone of an isotropic $α$-self-similar Markov process with a skew-product structure

In this paper we identify the asymptotic tail of the distribution of the exit time $τ_C$ from a cone $C$ of an isotropic $α$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component $Θ_t$. Under some additional regularity assumptions, the angular process $Θ_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asympto…

随机时间变化莱维粒子系统极值的研究

Maxima of stationary systems of randomly time-changed Lévy particles

本文研究了随机时间变化莱维粒子系统的极值，构建了无限维的平稳max-infinitely divisible过程类{Z^α}，通过状态依赖的时间变化扩展了经典Brown-Resnick过程的依赖结构，结合潜势理论和极值理论提出了新的平稳过程类。

关于马尔可夫链、擦除环路随机游走与环路汤的研究

A Study in Markov Chains, Loop-Erased Random Walk and Loop Soups

本文探讨了马尔可夫链、擦除环路随机游走及环路汤的相关概念，定义了带“杀灭”的暂态与常返链、格林函数等，揭示了LERW与UST的关系，并引入环路测度与“生长环路”模型。

Most transient random walks have infinitely many cut times

We prove that if $(X_n)_{n\geq 0}$ is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then $(X_n)_{n\geq 0}$ has infinitely many cut times almost surely. This condition applies in particular to any graph of spectral dimension strictly larger than $2$. In fact, our proof applies to general (possibly nonreversible) Markov chains satisfying a similar decay condition for the Green's function that is sharp for birth-death chains. We deduce that a conjecture of Diaconis and Freedman (Ann. Probab. 1980) holds for the same class of …

Heat kernel estimates for subordinate Markov processes and their applications

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.

Approximating Laplace transforms of meeting times for some symmetric Markov chains

We study distributions of meeting times for finite symmetric Markov chains. For Markov kernels defined on large state spaces which satisfy certain weak inhomogeneity in return probabilities of points up to large numbers of steps, we obtain approximation, with explicit error bounds, of the Laplace transforms of some meeting times (without scaling) by ratios of Green functions closely related to hitting times of points. In studying this approximation, we identify some key matrix power series in Markov kernels weighted with solutions to a discrete transport-like equation with explicit coefficient…

Time-changed CIR default intensities with two-sided mean-reverting jumps

The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process $(X,D)$ of a diffusion state variable $X$ driving default intensity and a default indicator process $D$ and time change it with a Lévy subordinator ${\mathcal{T}}$. We characterize the time-changed process $(X^ϕ_t,D^ϕ_t)=(X({\mathcal{T}}_t),D({\mathcal{T}}_t))$ as a Markovian--Itô semimartingale and show from the Doob--Meyer decomposition of $D^ϕ$ that the default time in the time-changed model has a j…

Capacity and Exit Time for Non-reversible Diffusions

Capacity is an important quantity in potential theory and in the study of Markov processes. We give equivalent conditions between the capacity, the mean exit time, and the Green function for non-reversible diffusions.

On stochastic equations with drift in $L_{d}$

For Itô stochastic equations in $\mathbb{R}^{d}$ with drift in $L_{d}$ several results are discussed such as the existence of weak solutions, the existence of the corresponding Markov process, Aleksandrov type estimates of their Green's functions, which yield their summability to the power of $d/(d-1)$, the Fabes-Stroock type estimates which show that Green's functions are summable to a higher degree, the Fanghua Lin type estimates, which are one of the main tools in the $W^{2}_{p}$-theory of fully nonlinear elliptic equations, the fact that Green's functions are in the class $A_{\infty}$ of M…

The Lamperti representation of real-valued self-similar Markov processes

In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative invariant processes. Doing so, we complete Kiu's work [Stochastic Process. Appl. 10 (1980) 183-191], following some ideas in Chybiryakov [Stochastic Process. Appl. 116 (2006) 857-872] in order to characterize the underlying processes in this representation. We provide some examples where the characteristics of the underlying processes can be computed explicitly.

Large deviation expansions for the coefficients of random walks on the general linear group

Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed elements of the general linear group $GL(d, \mathbb R)$. Consider the random walk $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansion for the coefficients $\langle f, G_n v \rangle$, where $f \in (\mathbb R^d)^*$ and $v \in \mathbb R^d$. In particular, our result implies the large deviation principle with an explicit rate function, thus improving significantly the large deviation bounds established earlier. Moreover, we establish Bahadur-Rao-Petrov type …

Global heat kernel estimates for symmetric Markov processes dominated by stable-like processes in exterior $C^{1,η}$ open sets

In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,η}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel intensity $$κ(x, y)ψ(|x-y|)^{-1}|x-y|^{-d-α}$$ where $α\in(0,2)$, $ψ$ is an increasing function on $[ 0, \infty)$ with $ψ(r)=1$ on $0<r\le 1$ and $c_1e^{c_2r^β}\le ψ(r)\le c_3e^{c_4r^β}$ on $r>1$ for $β\in[0, \infty]$. A symmetric function $κ(x, y)$ is bounded by two positive constants and $|κ(x, y)-κ(x,x)|\le c_5 |x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. As a coroll…

Ergodic properties for α-CIR models and a class of generalized Fleming-Viot processes

We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.

Stochastic Modelling with Randomised Markov Bridges

We consider the filtering problem of estimating a hidden random variable $X$ by noisy observations. The noisy observation process is constructed by a randomised Markov bridge (RMB) $(Z_t)_{t\in [0,T]}$ of which terminal value is set to $Z_T=X$. That is, at the terminal time $T$, the noise of the bridge process vanishes and the hidden random variable $X$ is revealed. We derive the explicit filtering formula, governing the dynamics of the conditional probability process, for a general RMB. It turns out that the conditional probability is given by a function of current time $t$, the current obser…

Ultrametric and tree potential

We study infinite tree and ultrametric matrices, and their action on the boundary of the tree. For each tree matrix we show the existence of a symmetric random walk associated to it and we study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. In the boundary, we construct the Markov kernel whose Green function is the extension of the matrix and we simulate it by using a cascade of killing independent exponential random variables and conditionally independent uniform vari…

Fractal Dimensions for Continuous Time Random Walk Limits

In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW scaling limits are time-changed processes that model anomalous diffusion. The outer process describes particle jumps, and the non-Markovian inner process (or time change) accounts for waiting times between jumps. This paper studies fractal properties of the sample functions of a time-changed process, and establishes some general results on the Hausdorff and packing dimensions of its range and graph. Then those results are applied to CTRW scaling limits.

First passage upwards for state dependent-killed spectrally negative Lévy processes

For a spectrally negative Lévy process (snLp) $X$, killed according to a rate that is a function $ω$ of its position, we analyse the exit probability of the one-sided upwards-passage problem. When $ω$ is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for $X$ that has been time-changed by the inverse of the additive functional $\int_0^\cdot ω(X_u)du$. In particular our findings thus shed extra light on related results concerning first passage times upwards (downwards) of spectrally negative positive self-similar Markov …

Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal B}(x,y) |x-y|^{-d-α}$ and killing potentials $κ(x)=cx_d^{-α}$, $0<α<2$. The boundary part ${\mathcal B}(x,y)$ is comparable to the product of three terms with parameters $β_1, β_2$, $β_3$ and $β_4$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $α$, ${\mathcal B}$ and a parameter $p\in ((α-1)_+, α+β_1)…

Isotropic Markov semigroups on ultra-metric spaces

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure μ on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,μ). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Lapla…

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree d, where d is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock, and by using it w…

Inference and rare event simulation for stopped Markov processes via reverse-time sequential Monte Carlo

We present a sequential Monte Carlo algorithm for Markov chain trajectories with proposals constructed in reverse time, which is advantageous when paths are conditioned to end in a rare set. The reverse time proposal distribution is constructed by approximating the ratio of Green's functions in Nagasawa's formula. Conditioning arguments can be used to interpret these ratios as low-dimensional conditional sampling distributions of some coordinates of the process given the others. Hence the difficulty in designing SMC proposals in high dimension is greatly reduced. We illustrate our method on es…

Stochastic stability of the classical Lorenz flow under impulsive type forcing

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in \cite{Ki}, \cite{Ke}, \cite{Me}. Namely, given a cross-section $\mathcal{M}$ for the unperturbed flow, each time the trajectory of the system crosses $\mathcal{M}$ the phase velocity field is changed with a new one sampled at random from a sui…

象限中随机游走格林函数渐近行为的分类

A Classification of Asymptotic Behaviors of Green Functions of Random Walks in the Quadrant

本文对象限中部分齐次随机游走格林函数的渐近行为进行了完整分类，确定了八个参数区域决定极限行为，并扩展了马丁边界和不变分布的极限定理。

Random walk with heterogeneous sojourn time

We introduce a discrete-time random walk model on a one-dimensional lattice with a nonconstant sojourn time and prove that the discrete density converges to a solution of a continuum diffusion equation. Our random walk model is not Markovian due to the heterogeneity in the sojourn time, in contrast to a random walk model with a nonconstant walk length. We derive a Markovian process by choosing appropriate subindexes of the time-space grid points, and then show the convergence of its discrete density through the parabolic-scale limit. We also find the Green's function of the continuum diffusion…