Laurent Decreusefond, David Nualart
We establish a general formula for the Laplace transform of the hitting times of a Gaussian process. Some consequences are derived, and particular cases like the fractional Brownian motion are discussed.
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Hitting time for one class of Gaussian proccesses
本文针对非Markov的Gaussian integrators，利用二次量子化方法求得其命中概率的级数表示。
In this short article, we will focus on the different links between some stochastic processes resulting from Brownian motion and two notions of probability theory (proportional increments and last hitting times).
For a certain class of asymmetric Lévy processes where the origin is regular for itself, the renormalized zero resolvent is proved to be harmonic for the killed process upon hitting zero.
We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $γ$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.
Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable Lévy processes. Itô's excursion theory plays a key role in this study.
We are concerned with the first hitting times of the Bessel processes. We give explicit expressions for the densities by means of the zeros of the Bessel functions and show their asymptotic behavior.
We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional $α$-stable processes $X_t$, where $α\in(1,2)$. We also find a spectral-type integral formula for the transition operators $P_0^t$ of $X_t$ killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for $P_0^t$.
A multiplicative identity in law connecting the hitting times of completely asymmetric $α-$stable Lévy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute fractional moments and to get series representations for the density. We also prove that the hitting times are unimodal as soon as $α\le 3/2.$ Analogous results are obtained, in a much simplified manner, for the first passage time across a positive level.
For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function of the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove optimal estimates of the transition density of the process killed after hitting B.
We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions and for the densities by means of the zeros of the Bessel functions. The results extend the classical ones and cover all the cases.
The asymptotic behavior of the tail probabilities for the first hitting times of the Bessel process with arbitrary index is shown without using the explicit expressions for the distribution function obtained in the authors' previous works.
We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The latter estimates are obtained from more general estimates of moments and tails established for convolutions of elementary mixtures of exponential distributions.
Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition inf z Psi"(z) / Psi'(z) > 0 is imposed on the Lévy-Khintchine exponent Psi. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin.
For birth-and-death processes, we show that every initial distribution is reproduced from the first hitting time distribution. The reproduction is done by applying to the distribution function a differential operator defined through the eigenfunction of the generator. Using the spectral theory for generalized second-order differential operators, we study asymmetric random walks.
The paper presents new asymptotic recurrent algorithms of phase space reduction for regularly and singularly perturbed semi-Markov processes. These algorithms give effective conditions of weak convergence for distributions and convergence of expectations for hitting times as well as recurrent formulas for computing the corresponding normalisation functions, Laplace transforms for limiting distributions and limits for expectations.
In this paper we pursue and complete the study of the simulation of the hitting time of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work, the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method was based mainly on the explicit formula for the distribution of the hitting time and on the connexion between the Bessel process and the Euclidean norm of the Brownian motion. This method does not apply anymore for a non inte…
We proved the explicit formulas in Laplace transform of the hitting times for the birth and death processes on a denumerable state space with $\ift$ the exit or entrance boundary. This extends the well known Keilson's theorem from finite state space to infinite state space. We also apply these formulas to the fastest strong stationary time for strongly ergodic birth and death processes, and obtain the explicit convergence rate in separation.
The probability that a max-stable process η in C[0, 1] with identical marginal distribution function F hits x \in R with 0 < F (x) < 1 is the hitting probability of x. We show that the hitting probability is always positive, unless the components of η are completely dependent. Moreover, we consider the event that the paths of standard MSP hit some x \in R twice and we give a sufficient condition for a positive probability of this event.
We show that the hitting times for points of real $α-$stable Lévy processes ($1<α\le 2$) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al., whereas in the completely asymmetric case we apply an identity of the second author. The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al.
For a scalar Gaussian process $B$ on $\mathbb{R}_{+}$ with a prescribed general variance function $γ^{2}\left(r\right) =\mathrm{Var}\left(B\left(r\right) \right) $ and a canonical metric $\mathrm{E}[\left(B\left(t\right) -B\left(s\right) \right) ^{2}]$ which is commensurate with $γ^{2}\left(t-s\right) $, we estimate the probability for a vector of $d$ iid copies of $B$ to hit a bounded set $A$ in $\mathbb{R}^{d}$, with conditions on $γ$ which place no restrictions of power type or of approximate self-similarity, assuming only that $γ$ is continuous, increasing, and concave, with $γ\left(0\righ…
We investigate the convergence of hitting times for jump-diffusion processes. Specifically, we study a sequence of stochastic differential equations with jumps. Under reasonable assumptions, we establish the convergence of solutions to the equations and of the moments when the solutions hit certain sets.
We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral condition \[ \int_0^{\infty} \frac{dξ}{1+\operatorname{Re} ψ(ξ)}<\infty. \] To this end, we first prove and then apply the global scale invariant Harnack inequality. Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processs which satisfy these assumptions.
Quasi-stationary distributions for time-changed symmetric $α$-stable processes killed upon hitting zero
该研究针对从无穷远进入、击中零点即被吸收的时间变换对称$\alpha$-稳定过程，证明了其拟平稳分布的存在唯一性，并在转移密度相关条件下证明任意初始分布指数收敛到该分布。
In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that e…
The Poisson process of order $i$ is a weighted sum of independent Poisson processes and is used to model the flow of clients in different services. In the paper below we study some extensions of this process, for different forms of the weights and also with the time-changed versions, with Bern\v stein subordinator playing the role of time. We focus on the analysis of hitting times of these processes obtaining sometimes explicit distributions. Since all the processes examined display a similar structure with multiple upward jumps sometimes they can skip all states with positive probability even…
We consider a system of $d$ non-linear stochastic fractional heat equations in spatial dimension $1$ driven by multiplicative $d$-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of $(u(s, y), u (t, x))$. From this result, we deduce optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualar…
In this article, we consider the properties of hitting times for $G$-martingale and the stopped processes. We prove that the stopped processes for $G$-martingales are still $G$-martingales and that the hitting times for a class of $G$-martingales including $G$-Brownian motion are quasi-continuous. As an application, we improve the $G$-martingale representation theorems in [Song10].
Hitting Probabilities of Finite Points for One-Dimensional Lévy Processes
本文研究了一维莱维过程中首次击中有限点集的概率，并推导了轨迹过程$Q$-矩阵的显式表达式。
We study the two-dimensional joint distribution of the first hitting time of a constant level by a continuous-state branching process with immigration and their primitive stopped at this time. We show an explicit expression of its Laplace transform. Using this formula, we study the polarity of zero and provide a necessary and sufficient criterion for transience or recurrence. We follow the approach of Shiga, T. (1990) [A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type. Probability Theory and Related Fields, 85(4), 425-447], by finding some $λ$-invariant functions for the g…
Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we…
We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the L{é}vy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.
We study a class of self-similar jump type SDEs driven by Hölder-continuous drift and noise coefficients. Using the Lamperti transformation for positive self-similar Markov processes we obtain a necessary and sufficient condition for almost sure extinction in finite time. We then show that for certain parameters pathwise uniqueness holds in a restricted sense, namely among solutions spending a Lebesgue-negligible amount of time at 0. A direct power transformation plays a key role.
Let $T_1^{(μ)}$ be the first hitting time of the point 1 by the Bessel process with index $μ\in \R$ starting from $x>1$. Using an integral formula for the density $q_x^{(μ)}(t)$ of $T_1^{(μ)}$, obtained in Byczkowski, Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the density of $T_1^{(μ)}$ which exibit the dependence both on time and space variables. Our result provides optimal estimates for the density of the hitting time of the unit ball by the Brownian motion in $\mathbb{R}^n$, which improve existing bounds. Another application is to provide sharp estimates for the …
Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes $(Z_{t}=X_{t}+iY_{t},t\geq0)$ including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:$(β_{t},t\geq0)$: for any fixed $u>0$: \sinh(β_{u}) \stackrel{(law)}{=} \hatβ_{(\int^{u}_{0}ds\exp(2β_{s}))}. These identities in law for 2-dimensional processes allow to study the distributions of hitting times $T^θ_{c}\equiv\inf\{t:θ_{t} =c \}, (c>0)$, $T^θ_{-d,c}\equiv\inf\{t:θ_{t}\notin(-d,c) \}, (c,d>0)$ and more specifically of $T^θ_{-c,…
The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors show that the mixing times and "worst-case" hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chai…
We begin with the study of some properties of the radial Dunkl process associated to a reduced root system $R$. It is shown that this diffusion is the unique strong solution for all $t \geq 0$ of a SDE with singular drift. Then, we study $T_0$, the first hitting time of the positive Weyl chamber : we prove, via stochastic calculus, a result already obtained by Chybiryakov on the finiteness of $T_0$. The second and new part deals with the law of $T_0$ for which we compute the tail distribution, as well as some insight via stochastic calculus on how root systems are connected with eigenvalues of…
Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that the hitting probabilities of this additive process at lower levels have a matrix exponential form. We here study the hitting probabilities at upper levels, which do not have a matrix exponential form in general. These probabilities give the ruin probab…
In a recent paper in the Journal of Theoretical Probability Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor's classical results on first-hitting times of a birth-death process on the nonnegative integers by establishing a representation for the Laplace transform $\mathbb{E}[e^{sT_{ij}}]$ of the first-hitting time $T_{ij}$ for $any$ pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{sT_{ij}}]$ when either $i$ or $j$ tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-pol…
We generalize the notion of strong stationary time and we give a representation formula for the hitting time to a target set in the general case of non-reversible Markov processes.
In this article we continue the study of the quenched distributions of transient, one-dimensional random walks in a random environment. In a previous article we showed that while the quenched distributions of the hitting times do not converge to any deterministic distribution, they do have a weak weak limit in the sense that - viewed as random elements of the space of probability measures - they converge in distribution to a certain random probability measure (we refer to this as a weak weak limit because it is a weak limit in the weak topology). Here, we improve this result to the path-valued…
For a finite state Markov process and a finite collection $\{ Γ_k, k \in K \}$ of subsets of its state space, let $τ_k$ be the first time the process visits the set $Γ_k$. We derive explicit/recursive formulas for the joint density and tail probabilities of the stopping times $\{ τ_k, k \in K\}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.
We consider two depending Wiener processes which have membranes at zero with different permeability coefficients. Starting from different points, the processes almost surely do not meet at any fixed point except that where membranes are situated. The necessary and sufficient conditions for the meeting of the processes are found. It is shown that the probability of meeting is equal to zero or one.
Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function $γ^2(r)=\operatorname{Var}\left(B_0(r)\right)$ and a canonical metric $δ(t,s):=(\mathbb{E}\left(B_0(t)-B_0(s)\right)^2)^{1/2}$ which is commensurate with $γ(t-s)$. We provide some general condition on $γ$ so that for any Borel set $E\subset [0,1]$, the Hausdorff dimension of the image $B(E)$ is constant a.s., and we explicit this constant. Also, we derive under some mild assumptions on $γ\,$ an upp…
Sharp approximation and hitting times for stochastic invasion processes
本文研究相互作用随机过程的突变入侵阶段，通过更精细的逼近刻画介观阶段及阈值到达时间的大方差，可用于癌症和流行病临界时间分析。
We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $δ$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and $δ$, provided $δ\le 0$. As an application, we show that the SLE($κ$) welding homeomorphism is continuous in $κ$ for $κ\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_κ$ in $κ$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $δ>…
Consider the high-order heat-type equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and minimum $m(t)$ up to time $t$; the hitting times $τ_a^+$ and $τ_a^-$ of the half lines $(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit expressions for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$, as well as those of the vectors $(τ_a^+,X(τ_a^+))$ and $(τ_a^-,X(τ_a^-))$.
We study the problem of characterizing the expected hitting times for a robust generalization of continuous-time Markov chains. This generalization is based on the theory of imprecise probabilities, and the models with which we work essentially constitute sets of stochastic processes. Their inferences are tight lower- and upper bounds with respect to variation within these sets. We consider three distinct types of these models, corresponding to different levels of generality and structural independence assumptions on the constituent processes. Our main results are twofold; first, we demonstrat…
There exists a simple, didactically useful one-to-one relationship between stopping times and adapted càglàd (LCRL) processes that are non-increasing and take the values 0 and 1 only. As a consequence, stopping times are always hitting times.
In this paper we consider a (reflected) Brownian motion with broken drift hitting a random boundary. Some dedicated calculations allow us to obtain the formula on the joint Laplace transform of the hitting time and hitting position. These develop the research on first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes by Perry et al. (2004).
Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.
In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using aga…
Spectral theory for the transition semigroup of one-dimensional symmetric Levy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction expansion of transition operators and the generator is given. As an application, integral fomulae for the transition density and the distribution of the hitting time of the origin are proved.
Generally the convergence rate in exponential ergodicity $λ$ is an upper bound for the convergence rate $κ$ in uniform ergodicity for a Markov process, that is $λ\geqslantκ$. In this paper, we prove that $κ\geqslant \inf \{lambda,1/M_H\}$, where $M_H$ is a uniform bound on the moment of the hitting time to a "compact" set $H$. In the case where $M_H$ can be made arbitrarily small for $H$ large enough, we obtain that $λ=κ$. The general results are applied to Markov chains, diffusion processes and solutions to SDEs driven by symmetric stable processes.
In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of…
New algorithms for computing power moments of hitting times and accumulated rewards of hitting type for semi-Markov processes. The algorithms are based on special techniques of sequential phase space reduction and recurrence relations connecting moments of rewards. Applications are discussed as well as possible generalisations of presented results and results of experimental studies.
In the boundary driven symmetric simple exclusion process consider an open set $\ms O$ of density profiles which does not contain the stationary density profile. We prove that the first time the empirical measure visits the set $\ms O$ converges to an exponential distribution.
Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu, rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) :=E(e^(-theta T_x - mu K_x ρL_x) 1_(T_x<+infinity)), where theta>=0, mu>=0, rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has finite exponential moments we exhibit an asymptotic expansion for F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At la…
In this paper we are concerned with hitting times of a family of density-dependent Markov chains. A moderate deviation principle of the hitting time is given. The proof of the main theorem relies heavily on moderate deviations of density-dependent Markov chains given in \cite{Xue2021} and upper bounds of large deviations of Markov processes given in \cite{Dupuis1991}. An analogue moderate deviation of the hitting time of the diffusion approximation of the density-dependent Markov chain introduced in \cite{Ethier1986} is also given.
For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are $ h $-transforms of the process stopped upon hitting zero, where $ h $'s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the $ h $-transforms are investigated.
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