Randomization in the first hitting time problem

In this paper, we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F(t), and a linear boundary, b(t)=μt, find a distribution of the initial state such that the distribution of the first hitting time is F(t). This problem has important applications in credit risk modeling where the process represents the so-called distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.     

Inverse Process with Independent Positive Increments: Finite-Dimensional Distributions

An inverse process with independent positive increments is considered. For such a process, the first hitting time τ_x of level x as a function of x ≥ 0 is a proper process with independent positive increments. In terms of first hitting times and their Levy measures, multidemensional distribution densities and Laplace transformations are derived. Stationary distributions of increments of the process are investigated. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 286–297.     

Complete matchings in random subgraphs of the cube

In this article it is shown that for almost every random cube process the hitting time of a complete matching equals the hitting time of having minimal degree (at least) one and also the hitting time of connectedness. It follows from this that if t = (n + c + o(1))2^n?2 for some constant c, then the probability that a random subgraph of the n-cube having precisely t edges has a complete matching tends to e.     

Discrete Quantum Walks Hit Exponentially Faster

This paper addresses the question: what processes take polynomial time on a quantum computer that require exponential time classically? We show that the hitting time of the discrete time quantum walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum walks. We provide the basic framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We outline a possible application to sequential packet routing.     

Hitting Time Problems of Sticky Brownian Motion and Their Applications in Optimal Stopping and Bond Pricing

This paper investigates the hitting time problems of sticky Brownian motion and their applications in optimal stopping and bond pricing. We study the Laplace transform of first hitting time over the constant and random jump boundary, respectively. The results about hitting the constant boundary serve for solving the optimal stopping problem of sticky Brownian motion. By introducing the sharpo ratio, we settle the bond pricing problem under sticky Brownian motion as well. An interesting result shows that the sticky point is in the continuation region and all the results we get are in closed form.

     

Joint distributions of first hitting time and first hitting location after explosion for birth and death processes

For a birth and death processX=|X(t),t <σ| with explosion and lifespanu distributions and joint distributions of first hitting time and first hitting location after explosion of setB _n = |0,1,...,n| ,n have been found.     

Some Properties of CIR Processes

Abstract

This article derives some properties of variants of squared Bessel processes known as CIR processes in the finance literature. We derive the transition probability density function of a square-root process and compute the resolvent density of CIR processes. As a consequence, we derive the density of CIR processes sampled at an independent exponential time. Moreover, we derive explicit expressions of the Laplace transforms (LTs) of first hitting times by martingale methods.     

Hitting time fork edge-disjoint spanning trees in a random graph

We show that in almost every random graph process, the hitting time for havingk edge-disjoint spanning trees equals the hitting time for having minimum degreek.     

Hitting, returning and the short correlation function

We consider a stochastic process with the weakest mixing condition: the so called α. For any fixed n-string we prove the following results. (1) The hitting time has approximately exponential law. (2) The return time has approximately a convex combination between a Dirac measure at the origin and an exponential law. In both cases the parameter of the exponential law is λ(A)ℙ(A) where ℙ(A) is the measure of the string and λ(A) is a certain autocorrelation function of the string. We show also that the weight of the convex combination is approximately λ(A). We describe the behavior of this autocorrelation function. Our results hold when the rate of mixing decays polinomially fast with power larger than the golden number.     

Hitting Half-spaces by Bessel-Brownian Diffusions

The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ ^n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ) ^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.     

First Hitting Time Analysis of the Independence Metropolis Sampler

In this paper, we study a special case of the Metropolis algorithm, the Independence Metropolis Sampler (IMS), in the finite state space case. The IMS is often used in designing components of more complex Markov Chain Monte Carlo algorithms. We present new results related to the first hitting time of individual states for the IMS. These results are expressed mostly in terms of the eigenvalues of the transition kernel. We derive a simple form formula for the mean first hitting time and we show tight lower and upper bounds on the mean first hitting time with the upper bound being the product of two factors: a “local” factor corresponding to the target state and a “global” factor, common to all the states, which is expressed in terms of the total variation distance between the target and the proposal probabilities. We also briefly discuss properties of the distribution of the first hitting time for the IMS and analyze its variance. We conclude by showing how some non-independence Metropolis–Hastings algorithms can perform better than the IMS and deriving general lower and upper bounds for the mean first hitting times of a Metropolis–Hastings algorithm.     

The hitting time for a Cox risk process

This paper investigates the hitting time of a Cox risk process. The relationship between the hitting time of the Cox risk process and the classical risk process is established and an explicit expression of the Laplace–Stieltjes transform of the hitting time is derived by the probability method. Similarly, we derive the explicit expression of the Laplace–Stieltjes transform of the last exit time. Further, we study the situation when the intensity process is an n$n$-state Markov process.     

Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs

We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k-dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment-generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

     

Hitting time results for Maker‐Breaker games

We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \begin{align*}\mathcal{P}\end{align*}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojakovi? and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A Fourier-Type Transform on the Semiaxis with an Arbitrary Phase

Integral equations on the semiaxis with kernels having the form of a linear combination of the Fourier sine and cosine transforms with arbitrary variable complex coefficients are considered. For the case in which the coefficients depend on only one variable, exact solutions are presented. Various generalizations and applications to integral equations are given.

     

Diffusion processes with delay at the endpoints of a segment

A continuous semi-Markov process with a segment as the range of values is considered. This process coincides with a diffusion process inside the segment, i.e., up to the first hitting time of the boundary of the segment and at any time when the process leaves the boundary. The class of such processes consists of Markov processes with reflection at the boundaries (instantaneously or with a delay) and semi-Markov processes with intervals of constancy on some boundary. We derive conditions of existence of such a process in terms of a semi-Markov transition generating function on the boundary. The method of imbedded alternating renewal processes is applied to find a stationary distribution of the process. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 284–297.     

Ruin problem for a generalized Poisson process with reflection

We consider a generalized Poisson process with reflection at the level T > 0. Under certain conditions on the distribution of the values of positive jumps of the process, we obtain representations for the characteristic functions of functionals associated with the exit of the indicated process to the negative semiaxis. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1465–1475, November, 2005.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

马氏链的离散分布(Ⅱ)

设X(ω)={x(t,ω), t≥0}是定义在完备概率空间(Ω,,p)上的马氏链。其状态空间1={0,1,2,…}。如不作特别声明都假定X(ω)具有标准转移矩阵,完全可分,Borel可测,状态稳定。令     

Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

The paper considers a particular family of fuzzy monotone set-valued stochastic processes. The proposed setting allows us to investigate suitable α-level sets of such processes, modeling birth-and-growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set-valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.