The first-passage density of a continuous gaussian process to a general boundary

Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.

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Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Advances in Applied Probability ◽

10.1017/s0001867800011952 ◽

2002 ◽

Vol 34 (04) ◽

pp. 869-887 ◽

Cited By ~ 7

Author(s):

Axel Lehmann

Keyword(s):

Integral Equation ◽

First Passage Time ◽

Passage Time ◽

Transition Function ◽

Moving Boundaries ◽

First Passage ◽

Uhlenbeck Process ◽

Sample Paths ◽

Time Density ◽

Strong Markov

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

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Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Advances in Applied Probability ◽

10.1239/aap/1037990957 ◽

2002 ◽

Vol 34 (4) ◽

pp. 869-887 ◽

Cited By ~ 11

Author(s):

Axel Lehmann

Keyword(s):

Integral Equation ◽

First Passage Time ◽

Passage Time ◽

Transition Function ◽

Moving Boundaries ◽

First Passage ◽

Uhlenbeck Process ◽

Sample Paths ◽

Time Density ◽

Strong Markov

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

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Extension of Wald's first lemma to Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200016831 ◽

1999 ◽

Vol 36 (01) ◽

pp. 48-59 ◽

Cited By ~ 1

Author(s):

George V. Moustakides

Keyword(s):

Integral Equation ◽

Markov Process ◽

Markov Processes ◽

Correction Term ◽

Stopping Time ◽

Poisson Integral ◽

Homogeneous Markov Process ◽

Partial Sum ◽

Poisson Integral Equation ◽

Scalar Nonlinearity

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let S n denote the partial sum S n = θ(ξ1) + … + θ(ξ n ), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼S N = [lim n→∞𝔼θ(ξ n )]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξ N ). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξ N ) = o(𝔼N).

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Hida-Cramér Multiplicity Theory for Multiple Markov Processes and Goursat Representations

Nagoya Mathematical Journal ◽

10.1017/s0027763000016639 ◽

1975 ◽

Vol 57 ◽

pp. 199-228 ◽

Cited By ~ 22

Author(s):

Loren D. Pitt

Keyword(s):

Markov Process ◽

Gaussian Process ◽

Markov Processes ◽

Representation Theorem ◽

Multiplicity One ◽

Degeneracy Condition ◽

False Result

This work grew out of an attempt to prove the false result that an n-ple Markov process in the sense of Hida (1960) or Lévy (1956a) has multiplicity one. Instead we proved the representation theorem (Theorem III. 1.) that a centered Gaussian process x(t) is n-ple Markov iff it can be written in the form(I.1) where is a Gaussian martingale with(I.2) sp {x(s): s ≤ t} ≡ sp {ai(s): s ≤ t and 1 ≤ i ≤ n}and A(t) and {ei(t)} satisfy some non-degeneracy condition. We also show (Corollary IV. 13.) that for any Gaussian martingale A(t) with simple left innovation spectrum, continuous ei(t) may be found so that the process x(t) given in (I.1) will satisfy (I.2).

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On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mathematics ◽

10.3390/math8112031 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2031

Author(s):

Mario Abundo ◽

Enrica Pirozzi

Keyword(s):

Dynamical System ◽

Stochastic Process ◽

Brownian Motion ◽

Markov Process ◽

Markov Processes ◽

Numerical Approximation ◽

Uhlenbeck Process ◽

Sample Paths ◽

Ornstein Uhlenbeck Process ◽

Application Fields

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

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Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

Journal of Applied Probability ◽

10.2307/3215298 ◽

1995 ◽

Vol 32 (2) ◽

pp. 429-442

Author(s):

A. N. Balabushkin

Keyword(s):

Brownian Motion ◽

Gaussian Process ◽

Gaussian Processes ◽

Simple Approximation ◽

Minimum Point ◽

Second Derivative ◽

Numerical Examples ◽

Exit Probability ◽

Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

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On an integral equation for first-passage-time probability densities

Journal of Applied Probability ◽

10.1017/s0021900200024694 ◽

1984 ◽

Vol 21 (02) ◽

pp. 302-314 ◽

Cited By ~ 19

Author(s):

L. M. Ricciardi ◽

L. Sacerdote ◽

S. Sato

Keyword(s):

Integral Equation ◽

Diffusion Process ◽

Continuous Time ◽

First Passage Time ◽

Passage Time ◽

Time Function ◽

Standard Wiener Process ◽

First Passage ◽

Probability Densities ◽

Bounded Derivative

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

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Distributional Properties of Fluid Queues Busy Period and First Passage Times

Mathematics ◽

10.3390/math8111988 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1988

Author(s):

Zbigniew Palmowski

Keyword(s):

Markov Process ◽

Busy Period ◽

First Passage Time ◽

Passage Time ◽

Fluid Queue ◽

First Passage ◽

Fluid Queues ◽

Distributional Properties ◽

Finite State ◽

Passage Times

In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has a IFR distribution and the busy period in the Anick-Mitra-Sondhi model has a DFR distribution.

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First-passage time for a particular stationary periodic Gaussian process

Journal of Applied Probability ◽

10.1017/s002190020004897x ◽

1976 ◽

Vol 13 (01) ◽

pp. 27-38 ◽

Cited By ~ 4

Author(s):

L. A. Shepp ◽

D. Slepian

Keyword(s):

Gaussian Process ◽

Infinite Series ◽

First Passage Time ◽

Passage Time ◽

Numerical Calculations ◽

Time Interval ◽

Two Dimensional ◽

First Passage ◽

Gaussian Density ◽

First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x 0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

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