Passage Time Of Brownian Motion Research Articles

On the excursions of drifted Brownian motion and the successive passage times of Brownian motion

By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)= a + bt.$ In the special case when $b = 0,$ we extend the result to a space-time transformation of Brownian motion.

First Passage Time for Brownian Motion and Piecewise Linear Boundaries

We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the first passage time distributions for general nonlinear boundaries. The numerical computation can be easily done by using the Monte Carlo integration, which is straightforward to implement. Some numerical examples are presented for illustration. This approach can be further extended to compute two-sided boundary crossing distributions.

Statistics of the first passage time of Brownian motion conditioned by maximum value or area

We derive the moments of the first passage time for Brownian motion conditioned by either the maximum value or the area swept out by the motion. These quantities are the natural counterparts to the moments of the maximum value and area of Brownian excursions of fixed duration, which we also derive for completeness within the same mathematical framework. Various applications are indicated.

Real-Time Estimation of Mean Remaining Life Using Sensor-Based Degradation Models

Advances in sensor technology have led to an increased interest in using degradation-based sensory information to predict the remaining lives of partially degraded components and systems. This paper presents a stochastic degradation modeling framework for computing and continuously updating remaining life distributions (RLDs) using in situ degradation signals acquired from individual components during their operational lives. Unfortunately, these sensory-updated RLDs cannot be characterized using parametric distributions and their moments do not exist. Such difficulties hinder the implementation of this sensor-based framework, especially from the standpoint of computational efficiency of embedded algorithms. In this paper, we identify an approximate procedure by which we can compute a conservative mean of the sensory-updated RLDs and express the mean and variance using closed-form expressions that are easy to evaluate. To accomplish this, we use the first passage time of Brownian motion with positive drift, which follows an inverse Gaussian distribution, as an approximation of the remaining life. We then show that the mean of the inverse Gaussian is a conservative lower bound of the mean remaining life using Jensen’s inequality. The results are validated using real-world vibration-based degradation information.

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

A reliability application of a mixture of inverse gaussian distributions

AbstractA mixture of inverse Gaussian distributions (IGDs) is examined as a model for the lifetime of components. The components differ in one of three ways: in their initial quality, rate of wear, or variability of wear. These three cases are well represented by the parameters of the IGD model. The mechanistic interpretation of the IGD as the first passage time of Brownian motion with positive drift is adopted. The parameters considered are either dichotomous or continuous random variables. Parameter estimation is also examined for these two cases. The model seems to be most appropriate when the single IGD model fails due to heterogeneity of the initial component quality.

First passage time statistics for some stochastic processes with superimposed shot noise

We present a method for finding statistical properties of the first passage time to exit an interval of general diffusion processes subject to random delta function impulses. Exact solutions are found for the mean first passage time for Brownian motion. Other special cases, detailed in the text, can also be solved in some generality.

The first passage time distribution of brownian motion and its applications

This paper traces the development of the first passage time distribution of Brownian motion (inverse Gaussian) together with its various applications in inventory problems, usage and storage times, detection theory, labor turnover, money supply, purchasing models, hospital inpatient stay times, strikes duration, biology, … etc. Besides, since the inverse Gaussian distribution arises as the distribution of first passage time of Brownian motion, its applicability to lifetime modeling is a natural consequence. The general characteristics of the inverse Gaussian model which make it attractive for various applications are also identified.

First-Passage Time Distribution of Brownian Motion as a Reliability Model

Since the inverse Gaussian distribution arises as the distribution of first passage time of Brownian motion, its applicability to lifetime or survival situations is a natural consequence. The failure rate for the inverse Gaussian distribution first increases until it reaches its maximum value somewhere to the right of the mode, then it decreases monotonically to a non-zero asymptotic value. This paper fits the inverse Gaussian distribution model to observed failure data of high speed steel tools in machining low carbon steel. Then the lognormal distribution model is hypothesized for the same failure data. The inverse Gaussian fits better.