Hitting Times for Diffusion Processes and Stochastic Models in Insurance

2013 ◽  
pp. 177-218
Author(s):  
Jacques Janssen ◽  
Oronzio Manca ◽  
Raimondo Manca
Keyword(s):  
Stochastic Models ◽  
Diffusion Processes ◽  
Hitting Times

scholarly journals Quantification of model uncertainty on path-space via goal-oriented relative entropy

2020 ◽  
Author(s):  
Jeremiah Birrell ◽  
Markos A. Katsoulakis ◽  
Luc Rey-Bellet
Keyword(s):  
Relative Entropy ◽  
Stochastic Models ◽  
Model Uncertainty ◽  
Diffusion Processes ◽  
Path Space ◽  
Hitting Times ◽  
Ergodic Averages ◽  
Information Theoretic ◽  
The Impact ◽  
Quantities Of Interest

Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, e.g., for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.

scholarly journals Limit theorems for hitting times of 1-dimensional generalized diffusions

1998 ◽  
Vol 152 ◽  
pp. 1-37
Author(s):  
Matsuyo Tomisaki ◽  
Makoto Yamazato
Keyword(s):  
Limit Theorems ◽  
Bessel Functions ◽  
Diffusion Processes ◽  
Hitting Times ◽  
Finite Dimensional ◽  
Independent Increments ◽  
Starting Point ◽  
Self Similar ◽  
The Mean ◽  
The One

Abstract.Limit theorems are obtained for suitably normalized hitting times of single points for 1-dimensional generalized diffusion processes as the hitting points tend to boundaries under an assumption which is slightly stronger than that the existence of limits γ + 1 of the ratio of the mean and the variance of the hitting time. Laplace transforms of limit distributions are modifications of Bessel functions. Results are classified by the one parameter {γ}, each of which is the degree of corresponding Bessel function. In case the limit distribution is degenerate to one point, by changing the normalization, we obtain convergence to the normal distribution. Regarding the starting point as a time parameter, we obtain convergence in finite dimensional distributions to self-similar processes with independent increments under slightly stronger assumption.

scholarly journals Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 155
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Stochastic Models ◽  
Maximum Likelihood Method ◽  
Diffusion Processes ◽  
Oscillatory Behavior ◽  
Likelihood Method ◽  
Systems Of Equations

Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is addressed and a strategy is established for this purpose. Finally, a simulation study is carried out.

scholarly journals On the Transient Behavior of Ehrenfest and Engset Processes

2012 ◽  
Vol 44 (02) ◽  
pp. 562-582 ◽  
Author(s):  
Mathieu Feuillet ◽  
Philippe Robert
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Heat Exchange ◽  
Communication Networks ◽  
Stochastic Models ◽  
Transient Behavior ◽  
Hitting Times ◽  
Kinetic Theory Of Gases ◽  
Two Bodies ◽  
Number Of Particles

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

scholarly journals Two Multi-Sigmoidal Diffusion Models for the Study of the Evolution of the COVID-19 Pandemic

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2409
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Juan José Serrano-Pérez ◽  
Francisco Torres-Ruiz
Keyword(s):  
Bayesian Methods ◽  
Stochastic Models ◽  
Diffusion Processes ◽  
Point Estimation ◽  
Practical Case ◽  
Polynomial Functions ◽  
First Passage ◽  
Specific Data ◽  
Passage Times

A proposal is made to employ stochastic models, based on diffusion processes, to represent the evolution of the SARS-CoV-2 virus pandemic. Specifically, two diffusion processes are proposed whose mean functions obey multi-sigmoidal Gompertz and Weibull-type patterns. Both are constructed by introducing polynomial functions in the ordinary differential equations that originate the classical Gompertz and Weibull curves. The estimation of the parameters is approached by maximum likelihood. Various associated problems are analyzed, such as the determination of initial solutions for the necessary numerical methods in practical cases, as well as Bayesian methods to determine the degree of the polynomial. Additionally, strategies are suggested to determine the best model to fit specific data. A practical case is developed from data originating from several Spanish regions during the first two waves of the COVID-19 pandemic. The determination of the inflection time instants, which correspond to the peaks of infection and deaths, is given special attention. To deal with this particular issue, point estimation as well as first-passage times have been considered.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (01) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

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