scholarly journals A unified approach for drawdown (drawup) of time-homogeneous Markov processes

2017 ◽  
Vol 54 (2) ◽  
pp. 603-626 ◽  
Author(s):  
David Landriault ◽  
Bin Li ◽  
Hongzhong Zhang
Keyword(s):  
Markov Processes ◽  
Financial Risk ◽  
First Passage Time ◽  
Passage Time ◽  
Regularity Conditions ◽  
Unified Approach ◽  
First Passage ◽  
Underlying Process ◽  
Short Time ◽  
General Time

AbstractDrawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

Identifying Coefficients in the Spectral Representation for First Passage Time Distributions

1987 ◽  
Vol 1 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Mark Brown ◽  
Yi-Shi Shao
Keyword(s):  
Markov Processes ◽  
Time Distribution ◽  
Infinitesimal Generator ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Spectral Approach ◽  
Generator Matrix ◽  
Eigenvalues And Eigenvectors ◽  
First Passage

The spectral approach to first passage time distributions for Markov processes requires knowledge of the eigenvalues and eigenvectors of the infinitesimal generator matrix. We demonstrate that in many cases knowledge of the eigenvalues alone is sufficient to compute the first passage time distribution.

scholarly journals Direct detection of molecular intermediates from first-passage times

2020 ◽  
Vol 6 (18) ◽  
pp. eaaz4642 ◽  
Author(s):  
Alice L. Thorneywork ◽  
Jannes Gladrow ◽  
Yujia Qing ◽  
Marc Rico-Pasto ◽  
Felix Ritort ◽  
...  
Keyword(s):  
Direct Detection ◽  
First Passage Time ◽  
Passage Time ◽  
Energy Landscapes ◽  
First Passage ◽  
Dna Hairpins ◽  
Experimental Systems ◽  
Intermediate States ◽  
Underlying Mechanisms ◽  
Short Time

All natural phenomena are governed by energy landscapes. However, the direct measurement of this fundamental quantity remains challenging, particularly in complex systems involving intermediate states. Here, we uncover key details of the energy landscapes that underpin a range of experimental systems through quantitative analysis of first-passage time distributions. By combined study of colloidal dynamics in confinement, transport through a biological pore, and the folding kinetics of DNA hairpins, we demonstrate conclusively how a short-time, power-law regime of the first-passage time distribution reflects the number of intermediate states associated with each of these processes, despite their differing length scales, time scales, and interactions. We thereby establish a powerful method for investigating the underlying mechanisms of complex molecular processes.

A computational approach to first-passage-time problems for Gauss–Markov processes

2001 ◽  
Vol 33 (2) ◽  
pp. 453-482 ◽  
Author(s):  
E. Di Nardo ◽  
A. G. Nobile ◽  
E. Pirozzi ◽  
L. M. Ricciardi
Keyword(s):  
Markov Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Computational Approach ◽  
First Passage ◽  
First Passage Time Problems

scholarly journals A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 279
Author(s):  
Enrica Pirozzi
Keyword(s):  
Probability Density ◽  
Markov Processes ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
First Passage ◽  
Transition Probability Density ◽  
Symmetry Properties ◽  
Time Density

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

Ageing first-passage times of Markov processes: a matrix approach

1997 ◽  
Vol 34 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Haijun Li ◽  
Moshe Shaked
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Critical Value ◽  
First Passage Times ◽  
Matrix Approach ◽  
First Passage ◽  
Passage Times

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

A computational approach to first-passage-time problems for Gauss–Markov processes

2001 ◽  
Vol 33 (2) ◽  
pp. 453-482 ◽  
Author(s):  
E. Di Nardo ◽  
A. G. Nobile ◽  
E. Pirozzi ◽  
L. M. Ricciardi
Keyword(s):  
Markov Processes ◽  
Brownian Bridge ◽  
First Passage Time ◽  
Accurate Method ◽  
Passage Time ◽  
Computational Approach ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
First Passage Time Problems

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

scholarly journals First-passage time of Markov processes to moving barriers

1984 ◽  
Vol 21 (4) ◽  
pp. 695-709 ◽  
Author(s):  
Henry C. Tuckwell ◽  
Frederic Y. M. Wan
Keyword(s):  
Differential Equation ◽  
Markov Processes ◽  
First Passage Time ◽  
Exit Time ◽  
Passage Time ◽  
First Exit Time ◽  
First Passage ◽  
Sample Paths ◽  
First Exit Times ◽  
First Passage Time Problems

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

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