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.

scholarly journals On the Simulation of a Special Class of Time-Inhomogeneous Diffusion Processes

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 818
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Probability Density ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Probability Density Functions ◽  
Periodic Functions ◽  
Initial State ◽  
Transition Probability Density ◽  
Inhomogeneous Diffusion

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed.

scholarly journals Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 555 ◽  
Author(s):  
Virginia Giorno ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Process ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Time Dependent ◽  
Transition Probability Density ◽  
Special Boundaries ◽  
Special Cases ◽  
Regulation Functions

We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.

scholarly journals First Passage Time Moments of Jump-Diffusions with Markovian Switching

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Jun Peng ◽  
Zaiming Liu
Keyword(s):  
Integral Equation ◽  
Recurrence Relations ◽  
Financial Models ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
Markovian Switching ◽  
First Passage ◽  
Transition Probability Density ◽  
Jump Diffusions

Using an integral equation associated with generalized backward Kolmogorov's equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of jump-diffusions with Markovian switching. The results are used to find the expectation of first passage time of some financial models.

scholarly journals First-passage Time Estimation of Diffusion Processes through Time-Varying Boundaries with an Application in Finance

2016 ◽  
Vol 6 (1) ◽  
pp. 59
Author(s):  
Imene Allab ◽  
Francois Watier
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Time Estimation ◽  
Local Approximation ◽  
Passage Time ◽  
First Passage ◽  
Linear System Of Equations ◽  
Brownian Bridges ◽  
Non Linear System ◽  
Time Density

In this paper, we develop a Monte Carlo based algorithm for estimating the FPT (first passage time) density of the solution of a one-dimensional time-homogeneous SDE (stochastic differential equation) through a time-dependent frontier. We consider Brownian bridges as well as local Daniels curve approximations to obtain tractable estimations of the FPT probability between successive points of a simulated path of the process. Under mild assumptions, a (unique) Daniels curve local approximation caneasily be obtained by explicitly solving a non-linear system of equations.

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.

First-passage-time densities for time-non-homogeneous diffusion processes

1997 ◽  
Vol 34 (3) ◽  
pp. 623-631 ◽  
Author(s):  
R. Gutiérrez ◽  
L. M. Ricciardi ◽  
P. Román ◽  
F. Torres
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Continuous Functions ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

On first passage times of sticky reflecting diffusion processes with double exponential jumps

2020 ◽  
Vol 57 (1) ◽  
pp. 221-236 ◽  
Author(s):  
Shiyu Song ◽  
Yongjin Wang
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Explicit Solutions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Double Exponential ◽  
Upper Level ◽  
First Passage Problem ◽  
Passage Times

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

scholarly journals A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

2011 ◽  
Vol 48 (02) ◽  
pp. 420-434 ◽  
Author(s):  
Peter J. Thomas
Keyword(s):  
Lower Bound ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Condition ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Fire Model ◽  
Integrate And Fire ◽  
Ornstein Uhlenbeck Process ◽  
Time Density

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck processX(t) obeying dX= -βXdt+ σdWto reach a fixed threshold θ from a suprathreshold initial conditionx0> θ > 0 has a lower bound of the form ρ(t) >kexp[-pe6βt] for positive constantskandpfor timestexceeding some positive valueu. We obtain explicit expressions fork,p, anduin terms of β, σ,x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

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