Exit problems for jump processes with applications to dividend problems

The distribution of the time elapsed before a random variable reaches a threshold value for the first time, called the first passage time (FPT) distribution, is a fundamental characteristic of stochastic processes. Here, by solving the standard macroscopic diffusion equation, we derive analytical expressions for the FPT distribution of a diffusing particle hitting a spherical object in two dimensions (2D) and three dimensions (3D) in the course of unrestricted diffusion in open space. In addition, we calculate, analytically, the angular dependence of the FPT, known as the hit distribution. The analytical results are also compared to simulations of the motions of a random walker on a discrete lattice. This topic could be of wide pedagogical interest because the FPT is important not only in physics but also in chemistry, biology, medicine, agriculture, engineering, and finance. Additionally, the central equations often appear in physics and engineering with only trivial variations, making the solution techniques widely applicable.

We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

This paper presents an extension of the generalized stochastic Petri net (GSPN) formalism that enables the computation of first passage time distributions. The tagged customer technique typical of queuing networks is adapted to the GSPN context by providing a formal definition and an automatic computation of the groups of tokens that can be identified as customers, i.e. classes of homogeneous entities behaving in a similar manner. Passage times are identified through the concept of events that correspond to the firing of transitions placed at the boundaries of a subnet. The extended model obtained with this specifications is translated into an ordinary GSPN by isolating a customer from the group and highlighting its path through the net thus obtaining a representation suited for the passage time analysis. Proofs are provided to show the equivalence between these models with respect to their steady-state distributions. An important and original aspect treated in this paper is the possibility of specifying several scheduling policies of tokens at places, an information not present in ordinary GSPN models, but that is vital for the precise computation of first passage time distributions as shown by a few results computed for a simple Flexible Manufacturing application.

Solutions of fractional nonlinear diffusion equation and first passage time: Influence of initial condition and diffusion coefficient

In this contribution we consider theory and associated computational tools to treat the kinetics associated with competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales and associated thermodynamic signatures in the heat capacity. Our focus here is on the first passage time distribution, which is encoded in a kinetic transition network containing all the locally stable states and the pathways between them. This network can be renormalised to reduce the dimensionality, while exactly conserving the mean first passage time and approximately conserving the full distribution. The structure of the reduced network can be visualised using disconnectivity graphs. We show how features in the first passage time distribution can be associated with specific kinetic traps, and how the appearance of competing relaxation time scales depends on the starting conditions. The theory is tested for two model landscapes and applied to an atomic cluster and a disordered peptide. Our most important contribution is probably the reconstruction of the full distribution for long time scales, where numerical problems prevent direct calculations. Here we combine accurate treatment of the mean first passage time with the reliable part of the distribution corresponding to faster time scales. Hence we now have a fundamental understanding of both thermodynamic and kinetic signatures of multifunnel landscapes.

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

This paper introduces two techniques for computing the distribution of the absorption or first passage time of the drifted Wiener diffusion subject to Poisson resetting times to an upper hard wall barrier and to a lower absorbing barrier. The first method, which we call the Padé-partial fraction approximation, starts with the Padé approximation to the Laplace transform of the first passage time distribution, which is then exactly inverted by means of the partial fraction decomposition. The second method, which we call the multiresolution algorithm, is a Monte Carlo technique that exploits the properties of the Wiener process to generate Brownian bridges at increasing levels of resolution. Our numerical study reveals that the multiresolution algorithm has higher efficiency than standard Monte Carlo, whereas the faster Padé-partial fraction method is accurate in various circumstances and provides an analytical formula. Also, a closed-form exact expression for the expected first passage time is derived.

In this paper, we present the analytical solution for the Laplace transform of the joint distribution of the first passage time and undershoot/overshoot value under a regime-switching jump-diffusion model. With the help of some martingale technique, the Laplace transform of the first passage time becomes the solution of a system of linear equations. The methodology discussed here is fairly elementary and can be applied to many stopping-time problems under a regime-switching model with jump risks. Some numerical examples are given to demonstrate the usefulness of our method.

In many physical situations, there appears the problem of reaching a single target that is spatially distributed. Here we analyse how stochastic resetting, also spatially distributed, can be used to improve the search process when the target location is quenched, i.e. it does not evolve in time. More specifically, we consider a model with minimal but sufficient ingredients that allows us to derive analytical results for the relevant physical quantities, such as the first passage time distribution. We focus on the minimisation of the mean first passage time (MFPT) and its fluctuations (standard deviation), which proves to be non-trivial. Our analysis shows that the no-disorder case is singular: for small disorder, the resetting rate distribution that minimises the MFPT leads to diverging fluctuations—which impinge on the practicality of this minimisation. Interestingly, this issue is healed by minimising the fluctuations: the associated resetting rate distribution gives first passage times that are very close to the optimal ones.

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistics of times required for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volumes, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volumes, we develop a scaling framework to study the first assembly time behavior as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have a surprisingly weak dependence on the total number of particles. We highlight how higher statistics (variance, distribution) of the first passage time may nevertheless help to infer key parameters, such as the size of the maximum cluster. Finally, we present a framework to quantify formation of macroscopic sized clusters, which are (asymptotically) very unlikely and occur as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

A method is developed for analyzing sudden injection dye (tracer) curves in order to obtain the distribution of first passage times in the mammalian circulation. The moments of the distribution yield useful quantities such as circulating blood volume. The method involves the use of Liouville-Neumann series. The calculation of the first passage time distribution (FPD) also requires a deconvolution through the use of Fourier transforms, but the calculation of moments does not.

Earthquakes on a specified fault (or fault segment) with magnitudes greater than a specified value have a statistical distribution of recurrence times. The mean recurrence time can be related to the rate of strain accumulation and the strength of the fault. Very few faults have a recorded history of earthquakes that define a distribution well. For hazard assessment, in general, a statistical distribution of recurrence times is assumed along with parameter values. Assumed distributions include the Weibull (stretched exponential) distribution, the lognormal distribution, and the Brownian passage-time (inverse Gaussian) distribution. The distribution of earthquake waiting times is the conditional probability that an earthquake will occur at a time in the future if it has not occurred for a specified time in the past. The distribution of waiting times is very sensitive to the distribution of recurrence times. An exponential distribution of recurrence times is Poissonian, so there is no memory of the last event. The distribution of recurrence times must be thinner than the exponential if the mean waiting time is to decrease as the time since the last earthquake increases. Neither the lognormal or the Brownian passage time distribution satisfies this requirement. We use the “Virtual California” model for earthquake occurrence on the San Andreas fault system to produce a synthetic distribution of earthquake recurrence times on various faults in the San Andreas system. We find that the synthetic data are well represented by Weibull distributions. We also show that the Weibull distribution follows from both damage mechanics and statistical physics.

Using biased molecular dynamics and Brownian dynamics in the study of fluorescent proteins

Reconstruction of Energy Profiles from Rupture Times: Application to Force Spectroscopy Experiments

Aging, the process of growing old or maturing, is one of the most widely seen natural phenomena in the world. For the stochastic processes, sometimes the influence of aging cannot be ignored. For example, in this paper, by analyzing the functional distribution of the trajectories of aging particles performing anomalous diffusion, we reveal that for the fraction of the occupation time of strong aging particles, with coefficient , having no relation with the aging time ta and α and being completely different from the case of weak (none) aging. In fact, we first build the models governing the corresponding functional distributions, i.e. the aging forward and backward Feynman–Kac equations; the above result is one of the applications of the models. Another application of the models is to solve the asymptotic behaviors of the distribution of the first passage time, . The striking discovery is that for weakly aging systems, , while for strongly aging systems, behaves as .