Introduction of first passage time (FPT) analysis for software reliability and network security

The problem of first passage times has been studied. Equations of probabilities of first passage times for Markovian stochastic differential systems subject to Wiener and Poisson inputs have been derived. The derivation is based on the theorems of Kac, Darling and Siegertt. A formulation of control problems of first passage times is given.

The problem of first passage time is mathematically equivalent to the tunnelling of a particle out of a potential well and hence can be treated by the multi-dimensional WKB technique. In general, the most probable tunnelling path (MPTP) is curved and is difficult to determine. The authors discuss a model potential for which the curved MPTP can be solved analytically. It is shown that the rate of tunnelling calculated along the straight path as usually assumed is unreliable. Numerical results of the mean switching time for the bistable two-mode laser and comparison with experiments and other theories are also discussed.

Many intracellular events trigger upon attaining critical concentrations of their corresponding regulatory proteins. How cells ensure precision in the timing of these events is a fundamental problem. Models used to study protein dynamics often do not explicitly account for cell growth and consider the dilution by the growth as equivalent to degradation. Here, we formulate the timing of protein threshold-crossing as a first-passage time (FPT) problem. We study the FPT statistics of three classical models of gene expression that do not explicitly consider cell growth: (birth-death process, discrete birth with continuous deterministic degradation, and Fokker-Planck approximation). These results were compared with a fourth model (growing cell) that comprises size-dependent expression in an exponentially elongating cell. This article specifically focuses on how the mean FPT and its fluctuations depend on the threshold concentration. The first three models offer similar FPT statistics, while the fourth one shows a transition between noisy dynamics in small sizes to an almost deterministic behavior when cells grow enough. While in the first three models, cells can reach concentrations higher than the steady-state concentration. In the growing cell model, although some cells can reach steady level at small sizes, the mean FPT diverges if the threshold concentration is higher than the steady one. We use this threshold as parameter to estimate the mean time that minimizes the FPT variability. The fourth model predicts higher FPT that minimizes the noise and fewer fluctuations on the FPTs than the other three models. We discuss the origin of these divergences and some applications of this model when cell growth is the leading cause of the decreasing molecule concentration within a cell.

Randomly connected populations of spiking neurons display a rich variety of dynamics. However, much of the current modeling and theoretical work has focused on two dynamical extremes: on one hand homogeneous dynamics characterized by weak correlations between neurons, and on the other hand total synchrony characterized by large populations firing in unison. In this paper we address the conceptual issue of how to mathematically characterize the partially synchronous "multiple firing events" (MFEs) which manifest in between these two dynamical extremes. We further develop a geometric method for obtaining the distribution of magnitudes of these MFEs by recasting the cascading firing event process as a first-passage time problem, and deriving an analytical approximation of the first passage time density valid for large neuron populations. Thus, we establish a direct link between the voltage distributions of excitatory and inhibitory neurons and the number of neurons firing in an MFE that can be easily integrated into population-based computational methods, thereby bridging the gap between homogeneous firing regimes and total synchrony.

Contingent convertible bonds (CoCos) are typical form of contingent capital that converts into equity of issuing firm or writes down if a prespecified trigger occurs. This paper proposes a general Lévy framework for pricing CoCos. The Lévy framework indicates that the difficulty in giving closed-form expression for CoCos price is the possible introduction of the Lévy process whose first-passage time problem has not been solved. According to characteristics of new Lévy measure after the measure transform, three specific Lévy models driven by drifted Brownian motion, spectrally negative Lévy process, and double exponential jump diffusion process are proposed to give the solution keeping the form of the driving process unchanged under the measure transform. These three Lévy models provide closed-form expressions for CoCos price while the latter two possess them up to Laplace transform, whose pricing results are given by combining with numerical Fourier inversion and Laplace inversion. Numerical results show that negative jumps have large influence on CoCos pricing and the Black-Scholes model would overestimate CoCos price by simply compressing jumps information into volatility while the other two models would give more accurate CoCos price by taking jump risk into consideration.

Recent developments in the measurement of radioactive gases in passive diffusion motivate the analysis of Brownian motion of decaying particles, a subject that has received little previous attention. This paper reports the derivation and solution of equations comparable to the Fokker-Planck and Langevin equations for one-dimensional diffusion and decay of unstable particles. In marked contrast to the case of stable particles, the two equations are not equivalent, but provide different information regarding the same stochastic process. The differences arise because Brownian motion with particle decay is not a continuous process. The discontinuity is readily apparent in the computer-simulated trajectories of the Langevin equation that incorporate both a Wiener process for displacement fluctuations and a Bernoulli process for random decay. This paper also reports the derivation of the mean time of first passage of the decaying particle to absorbing boundaries. Here, too, particle decay can lead to an outcome markedly different from that for stable particles. In particular, the first-passage time of the decaying particle is always finite, whereas the time for a stable particle to reach a single absorbing boundary is theoretically infinite due to the heavy tail of the inverse Gaussian density. The methodology developed in this paper should prove useful in the investigation of radioactive gases, aerosols of radioactive atoms, dust particles to which adhere radioactive ions, as well as diffusing gases and liquids of unstable molecules.

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions.

Waiting for Returns: Using Space-Time Duality to Calibrate Financial Diffusions

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options Menghui Cao This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of βσ √ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one

Computation of time probability distributions for the occurrence of uncertain future events

An essential part of any disease containment and eradication policy is the implementation of restricted zones, but determining the appropriate size of these zones can be challenging for managers. We designed a new method, based on animal movement, to help assess how large restricted zones should be after a spontaneous outbreak to successfully control infectious diseases in wildlife. Our approach uses first-passage time (FPT) analysis and Cox proportional hazard (CPH) models to calculate and compare the risk of an animal leaving different-sized areas. We illustrate our approach using the example of the African swine fever (ASF) virus and its wild pig reservoir host species, the wild boar (Sus scrofa), and we investigate the feasibility of applying this method to other systems. Using GPS data from 57 wild boar living in the Hainich National Park, Germany, we calculate the time spent by each individual in areas of different sizes using FPT analysis. We apply CPH models on the derived data to compare the risk of leaving areas of different sizes and to assess the effects of season and the sex of the wild boar on the risk of leaving. We conduct survival analyses to estimate the risk of leaving an area over time. Our results indicate that the risk of leaving an area decreases exponentially by 10% for each 100 m increase in radius size so that the differences were more pronounced for small sizes. Furthermore, the probability of leaving increases exponentially with time. Wild boar had a similar risk of leaving an area of a given size throughout the year, except in spring and winter, when females had a much lower risk of leaving. Our findings are in agreement with the literature on wild boar movement, further validating our method, and repeated analyses with location data resampled at different rates gave similar results. Our results may be applicable only to our study area, but they demonstrate the applicability of the proposed method to any ecosystem where wild boar populations are likely to be infected with ASF and where restricted zones should be established accordingly. The outlined approach relies solely on the analysis of movement data and provides a useful tool to determine the optimal size of restricted zones. It can also be applied to future outbreaks of other diseases.

A molecular motor is made of either a single macromolecule or a macromolecular complex.Just like their macroscopic counterparts, molecular motors "transduce" input energy into mechanical work.Their diffusion is biased by the energy pumped in by the "burning fuel".In this talk I'll begin by giving an elementary introduction to the most essential and relevant features of molecular motors [1, 2] and then present three sets of examples to establish the ubiquity of first-passage times in intracellular processes.Some motors walk along filamentous proteins carrying molecular cargo.The stochastic pause-and-translocation of such a porter is characterized by the distribution of the times of its dwell at the successive positions on its track.The dwell times are essentially first-passage times and their fluctuation provides a lower bound on the number of kinetic states of the motor.For a motor that can step both forward and backward the kinetics of translocation is characterized by the distributions of four distinct conditional dwell times.DNA polymerase is a motor that can not only step forward and backward mechanically, but also step backward by catalyzing a different chemical reaction; its stepping kinetics is characterized by nine distinct conditional dwell times.As the first example of first passage times, I'll define these conditional dwell times and summarize our analytical results as well as their physical implications [1, 2].In the nucleus DNA is stored in a hierarchical structure called chromatin.The basic unit of chromatin consists of 146 base pairs of a double stranded DNA (dsDNA) wrapped around a spool formed by proteins.The helical path of the dsDNA on a spool is also called its "footprint".We have investigated the mechanism of footprint traversal by a chromatin-remodeling enzyme (CRE), a motor that translocates along the dsDNA thereby unwrapping it from the spool.A CRE-induced biased diffusion of a small DNA loop along the footprint can lead to the sliding of the spool along the DNA.The average time needed for the traversal of the footprint, for the first time, by sliding is essentially the corresponding mean first passage time.As an example of second class of first-passage problems, I'll present our model and fuel-dependence of the mean foot-print traversal time [3].The third type of example I'll present occurs in chromosome segregation before cell division.Each of the sister chromatids, that result from chromosome replication, is bound to a structure, called kinetochore (kt) that, in turn, is coupled to the plus ends of stiff protein filaments called microtubules (MT).The lifetime of a kt-MT attachment is also a first-passage time.I'll present our recent results that explain the experimentally observed, apparently counter-intuitive, dependence of the mean lifetime on the externally applied tension that tends to detach MT from the kinetochore [4,5].

To describe the nonequilibrium states of a system we introduce a new thermodynamic parameter – the lifetime (the first passage time) of a system. The statistical distributions that can be obtained out of the mesoscopic description characterizing the behaviour of a system by specifying the stochastic processes are written. Superstatistics, introduced in [Beck C., Cohen E.G.D., Physica A, 2003, 322A, 267] as fluctuating quantities of intensive thermodynamical parameters, are obtained from statistical distribution with lifetime (random time to system degeneracy) as thermodynamical parameter (and also generalization of superstatistics).

Diffusion in fluctuating media: first passage time problem

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.