Passage Time Model Research Articles

The Generalized Long-Term Fault Memory Model and Applications to Paleoseismic Records

ABSTRACT Paleoseismic studies show large variability in earthquake interevent times along a fault, with short intervals often separated by long quiescent periods. Some paleoseismologists have interpreted this variability as a product of an earthquake’s partial strain release with the next earthquake occurring sooner than expected because of the remaining residual strain. However, commonly used probabilistic large earthquake recurrence models attribute this variability purely to chance, not the state of strain on the fault. Here, we present an alternative probabilistic model, built on the long-term fault memory model framework that better reflects the strain accumulation and release process. This generalized long-term fault memory model (GLTFM) predicts that this interevent time variability arises from both chance and the state of strain on the fault. Specifically, it estimates when residual strain is likely present and its impact on the timing of the next earthquake in the sequence. In addition, GLTFM assumes that additional accumulated strain always increases earthquake probability. In contrast, the commonly used lognormal and Brownian passage time models predict that the probability of a large earthquake stays constant or even decreases after it is “overdue” (past the observed average recurrence interval) so additional accumulated strain does not make an earthquake more likely. GLTFM’s simple implementation and versatility should make it a powerful tool in earthquake forecasting.

Defaultable perpetual American put option in a last passage time model

In this article we provide a valuation formula for a defaultable perpetual American put option in the Black-Schole’s market where the default time is modelled using the last passage time of the set [L,∞) by the stock price dynamics. In other words, we suppose that financial default happens if the stock price can no longer recover and rebound above the level L.

Recurrence and Long-Term Evaluation of Kanto Earthquakes

ABSTRACT The 1923 Taisho Kanto earthquake—an interplate event along the Sagami trough where the Philippine Sea plate subducts beneath the Kanto region—produced ground shaking, crustal deformation, landslides, and tsunamis, and caused the worst earthquake disaster in Japan. Based on seismological and geodetic data, many fault models have been proposed, extending ∼100 km from the epicenter, with a moment magnitude (Mw) of 7.8–8.2, and large slips of ∼8 m located near the epicenter and beneath the Miura Peninsula. The penultimate 1703 Genroku Kanto earthquake produced similar macroseismic effects around Sagami Bay and the Miura Peninsula, but larger coastal uplift and tsunami in the Boso Peninsula. The proposed fault models extend off the Boso Peninsula with Mw of 8.1–8.5. In 2004, the Earthquake Research Committee (ERC) classified the Kanto earthquakes as “Taisho type” and “Genroku type” with recurrence intervals of 200–400 yr and 2300 yr, respectively. In 2014, the ERC revised the long-term evaluation to a recurrence interval of 180–590 yr and a 30 yr probability of 0%–5% based on the Brownian passage time model. With the Cabinet Office, the ERC considered the source area of the maximum possible earthquake of Mw 8.6–8.7. The recent historiographical and paleoseismological studies have identified other candidates for the past Kanto earthquakes in 1495, 1433, 1293, and 878. Various combinations of these candidates give a mean recurrence interval of 210–315 yr, an aperiodicity parameter of 0.04–0.76, and a 30 yr probability of 0.0%–19%. The Cabinet Office has calculated the seismic intensity and tsunami heights of various types of Kanto earthquakes. National and local governments estimate the damage from these hazards. For the Tokyo metropolitan area, the estimated damage and occurrence probability are more significant for M ∼7 earthquakes with various types and depths, and most mitigation efforts are directed at such events.

First Passage Time Model Based on Lévy Process for Contingent Convertible Bond Pricing

This paper develops a general Lévy framework to reduce the pricing problem of contingent convertible (CoCos) bonds to the problem of the first pass time of the triggering process. We consider two Lévy models driven by the derived Brownian motion and the spectrally negative Lévy process. These two Lévy models keep the form of the Lévy process unchanged under the measure transform, which avoids the difficulty that only rare forms of Lévy processes solved the first passage time problem. We use single and double Laplace transform in combination with numerical Fourier inversion to find closed form expressions for the price of CoCos bonds. The results show that the model driven by the spectrally negative Lévy process would provide a more accurate CoCos bonds price when taking into account the phenomenon of jumps in the financial market. Indeed, negative jumps play a much critical role in the pricing of CoCos bonds. This paper underlines the importance of the evaluation of the CoCos bonds by the Lévy process.

Is the Long-Term Probability of the Occurrence of Large Earthquakes along the Nankai Trough Inflated?—Scientific Review

Abstract The Earthquake Research Committee (ERC), the Headquarters for Earthquake Research Promotion, Japan, issued a new report on long-term evaluation of the occurrence of large earthquakes along the Nankai trough, southwestern Japan, in 2013, which was made along the line of the revision of national earthquake counter measures according to the 2011 Great Eastern Japan Earthquake Disaster. The central issue was the 30-yr probability, but the method to estimate probability was not changed from the previous one issued in 2001 based on the time-predictable model, which led to a 60%–70% chance of M ≥ 8 event during the coming 30 yr. However, there are several drawbacks in this evaluation such as neglect of errors in uplift data, inconsistency between uplift rate and geodetic and geomorphological data, mechanical interpretation of time-predictable model, and contradiction between the assumptions in Brownian passage time model and time-predictable model. Because ERC calculates probability with average recurrence intervals for other subduction and active fault zones, the same scheme should be adopted to the Nankai trough.

Probabilistic long-term resilience of bridges under seismic and deterioration processes

Accurate long-term risk and resilience assessments of bridges are of paramount importance to aid rational decision making under seismic hazards. Both earthquakes and structural deterioration exhibit time-varying features. It has been found that the occurrence of large earthquakes is dependent on time, due to energy accumulation, whereas the widely adopted homogeneous Poisson process assumes a time-independent hazard occurrence. Bridges can deteriorate over time due to environmental exposure, resulting in increased seismic vulnerability. The time-varying characteristics associated with earthquakes and deterioration, which have compound effects on structures, should thus be incorporated in long-term seismic risk and resilience assessments. An approach for assessing the long-term resilience of bridges incorporating the time-varying characteristics of earthquakes and deterioration is presented. The Brownian passage time model, capturing energy accumulation and release, is used to model the time-varying characteristics of earthquakes. Bridge seismic vulnerability is computed in a time-varying manner considering deterioration. Subsequently, long-term bridge resilience loss is computed by considering earthquakes and deterioration over the entire service life of the bridge. The proposed approach is illustrated on a highway bridge subjected to time-dependent seismic hazard and structural deterioration.

Estimating Earthquake Probabilities by Jaynes’s Method of Maximum Entropy

ABSTRACT An important midcentury statistical method, already applied in several other scientific disciplines, enables epistemic ignorance to be incorporated more fully in multidecadal earthquake forecasts. Codified by the physicist E. T. Jaynes in the 1950s, the method of maximum entropy (MaxEnt) is suited for settings of great epistemic uncertainty. Shortly after its initial formulation it was being applied by industrial engineers to rates of failure and expected lifetimes—engineering problems analogous to fault ruptures and their recurrence intervals. Here, I show how MaxEnt can improve upon previous estimates of time-dependent conditional probabilities in Cascadia, California, and New Zealand. The method formalizes probabilistic inferences from long geology-based earthquake histories as truncated-Gaussian curves of time-dependent hazard that match the observed mean and mean square of the historical recurrence data. For each example, rupture forecasts by Brownian passage time, lognormal, Weibull, or Poisson methods have been published previously. In the Cascadia example, the MaxEnt estimate for a 50 yr exposure, 8.3%, compares with previous estimates of up to 14% for passage time or lognormal models. For other datasets drawn from the third Uniform California Earthquake Rupture Forecast study of California earthquakes and from the southern Alpine fault of New Zealand, MaxEnt recurrence probabilities similarly differ by up to a factor of nearly 2.0 from estimates published earlier. The lognormal approach is a variant of the MaxEnt model in which the estimate’s constraints concern the logarithms of the observed recurrence intervals; there is a continuum of methods in between. The discrepancies between MaxEnt and these other probability claims are not trivial; they arise from assumptions built into other methods that do not correspond to the actual information content of the data at hand. Discrepancies like these are directly relevant to the communication of these aspects of earthquake hazards to stakeholders concerned with the consequent risks.

Probabilistic seismic hazard assessment for Taiwan: TEM PSHA2020

The Taiwan Earthquake Model (TEM) published the first version of the Taiwan probabilistic seismic hazard assessment (named TEM PSHA2015) 5 years ago. For updating to the TEM PSHA2020, we considered an updated seismogenic structure database, including the structures newly identified with 3D geometry, an earthquake catalog made current to 2016, state-of-the-art seismic models, a new set of ground motion prediction equations, and site amplification factors. In addition to earthquakes taking place on each individual seismogenic structure, the updated seismic model included the possibility of an earthquake occurring on multiple structures. To include fault memory for illustrating activity on seismogenic structure sources, we incorporated the Brownian passage time model. For the crustal seismicity that cannot be attributed to any specific structure, we implemented both area source and smoothing kernel models. A new set of ground motion prediction equations is incorporated. In addition to the calculation of hazard at engineering bedrock, our assessment included site amplification factors that competent authorities of governments and private companies could use to implement hazard prevention and reduction strategies.

Time-dependent probabilistic seismic hazard assessment in Kerman and adjacent areas in the west of Lut block, Central-East Iran

This study is the first attempt to apply fault-based and time-dependent seismicity models cooperatively for probabilistic seismic hazard assessment (PSHA) of the Kerman and adjacent areas (54–59° E, 28.5–34° N). For this study, almost all available literature and databases have been investigated to compile reliable information reported as instrumental, historical, and paleoseismological events, together with the geometry and kinematics information for active faults. This information systematically is integrated and contributed to PSHA. Conditional probabilities of occurrence for earthquakes greater than the maximum earthquake from which its standard deviation is deducted (M > Mmax − σMmax) are computed based on a Brownian passage time (BPT) model. The spatial distribution of the time-dependent peak ground acceleration (PGA) has been calculated, and spectral accelerations have been computed for three selected cities (Kerman, Rafsanjan, Ravar), with 10% probability of exceedance in 50 years. The time-dependent PSHA map for PGA shows variations from − 21 along the Gowk fault to + 32% near the Sabzevaran fault segments in comparison with the corresponding time-independent PGAs. The difference between the time-dependent and time-independent PGAs at most parts of the study area is negligible, especially at sites farther away from the faults, where the differences vary from − 7 to + 7%. Moreover, the difference between the time-dependent and time-independent spectral accelerations for Ravar and Rafsanjan cities is insignificant. However, this difference at periods of 0.1–0.3 s is significant in Kerman city. The results of this study provide a stimulus for considering time-dependency assumption and fault-based approach in future earthquake-resistant constructions required in Kerman and adjacent areas.

Earthquake recurrence models and occurrence probabilities of strong earthquakes in the North Aegean Trough (Greece)

The determination of strong earthquakes’ recurrence time above a predefined magnitude, associated with specific fault segments, is an important component of seismic hazard assessment. The occurrence of these earthquakes is neither periodic nor completely random but often clustered in time. This fact in connection with their limited number, due to shortage of the available catalogs, inhibits a deterministic approach for recurrence time calculation, and for this reason, application of stochastic processes is required. In this study, recurrence time determination in the area of North Aegean Trough (NAT) is developed by the application of time-dependent stochastic models, introducing an elastic rebound motivated concept for individual fault segments located in the study area. For this purpose, all the available information on strong earthquakes (historical and instrumental) with Mw ≥ 6.5 is compiled and examined for magnitude completeness. Two possible starting dates of the catalog are assumed with the same magnitude threshold, Mw ≥ 6.5 and divided into five data sets, according to a new segmentation model for the study area. Three Brownian Passage Time (BPT) models with different levels of aperiodicity are applied and evaluated with the Anderson–Darling test for each segment in both catalog data where possible. The preferable models are then used in order to estimate the occurrence probabilities of Mw ≥ 6.5 shocks on each segment of NAT for the next 10, 20, and 30 years since 01/01/2016. Uncertainties in probability calculations are also estimated using a Monte Carlo procedure. It must be mentioned that the provided results should be treated carefully because of their dependence to the initial assumptions. Such assumptions exhibit large variability and alternative means of these may return different final results.

Search and foraging behaviors from movement data: A comparison of methods.

Search behavior is often used as a proxy for foraging effort within studies of animal movement, despite it being only one part of the foraging process, which also includes prey capture. While methods for validating prey capture exist, many studies rely solely on behavioral annotation of animal movement data to identify search and infer prey capture attempts. However, the degree to which search correlates with prey capture is largely untested. This study applied seven behavioral annotation methods to identify search behavior from GPS tracks of northern gannets (Morus bassanus), and compared outputs to the occurrence of dives recorded by simultaneously deployed time–depth recorders. We tested how behavioral annotation methods vary in their ability to identify search behavior leading to dive events. There was considerable variation in the number of dives occurring within search areas across methods. Hidden Markov models proved to be the most successful, with 81% of all dives occurring within areas identified as search. k‐Means clustering and first passage time had the highest rates of dives occurring outside identified search behavior. First passage time and hidden Markov models had the lowest rates of false positives, identifying fewer search areas with no dives. All behavioral annotation methods had advantages and drawbacks in terms of the complexity of analysis and ability to reflect prey capture events while minimizing the number of false positives and false negatives. We used these results, with consideration of analytical difficulty, to provide advice on the most appropriate methods for use where prey capture behavior is not available. This study highlights a need to critically assess and carefully choose a behavioral annotation method suitable for the research question being addressed, or resulting species management frameworks established.

US Bank Credit Spreads During the Financial Crisis

This paper argues that first passage time models are likely to better than affine hazard rate models in modelling stressed credit markets and confirms their superior performance in explaining the behavior of Credit Default Swap rates for the major US banking groups over the period of the financial crisis. Affine models find it hard to deal with periods of exceptionally high or low default risk given their assumption of a constant rate of mean reversion in the hazard rate. In contrast, first passage time models are specified in terms of the distance to default rather than the hazard rate. The persistence of shocks varies with the distance to default, allowing the default curve to invert sharply (compress) when the distance to default is low (high). I use an empirical version of the Collin-Dufresne et al. (2003) model, which contains a smoothing parameter that allows it to control the relative effect of these shocks on the short spreads and can be interpreted as an information lag.

Gaussian counter models for visual identification of briefly presented, mutually confusable single stimuli in pure accuracy tasks

When identifying confusable visual stimuli, accumulation of information over time is an obvious strategy of the observer. However, the nature of the accumulation process is unresolved: for example it may be discrete or continuous in terms of the information encoded. Another unanswered question is whether or not stimulus sampling continues after the stimulus offset. In the present paper we propose various continuous Gaussian counter models of the time course of visual identification of briefly presented, mutually confusable single stimuli in a pure accuracy task. During stimulus analysis, tentative categorizations that stimulus i belongs to category j are made until a maximum time after the stimulus disappears. Two classes of models are proposed. First, the overt response is based on the categorization that had the highest value at the time the stimulus disappears (race models). Second, the overt response is based on the categorization that made the minimum first passage time through a constant boundary (first passage time models). Within this framework, multivariate Wiener and Ornstein–Uhlenbeck counter models are considered under different parameter regimes, assuming either that the stimulus sampling stops immediately or that it continues for some time after the stimulus offset. Each type of model was evaluated by Monte Carlo tests of goodness of fit against observed probability distributions of responses in two extensive experiments. A comparison of these continuous models with a simple discrete Poisson counter model proposed by Kyllingsbæk, Markussen, and Bundesen (2012) was carried out, together with model selection among the competing candidates. Both the Wiener and the Ornstein–Uhlenbeck race models provide a close fit to individual data on identification of both digits and Landolt rings, outperforming the first passage time model and the Poisson counter race model.

When time and faults matter: towards a time-dependent probabilistic SHA in Calabria, Italy

In this study, we attempt to improve the standards in Probabilistic Seismic Hazard Assessment (PSHA) towards a time-dependent hazard assessment by using the most advanced methods and new databases for the Calabria region, Italy. In this perspective we improve the knowledge of the seismotectonic framework of the Calabrian region using geologic, tectonic, paleoseismological, and macroseismic information available in the literature. We built up a PSHA model based on the long-term recurrence behavior of seismogenic faults, together with the spatial distribution of historical earthquakes. We derive the characteristic earthquake model for those sources capable of rupturing the entire fault segment (full-rupture) independently with a single event of maximum magnitude. We apply the floating rupture model to those earthquakes whose location is not known sufficiently constrained. We thus associate these events with longer fault systems, assuming that any such earthquake can rupture anywhere within the particular fault system (floating partial-rupture) with uniform probability. We use a Brownian Passage Time (BPT) model characterized by mean recurrence, aperiodicity, or uncertainty in the recurrence distribution and elapsed time since the last characteristic earthquake. The purpose of this BPT model is to express the time-dependence of the seismic processes to predict the future ground motions in the region. Besides, we consider the influence on the probability of earthquake occurrence controlled by the change in static Coulomb stress (ΔCFF) due to fault interaction; to pursue this, we adopt a model built on the fusion of BPT model (BPT + ΔCFF). We present our results for both time-dependent (renewal) and time-independent (Poisson) models in terms of Peak Ground Acceleration (PGA) maps for 10% probability of exceedance in 50 years. The hazard may increase by more than 20% or decrease by as much as 50% depending on the different occurrence model. Seismic hazard in terms of PGA decreases about 20% in the Messina Strait, where a recent major earthquake took place, with respect to traditional time-independent estimates. PGA near the city of Cosenza reaches ~ 0.36 g for the time-independent model and 0.40 g for the case of the time-dependent one (i.e. a 15% increase). Both the time-dependent and time-independent models for the period of 2015–2065 demonstrate that the city of Cosenza and surrounding areas bear the highest seismic hazard in Calabria.

US bank credit spreads during the financial crisis

This paper argues that first passage time models are likely to better than affine hazard rate models in modelling stressed credit markets and confirms their superior performance in explaining the behavior of Credit Default Swap rates for the major US banking groups over the period of the financial crisis. Affine models find it hard to deal with periods of exceptionally high or low default risk given their assumption of a constant rate of mean reversion in the hazard rate. In contrast, first passage time models are specified in terms of the distance to default rather than the hazard rate. The persistence of shocks varies with the distance to default, allowing the default curve to invert sharply (compress) when the distance to default is low (high). I use an empirical version of the Collin-Dufresne et al. (2003) model, which contains a smoothing parameter that allows it to control the relative effect of these shocks on the short spreads and can be interpreted as an information lag.

First Passage Time Model Driven by LLvy Process for Pricing CoCos

First Passage Time Model Driven by LLvy Process for Pricing CoCos

Maximum‐Likelihood Recurrence Parameters and Conditional Probability of a Ground‐Rupturing Earthquake on the Southern Alpine Fault, South Island, New Zealand

Abstract The Alpine fault in south Westland, New Zealand, releases strains of Pacific–Australian relative plate motion in large earthquakes with an average interevent spacing of ∼330 years. A new record of earthquake recurrence has been developed at Hokuri Creek, with evidence for 22 events. The youngest Hokuri Creek earthquake overlaps in time and is believed to be the same as the oldest of another site about 100 km to the northwest near Haast. The combined record spans the last 7900 years and includes 24 events. We study the recurrence rate and conditional probability of ground ruptures from this record using a new likelihood‐based approach for estimation of recurrence model parameters. Paleoseismic parameter estimation includes both dating and natural recurrence uncertainties. Lognormal and Brownian passage time (BPT) models are considered. The likelihood surface has distribution location and width parameters as axes, the mean and standard deviation of the log recurrence for the lognormal, and the mean and coefficient of variation for the BPT. The maximum‐likelihood (ML) point gives the parameters most likely to have given rise to the data. The ML point, 50‐year conditional probabilities of a ground‐rupturing earthquake are 26.8% and 26.1% for the lognormal and BPT models, respectively. Contours of equal likelihood track the parameter pairs that are equally probable to have given rise to the observed data. Conditional probabilities on the lognormal 95% boundary around the ML point range from 18.2% to 35.8%. An empirical distribution model completely based on past recurrence times gives a similar conditional probability of 27.1% (9.6%–50.2%). In contrast, the time‐independent conditional probability estimate of 13.6% (8.8%–19.1%) is about half that of the time‐dependent models. A nonparametric test of earthquake recurrence at Hokuri Creek indicates that time‐dependent recurrence models best represent the southern Alpine fault of the South Island, New Zealand.

The Mills Ratio and the Behavior of Redeemable Bond Prices in the Gaussian Structural Model of Corporate Default

This paper shows that forward default intensities in the Black and Cox (1976) model of corporate default can be expressed in terms of the Mills Ratio (Mills, 1926). The behavior of the forward default intensity and hence the survivorship functions then follows from inequalities that are satisfied by this ratio. This allows me to analyze the effect of the firm’s distance to default, growth rate and volatility upon the value of its debt. These results can be used to analyze the comparative static properties of other models of corporate default and perhaps other first passage time models.

Novel last passage time based jitter model with application to low slew rate/high noise ring oscillator

In a ring oscillator, the behaviour in which the output voltage ramp (up/down) of its individual delay cell crosses the threshold, which triggers the next stage delay cell, is crucial in determining the timing jitter, and hence phase noise. Specifically, as the slew rate of the ramp decreases and/or the amount of noise contribution from the transistors in the delay cell increases, the voltage ramp (up or down) has a higher probability of crossing the threshold multiple times, before finally passing it at a time denoted as last passage time, which is more accurate than the conventional first passage time model. This multiple crossing results in a higher jitter. In the past, investigation in last passage time jitter model results in jitter expression that can only be calculated numerically and thus no design guidelines/insights are apparent. In this paper, a novel model is presented with a simple closed form formula, which shows the extra jitter, due to multiple crossing, adds a term that increases as a function of the fourth power of the noise strength/slew rate ratio. The formula is applied to a real life practical low slow rate/high noise ring oscillator which finds application, for example, in random number generator implementation. Corresponding transistor level simulation results agree reasonably well with the model. Furthermore, it is shown that, on example designs, the last passage time approach in this paper can lead to time jitter that is 100 % larger than that due to conventional first passage time model.

The Mills Ratio and the behavior of redeemable bond prices in the Gaussian structural model of corporate default

This paper shows that forward default intensities in the Black and Cox (1976) model of corporate default can be expressed in terms of the Mills Ratio (Mills, 1926). The behaviour of the forward default intensity and hence the survivorship functions then follows from inequalities that are satisfied by this ratio. This allows me to analyze the effect of the firm’s distance to default, growth rate and volatility upon the value of its debt. These results can be used to analyze the comparative static properties of other models of corporate default and perhaps other first passage time models.