Passage Time Model Research Articles

Long term probability of a Magnitude 8 Kanto earthquake along the Sagami Trough, central Japan

We attempt to estimate the long-term probability of a Magnitude (M) 8 earthquake along the Sagami Trough in the Kanto subduction zone, central Japan. A Brownian passage time model is applied to sets of historical earthquakes identified in previous studies. An optimal model is obtained by the maximum likelihood method for each data set. The optimal parameters are not well constrained since each data set includes a small number of earthquakes. To obtain reliable probabilities, two weighting methods are introduced. First, we apply the weighted log-likelihood method, where the model parameters are estimated from the log-likelihood function, summed up with each log-likelihood weighted in proportion to the reliability of the data set. Second, probabilities are estimated as the weighted average of every alternative model. The weight of each model represents the normalized relative likelihood of the model. The probabilities of the weighted log-likelihood function are within the ranges of those obtained for each set. In averaging the proposed sequence and over probable parameter values, the probability of an M 8 earthquake occurring in the next 30 years is estimated to be 2.0 to 4.6%, depending on the cutoff value of the weight.

A Bayesian Framework to Rank and Combine Candidate Recurrence Models for Specific Faults

Abstract We propose a probabilistic framework in which different types of information pertaining to the recurrence of large earthquakes on a fault can be combined in order to constrain the parameter space of candidate recurrence models and provide the best combination of models knowing the chosen data set and priors. We use Bayesian inference for parameter and error estimation, graphical models (Bayesian networks) for modeling, and stochastic modeling to link cumulative offsets (CO) to coseismic slip. The cumulative offset‐based Bayesian approach (COBBRA) method (Fitzenz et al. , 2010) was initially developed to use CO data to further constrain and discriminate between recurrence models built from historical and archaeological catalogs of large earthquakes (CLE). We discuss this method and present an extension of it that incorporates trench data (TD). For our case study, the Jordan Valley fault (JVF), the relative evidence of each model slightly favors the Brownian passage time (BPT) and lognormal models. We emphasize that (1) the time variability of fault slip rate is critical to constrain recurrence models; (2) the shape of the probability density functions (PDF) of paleoseismic events is very important, in most cases not Gaussian, and should be reported in its complexity; (3) renewal models are in terms of intervals between consecutive earthquakes, not dates, and the algorithms should account for that fact; and (4) maximum‐likelihood methods are inadequate for parameter uncertainty evaluation and model combination or ranking. Finally, more work is needed to define proper priors and to model the relationship between cumulative slip and coseismic slip, in particular, when the fault behavior is more complex.

How linear features alter predator movement and the functional response.

In areas of oil and gas exploration, seismic lines have been reported to alter the movement patterns of wolves (Canis lupus). We developed a mechanistic first passage time model, based on an anisotropic elliptic partial differential equation, and used this to explore how wolf movement responses to seismic lines influence the encounter rate of the wolves with their prey. The model was parametrized using 5 min GPS location data. These data showed that wolves travelled faster on seismic lines and had a higher probability of staying on a seismic line once they were on it. We simulated wolf movement on a range of seismic line densities and drew implications for the rate of predator–prey interactions as described by the functional response. The functional response exhibited a more than linear increase with respect to prey density (type III) as well as interactions with seismic line density. Encounter rates were significantly higher in landscapes with high seismic line density and were most pronounced at low prey densities. This suggests that prey at low population densities are at higher risk in environments with a high seismic line density unless they learn to avoid them.

Bayesian forecasting of recurrent earthquakes and predictive performance for a small sample size

[1] This paper presents a Bayesian method of probability forecasting for a renewal of earthquakes. When only limited records of characteristic earthquakes on a fault are available, relevant prior distributions for renewal model parameters are essential to computing unbiased, stable time-dependent earthquake probabilities. We also use event slip and geological slip rate data combined with historical earthquake records to improve our forecast model. We apply the Brownian Passage Time (BPT) model and make use of the best fit prior distribution for its coefficient of variation (the shape parameter, alpha) relative to the mean recurrence time because the Earthquake Research Committee (ERC) of Japan uses the BPT model for long-term forecasting. Currently, more than 110 active faults have been evaluated by the ERC, but most include very few paleoseismic events. We objectively select the prior distribution with the Akaike Bayesian Information Criterion using all available recurrence data including the ERC datasets. These data also include mean recurrence times estimated from slip per event divided by long-term slip rate. By comparing the goodness of fit to the historical record and simulated data, we show that the proposed predictor provides more stable performance than plug-in predictors, such as maximum likelihood estimates and the predictor currently adopted by the ERC.

A First Passage Time Model for Long-Term Survivors with Competing Risks

We investigate a competing risks model, using the specification of the Gompertz distribution for failure times from competing causes and the inverse Gaussian distribution for failure times from the cause of interest. The expectation-maximization algorithm is used for parameter estimation and the model is applied to real data on breast cancer and melanoma. In these applications, our models compare favourably with existing techniques. The proposed method provides a useful technique that may be more broadly applicable than existing alternatives.

Credit gap risk in a first passage time model with jumps

The payoff of many credit derivatives depends on the level of credit spreads. In particular, credit derivatives with a leverage component are subject to gap risk, a risk associated with the occurrence of jumps in the underlying credit default swap spreads. In the framework of first passage time models, we consider a model that addresses these issues. The principal idea is to model a credit quality process as an Itô integral with respect to a Brownian motion with a stochastic volatility. Using a representation of the credit quality process as a time-changed Brownian motion, one can derive formulas for conditional default probabilities and credit spreads. An example for a stochastic volatility process is the square root of a Lévy-driven Ornstein–Uhlenbeck process. The model can be implemented efficiently using a technique called Panjer recursion. Calibration to a wide range of dynamics is supported. We illustrate the effectiveness of the model by valuing a leveraged credit-linked note.

Credit Gap Risk in a First Passage Time Model with Jumps

The payoff of many credit derivatives depends on the level of credit spreads. In particular, credit derivatives with a leverage component are subject to gap risk, a risk associated with the occurrence of jumps in the underlying credit default swaps. In the framework of first passage time models, we consider a model that addresses these issues. The principal idea is to model a credit quality process as an Ito integral with respect to a Brownian motion with a stochastic volatility. Using a representation of the credit quality process as a time-changed Brownian motion, one can derive formulas for conditional default probabilities and credit spreads. An example for a volatility process is the square root of a Levy-driven Ornstein-Uhlenbeck process. The model can be implemented efficiently using a technique called Panjer recursion. Calibration to a wide range of dynamics is supported. We illustrate the effectiveness of the model by valuing a leveraged credit-linked note.

Credit Dynamics in a First Passage Time Model with Jumps

The payoff of many credit derivatives depends on the level of credit spreads. In particular, the payoff of credit derivatives with a leverage component is sensitive to jumps in the underlying credit spreads. In the framework of first passage time models we address these issues by specifying a credit quality process to be driven by an Ito integral with respect to a Brownian motion with stochastic volatility. We derive formulas for conditional default probabilities and credit spreads. An example for a volatility process is the square root of a Levy-driven Ornstein-Uhlenbeck process, for which we show that jumps in the volatility translate into jumps in credit spreads. We examine the dynamics of the model and provide examples.

Reviews

Reviews

Diffusive behavior and the modeling of characteristic times in limit order executions

We present an empirical study of the first passage time (FPT) of order book prices needed to observe a prescribed price change Δ, the time to fill (TTF) for executed limit orders and the time to cancel (TTC) for canceled orders in a double auction market. We find that the distribution of all three quantities decays asymptotically as a power law, but that of FPT has significantly fatter tails than that of TTF. Thus a simple first passage time model cannot account for the observed TTF of limit orders. We propose that the origin of this difference is the presence of cancelations. We outline a simple model that assumes that prices are characterized by the empirically observed distribution of the first passage time and orders are canceled randomly with lifetimes that are asymptotically power law distributed with an exponent λLT. In spite of the simplifying assumptions of the model, the inclusion of cancelations is sufficient to account for the above observations and enables one to estimate characteristics of the cancelation strategies from empirical data.

Comparison of phase noise models on a class of ring oscillators using low voltage swing fully differential delay cells

This paper compares the phase noise models on a class of differential saturated ring oscillator with low voltage swing. Variance parameter/single stage jitter is used as the metric for comparison. When first passage time (fpt) is employed, first for the simple case of the RC representation of the differential pair, an exact expression involving the parabolic cylinder function is used in this comparison. Then for the general case, the expression using time scaling is used for comparison. On the last passage time (lpt) model, the probability distribution function (pdf) of lpt is used to further differentiate the models. The fpt part in the complete lpt pdf is identified. Comparison of the phase noise's power spectral density expression is then presented. The phase noise expression of this author does not go to infinity as offset frequency approaches zero. Finally simulation results on example designs are presented, where it is shown that the model by this author, when compared to other models, can improve accuracy by 7-10 dBc/Hz.

Non-Poissonian earthquake occurrence in coupled stress release models and its effect on seismic hazard

SUMMARY Most seismic hazard estimations are based on the assumption of a Poisson process for earthquake occurrence, even though both observations and models indicate a departure of real seismic sequences from this simplistic assumption. Instrumental earthquake catalogues show earthquake clustering on regional scales while the elastic rebound theory predicts a periodic recurrence of characteristic earthquakes on longer timescales for individual events. Recent implementations of time-dependent hazard calculations in California and Japan are based on quasi-periodic recurrences of fault ruptures according to renewal models such as the Brownian Passage Time model. However, these renewal models neglect earthquake interactions and the dependence on the stressing history which might destroy any regularity of earthquake recurrences in reality. To explore this, we investigate the (coupled) stress release model, a stochastic version of the elastic rebound hypothesis. In particular, we are interested in the time-variability of the occurrence of large earthquakes and its sensitivity to the occurrence of Gutenberg‐Richter type earthquake activity and fault interactions. Our results show that in general large earthquakes occur quasi-periodically in the model: the occurrence probability of large earthquakes is strongly decreased shortly after a strong event and becomes constant on longer timescales. Although possible stress-interaction between adjacent fault zones does not affect the recurrence time distributions in each zone significantly, it leads to a temporal clustering of events on larger regional scales. The non-random characteristics, especially the quasi-periodic behaviour of large earthquakes, are even more pronounced if stress changes due to small earthquakes are less important. The recurrence-time distribution for the largest events is characterized by a coefficient of variation from 0.6 to 0.84 depending on the relative importance of small earthquakes.

The Credit Risk and Pricing of OTC Options

In the over-the-counter (OTC) markets, the options traded are always subject to credit risk. Therefore the counterparty’s credit risk is a striking factor when pricing options, whereas it is not considered in the classic Black-Scholes models. Based on the first passage time models, this paper develops the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism. The default probability and the pricing formulae of the OTC options are obtained by using partial differential equation (PDE) techniques, especially Green’s function.

Updating seismic hazard at Parkfield

The occurrence of the September 28, 2004 M w = 6.0 mainshock at Parkfield, California, has significantly increased the mean and aperiodicity of the series of time intervals between mainshocks in this segment of the San Andreas fault. We use five different statistical distributions as renewal models to fit this new series and to estimate the time-dependent probability of the next Parkfield mainshock. Three of these distributions (lognormal, gamma and Weibull) are frequently used in reliability and time-to-failure problems. The other two come from physically-based models of earthquake recurrence (the Brownian Passage Time Model and the Minimalist Model). The differences resulting from these five renewal models are emphasized.

Viscoelasticity, Postseismic Slip, Fault Interactions, and the Recurrence of Large Earthquakes

The Brownian Passage Time (bpt) model for earthquake recurrence is modified to include transient deformation due to either viscoelasticity or deep postseismic slip. Both of these processes act to increase the rate of loading on the seismogenic fault for some time after a large event. To approximate these effects, a decaying exponential term is added to the bpt model’s uniform loading term. The resulting interevent time distributions remain approximately lognormal, but the balance between the level of noise (e.g., unknown fault interactions) and the coefficient of variability of the interevent time distribution changes depending on the shape of the loading function. For a given level of noise in the loading process, transient deformation has the effect of increasing the coefficient of variability of earthquake interevent times. Conversely, the level of noise needed to achieve a given level of variability is reduced when transient deformation is included. Using less noise would then increase the effect of known fault interactions modeled as stress or strain steps because they would be larger with respect to the noise. If we only seek to estimate the shape of the interevent time distribution from observed earthquake occurrences, then the use of a transient deformation model will not dramatically change the results of a probability study because a similar shaped distribution can be achieved with either uniform or transient loading functions. However, if the goal is to estimate earthquake probabilities based on our increasing understanding of the seismogenic process, including earthquake interactions, then including transient deformation is important to obtain accurate results. For example, a loading curve based on the 1906 earthquake, paleoseismic observations of prior events, and observations of recent deformation in the San Francisco Bay region produces a 40% greater variability in earthquake recurrence than a uniform loading model with the same noise level.

Earthquake probabilities and magnitude distribution (M≥6.7) along the Haiyuan fault, northwestern China

In recent years, some researchers have studied the paleoearthquake along the Haiyuan fault and revealed a lot of paleoearthquake events. All available information allows more reliable analysis of earthquake recurrence interval and earthquake rupture patterns along the Haiyuan fault. Based on this paleoseismological information, the recurrence probability and magnitude distribution for M≥6.7 earthquakes in future 100 years along the Haiyuan fault can be obtained through weighted computation by using Poisson and Brownian passage time models and considering different rupture patterns. The result shows that the recurrence probability of MS≥6.7 earthquakes is about 0.035 in future 100 years along the Haiyuan fault.

Application of Brownian model in the northwestern Beijing, China

The mathematic theory of Brownian passage-time model and its difference from other recurrence models such as Poisson, lognormal, gamma and Weibull, were introduced. We assessed and analyzed the earthquake probabilities of the major faults with the elapsed time much greater than the recurrence interval in the northwest region of Beijing (China) in 100-year by using both Brownian passage-time model and Poisson model, and concluded that the calculated results obtained from Brownian passage-time model is more reasonable.

Slip‐size‐dependent renewal processes and Bayesian inferences for uncertainties

This paper is initially concerned with uncertainty of hazard estimates using renewal process models where the parameters are poorly constrained because of the scarcity of large earthquakes occurring along the same fault segment. A Bayesian inference is adopted, taking account of the whole likelihood function of the parameters. Also, new statistical models are considered which make use of knowledge on the slip associated with earthquake ruptures, may help to improve the hazard estimate in a positive way. Including these models, the predictive efficiency is compared by using the Akaike's Bayesian information criterion (ABIC). Three data sets are analyzed for the illustrations. The Brownian Passage Time (BPT) model is selected to fit the first data set, consisting of 10 historical great earthquakes from Nanaki trough in Japan. However, its predictive hazard function shows a large uncertainty (>100 years) of likely occurrence time around 2070. The second data set consists of the last three events of the first data set but associated with record of slip sizes, for which the ABIC selected the extended lognormal renewal process model where the time intervals between successive events are normalized by the corresponding slip sizes of the starting events of the intervals. The estimated predictive hazard function implies that the next event is likely to occur around 2040 ± 10. The last data set, consisting of 4 events with estimated occurrence times and slip sizes from a submarine fault. The ABIC selected the slip‐size‐dependent BPT model for this data. This model indicates that the likelihood of occurrence time of the next event is decreasing from now, and the period of its half decay is more than several hundred years. For the data sets of the last two examples, it was also shown that the slip size records are useful for better prediction of the next event.

Probability representations of a class of two-way diffusions

There has been little progress in the analysis of two-way diffusion in the last few decades due to the difficulties brought by the interface section similar to a free boundary condition. In this paper, however, the equivalent probability model is considered and the interface section is precisely described by an integral equation. The solution of two-way diffusion is then expressed in an integral form with the integrand being the solution of a classical first passage time model and the solution of a one-dimensional integral equation which is relatively easier to solve. The exact expression of the two-way diffusion enables us to find the explicit solution of the model with infinite horizontal boundaries and without drifting.

The Impact of Default Correlations on the Prices of Collateralized Bond Obligat

This paper presents a numerical procedure for pricing collateralized bond obligations (CBO) and analyze the impact of default correlations for the prices of collateralized bond obligations. Specifically, we adopt default correlation model of Zhou (2001) and first passage time model of Black and Cox (1976). The model of Black and Cox is used for estimating the value of the firm and the volatility of the firm value which are unobservable variables. We find that the impact of default correlations on the prices of collateralized bond obligations is generally quite large. This can be tested by carrying out Monte-Carlo simulations for firm value processes, assuming first no default correlations and second modeling default correlations between the processes. We also compare the model prices and recently issued CBO market price and find that no default correlation model over prices the issued CBO and default correlation model under prices the issued CBO. These results in this paper emphasize that modeling default correlations is very important in analyzing CBO and a more complicated further analysis is required.