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This study proposes an updated renewal model to estimate active patent value and technical depreciation in five major Indian technology fields. According to a study of 27,000 patents issued by the Indian Patent Office between 1998 and 2018, Indian patents have a lower value than foreign patents. This suggests that India's patenting and R&D qualities differ from those of more advanced countries. Technology-specific factors are important because technology fields depreciate at different rates. The top 5% of patents accounted for most international and domestic patent values. A genetic algorithm is used to optimise the renewal model accuracy. This study has significant implications for Indian policymakers, patent holders, international investors, and technology market participants. This model allows startups and new businesses who want to value their IP at the very early stage to negotiate with their counterpart when licensing or selling the patents. [ABSTRACT FROM AUTHOR]

The construction of operation models for complex technical systems is considered. It is shown that the issues of optimizing systems with metrological support are not fully covered in the literature. We propose a model for managing the processes of operation and updating the fleet of measuring equipment related to several degradation groups. A functional dependence of the stationary readiness coefficient on the metrological parameters of measuring equipment is built — the intervals between certifications, relative operating tolerances for monitored parameters and relative measurement errors, as well as on organizational, technical and engineering parameters. The developed model makes it possible to calculate the optimal (rational) parameters of operation and renewal of the measuring equipment fleet. The model can be used in the classification of complex technical systems in order to set requirements for their metrological support. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniform asymptotic behavior for the ruin probability in a continuous time renewal counting process. For the proposed model, we assume that the financial claims for each extreme event are compensated by a finite number of independent insurance companies. Moreover, it is assumed that the claims of each insurance company is a sequence of random variables the tail distribution of which belongs to the class of subexponential distributions with finite mean. More specifically the objective of this work is the study of the uniform asymptotic behavior of ruin probability within the class of strongly subexponential distributions, a subclass of subexponential distributions, which provides a convenient framework for investigating heavy tailed distributions. The results are established for two types of asymptotic relations, namely for the common ruin probability of insurance companies and also for the ruin probability of at least one insurance company.

Patents contribute to economic development; therefore, estimations of patent value are significant in measuring their impact on the economy. With this in mind, this article discusses estimations of patent value with the renewal model developed by several economists. Targeted patents are those issued in the United States and Japan from 1990 to 2010. In addition, the study examines patent values for several technology fields and considers which technology field should be the focus for the United States and Japan. Moreover, this article compares estimations based on the renewal model versus those based on other data such as sales, trade prices, or trials and analyzes how close this model's estimations are to the real values. This article's results demonstrate that the renewal model is useful for evaluating patents overall and analyzing technological trends from a patent perspective. [ABSTRACT FROM AUTHOR]

We consider a renewal risk model with general Erlang distributed interarrival times. We treat this as a Markov modulated risk model and assume, for simplicity, that the states are observable. The insurer can pay dividends and has to inject capital in order to keep the surplus positive. We determine the optimal dividend/capital injection strategy. [ABSTRACT FROM AUTHOR]

In this work, we study a multiscale inverse problem associated with a multi-type model for age structured cell populations. In the single type case, the model is a McKendrick-VonFoerster like equation with a mitosis-dependent death rate and potential migration at birth. In the multi-type case, the migration term results in an unidirectional motion from one type to the next, so that the boundary condition at age 0 contains an additional extrinsic contribution from the previous type. We consider the inverse problem of retrieving microscopic information (the division rates and migration proportions) from the knowledge of macroscopic information (total number of cells per layer), given the initial condition. We first show the well-posedness of the inverse problem in the single type case using a Fredholm integral equation derived from the characteristic curves, and we use a constructive approach to obtain the lattice division rate, considering either a synchronized or non-synchronized initial condition. We take advantage of the unidirectional motion to decompose the whole model into nested submodels corresponding to self-renewal equations with an additional extrinstic contribution. We again derive a Fredholm integral equation for each submodel and deduce the well-posedness of the multi-type inverse problem. In each situation, we illustrate numerically our theoretical results.

In this article, we investigate the uniform asymptotic behavior for the tail probability of maxima of a random walk { S n , n ≥ 0 } with S 0 = 0 and negative drift. We establish the tail probability in the frame of a time-dependent renewal risk model. In this model, we assume that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs. Furthermore, we suppose the tail probability distribution belongs to the strongly subexponential class. [ABSTRACT FROM AUTHOR]

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation.

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

The problem on the early fluctuation of instantaneous availability (IA) is considered in this paper, where it is for a one-unit repairable system. By transforming the renewal equation into differential equations, the instantaneous availability can be given under specific distribution. We use a fluctuation definition and several decision lemmas to analyze the fluctuation of instantaneous availability. Then we make a further research to give some methods of fluctuation suppression. The simulation results are in good agreement with the theoretical results. [ABSTRACT FROM AUTHOR]

Given the probability distribution function relating to the recurrence interval and the occurrence time of the previous occurrence of a fault, a time-dependent model of a particular fault for seismic hazard assessment was developed that takes into account the active fault rupture cyclic characteristics during a particular lifetime up to the present time. The Gutenberg and Richter (1944) exponential frequency-magnitude relation uses to describe the earthquake recurrence rate for a regional source. It is a reference for developing a composite procedure modelled the occurrence rate for the large earthquake of a fault when the activity information is shortage. The time-dependent model was used to describe the fault characteristic behavior. The seismic hazards contribution from all sources, including both time-dependent and time-independent models, were then added together to obtain the annual total lifetime hazard curves. The effects of time-dependent and time-independent models of fault [e.g., Brownian passage time (BPT) and Poisson, respectively] in hazard calculations are also discussed. The proposed fault model result shows that the seismic demands of near fault areas are lower than the current hazard estimation where the time-dependent model was used on those faults, particularly, the elapsed time since the last event of the faults (such as the Chelungpu fault) are short. [ABSTRACT FROM AUTHOR]

We propose and analyze a simplified version of a partial differential equation (PDE) model for chronic myeloid leukemia (CML) derived from an agent-based model proposed by Roeder et al. This model describes the proliferation and differentiation of leukemic stem cells in the bone marrow and the effect of the drug Imatinib on these cells. We first simplify the PDE model by noting that most of the dynamics occurs in a subspace of the original 2D state space. Then we determine the dominant eigenvalue of the corresponding linearized system that controls the long-term behavior of solutions. We mathematically show a non-monotonous dependence of the dominant eigenvalue with respect to treatment dose, with the existence of a unique minimal negative eigenvalue. In terms of CML treatment, this shows that there is a unique dose that maximizes the decay rate of the CML tumor load over long time scales. Moreover this unique dose is lower than the dose that maximizes the initial tumor load decay. Numerical simulations of the full model confirm that this phenomenon is not an artifact of the simplification. Therefore, while optimal asymptotic dosage might not be the best one at short time scales, our results raise interesting perspectives in terms of strategies for achieving and improving long-term deep response. [ABSTRACT FROM AUTHOR]

This article presents an overview of methods developed for the modeling and control of local coronavirus outbreaks. The article reviews early transmission dynamics featuring exponential growth in infections, and links this to a renewal epidemic model where the current incidence of infection depends upon the expected value of incidence randomly lagged into the past. This leads directly to simple formulas for the fraction of the population infected in an unmitigated outbreak, and reveals herd immunity as the solution to an optimization problem. The model also leads to direct and easy-to-understand formulas for aligning observable epidemic indicators such as cases, hospitalizations and deaths with the unobservable incidence of infection, and as a byproduct leads to a simple first-order approach for estimating the effective reproduction number R t . The model also leads naturally to direct assessments of the effectiveness of isolation in preventing the spread of infection. This is illustrated with application to repeat asymptomatic screening programs of the sort utilized by universities, sports teams and businesses to prevent the spread of infection., (© 2021 The Authors.)

Abstract: Many spatio-temporal models have been proposed for forecasting the location and timing of the next eruption in a monogenetic volcanic field. These have almost invariably been fitted retrospectively. That is, the model has been tuned to all of the data, and hence an assessment of the goodness of fit has not been carried out on independent data. The low rate of eruptions in monogenetic fields means that there is not the opportunity to carry out a purely prospective test, as thousands of years would be required to accumulate the necessary data. This leaves open the possibility of a retrospective sequential test, where the parameters are calculated only on the basis of prior events and the resulting forecast compared statistically with the location and time of the next eruption. In general, events in volcanic fields are not dated with sufficient accuracy and precision to pursue this line of investigation; An exception is the Auckland Volcanic Field (New Zealand), consisting of c. 50 centers formed during the last c. 250kyr, for which an age-order model exists in the form of a Monte Carlo sampling algorithm, facilitating repeated sequential testing. I examine a suite of spatial, temporal and spatio-temporal hazard models, comparing the degree of fit, and attempt to draw lessons from how and where each model is particularly successful or unsuccessful. A relatively simple (independent) combination of a renewal model (temporal term) and a spatially uniform ellipse (spatial term) performs as well as any other model. Both avoid over fitting the data, and hence large errors, when the spatio-temporal occurrence pattern changes. [Copyright &y& Elsevier]

One of the uses of Markov Chains is the simulation of the seismic cycle in a fault, i.e. as a renewal model for the repetition of its characteristic earthquakes. This representation is consistent with Reid's elastic rebound theory. We propose a general one-way Markovian model in which the waiting time distribution, its first moments, coefficient of variation, and functions of error and alarm (related to the predictability of the model) can be obtained analytically. The fact that in any one-way Markov cycle the coefficient of variation of the corresponding distribution of cycle lengths is always lower than one concurs with observations of large earthquakes in seismic faults. The waiting time distribution of one of the limits of this model is the negative binomial distribution; as an application, we use it to fit the Parkfield earthquake series in the San Andreas fault, California. [ABSTRACT FROM AUTHOR]

Abstract: Historical eruption records are often incomplete, a problem which is exacerbated in dealing with catalogs derived from geologic records. We examine the problem of estimating the true (adjusted for missing observations) parameters and hence the hazard in a Weibull (or gamma) renewal model, which is commonly used to model time series of volcanic onsets. Robust regression, robust estimation using repeated medians, and results from the theory of inverses of thinned renewal processes failed to provide consistent estimates from simulated data. Hence we adopted a hidden Markov model framework, where the hidden state is a reflection of the number of missing onsets. This also allows for the completeness level of the record to be estimated, and offers a means of determining where in the record the missing observations are likely to be found. Tested on data from the Holocene record of Mt Taranaki, the preliminary estimates of completeness are 86–87% complete (record 7ka — present) and 78–80% complete beyond that. These figures were independently verified using a model of tephra dispersal. The estimated present hazard is approximately 20% higher than estimated without allowing for missing data. [Copyright &y& Elsevier]

Abstract: Because paleoseismology can extend the record of earthquakes back in time up to several millennia, it represents an opportunity to study how earthquakes recur through time and thus to provide innovative contributions to seismic hazard assessment. Based on a database of recurrence from paleoseismology we collected 19 sequences with 5 up to 14 dated events on a single fault. By using the age of the paleoearthquakes, with their associated uncertainty, and the historical earthquakes, we tested the null hypothesis that the observed inter-event times come from a uniform random distribution (Poisson model). We used the concept of likelihood for a specific sequence of events under a given occurrence model. The difference dlnL of the likelihoods estimated under two hypotheses gives an indication of which between the two hypotheses fits better the observations. To take into account the uncertainties, we used a Monte Carlo procedure computing the average and the standard deviation of dlnL for 1000 inter-event sets by choosing the occurrence time of each event within the limits of uncertainty and estimating the probability that a value equal to or larger than an observed dlnL comes by chance from a Poisson distribution of inter-event times. These tests were carried out for the Log-normal, Gamma, Weibull, Double-exponential and Brownian Passage Time (BPT) distributions. Our results show that a renewal model, associated with a time dependent hazard, and some kind of predictability of the next large earthquake on a fault is significantly better than a plain time-independent Poisson model only for four, out of the 19 sites examined in this study. The lack of regularity in the earthquake occurrence for more than 30% of the examined faults can be explained either by the large uncertainties in the estimate of paleoseismological occurrence times or by physical interaction between neighboring faults. [Copyright &y& Elsevier]

We investigate precise large deviations for heavy-tailed random sums. We prove a general asymptotic relation in the compound renewal risk model for consistently varying-tailed distributions. This model was introduced in [Q. Tang, C. Su, T. Jiang, and J.S. Zang, Large deviation for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52:91-100, 2001] as a more practical risk model. The proof is based on the inequality found in [D. Fuk and S.V. Nagaev, Probability for sums of independent random variables, Theory Probab. Appl., 16:600-675, 1971]. [ABSTRACT FROM AUTHOR]

In 2007, Chen and Ng investigated infinite-time ruin probability with constant interest force and negatively quadrant dependent and extended regularly varying-tailed claims. Following this work, the authors obtain a weakly asymptotic equivalent formula for the finite-time and infinite-time ruin probability with constant interest force, negatively quadrant dependent, and dominated varying-tailed claims and negatively lower orthant dependent inter-arrival times. In particular, when the claims are consistently varying-tailed, an asymptotic equivalent formula is presented. [ABSTRACT FROM AUTHOR]

Abstract: The renewal model for cost-benefit optimization including maintenance is presented. Three types of maintenance strategies are proposed, i.e. age-dependent maintenance, block maintenance and maintenance by inspection and (perfect) repair. Then, new concepts for risk acceptability based on the life quality index philosophy are introduced. It is demonstrated that in limiting the failure rate three concepts can be used, the first addressing the maximum failure rate, a second using a time-average and the third an asymptotic value which may be proposed for practical use. All concepts, however, require computation of the exact failure rate (renewal intensity) or, at least, the asymptotic failure rate. A suitable asymptotic failure rate related to all corrective renewals after failure is given. A numerical example is given showing that the inspection/repair maintenance strategy is more cost-effective than the age-dependent strategy. [Copyright &y& Elsevier]