scholarly journals Asymptotics and Approximations of Ruin Probabilities for Multivariate Risk Processes in a Markovian Environment

2019 ◽  
Vol 22 (3) ◽  
pp. 927-948
Author(s):  
G. A. Delsing ◽  
M. R. H. Mandjes ◽  
P. J. C. Spreij ◽  
E. M. M. Winands
Keyword(s):  
Numerical Experiments ◽  
Ruin Probabilities ◽  
Time Interval ◽  
Transition Rates ◽  
Markovian Environment ◽  
Multivariate Risk ◽  
Environmental Process ◽  
The Individual ◽  
Risk Processes ◽  
Arrival Intensity

Abstract This paper develops asymptotics and approximations for ruin probabilities in a multivariate risk setting. We consider a model in which the individual reserve processes are driven by a common Markovian environmental process. We subsequently consider a regime in which the claim arrival intensity and transition rates of the environmental process are jointly sped up, and one in which there is (with overwhelming probability) maximally one transition of the environmental process in the time interval considered. The approximations are extensively tested in a series of numerical experiments.

scholarly journals Multivariate risk processes with interacting intensities

2008 ◽  
Vol 40 (02) ◽  
pp. 578-601
Author(s):  
Nicole Bäuerle ◽  
Rudolf Grübel
Keyword(s):  
Single Type ◽  
New Class ◽  
Multivariate Risk ◽  
Large Numbers ◽  
Classical Models ◽  
Component Processes ◽  
Arrival Processes ◽  
The Individual ◽  
Risk Processes ◽  
Birth Type

The classical models in risk theory consider a single type of claim. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous-time Markov chain of pure-birth type and the dependency of the components arises from the fact that the birth rate for a specific claim type may depend on the number of claims in the other component processes. Under certain conditions, we obtain a fluid limit, i.e. a functional law of large numbers for these processes. We also investigate the consequences of such results for questions of interest in insurance applications. Several specific subclasses of the general model are discussed in detail and the Cramér asymptotics of the ruin probabilities are derived in particular cases.

scholarly journals On the Calculation of the Ruin Probability for a finite time Period

1971 ◽  
Vol 6 (2) ◽  
pp. 129-133 ◽  
Author(s):  
R. E. Beard
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Insurance Companies ◽  
Frequency Function ◽  
Ruin Probabilities ◽  
Time Interval ◽  
Computational Aspects ◽  
The Individual ◽  
Negative Exponential ◽  
Image Position

In many risk theoretical questions a central problem is the numerical evaluation of a convolution integral and much effort has been devoted over the years to mathematical and computational aspects. The paper presented to this colloquium by O. Thorin shows the subject to be topical but the present note stems from a recent paper by H. L. Seal, “Simulation of the ruin potential of non-life insurance companies”, published in the Transactions of the Society of Actuaries, Volume XXI page 563.In this paper, amongst other topics, Seal has presented some simulations of ruin probabilities over a finite time interval. Some years ago (Journal Institute of Actuaries Students' Society, Volume 15, 1959). I pointed out that the form of ruin probability could be expressed as a successive product of values of a distribution function. To my knowledge no one has attempted to see if this product form was capable of development and the numerical values in Seal's paper prompted me to spend a little time on the problem. In the time I had available it has not been possible to do more than experiment, but the conclusions reached may be of interest to other workers in this field. They showed that calculation is feasible but laborious. However, the knowledge that it can be done may suggest methods of improving the techniques.Seal's first simulation example is the calculation of ruin probabilities when the distribution of the interval of time between the claims is negative exponential and the individual claim distribution is also negative exponential. Instead of following Seal's method we can determine the “gain per interval” and find that if λ is the security loading the frequency function for the gain z isi.e. a Laplace distribution.

scholarly journals Multivariate risk processes with interacting intensities

2008 ◽  
Vol 40 (2) ◽  
pp. 578-601 ◽  
Author(s):  
Nicole Bäuerle ◽  
Rudolf Grübel
Keyword(s):  
Single Type ◽  
New Class ◽  
Multivariate Risk ◽  
Large Numbers ◽  
Classical Models ◽  
Component Processes ◽  
Arrival Processes ◽  
The Individual ◽  
Risk Processes ◽  
Birth Type

The classical models in risk theory consider a single type of claim. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous-time Markov chain of pure-birth type and the dependency of the components arises from the fact that the birth rate for a specific claim type may depend on the number of claims in the other component processes. Under certain conditions, we obtain a fluid limit, i.e. a functional law of large numbers for these processes. We also investigate the consequences of such results for questions of interest in insurance applications. Several specific subclasses of the general model are discussed in detail and the Cramér asymptotics of the ruin probabilities are derived in particular cases.

DIRECTED MOTION OF BROWNIAN MOTORS: A RATCHET MODEL OF TWO-COUPLED MOLECULAR MOTORS

2007 ◽  
Vol 21 (28) ◽  
pp. 1915-1921 ◽  
Author(s):  
SHUTANG WEN ◽  
HONGWEI ZHANG ◽  
LEIAN LIU ◽  
XIAOFENG SUN ◽  
YUXIAO LI
Keyword(s):  
Molecular Motors ◽  
Particle System ◽  
Langevin Equations ◽  
Transition Rates ◽  
Neck Length ◽  
Directed Motion ◽  
Mean Velocity ◽  
Positive Direction ◽  
Brownian Motors ◽  
The Individual

We investigated the motion of two-head Brownian motors by introducing a model in which the two heads coupled through an elastic spring is subjected to a stochastic flashing potential. The ratchet potential felt by the individual head is anti-correlated. The mean velocity was calculated based on Langevin equations. It turns out that we can obtain a unidirectional current. The current is sensitive to the transition rates and neck length and other parameters. The coupling of transition rate and neck length leads to variations both in the values and directions of currency. With a larger neck length, the bi-particle system has a larger velocity in one direction, while with a smaller neck length, it has a smaller velocity in the other direction. This is very likely the case of myosins with a larger neck length and larger velocity in the positive direction of filaments and kinesins with a smaller neck length and smaller velocity in the negative direction of microtubules. We also further investigated how current reversal depended on the neck length and the transition rates.

LONG‐TIME RESPONSE PREDICTED BY EXACT ELASTIC RAY THEORY

Geophysics ◽  
1965 ◽  
Vol 30 (3) ◽  
pp. 363-368 ◽  
Author(s):  
T. W. Spencer
Keyword(s):  
Formal Solution ◽  
Intermediate Stage ◽  
Time Limit ◽  
Time Interval ◽  
Individual Response ◽  
Axially Symmetric ◽  
Long Time ◽  
The Difference ◽  
The Individual ◽  
Significant Figures

The formal solution for an axially symmetric radiation field in a multilayered, elastic system can be expanded in an infinite series. Each term in the series is associated with a particular raypath. It is shown that in the long‐time limit the individual response functions produced by a step input in particle velocity are given by polynomials in odd powers of the time. For rays which suffer m reflections, the degree of the polynomials is 2m+1. The total response is obtained by summing all rays which contribute in a specified time interval. When the rays are selected indiscriminately, the difference between the magnitude of the partial sum at an intermediate stage of computation and the magnitude of the correct total sum may be greater than the number of significant figures carried by the computer. A prescription is stated for arranging the rays into groups. Each group response function varies linearly in the long‐time limit and goes to zero when convolved with a physically realizable source function.

Time Estimation Performance Before, during and after Physical Activity

1988 ◽  
Vol 32 (15) ◽  
pp. 985-989 ◽  
Author(s):  
T. Mihaly ◽  
P.A. Hancock ◽  
M. Vercruyssen ◽  
M. Rahimi
Keyword(s):  
Interval Estimation ◽  
Time Estimation ◽  
Cycle Ergometer ◽  
Estimation Accuracy ◽  
Physical Work ◽  
Time Interval ◽  
Individual Subject ◽  
Knowledge Of Results ◽  
The Individual ◽  
Experimental Findings

An experiment is reported which evaluated performance on a 10-sec time interval estimation task before, during and after physical work on cycle ergometer at intensities of 30 and 60% VO2max, as scaled to the individual subject. Results from the eleven subjects tested indicate a significant increase in variability of estimates during exercise compared to non-exercise phases. Such a trend was also seen in the mean of estimates, where subjects significantly underestimated the target interval (10 seconds) during exercise. Subjects also performed more accurately with information feedback than without knowledge of results, but they were still not able to overcome the effects of exercise. As suggested by the experimental findings, decreased estimation accuracy and increased variability can be expected during physical work and is part of a body of evidence which indicates that exercise and its severity has a substantive impact on perceptual and cognitive performance.

scholarly journals On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

1984 ◽  
Vol 14 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Jean-Marie Reinhard
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Risk Models ◽  
Markovian Environment ◽  
Necessary And Sufficient ◽  
Natural Way ◽  
Quantities Of Interest

AbstractWe consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

scholarly journals An algorithm for the estimation of road traffic space mean speeds from double loop detector data

2016 ◽  
Author(s):  
Margarita Martínez-Díaz ◽  
Ignacio Pérez Pérez
Keyword(s):  
Traffic Management ◽  
Road Traffic ◽  
Time Interval ◽  
Double Loop ◽  
Negative Effects ◽  
Loop Detectors ◽  
The Common ◽  
Log Normal ◽  
The Individual ◽  
Log Normal Distribution

Most algorithms trying to analyze or forecast road traffic rely on many inputs, but in practice, calculations are usually limited by the available data and measurement equipment. Generally, some of these inputs are substituted by raw or even inappropriate estimations, which in some cases come into conflict with the fundamentals of traffic flow theory. This paper refers to one common example of these bad practices. Many traffic management centres depend on the data provided by double loop detectors, which supply, among others, vehicle speeds. The common data treatment is to compute the arithmetic mean of these speeds over different aggregation periods (i.e. the time mean speeds). Time mean speed is not consistent with Edie’s generalized definitions of traffic variables, and therefore it is not the average speed which relates flow to density. This means that current practice begins with an error that can have negative effects in later studies and applications. The algorithm introduced in this paper enables easily the estimation of space mean speeds from the data provided by the loops. It is based on two key hypotheses: stationarity of traffic and log-normal distribution of the individual speeds in each time interval of aggregation. It could also be used in case of transient traffic as a part of any data fusion methodology.DOI: http://dx.doi.org/10.4995/CIT2016.2016.3208 

A PROPOSED FORMULA FOR HORIZONTAL REVENUE ALLOCATION IN NIGERIA

2012 ◽  
Vol 29 (06) ◽  
pp. 1250033
Author(s):  
VIRTUE U. EKHOSUEHI ◽  
AUGUSTINE A. OSAGIEDE
Keyword(s):  
Optimal Control ◽  
Federal Government ◽  
Optimal Control Theory ◽  
Fixed Time ◽  
Tax Revenue ◽  
Time Interval ◽  
Fixed Time Interval ◽  
Hypothetical Data ◽  
Revenue Allocation ◽  
The Individual

In this study, we have applied optimal control theory to determine the optimum value of tax revenues accruing to a state given the range of budgeted expenditure on enforcing tax laws and awareness creation on the payment of the correct tax. This is achieved by maximizing the state's net tax revenue over a fixed time interval subject to certain constraints. By assuming that the satisfaction derived by the Federal Government of Nigeria on the ability of the individual states to generate tax revenue which is as near as the optimum tax revenue (via the state's control problem) is described by the logarithmic form of the Cobb–Douglas utility function, a formula for horizontal revenue allocation in Nigeria in its raw form is derived. Afterwards, we illustrate the use of the proposed horizontal revenue allocation formula using hypothetical data.

scholarly journals Ruin Probabilities for Two Classes of Risk Processes

2005 ◽  
Vol 35 (1) ◽  
pp. 61-77 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Counting Processes ◽  
Arrival Times ◽  
The Laplace Transform ◽  
Sparre Andersen Risk Model ◽  
Risk Processes

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

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