scholarly journals Three Methods to Calculate the Probability of Ruin

1989 ◽  
Vol 19 (1) ◽  
pp. 71-90 ◽  
Author(s):  
François Dufresne ◽  
Hans U. Gerber
Keyword(s):  
Stationary Distribution ◽  
Geometric Distribution ◽  
Random Variable ◽  
Claim Amount ◽  
Adjustment Coefficient ◽  
Exponential Distributions ◽  
Probability Of Ruin ◽  
The Third ◽  
Compound Geometric Distribution ◽  
Aggregate Loss

AbstractThe first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

scholarly journals THE FULL TAILS GAMMA DISTRIBUTION APPLIED TO MODEL EXTREME VALUES

2017 ◽  
Vol 47 (3) ◽  
pp. 895-917 ◽  
Author(s):  
Joan del Castillo ◽  
Jalila Daoudi ◽  
Isabel Serra
Keyword(s):  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Extreme Values ◽  
Dispersion Model ◽  
Value Theory ◽  
Quantitative Modelling ◽  
Exponential Distributions ◽  
Exponential Dispersion Model ◽  
Aggregate Loss ◽  
Very High

AbstractIn this paper, we introduce the simplest exponential dispersion model containing the Pareto and exponential distributions. In this way, we obtain distributions with support (0, ∞) that in a long interval are equivalent to the Pareto distribution; however, for very high values, decrease like the exponential. This model is useful for solving relevant problems that arise in the practical use of extreme value theory. The results are applied to two real examples, the first of these on the analysis of aggregate loss distributions associated to the quantitative modelling of operational risk. The second example shows that the new model improves adjustments to the destructive power of hurricanes, which are among the major causes of insurance losses worldwide.

scholarly journals Distributions stationnaires d'un système bonus–malus et probabilité de ruine

1988 ◽  
Vol 18 (1) ◽  
pp. 31-46 ◽  
Author(s):  
Par François Dufresne
Keyword(s):  
Theoretical Model ◽  
Stationary Distribution ◽  
Recursive Algorithm ◽  
Third Party ◽  
Ruin Probabilities ◽  
Stationary Distributions ◽  
Probability Of Ruin

AbstractIt is shown how the stationary distributions of a bonus–malus system can be computed recursively. It is further shown that there is an intrinsic relationship between such a stationary distribution and the probability of ruin in the risk-theoretical model. The recursive algorithm is applied to the Swiss bonus–malus system for automobile third-party liability and can be used to evaluate ruin probabilities.

scholarly journals Stein's Method for Compound Geometric Approximation

2010 ◽  
Vol 47 (1) ◽  
pp. 146-156 ◽  
Author(s):  
Fraser Daly
Keyword(s):  
Generating Functions ◽  
Analytical Approach ◽  
Geometric Distribution ◽  
Stein’S Method ◽  
Hitting Times ◽  
Stein's Method ◽  
Geometric Approximation ◽  
Compound Geometric Distribution ◽  
Probabilistic Approximation ◽  
Sequence Patterns

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

scholarly journals Optimal Coordination of Last Trains for Maximum Transfer Accessibility with Heterogeneous Walking Time

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Yao Chen ◽  
Baohua Mao ◽  
Yun Bai ◽  
Tin Kin Ho ◽  
Zhujun Li
Keyword(s):  
Dwell Time ◽  
Random Variable ◽  
Minimal Extension ◽  
The Third ◽  
Train Timetable ◽  
Urban Rail ◽  
Optimal Coordination ◽  
Transfer Stations ◽  
The Given ◽  
Walking Time

Last train coordination aims to synchronize the arrival and departure times of the last feeder trains and the last connecting trains at transfer stations to improve the transfer accessibility of urban rail networks. This study focuses on the transfer accessibility between last trains with considering heterogeneous transfer walking time. Three mathematical models are developed on the last train timetable optimization. The first model fine-tunes the last train timetable under the given bound of the dwell time. The second one aims to allow the mutual transfers with the prolonged dwell time to maximize the transfer accessibility. A biobjective function is proposed to seek the trade-off between the maximal transfer accessibility and the minimal extension of dwell time. The third model considers the heterogeneity of transfer walking time that is represented as a random variable following a probability distribution. A discrete approximation method is proposed to reformulate the nonlinear model. The embedded Branch & Cut algorithm of CPLEX is applied to solve the models. A real case on the Shenzhen metro network is conducted to demonstrate the performance of the models. The three models all provide better last train timetable than the current timetable in practice. The sensitivity analysis manifests that the third model are always advantageous in the optimization of successful transfer passengers.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (02) ◽  
pp. 329-337
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj ) where p ij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

COMPARISON OF APPROXIMATIONS FOR COMPOUND POISSON PROCESSES

2015 ◽  
Vol 45 (3) ◽  
pp. 601-637 ◽  
Author(s):  
Raffaello Seri ◽  
Christine Choirat
Keyword(s):  
Random Variable ◽  
Cumulative Distribution ◽  
Claim Amount ◽  
Approximation Methods ◽  
Total Claim Amount ◽  
Aggregate Claim ◽  
Time Period ◽  
Total Claim ◽  
The Difference ◽  
Object Of Study

AbstractIn this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

Likelihood ratios for the infinite alleles model

1994 ◽  
Vol 31 (03) ◽  
pp. 595-605 ◽  
Author(s):  
Paul Joyce
Keyword(s):  
Likelihood Ratio ◽  
Stationary Distribution ◽  
Random Sequence ◽  
Random Variable ◽  
Single Locus ◽  
Likelihood Ratios ◽  
Gene Frequencies ◽  
Random Samples ◽  
Infinite Alleles Model

The stationary distribution for the population frequencies under an infinite alleles model is described as a random sequence (x 1, x 2, · ··) such that Σxi = 1. Likelihood ratio theory is developed for random samples drawn from such populations. As a result of the theory, it is shown that any parameter distinguishing an infinite alleles model with selection from the neutral infinite alleles model cannot be consistently estimated based on gene frequencies at a single locus. Furthermore, the likelihood ratio (neutral versus selection) converges to a non-trivial random variable under both hypotheses. This shows that if one wishes to test a completely specified infinite alleles model with selection against neutrality, the test will not obtain power 1 in the limit.

scholarly journals Natural Test for Random Numbers Generator Based on Exponential Distribution

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 920 ◽  
Author(s):  
Tanackov ◽  
Sinani ◽  
Stanković ◽  
Bogdanović ◽  
Stević ◽  
...  
Keyword(s):  
Standard Deviation ◽  
Mathematical Expectation ◽  
Random Number ◽  
Random Variable ◽  
Random Numbers ◽  
The Third ◽  
Atmospheric Noise ◽  
Constant E ◽  
Fourth Series ◽  
Natural Test

We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable Ex(λ). For a set of n random numbers, the parameters of mathematical expectation and standard deviation is λ =n−1. The theorem was verified on four series of 200 sets of 101 random numbers each. The first series was obtained on the basis of decimals of the constant e=2.718281…, the second on the decimals of the constant π =3.141592…, the third on a Pseudo Random Number generated from Excel function RAND, and the fourth series of True Random Number generated from atmospheric noise. The obtained results confirm the application of the derived theorem in practice.

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (4) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
Fixed Interval ◽  
The Real ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

scholarly journals A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME INCREMENTS

2011 ◽  
Vol 11 (01) ◽  
pp. 5-48
Author(s):  
JAY ROSEN
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Random Variable ◽  
Normal Random Variable ◽  
Higher Moments ◽  
Second Moment ◽  
The Third ◽  
Brownian Local Time

Let [Formula: see text] denote the local time of Brownian motion. Our main result is to show that for each fixed t[Formula: see text] as h → 0, where η is a normal random variable with mean zero and variance one, that is independent of [Formula: see text]. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments.

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