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 Market Choices Driven by Reference Groups: A Comparison of Analytical and Simulation Results on Random Networks

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1007
Author(s):  
Michał Ramsza
Keyword(s):  
Theoretical Model ◽  
Reference Group ◽  
Third Party ◽  
Qualitative Behavior ◽  
Reference Groups ◽  
Group Influence ◽  
Simple Theoretical Model ◽  
High Connectivity ◽  
Simulation Results ◽  
Two Equilibria

The present paper reports simulation results for a simple model of reference group influence on market choices, e.g., brand selection. The model was simulated on three types of random graphs, Erdos–Renyi, Barabasi–Albert, and Watts–Strogatz. The estimates of equilibria based on the simulation results were compared to the equilibria of the theoretical model. It was verified that the simulations exhibited the same qualitative behavior as the theoretical model, and for graphs with high connectivity and low clustering, the quantitative predictions offered a viable approximation. These results allowed extending the results from the simple theoretical model to networks. Thus, by increasing the positive response towards the reference group, the third party may create a bistable situation with two equilibria at which respective brands dominate the market. This task is easier for large reference groups.

scholarly journals Basing population genetic inferences and models of molecular evolution upon desired stationary distributions of DNA or protein sequences

2008 ◽  
Vol 363 (1512) ◽  
pp. 3931-3939 ◽  
Author(s):  
Sang Chul Choi ◽  
Benjamin D Redelings ◽  
Jeffrey L Thorne
Keyword(s):  
Molecular Evolution ◽  
Markov Model ◽  
Stationary Distribution ◽  
Population Genetic ◽  
Amino Acid Level ◽  
Protein Sequences ◽  
Relative Fitness ◽  
Evolutionary Models ◽  
Effective Population ◽  
Stationary Distributions

Models of molecular evolution tend to be overly simplistic caricatures of biology that are prone to assigning high probabilities to biologically implausible DNA or protein sequences. Here, we explore how to construct time-reversible evolutionary models that yield stationary distributions of sequences that match given target distributions. By adopting comparatively realistic target distributions, evolutionary models can be improved. Instead of focusing on estimating parameters, we concentrate on the population genetic implications of these models. Specifically, we obtain estimates of the product of effective population size and relative fitness difference of alleles. The approach is illustrated with two applications to protein-coding DNA. In the first, a codon-based evolutionary model yields a stationary distribution of sequences, which, when the sequences are translated, matches a variable-length Markov model trained on human proteins. In the second, we introduce an insertion–deletion model that describes selectively neutral evolutionary changes to DNA. We then show how to modify the neutral model so that its stationary distribution at the amino acid level can match a profile hidden Markov model, such as the one associated with the Pfam database.

scholarly journals Erlangian Approximations for Finite-Horizon Ruin Probabilities

2002 ◽  
Vol 32 (2) ◽  
pp. 267-281 ◽  
Author(s):  
Soren Asmussen ◽  
Florin Avram ◽  
Miguel Usabel
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Fluid Model ◽  
Ruin Probabilities ◽  
Approximation Procedure ◽  
Exponential Form ◽  
Probability Of Ruin ◽  
Phase Type ◽  
The Matrix ◽  
The Mean

AbstractFor the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

Stationary distributions for dams with additive input and content-dependent release rate

1977 ◽  
Vol 9 (03) ◽  
pp. 645-663 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Stationary Distribution ◽  
Release Rate ◽  
Zero Set ◽  
Stationary Distributions ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Limiting Case ◽  
Finite Dam

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt } of an infinite dam whose cumulative input {At } is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = x α and {A t } is stable with index β ∊ (0, 1). In general if EAt , < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA 1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA 1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt } as t → ∞. For {At } stable with index β and r(x) = x α , α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt } in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.

scholarly journals On Truncation of the Matrix-Geometric Stationary Distributions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 798 ◽  
Author(s):  
Naumov ◽  
Gaidamaka ◽  
Samouylov
Keyword(s):  
Finite Number ◽  
Stationary Distribution ◽  
Queueing Systems ◽  
Death Process ◽  
Birth And Death Process ◽  
Loss System ◽  
Stationary Distributions ◽  
Open Network ◽  
The Matrix

In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed.

Level crossings and stationary distributions for general dams

1980 ◽  
Vol 17 (1) ◽  
pp. 218-226 ◽  
Author(s):  
Michael Rubinovitch ◽  
J. W. Cohen
Keyword(s):  
Explicit Expression ◽  
Stationary Distribution ◽  
Basic Relation ◽  
Expected Number ◽  
Stationary Distributions ◽  
Level Crossings ◽  
Fixed Level

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

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.

Quasi-stationary distributions and one-dimensional circuit-switched networks

1987 ◽  
Vol 24 (04) ◽  
pp. 965-977 ◽  
Author(s):  
Ilze Ziedins
Keyword(s):  
Communication Networks ◽  
Stationary Distribution ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Switched Networks ◽  
One Dimensional ◽  
Special Cases ◽  
Quasi Stationary Distribution

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

scholarly journals A Stable Recursive Algorithm for Evaluation of Ultimate Ruin Probabilities

1984 ◽  
Vol 14 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Marc Goovaerts ◽  
Florian de Vylder
Keyword(s):  
Recursive Algorithm ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Tolerance Level ◽  
Quadrature Formulae

AbstractProbabilities of ruin are solutions of differential or integrodifferential equations. Solving such equations numerically can be performed by means of approximate quadrature formulae for the convolution part of the equation. In this contribution it is shown how applicable recursion formulae, giving results within a prescribed tolerance level, can be obtained. Some numerical results are displayed.

Explicit formulae for stationary distributions of stress release processes

2000 ◽  
Vol 37 (2) ◽  
pp. 315-321 ◽  
Author(s):  
K. Borovkov ◽  
D. Vere-Jones
Keyword(s):  
Closed Form ◽  
Stationary Distribution ◽  
Exponential Function ◽  
Markov Models ◽  
Risk Process ◽  
Stress Release ◽  
Stationary Distributions ◽  
Explicit Formulae ◽  
Tectonic Forces ◽  
Release Processes

Stress release processes are special Markov models attempting to describe the behaviour of stress and occurrence of earthquakes in seismic zones. The stress is built up linearly by tectonic forces and released spontaneously when earthquakes occur. Assuming that the risk is an exponential function of the stress, we derive closed form expressions for the stationary distribution of such processes, the moments of the risk, and the autocovariance function of the reciprocal risk process.

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