scholarly journals Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

2020 ◽  
Vol 57 (2) ◽  
pp. 513-530
Author(s):  
Hansjörg Albrecher ◽  
Bohan Chen ◽  
Eleni Vatamidou ◽  
Bert Zwart
Keyword(s):  
Finite Time ◽  
Heavy Tails ◽  
Asymptotic Expression ◽  
Sample Path ◽  
Rare Event ◽  
Large Claim ◽  
Simulation Techniques ◽  
Insurance Portfolio ◽  
Regularly Varying ◽  
Heavy Tailed

AbstractWe investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. Finally, we assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniques.

State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (04) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals Improved algorithms for rare event simulation with heavy tails

2006 ◽  
Vol 38 (2) ◽  
pp. 545-558 ◽  
Author(s):  
Søren Asmussen ◽  
Dirk P. Kroese
Keyword(s):  
Monte Carlo ◽  
Relative Error ◽  
Heavy Tails ◽  
Numerical Study ◽  
Rare Event ◽  
Random Variable ◽  
Event Simulation ◽  
Conditional Monte Carlo ◽  
Heavy Tailed ◽  
Geometric Random Variable

The estimation of P(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

scholarly journals Improved algorithms for rare event simulation with heavy tails

2006 ◽  
Vol 38 (02) ◽  
pp. 545-558 ◽  
Author(s):  
Søren Asmussen ◽  
Dirk P. Kroese
Keyword(s):  
Monte Carlo ◽  
Relative Error ◽  
Heavy Tails ◽  
Numerical Study ◽  
Rare Event ◽  
Random Variable ◽  
Event Simulation ◽  
Conditional Monte Carlo ◽  
Heavy Tailed ◽  
Geometric Random Variable

The estimation of P(S n >u) by simulation, where S n is the sum of independent, identically distributed random varibles Y 1 ,…,Y n , is of importance in many applications. We propose two simulation estimators based upon the identity P(S n >u)=nP(S n >u, M n =Y n ), where M n =max(Y 1 ,…,Y n ). One estimator uses importance sampling (for Y n only), and the other uses conditional Monte Carlo conditioning upon Y 1 ,…,Y n−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

scholarly journals Transient Asymptotics of Lévy-Driven Queues

2010 ◽  
Vol 47 (1) ◽  
pp. 109-129 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Stable Process ◽  
Sample Path ◽  
Rare Event ◽  
Compound Poisson Process ◽  
Input Process ◽  
Deterministic Function ◽  
Heavy Tailed ◽  
Stationary Workload ◽  
Sample Path Large Deviations

With (Qt)t denoting the stationary workload process in a queue fed by a Lévy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB, QTB > qB) for given positive numbers p and q, and a positive deterministic function TB. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e. TB/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e. TB / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB / B2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

scholarly journals State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (4) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (Xk:k≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (Sn:n≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail ofSnin a large deviation regime asn↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals Transient Asymptotics of Lévy-Driven Queues

2010 ◽  
Vol 47 (01) ◽  
pp. 109-129 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Stable Process ◽  
Sample Path ◽  
Rare Event ◽  
Compound Poisson Process ◽  
Input Process ◽  
Deterministic Function ◽  
Heavy Tailed ◽  
Stationary Workload ◽  
Sample Path Large Deviations

With (Q t ) t denoting the stationary workload process in a queue fed by a Lévy input process (X t ) t , this paper focuses on the asymptotics of rare event probabilities of the type P(Q 0 > pB, Q T B > qB) for given positive numbers p and q, and a positive deterministic function T B . We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to T B being sublinear (i.e. T B /B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that T B is superlinear (i.e. T B / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to T B / B 2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T B increasing superlinearly but subquadratically. We pay special attention to the case T B = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

scholarly journals Two-node fluid network with a heavy-tailed random input: the strong stability case

2014 ◽  
Vol 51 (A) ◽  
pp. 249-265
Author(s):  
Sergey Foss ◽  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Heavy Tails ◽  
Sample Path ◽  
Strong Stability ◽  
Tail Asymptotics ◽  
Two Dimensional ◽  
Batch Arrivals ◽  
One Dimensional ◽  
Fluid Network ◽  
Heavy Tailed

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

scholarly journals Heavy-Tailed Distributions and Rating

2001 ◽  
Vol 31 (1) ◽  
pp. 37-58 ◽  
Author(s):  
J. Beirlant ◽  
G. Matthys ◽  
G. Dierckx
Keyword(s):  
Parametric Model ◽  
Extreme Value ◽  
Small Sample ◽  
Data Set ◽  
Large Claim ◽  
Heavy Tailed Distributions ◽  
Fire Insurance ◽  
Insurance Portfolio ◽  
Claim Size Distribution ◽  
Heavy Tailed

AbstractIn this paper we consider the problem raised in the Astin Bulletin (1999) by Prof. Benktander at the occasion of his 80th birthday concerning the choice of an appropriate claim size distribution in connection with reinsurance rating problems. Appropriate models for large claim distributions play a central role in this matter. We review the literature on extreme value methodology and consider its use in reinsurance. Whereas the models in extreme-value methods are non-parametric or semi-parametric of nature, practitioners often need a fully parametric model for assessing a portfolio risk both in the tails and in more central portions of the claim distribution. To this end we propose a parametric model, termed the generalised Burr-gamma distribution, which possesses such flexibility. Throughout we consider a Norwegian fire insurance portfolio data set in order to illustrate the concepts. A small sample simulation study is performed to validate the different methods for estimating excess-of-loss reinsurance premiums.

scholarly journals Two-node fluid network with a heavy-tailed random input: the strong stability case

2014 ◽  
Vol 51 (A) ◽  
pp. 249-265 ◽  
Author(s):  
Sergey Foss ◽  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Heavy Tails ◽  
Sample Path ◽  
Strong Stability ◽  
Tail Asymptotics ◽  
Two Dimensional ◽  
Batch Arrivals ◽  
One Dimensional ◽  
Fluid Network ◽  
Heavy Tailed

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Rare event simulation

2017 ◽  
Author(s):  
Michael P. Allen ◽  
Dominic J. Tildesley
Keyword(s):  
Rare Event ◽  
Path Sampling ◽  
Rapid Progress ◽  
Computationally Efficient ◽  
Transition Path ◽  
Transition Path Sampling ◽  
Event Simulation ◽  
Reaction Coordinates ◽  
Simulation Techniques ◽  
Transition Interface

The development of techniques to simulate infrequent events has been an area of rapid progress in recent years. In this chapter, we shall discuss some of the simulation techniques developed to study the dynamics of rare events. A basic summary of the statistical mechanics of barrier crossing is followed by a discussion of approaches based on the identification of reaction coordinates, and those which seek to avoid prior assumptions about the transition path. The demanding technique of transition path sampling is introduced and forward flux sampling and transition interface sampling are considered as rigorous but computationally efficient approaches.

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