scholarly journals The De Vylder–Goovaerts conjecture holds within the diffusion limit

2019 ◽  
Vol 56 (2) ◽  
pp. 546-557
Author(s):  
Stefan Ankirchner ◽  
Christophette Blanchet-Scalliet ◽  
Nabil Kazi-Tani
Keyword(s):  
Open Problem ◽  
Diffusion Approximation ◽  
Ruin Probability ◽  
Risk Model ◽  
Risk Theory ◽  
Diffusion Limit ◽  
Diffusion Approximations ◽  
Initial Model ◽  
Terminal Time ◽  
Standard Risk

AbstractThe De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

On the De Vylder and Goovaerts Conjecture About Ruin for Equalized Claims

2014 ◽  
Vol 51 (03) ◽  
pp. 874-879 ◽  
Author(s):  
C. Y. Robert
Keyword(s):  
Open Problem ◽  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Ruin Probabilities ◽  
Ruin Theory ◽  
Finite Time Ruin Probability

In ruin theory, the conjecture given in De Vylder and Goovaerts (2000) is an open problem about the comparison of the finite time ruin probability in a homogeneous risk model and the corresponding ruin probability evaluated in the associated model with equalized claim amounts. In this paper we consider a weaker version of the conjecture and show that the integrals of the ruin probabilities with respect to the initial risk reserve are uniformly comparable.

scholarly journals Tree-indexed processes: a high level crossing analysis

2003 ◽  
Vol 16 (2) ◽  
pp. 127-139 ◽  
Author(s):  
Mark Kelbert ◽  
Yuri Suhov
Keyword(s):  
Diffusion Process ◽  
Ruin Probability ◽  
Multiple Choice ◽  
Risk Theory ◽  
Level Crossing ◽  
Branching Diffusion ◽  
Logical Tree ◽  
Level Crossing Analysis ◽  
High Level ◽  
Standard Risk

Consider a branching diffusion process on R1 starting at the origin. Take a high level u>0 and count the number R(u,n) of branches reaching u by generation n. Let Fk,n(u) be the probability P(R(u,n)<k),k=1,2,…. We study the limit limn→∞Fk,n(u)=Fk(u). More precisely, a natural equation for the probabilities Fk(u) is introduced and the structure of the set of solutions is analysed. We interpret Fk(u) as a potential ruin probability in the situation of a multiple choice of a decision taken at vertices of a ‘logical tree’. It is shown that, unlike the standard risk theory, the above equation has a manifold of solutions. Also an analogue of Lundberg's bound for branching diffusion is derived.

scholarly journals On the De Vylder and Goovaerts Conjecture About Ruin for Equalized Claims

2014 ◽  
Vol 51 (3) ◽  
pp. 874-879 ◽  
Author(s):  
C. Y. Robert
Keyword(s):  
Open Problem ◽  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Ruin Probabilities ◽  
Ruin Theory ◽  
Finite Time Ruin Probability

In ruin theory, the conjecture given in De Vylder and Goovaerts (2000) is an open problem about the comparison of the finite time ruin probability in a homogeneous risk model and the corresponding ruin probability evaluated in the associated model with equalized claim amounts. In this paper we consider a weaker version of the conjecture and show that the integrals of the ruin probabilities with respect to the initial risk reserve are uniformly comparable.

scholarly journals Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 982
Author(s):  
Yujuan Huang ◽  
Jing Li ◽  
Hengyu Liu ◽  
Wenguang Yu
Keyword(s):  
Fourier Series ◽  
Ruin Probability ◽  
Risk Model ◽  
Expansion Method ◽  
Nonparametric Estimator ◽  
Insurance Risk ◽  
Numerical Examples ◽  
Complex Fourier Series ◽  
Premium Income ◽  
Insurance Risk Model

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (4) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

scholarly journals Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Yan Li ◽  
Guoxin Liu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Optimal Strategies ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman ◽  
Poisson Risk Model

We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.

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