On a Sparre-Andersen risk model with PH(n)

46 Pages·2013·0.36 MB·English

n
On a Sparre-Andersen risk model with PH( )
interclaim times
Agnieszka I. Bergel, Alfredo D. Egidio dos Reis
CEMAPREandISEG,TechnicalUniversityofLisbon,Portugal
May 2013
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
2 De(cid:28)nitions
3 Motivation
4 The multiple roots of the fundamental Lundberg’s equation
5 The Lundberg’s Matrix and its eigenvectors
6 Applications
7 Conclusions
Table of contents
1 Model
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
3 Motivation
4 The multiple roots of the fundamental Lundberg’s equation
5 The Lundberg’s Matrix and its eigenvectors
6 Applications
7 Conclusions
Table of contents
1 Model
2 De(cid:28)nitions
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
4 The multiple roots of the fundamental Lundberg’s equation
5 The Lundberg’s Matrix and its eigenvectors
6 Applications
7 Conclusions
Table of contents
1 Model
2 De(cid:28)nitions
3 Motivation
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
5 The Lundberg’s Matrix and its eigenvectors
6 Applications
7 Conclusions
Table of contents
1 Model
2 De(cid:28)nitions
3 Motivation
4 The multiple roots of the fundamental Lundberg’s equation
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
6 Applications
7 Conclusions
Table of contents
1 Model
2 De(cid:28)nitions
3 Motivation
4 The multiple roots of the fundamental Lundberg’s equation
5 The Lundberg’s Matrix and its eigenvectors
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
7 Conclusions
Table of contents
1 Model
2 De(cid:28)nitions
3 Motivation
4 The multiple roots of the fundamental Lundberg’s equation
5 The Lundberg’s Matrix and its eigenvectors
6 Applications
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
Table of contents
1 Model
2 De(cid:28)nitions
3 Motivation
4 The multiple roots of the fundamental Lundberg’s equation
5 The Lundberg’s Matrix and its eigenvectors
6 Applications
7 Conclusions
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
Model
Sparre(cid:21)Andersen model
N(t)
(cid:88)
U(t)=u+ct− X , X ≡0, t ≥0, u ≥0,
i 0
i=0
• u initial capital,
• t time variable,
• c >0, loading factor,
•{X }∞ , single claim amount sequence,
i i=1
•{W }∞ , interclaim time sequence,
i i=1
•N(t)=max{k : W +W +···+W ≤t}, number of claims.
1 2 k
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes
Assumptions on the model:
•X(cid:48)s i.i.d., X ∼P(x) with density p(x),
i i
•pˆ(s)=(cid:82)∞e−sxp(x)dx, Laplace transform of p(x),
0
•µ =E[Xk] the k-th moment of X ,
k 1 1
•W(cid:48)s i.i.d. and independent from the X (cid:48)s ,
i i
•W ∼K(t), W ∼ PH(n), with representation (α,B), density k(t)
i i
and Laplace transform kˆ(s),
•cE(W )>E(X ) positive loading factor.
1 1
AgnieszkaI.Bergel,AlfredoD.EgidiodosReis OnaSparre-AndersenriskmodelwithPH(n)interclaimtimes