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ASYMPTOTIC FORMULAS IN DISCRETE TIME RISK MODEL WITH DEPENDENCE OF FINANCIAL AND INSURANCE RISKS
G.Tsitsiashvili •
640041, Russia, Vladivostok, Radio str. 7, IAM, FEB RAS, e-mail: [email protected]
ABSTRACT
Asymptotic formulas for a ruin probability in discrete time risk model with a dependence of financial and insurance risks are obtained. These formulas are constructed in a suggestion which is adequate to economical crisis: the larger is a financial risk the larger is an insurance risk.
1. INTRODUCTION
In this paper we obtain asymptotic formulas for a ruin probability in discrete time risk model with a dependence of financial and insurance risks. Earlier simple asymptotic formulas for the ruin probability in a case of independent financial and insurance risks have been obtained in [1]. More complicated cases with special restrictions on insurance risks dependence are considered for example in [2,3]. Nevertheless until recently an asymptotic analysis of risk model with a dependence of insurance and financial risks is not made. But in modern period of strong economical crisis such dependence may be recognized easily in different large anthropogenic catastrophes. So a problem to analyze asymptotically this dependence is actual now.
In this paper we consider special model of insurance and financial risks dependence based on suggestion that a financial risk has a finite number of meanings and for each meaning an insurance risk has its own distribution. Then Pareto-tailed and Weibull-tailed asymptotics of insurance risks distributions are considered. In frames of this suggestion we assume that the larger is the financial risk the larger is the insurance risk. This stochastic modeling approach allows to obtain new asymptotic formulas for ruin probability in risk models.
2 PRELIMINARIES
Classes of distributions. Throughout, for a given random variable (r.v.) X concentrated
on (-w, w) with a distribution function (d.f.) F then its right tail F(x) = P(X > x). For two d.f.'s
F1 and F2 concentrated on (-w,w) we write by F1 * F2 (x) the convolution of F1 and F2 and write
by F*2 = F1* F1 the convolution of F1 with itself. All limiting relationships, unless otherwise
stated, are for x ^w. Let a (x) > 0 and b (x) > 0 be two infinitesimals, satisfying
. „a(x) a(x) ,,
l" < liminf^^< limsup^^ < l+ .
x^w b (x) x^w b (x)
We write a (x) = O (b (x)), if l + <w and a (x) ~ b (x) if l + = l- = 1.
Introduce the following classes of d.f.'s concentrated on [0, w):
5 = J F (x): lim ^^ = 2!, L = i F (x): V? lim - ?) = 1},
| v > x^w F (x) J [ W x^w F (x) J
R-a=JF(x): V9> 0 lim F(Qx) = 9-a !, 0 <a<w , ^ = u R a.. 5 is called the class of
[ x ^w F (x) J 0<a<w
subexponential d.f.'s. l is called the class of long tailed d.f.'s. r (or a) is called the class of
regular varying d.f/s (with index a). More generally, d.f. F concentrated on (-<»,<») is also said to belong to these classes if its right-hand distribution F (x) = F (x )□( x > 0) does.
Proposition 1. The classes r,s,l satisfy the formula [4] r c s c l . If for some a,b,0 < a,0 < b < 1 d.f. F satisfies the equivalence F(t) ~exp(-atb), t ^ w then f e s .
Proposition 2. Let F1 and F2 be two d.f.'s concentrated on (-<», <»). If f2 e l, F1 e s and F2(x) = O(Fi(x)),then [1, Lemma 3.2] F1 *F2 e s and F1 *F2(x)~F1 (x) + Y2(x).
3 DISCRETE TIME RISK MODEL AND ITS PROPERTIES
Consider discrete time risk model (with annual step) with initial capital x, x > 0and nonnegative losses
Zn , n = 1,2,..., P(Zn < t) = F(t).
Suppose that income An, n = l,2,...to end of n-th year is defined as difference between unit premium sum and loss An = 1 -Zn. Assume that Rn >lis inflation factor from n -1 to n year, n = 1,2,.... In [5] Xn = -An is called insurance risk and Yn = R— is called financial risk. Suppose that the following condition is true:
(A). {(An,Rn), n > 1} is sequence of independent and identically distributed random vectors (i.i.d.r.) vectors
¿0 = x, s„ = R„S„-1 + An , n = 1,2... (1)
In this model with initial capital x ruin time is defined by formula
t(x) = inf {n = 1,2,...: Sn < 0|S0 = x} , and finite time ruin probability x, n) - by formula
y(x, n ) = P (t( x )< n) .
So the sum Sn money accumulated by insurance company to n- th year end satisfies recurrent formula
S0 = x ,Sn = xn Bj + £ A, n B, , n = 1,2..., (2)
j=1 ,=1 j=,+1
where n=n+1 = 1by convention.
According to the notation above, we can rewrite the discounted value of the surplus Sn in (2)
as
§0 = x, S n = Sn Y\ Y, = x-¿X, n Y, = x - Wn . j=1 ,=1 j=1
Hence, we easily understand that, for each n = 0,1,...,
v(x,n) = P(Un > x), (3)
where
Un = max J 0,max Wk i, with U0 = 0 . (4)
I 1<k< n J
Define another Markov chain as
V0 = 0, V„ = Y„ (0,X„ + V„-1), n = 1,2.... (5)
Theorem 1. Suppose that the condition (A) is true.
1. Random variables Un and Vn coincide by distribution
Un =Vn , . n = 0,1,... (6)
2. Equality
x,n) = P (Vn > x) (7)
is true.
Proof: The result (6) is trivial for the case when n = 0 . Now we aim at (6) for each n = 1,2.... Let n > 1 be fixed. It is easy to obtain the equality
U„ = max|o,max | Xt n Y,} = Tn ((Xl,Yl),...,(Xn,Yn)) .
[ 1<k<n t=1 j=1 J
Here Tn is a deterministic function. From the condition (A) we obtain that and consequently
(d )
((X1,Y1 ),..., (Xn ,Yn )) = ((X„, Yn ),..., ((X 1,Y1 )).
(d )
Un = Tn ((X1,Y ),...,(Xn ,Yn )) = Tn ((Xn ,Yn ),...,(X 1,Y1 )) =
| k i II n n I
= max J 0, max 2 Xn+i_, n Y„+i_. \ = max \ 0, max 2 Xi* n Y.. } =
L 1sk ^ n i=1 j=1 J I ^k ^n i*=n+1_k j'=i' 1 J
= max J 0, max X,* n Y.. I = Vn . (8)
L 1<k <n i*=k* y*=i' j j
Here
I%„ = Yn (0,X„ + V„_! )+ , n = 1,2...,
which is just the same as (5). So we immediately conclude that V%n = Vn for each n = 1,2.... Finally, it follows from (8) that (6) holds for each n = 1,2.... The formula (7) is a sequence of the formulas (3), (6). This ends the proof of Theorem 1.
Remark 1. Theorem 1 proof practically repeats the proof of [1,Theorem 2.1]. A single difference is that the condition of r.v/s X1,Y1,...,Xn,Yn independence is replaced by more weak condition (C).
Introduce the finite set Q = {1,...,m} and for any qeQ define d.f/s Fq(t) and i.i.d.r.v/s Xnq , P(Xnq > t) = Fq (t) and positive constants Rq, R1 < Rq, q ^ 1, qeQ. Suppose that 0 < pq < 1,
2 Pq = 1.
qeQ
(B). Random vector ( Xn ,Yn ) satisfies the condition
P[(Xn,Yn) = ^Xnq),r1)= Pq , qeQ. (9)
From the formula (2) and the condition (9) we have
P (Vn > t)= 2 PqP (Xiq) + Vn_i > R(q)t), t > 0. (10)
qeQ
(C). Suppose that Fq(t)es, qeQ and for any q1,q2 e Q ,q1 ^ q2 one of the following equalities is
K(t) = O (((t)) or f;2 (t) = O ((t)) (11)
true
4 ASYMPTOTIC ANALYSIS OF RUIN PROBABILITY
Theorem 2. If the conditions (A), (B), (C) are true then for t ^
P V > t)~ 2 pqi Fqi (R( q V)+ Z Pqi Pqi Fqi (R(q1 )R(q2 V) + ...+
®eQ qu^i^Q (12)
+ Z Pq1 • ... * PqnFq1 (R(q )• ... • R(q» V).
qi>...>q„GQ
Proof: Suppose that n = l then
P (Vl > t )= Z P qP (( )> R (q )t )= Z PqFqi (R )t, t > 0 ) .
qeQ qeQ
So for n = l the asymptotic formula (12) is true. Suppose that the formula (12) takes place for fixed n. Then from the formula (10) we obtain
P (Vn+1 > t)= Z Pqn+1P (xn+r) + Vn > R(qn+1 )t), t > 0.
So from the formula 12) and from Propositions 1, 2 and from the conditions A), (B), (C) we have for t that
P (Vn+i > t)~ Z Pq++l
qn+ie2
Z Pqn+1 Fqi ( (qn+1 )Ä(q)l t)+ Z Pq1 Pq2 Fqi ( ^ ^ W^t)+... -_ qi^g qi,q2^Q
+ Z pqi •...PqnFqi (R^W*1) •...• R(q»V) + F^ ((^)i)
qi,...,q»eg
= Z Pqn+1 Pqi •Fqi (R (q"+1 )R(qi t) + Z Pqn+1 Pqi Pq2 F q> (R(fci )R(qi ^ ) + ... + q»+i,qieg q„+i,qi,q2 eg
+ Z QPq„+i Pqi,.^ Pq„"Fqi ((+1 )R(qi} ,., ^ >*) + 2! q Pq„+i Fq„+i (»+1 V) = q»+i,qi,...,q„e2 q„+ieg
= Z Pq1 Fqi (R(qi t) + Z Pq1 Pq2 Fqi (R(qi t) + ... +
qieg qi,qi^Q
+ Z QPqi- Pq»+i Fqi (R (* J •...• R (q»+1 ^ ) . qf...' q» ' q»+ieg
Last equality is obtained by a replacement of indexes 1,..., » +1 in its summands. So the formula (12) is proved for index » +1 also.
Consider the following asymptotic conditions for t ^ œ .
(Di). There are positive numbers cq,aq, qeg, a1 <a , 1 <q<m so that Fq (t)~ c t~aq (D2). There are positive numbers cq, qeg, a so that Fq (t) ~ cqt~a
(D3). There are positive numbers cq,pq, qeg, p1 <pq, 1 <q< m so that Fq (t) ~exp(-cqtPq ).
(D4). There are positive numbers cq, q eg, p so that Fq (t) — exp (-cqtp)
It is easy to prove that the family Fq (t ), q eg under each of the conditions (D1), (D2), (D3), (D4) satisfies the condition (C).
In the condition (D1) the formula (12) may be represented in the following form
v(t, n) ~ 2 PikF 1 (Rikt)~ c,t-ai 2
Pi
k=1
k=1 Rk
ka
and consequently
v(t, n )'
c,t'
1 _ Pn+1 R-(n+k)a
_aik Pi Ri-, piR1_a ^ i
i _ pR_a
nc1t "ai, p1R1_a = i.
In the condition (D2) the formula (12) may be represented in the following form
V(t, n) ~ ^ ]TQCqi pqiR_a + 2 q ^ p^ p^R_aR_a +
_ qieQ qi,qieQ
1_ sn
K + = t ~ a s
qi^qi ••• fqn"qi •••
+ 2 c^, •...• PqR-_a-...• R_a +
qi,...,q„eQ
= t_ aSk
_
i_ s2
with
S1= 2 Cq,pq,Rq_a , S1= 2 Cq, pq,
In the condition (D3) the formula (12) may be represented in the following form
v(t,n) ~ 2 pikFi (Rikt) = 22 p,k exp(_c, (Riktf )
k=i k=i v '
and so
V(t, n )
pn (_ci (Rintf ), Ri < i, p, (_ci (Ri tf ), Ri > i,
exp
(_ c, (R, t f)1, pî
n+1
p,
, R, = i.
In the condition (D4) the formula (12) may be represented in the following form
v(t, n ) ~ 22
k=i
2 Qpq, • ... • pqk exp ( Cq, (Rq, • ... • Rqkt)P)
_q,,...,qkeQ v ' _
Suppose that there is the constant q' satisfying the inequalities
Cq'Rq' < Cq'RqP , q * q', q eQ .
The equivalences (16) may be rewritten as follows
v(t,n) ~ pq, 22 pk-1 exp(-Cq, (Rq,Rk-1t)) .
k=1 ^ '
and so
V(t, n )'
pq. pî_ exp ( Cq' (Rq' Rl"_1t )P), Ri < 1,
pq' exp ( Cq' (Rq't ) ) ,R1 < 1,
((V )) Ri = k.
1 _ pn
pq'--L exp
1 - p1
(13)
(14)
(15)
(16)
(17)
5 CONCLUSION
A comparison of the asymptotic formulas (13), (14), (15), (17) with the results of [1] shows that a dependence of financial and insurance risks introduces significant changes into asymptotic formulas for ruin probability of discrete time risk model.
REFERENCES
1. Tang Q.; Tsitsiashvili G. 2003. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochast.Process. appi. Vol. 108, no. 2, 299--325.3.
2 Tang Q.; Tsitsiashvili G. 2003. Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory. Extremes. Vol. 6, pp. 171-188.
3. Tang Q., Wang G.,_ Yuen K.C. Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. 2010. Insurance: Mathematics and Economics. Vol. 46, pp. 362-370.
4. Embrechts P., Kluppelberg C., Mikosch T. 1997. Modelling Extremal Events in Finance and Insurance. Springer.
5. Norberg R. 1999. Ruin problems with assets and liabilities of diffusion type. Stochastic Process. appi. Vol. 81. No. 2.
