ON THE ASYMPTOTIC RUIN PROBABILITY WITH SOME SPECIFIC STOCHASTIC

ON THE ASYMPTOTIC RUIN PROBABILITY WITH SOME SPECIFIC STOCHASTIC

DISCOUNT FACTORS

RALUCA VERNIC

We consider an insurance context in which the discounted sum of losses can be described as a randomly weighted sum of a sequence of independent random vari- ables. We introduce a specific dependence structure for the discount factors based on their marginal distributions, and apply the results obtained by Goovaerts et al. (2005) on approximating ruin probabilities. The particular case of the Farlie- Gumbel-Morgenstern distribution is underlined.

AMS 2000 Subject Classification: 62P05, 62E20.

Key words: asymptotics, tail probability, finite and infinite time ruin probabilities, Farlie-Gumbel-Morgenstern distributions.

1. INTRODUCTION

Recently, in a general probabilistic setting, Goovaerts et al. [5] investi- gated the tail probabilities of the randomly weighted sums

(1) S_n=

n k=1

θ_kX_k, n≥1,

and their maxima. Here{X_n, n≥1} is a sequence of independent and iden- tically distributed (i.i.d.) real-valued random variables with generic random variableXand common distribution functionF, and{θ_n, n≥1}is another se- quence of positive random variables, independent of the sequence{Xn, n≥1}. We will present an actuarial model that reduces to (1).

As in the recent work of Nyrhinen ([10], [11]) and Tang and Tsitsiashvili ([12], [13]), we consider the discrete time risk model

(2) S₀ = 0, S_n=

n i=1

X_i i j=1

Y_j, n= 1,2, . . . ,

where{X_n, n≥1}is as in model (1), and{Y_n, n≥1}is another sequence of nonnegative random variables distributed on [0,∞), the two sequences being

REV. ROUMAINE MATH. PURES APPL.,52(2007),4, 479-495

mutually independent. In this model, the random variableXnis the total loss during periodnand the random variableY_n is the discount factor from time n to time n−1, n = 1,2, . . .. Thus, the sum Sn represents the aggregated discounted losses by timenof an insurer in a stochastic economic environment.

In the terminology of Norberg [9],{X_n, n≥1} are called insurance risks and {Y_n, n≥1} financial risks.

We notice that (2) reduces to model (1) with θ_k = _k

i=1Y_i, a product of positive random variables. For model (2), the finite and infinite time ruin probabilities are defined for an insurer whose initial wealth isx≥0 as

ψ(x;n) = Pr

0≤maxk≤nS_k > x

and ψ(x) = Pr

0≤maxk<∞S_k> x , respectively.

Goovaerts et al. [5] derived asymptotic expressions for the tail proba- bilities of (1) and its maxima and, based on them, considered two particular distributions for {θ_k, k ≥ 1}, for which the finite time ruin probability can be asymptotically evaluated. After recalling the main results in Section 2, in Section 3 we will present other such distributions with given marginals and a specific dependence structure. The particular case of the Farlie-Gumbel- Morgenstern (FGM) distribution is investigated, and for this distribution we also obtain an explicit asymptotic form of the infinite time ruin probability.

In what follows we will assume that summing over an empty domain gives 0.

For two positive infinitesimals a(x) and b(x), we write a(x) ∼ b(x) if lim sup

x→∞

a(x) b(x) = 1.

2. A RECENT RESULT

As in Goovaerts et al. [5], we shall restrict ourselves to the case of regularly varying tailed loss distributions. We say that the right tail of F is regularly varying with regularity index α > 0, denoted by F ∈ R₋α, if F(x) = 1−F(x)>0 for all xand

F(xy)∼y^−αF(x) for all y >0.

This class contains the famous Pareto distributions, widely used in insurance to model losses. For more details on this class, see Bingham et al. [1].

Goovaerts et al. [5] proved the result below.

Proposition 2.1. Consider model (1) with F ∈ R₋α for some α > 0.

We have

Pr

1≤maxm≤nS_m > x

∼F(x) n k=1

Eθ^α_k

if there exists some δ >0 such thatEθ_k^α+δ<∞ for each 1≤k≤n.

We also have Pr

1≤maxn<∞S_n> x

∼F(x)^∞

k=1

Eθ^α_k if one of the following assumptions holds:

(1) 0< α <1, _∞

k=1Eθ_k^α+δ<∞ and_∞

k=1Eθ^α_k^−δ <∞ for some δ >0;

(2) 1 ≤ α < ∞, _∞

k=1

Eθ^α_k^+δ_α+δ¹

< ∞ and _∞

k=1

Eθ_k^α−δ_α+δ¹

< ∞ for someδ >0.

From this proposition, the result below is immediate.

Corollary 2.1.In particular, under the assumptions of Proposition2.1, we have

(3) ψ(x;n)∼F(x)

n k=1

Eθ_k^α and

(4) ψ(x)∼F(x)

∞ k=1

Eθ_k^α.

3. SPECIFIC CASES

In order to evaluate the asymptotic ruin probability given in Corol- lary 2.1, we need to calculate the expectations Eθ^α_k for k = 1,2, . . .. In this section, we give some special cases for which such calculation can be per- formed. We start by presenting some multivariate distribution families with given marginals.

3.1. A family of multivariate density functions with preassigned marginals

3.1.1. The general family

A variety of joint distribution functions with given marginal distributions have been developed over the years. In what follows we will consider a general family given in Kotz et al. [6], Section 44.13. Let fX₁, . . . , fX_n be univariate density functions corresponding to the random variables X₁, . . . , X_n, respec- tively, and let φi, i = 1, . . . , n, be a set of bounded non-constant functions

such that

(5) Rφ_i(x)f_Xi(x) dx= 0 for all i= 1, . . . , n.

Then the function

(6) f(x₁, . . . , x_n) = _n

i=1

f_Xi(x_i) 1 + 1

αnR_{φ1,...,φn,Ωn}(x₁, . . . , x_n)

is a multivariate density function, where R_{φ1,...,φn,Ωn}(x₁, . . . , x_n) =

1≤i₁<i₂≤n

ω_i1i2φ_i1(x_i1)φ_i2(x_i2) +

+

1≤i₁<i₂<i₃≤n

ω_i1i2i3φ_i1(x_i1)φ_i2(x_i2)φ_i3(x_i3) +· · ·+ω_12...n n i=1

φ_i(x_i), and Ω_n ={ω_i1i2, ω_i1i2i3, . . . , ω_12...n}.The sets Ω_n and α_n of real numbers are chosen such that

|R_{φ1,...,φn,Ωn}(x₁, . . . , x_n)| ≤α_n

for allx_i ∈R,i= 1, . . . , n.Thenf_X1, . . . , f_Xnare the marginal densities of (6).

If |φ_i(x)| ≤ C_i for all x ∈ R, i= 1, . . . , n,then Ωn can be chosen such that

(7) |ω_i1i2| ≤ 1

C_i1C_i2, . . . ,|ω_12...n| ≤ 1 C₁·. . .·C_n,

and α_n can be chosen as the number of nonzero ω’s in the set Ω_n, with 1 ≤ α_n ≤ 2ⁿ−n−1. If all ω’s are taken to be 0, the density (6) reduces to the independent case.

Restricting to the case|φ_i(x)| ≤1,i= 1, . . . , n,it can be shown without loss of generality that for any subset of (X₁, . . . , Xn),say (Xi₁, . . . , Xi_l) where 1≤i₁ <· · ·< i_l≤n, the corresponding joint density function is

f(x_i1, . . . , x_il) = l

j=1

f_Xij

x_ij 1 + 1

α_nR_φi

1,...,φ_il,Ωl(x_i1, . . . , x_il)

, where Rφ₁,Ω1 = 0 and Ωl is the subset of Ωn such that the subscripts of ω’s involve only combinations of the integersi₁, . . . , i_l.

A particular case of (6) often encountered in the literature is the Farlie- Gumbel-Morgenstern distribution, obtained whenφi = 1−2FX_i,whereFX_i is the distribution function ofX_i,i= 1, . . . , n. This case will be discussed below in detail.

3.1.2. The Farlie-Gumbel-Morgenstern family

As in Kotz et al. [6], the general n-dimensional FGM distribution has the form

(8) F_X1,...,Xn(x₁, . . . , x_n) = n

i=1

F_Xi(x_i)

×

×

1 +

1≤i<j≤n

a_ij[1−F_Xi(x_i)]

1−F_Xj(x_j)

+· · ·+a_12...n n i=1

[1−F_Xi(x_i)]

, whereFX_i,i= 1, . . . , n, are the marginal distributions. The coefficientsai₁...i_l

are real numbers with constraints in order to ensure that F_X1,...,Xn in (8) is non-decreasing in each ofx₁, . . . , x_n.These constraints are

(9) 1 +

1≤i<j≤n

ε_iε_ja_ij +· · ·+ε₁. . . ε_na_12...n≥0

for allε_i =−M_ior 1−m_i,whereM_iandm_iare the supremum and the infimum of the set {F_Xi(x) ;−∞< x <∞} \ {0,1}. If F_Xi is absolutely continuous, we have Mi = 1 and mi = 0, hence εi = ±1. This will be the case in the next section.

In order to explain the meaning of the coefficients aij,denote c_i =

RF_Xi(x) [1−F_Xi(x)] dx, i= 1,2, . . . . Then from Cambanis [2] we have that

E [(Xj₁ −µj₁)·. . .·(Xj_k−µj_k)] = (−1)^kcj₁ ·. . .·cj_k aj₁...j_k,

where µ_i = EX_i. In particular, cov (X_i, X_j) = c_ic_ja_ij. This shows that if X_i andX_j are uncorrelated, then they are independent.

As a remark, the joint distribution (8) can also be obtained using a Farlie-Gumbel-Morgenstern multivariate copula function, namely,

C(u₁, . . . , u_n) = n

i=1

u_i

×

×

1 +

1≤i<j≤n

a_ij(1−u_i)(1−u_j) +· · ·+a_12...n n i=1

(1−u_i)

.

From a mathematical point of view, a copulaCis a distribution function on the hypercube [0,1]ⁿwith (0,1)-uniform marginals. A copula provides the natural link between then-dimensional distribution F and its marginals (F₁, . . . , Fn) asF(x₁, . . . , x_n) = C(F₁(x₁), . . . , F_n(x_n)).For more details on copulas see Nelsen [8]. In the actuarial literature, there is a growing interest for the use

of copulas to model risk dependency. See e.g. Frees and Valdez [4], Klugman and Parsa [7], Embrechts et al. [3] etc.

If the marginal densities fX_i, i = 1, . . . , n, exist, then the joint density of the FGM distribution given in (8) is

(10) f_X1,...,Xn(x₁, . . . , x_n) = n

i=1

f_Xi(x_i)

×

×

1+

1≤i<j≤n

a_ij[1−2F_Xi(x_i)]

1−2F_Xj(x_j)

+· · ·+a_12...n n i=1

[1−2F_Xi(x_i)]

. As mentioned at the end of the last section, this corresponds to (6) forφi= 1− 2F_Xi anda_i1...il=ω_i1...il/α_n. Hence the k-dimensional marginal distributions of (8) are of the same type and with the same coefficients. In particular, (11) f_X1,...,Xk(x₁, . . . , x_k) =

k i=1

f_Xi(x_i)

×

×

1+

1≤i<j≤k

a_ij[1−2F_Xi(x_i)]

1−2F_Xj(x_j)

+· · ·+a_12...k k i=1

[1−2F_Xi(x_i)]

.

3.2. The asymptotic ruin probability

The calculation involved in this section will need the following result.

Lemma 3.1. If N is a positive integer,then (i)

N k=1

km^k−1 = (N + 1)m^N

m−1 +1−m^N+1

(m−1)² = (Nm−N −1)m^N+ 1 (1−m)² ;

(ii)

∞ k=1

km^k−1= 1

(1−m)², where 0< m <1;

(iii)

1≤i₁<···<i_l≤N

h(i₂−i₁, . . . , i_l−i_l−1) =

=

1≤i₁,...,i_l−1≤N i₁+···+i_l−1≤N

[N−(i₁+· · ·+i_l−1)]h(i₁, . . . , i_l−1),

where l≤N and h:R^l₊⁻¹ →R.

Proof. We will not give a detailed proof of the first two formulas because they are based on elementary algebra. As a hint, (i) can be obtained con- sidering the first derivative of _N

k=0m^k, with respect to m while (ii) results from (i) by lettingN → ∞.

In order to show (iii), we rewrite it as (12)

1≤i₁<···<i_l≤N

h(i₂−i₁, . . . , i_l−i_l−1) = N i_l=l

1≤i₁<···<i_l−1<i_l

h(i₂−i₁, . . . , i_l−i_l−1).

Let us fix the values 1≤j₁, . . . , j_l−1 ≤N.We want to count how many times h(j₁, . . . , j_l−1) appears in the above sum. We are looking for alli₁, . . . , i_l such that 1≤i₁ <· · ·< i_l ≤N and h(i₂−i₁, . . . , i_l−i_l−1) =h(j₁, . . . , j_l−1),i.e.

j_k=i_k+1−i_k,k= 1, . . . , l−1.This gives









i₂ =j₁+i₁

i₃ =j₂+i₂ =j₂+j₁+i₁ . . .

i_l=j_l−1+i_l−1 =j_l−1+· · ·+j₁+i₁.

This means thath(j₁, . . . , j_l−1) appears in (12) forj_l−1+· · ·+j₁+ 1≤i_l≤N, i.e. N−(j_l−1+· · ·+j₁) times. This completes the proof of (iii).

3.2.1. The general case

We will now assume that the random vectorY = (Y₁, . . . , Yn) introduced in (2) has ann-variate distribution with density (6). In order to find an explicit form for ψ(x;n) from Corollary 2.1, we should evaluate _n

k=1Eθ^α_k, k ≤ n, whereθ_k=_k

i=1Y_i.Starting with Eθ^α_k, we have the following result.

Proposition 3.1. If the random vector Y has density (6), then (13) Eθ^α_k =

k i=1

EY_i^α+ 1 α_n

1≤i<j≤k

ωij

k l=1 l=i,j

EY_l^α

E [Y_i^αφi(Yi)] E

Y_j^αφj(Yj)

+· · ·+ 1 α_nω_12...k

k i=1

E [Y_i^αφ_i(Y_i)]. Proof. The formula follows from

Eθ^α_k = E k

i=1

Y_i^α (6)

= · · ·

R^k

k i=1

x^α_if_Yi(x_i)×

×

1 + 1

αnR_{φ1,...,φk,Ωk}(x₁, . . . , x_k)

dx₁. . .dx_k.

Some extraassumptions will simplify this complex formula. A usual as- sumption is that {Yk, k ≥1} is a sequence of identically distributed random variables, with common density f_Y and distribution function F_Y. We will also consider that φ₁ = · · · = φ_n = φ. Then, denoting m_α = EY^α and s_α = E [Y^αφ(Y)],Proposition 3.1 reduces to

Corollary 3.1. Under the above assumptions, (14)

Eθ_k^α=m^k_α+ 1 α_n

m^k_α⁻²s²_α

1≤i<j≤k

ω_ij+m^k_α⁻³s³_α

1≤i<j<l≤k

ω_ijl+· · ·+ω_12...ks^k_α

. For this particular case, by taking α= 1 and k= 2 in (14), we can also evaluate the covariance of any (Y_i, Y_j),i < j, as

(15) cov (Y_i, Y_j) = E (Y_iY_j)−EY_iEY_j =m²₁+ω_ij

αns²₁−m²₁ = ω_ij αns²₁. We will now choose some special forms for the coefficientsω. There are many interesting choices forωand we will try to give a general idea on some of them.

1. We assume that there exist some functions h_l :R^l₊⁻¹ → R such that ω_i1...il =h_l(i₂−i₁, . . . , i_l−i_l−1), l= 2, . . . , n.Then, from Lemma 3.1,

1≤i₁<···<i_l≤k

ω_i1...il =

1≤i₁,...,i_l−1≤k i₁+···+i_l−1≤k

[k−(i₁+· · ·+i_l−1)]h_l(i₁, . . . , i_l−1),

and, denotingi_l+=i₁+· · ·+i_l−1,from (14) we have Eθ^α_k =m^k_α+ 1

αn

k l=2

m^k_α^−ls^l_α

1≤i₁,...,i_l−1≤k i_l+≤k

(k−i_l+)h_l(i₁, . . . , i_l−1).

Hence, applying again Lemma 3.1, (16)n

k=1

Eθ^α_k= n k=1

m^k_α+ 1 αn

n l=2

s^l_α n k=l

m^k_α^−l

1≤i₁,...,i_l−1≤k i_l+≤k

(k−i_l+)h_l(i₁, . . . , i_l−1) =

= n k=1

m^k_α+ 1 α_n

n l=2

s^l_α

1≤i₁,...,i_l−1≤n i_l+≤n

h_l(i₁, . . . , i_l−1) n k=i_l++1

(k−i_l+)m^k_α^−l=

= n k=1

m^k_α+ 1 α_n

n l=2

s^l_α

1≤i₁,...,i_l−1≤n i_l+≤n

hl(i₁, . . . , il−1)

n−i_l+

j=1

jm^jα^+il+−l=

= n k=1

m^k_α+m_α α_n

n l=2

s_α m_α

l

1≤i₁,...,i_l−1≤n i_l+≤n

h_l(i₁, . . . , i_l−1)mⁱα^l+

n−i_l+

j=1

jm^j_α⁻¹ =

(i)= mα(1−mⁿ_α)

1−m_α + mα

α_n(1−m_α)² n

l=2

sα

m_α

l

1≤i₁,...,i_l−1≤n i_l+≤n

h_l(i₁, . . . , i_l−1)mⁱα^l+×

×

((n−i_l+)m_α−n+i_l+−1)mⁿα^−il+ + 1

=

= mα(1−mⁿ_α)

1−m_α + mα

α_n(1−m_α)² n l=2

sα

m_α l

1≤i₁,...,i_l−1≤n i_l+≤n

h_l(i₁, . . . , i_l−1)mⁱα^l++

+mⁿ_α

1≤i₁,...,i_l−1≤n i_l+≤n

h_l(i₁, . . . , i_l−1) ((n−i_l+)m_α−n+i_l+−1)

. 2. We will now assume that there exist some nonincreasing functions gl : R+ → R such that ωi₁...i_l = gl(il−i₁), l = 2, . . . , n. We notice that we are in fact in case 1 with h_l(j₁, . . . , j_l−1) =g_l(j₁ +· · ·+j_l−1), sinceω_i1...il = h_l(i₂ −i₁, . . . , i_l−i_l−1) = g_l((i₂−i₁) + (i₃−i₂) +· · ·+ (i_l−i_l−1)) =g_l(i_l− i₁). Hence (16) reduces to

(17)

n k=1

Eθ^α_k = m_α(1−mⁿ_α)

1−m_α + m_α αn(1−mα)²

n l=2

s_α m_α

l

×

×



 ⁿ

i_l+=l−1

g_l(i_l+)mⁱ_α^l+ +mⁿ_α n i_l+=l−1

g_l(i_l+) ((n−i_l+)m_α−n+i_l+−1)



=

= mα(1−mⁿ_α)

1−m_α + mα

α_n(1−m_α)² n

l=2

sα

m_α l

×

×[S_gl;mα +mⁿ_α(1−m_α)S_gl;•+mⁿ_α(nm_α−n−1)S_gl], where

(18) Sg_l= n i=l−1

g_l(i), Sg_l;•= n i=l−1

ig_l(i), Sg_l;m_α = n i=l−1

mⁱ_αg_l(i).

The above sums can all go till n−1 since for i=n, the quantity within the square brackets of (17) is zero. Also, in this case, from (15) fori < j we have

cov (Y_i, Y_j) = g₂(j−i) α_n s²₁.

Sinceg₂is nonincreasing, cov(Y_i, Y_j)≥cov(Y_i, Y_j) whenever|j−i|<|j−i|, i.e., the correlation between the random variables is not increasing in time.

This is very good, because it is normal for two random discount factors more distanced in time to be less correlated.

Let us have a look at condition (7) that must be fulfilled by the g’s.

Assuming that there exists a positive C such that |φ(x)| ≤ C for all x ∈ R, since the g’s are nonincreasing and we need just g_l(l−1), g_l(l), . . ., it is sufficient that

(19) |g_l(l−1)| ≤ 1

C^l

for any l= 2, . . . , n. We will now consider two particular cases of such g’s.

3. Letωi₁...i_l =ω for any 1≤i₁<· · ·< il≤n.We notice that we are in the above case forg_l(x) =ω, l= 2, . . . , n,x∈R. We could replace these g’s into (18), but straightforward calculation seems simpler in this case. We have

1≤i₁<···<i_l≤k

ωi₁...i_l=ω k

l

,

and, from (14), Eθ_k^α=m^k_α+ ω

αn

k l=2

k l

m^k_α^−ls^l_α=m^k_α+ ω αn

(mα+sα)^k−m^k_α−km^k_α⁻¹sα

=

=

1− ω α_n

m^k_α+ ω α_n

(m_α+s_α)^k−km^k_α⁻¹s_α . Hence, using (i) in Lemma 3.1,

n k=1

Eθ_k^α=

1− ω αn

n k=1

m^k_α+ ω αn

n k=1

(m_α+s_α)^k−km^k_α⁻¹s_α

=

=

1− ω α_n

m_α(1−mⁿ_α) 1−m_α + ω

α_n

(m_α+s_α)1−(m_α+s_α)ⁿ 1−(m_α+s_α) −

−s_α

(n+ 1)mⁿ_α

mα−1 + 1−mⁿ_α⁺¹ (mα−1)² .

Also, we see from (19) that it is sufficient thatω ≤min C⁻², C⁻ⁿ! .

4. Let now ωi₁...i_l = ω^l for any 1 ≤i₁ <· · · < i_l ≤n. We are again in the second case forg_l(x) =ω^l,but we will use direct calculation as

1≤i₁<···<i_l≤k

ωi₁...i_l =ω^l k

l

,

and, from (14),

Eθ_k^α=m^k_α+ 1 α_n

k l=2

k l

m^k_α^−l(ωsα)^l =

=

1− 1 α_n

m^k_α+ 1 α_n

(m_α+ωs_α)^k−km^k_α⁻¹ωs_α . Then, just as before,

n k=1

Eθ_k^α=

1− 1 α_n

mα(1−mⁿ_α) 1−m_α + 1

α_n

(mα+ωsα)1−(mα+ωsα)ⁿ 1−(m_α+ωs_α) −

−ωs_α

(n+ 1)mⁿ_α

mα−1 + 1−mⁿ_α⁺¹ (m_α−1)² . From (19), a sufficient condition forω is that ω≤C⁻¹.

5. We will now assume thatω_i1...il = 0 for any l >2,and ω_ij =h(j−i) for 1≤i < j ≤n.We are in the second case withg_l≡0 forl >2,and g₂ =h.

Then (17) reduces to n k=1

Eθ^α_k = m_α(1−mⁿ_α)

1−mα + s²_α

α_nm_α(1−m_α)²×

×[Sh;m_α+mⁿ_α(1−mα)Sh;•+mⁿ_α(nmα−n−1)Sh], where, from (18),

S_h = n i=1

h(i), S_h;• = n

i=1

ih(i), S_h;mα = n

i=1

mⁱ_αh(i).

Also,h must fulfill the condition|h(l)| ≤C⁻² forl= 1, . . . , n.Taking h nonincreasing, it will thus be sufficient that|h(1)| ≤C⁻².

3.2.2. The Farlie-Gumbel-Morgenstern case

We will now assume that the distribution of Y is of the form (8) with density (10). As before, in order to apply Corollary 2.1, we need to evaluate Eθ_k^α,whereθ_k=_k

i=1Y_i.Since in fact this is a particular case of the general one, in the following we will only consider some situations in which we can