Asymptotic expansions for distributions of the surplus prior and at the time of ruin
Asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin are given for nonlinearly perturbed risk processes.
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Інститут математики НАН України
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| Цитувати: | Asymptotic expansions for distributions of the surplus prior and at the time of ruin / D.S. Silvestrov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 183–188. — Бібліогр.: 15 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859671116735315968 |
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| author | Silvestrov, D.S. |
| author_facet | Silvestrov, D.S. |
| citation_txt | Asymptotic expansions for distributions of the surplus prior and at the time of ruin / D.S. Silvestrov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 183–188. — Бібліогр.: 15 назв.— англ. |
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| description | Asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin are given for nonlinearly perturbed risk processes.
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| first_indexed | 2025-11-30T13:55:10Z |
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Theory of Stochastic Processes
Vol.13(29), no.4, 2007, pp.183–188
DMITRII S. SILVESTROV
ASYMPTOTIC EXPANSIONS
FOR DISTRIBUTIONS OF THE SURPLUS
PRIOR AND AT THE TIME OF RUIN
Asymptotic expansions for the distribution of the surplus prior to
and at the time of a ruin are given for nonlinearly perturbed risk
processes.
1. Introduction
Let X
(ε)
u (t), t ≥ 0, be a standard risk process defined for every ε ≥ 0
(perturbation parameter) in the following way:
X(ε)
u (t) = u + c(ε)t −
N(ε)(t)∑
k=1
Z
(ε)
k , t ≥ 0. (1)
The process X
(ε)
u (t) is a classical model used to describe functioning of an
insurance company. Here, (a) u is a nonnegative constant that denotes an
initial capital of a company; (b) a positive constant c(ε) denotes the gross
premium rate; (c) N (ε)(t), t ≥ 0 is a Poisson process (with a parameter
λ(ε)) that counts the number of claims to the company in the time-intervals
[0, t], t ≥ 0; (d) Z
(ε)
k , k = 1, 2, . . ., is a sequence of nonnegative i.i.d. random
variables (Z
(ε)
k is the amount of the kth claim) with a distribution function
G(ε)(u) which have a finite positive mean μ(ε) =
∫ ∞
0
uG(ε)(du) ∈ (0,∞);
(e) the sequence of the random variables Z
(ε)
k , k = 1, 2, . . ., and the process
N (ε)(t), t ≥ 0, are independent.
We assume to hold the following condition, which let us consider the
risk process X
(ε)
u (t), for ε > 0, as a perturbed version of the risk process
X
(0)
u (t):
2000 Mathematics Subject Classifications: 60K15, 60F17, 60K20.
Key words and phrases: risk process, ruin probability, surplus prior, surplus at the
time of ruin, nonlinear perturbation, asymptotic expansion
183
184 DMITRII S. SILVESTROV
A: (a) c(ε) → c(0) as ε → 0; (b) λ(ε) → λ(0) as ε → 0; (c) G(ε)(·) ⇒ G(0)(·)
as ε → 0.
Let us introduce, for u ≥ 0, a random variable, which is the time of a
ruin,
τ (ε)
u = inf(t ≥ 0 : X(ε)
u (t) < 0).
This random variable takes values in the interval (0,∞] with probability
1. By the definition, the probability of ruin can be expressed as,
P(ε)(u) = P{τ (ε)
u < ∞}, u ≥ 0.
Let us introduce parameter α(ε) = λ(ε)μ(ε)
c(ε)
referred usually as a safety
loading coefficient. If α(ε) ≥ 1 then the ruin probability P(ε)(u) = 1. That
is why, we also assume the following condition:
B: α(ε) ∈ (0, 1] for every ε ≥ 0.
Let us now introduce, for u, x, y ≥ 0, random variables X
(ε)
u (τ
(ε)
u − 0)
and −X
(ε)
u (τ
(ε)
u ) that are, respectively, the surplus (of capital) prior to and
at the time of the ruin. These random variables are well defined if τ
(ε)
u < ∞.
Let us also assign them value +∞ if τ
(ε)
u = ∞.
Let us now introduce the following surplus probabilities,
P(ε)
x,y(u) = P{τ (ε)
u < ∞,−X(ε)
u (τ (ε)
u ) > x, X(ε)
u (τ (ε)
u − 0) > y}, u, x, y ≥ 0.
Note that, by the definition, both random variables, the surplus at the
time of the ruin, −X
(ε)
u (τ
(ε)
u ), and the surplus prior to the ruin, X
(ε)
u (τ
(ε)
u −0),
are positive random variables with probability 1. This implies that P(ε)(u) =
P
(ε)
0,0(u), u ≥ 0, i.e., the ruin probability P(ε)(u) is a particular case of the
surplus probability P
(ε)
x,y(u).
The surplus prior and at the time of ruin and its asymptotics have
been studied in Gerber, Goovarerts, and Kaas (1987), Dufresne and Gerber
(1988), Dickson (1992), Dickson, Dos Reis, and Waters (1995), Willmot and
Lin (1998), Rolski, Schmidli, Schmidt, and Teugels (1999), Schmidli (1999),
and Badescu, Breuer, Drekic, Latouche, and Stanford (2005).
In this paper, we give asymptotic exponential expansions for the distri-
bution of the surplus prior to and at the time of a ruin. These expansions
generalise to the case of surplus probabilities results on Cramér-Lundberg
and diffusion approximations of ruin probabilities for nonlinearly perturbed
risk processes obtained in Gyllenberg and Silvestrov (1999, 2000a).
NONLINEARLY PERTURBED RISK PROCESSES 185
2. Main results
A starting point in our asymptotic analysis is the following renewal
equation for the surplus probability P
(ε)
x,y(u), as function in u ≥ 0, given in
Schmidli (1999):
P(ε)
x,y(u) = α(ε)(1− Ḡ(ε)(u∨y +x))+α(ε)
∫ u
0
P(ε)
x,y(u−s)Ḡ(ε)(ds), u ≥ 0, (2)
where the distribution function Ḡ(ε)(s), referred usually as a steady claim
distribution, is defined by the following formula,
Ḡ(ε)(s) =
1
μ(ε)
∫ s
0
(1 − G(ε)(v))dv, s ≥ 0.
This renewal equation (2) reduces to the well known renewal equation
satisfied by the ruin probabilities P(ε)(u) in the case where x, y = 0.
We apply to this equation asymptotic results concerned perturbed re-
newal equation obtained in Silvestrov (1978, 1979, 1995) and Gyllenberg
and Silvestrov (2000a, 2000b).
Consider the following moment generating function:
ϕ(ε)(ρ) =
∫ ∞
0
eρs(1 − G(ε)(s))ds = μ(ε)
∫ ∞
0
eρsḠ(ε)(ds), ρ ∈ R1.
The following condition is the Cramér type condition for the claim dis-
tribution:
C: There exists δ > 0 such that
(a) lim0≤ε→0ϕ
(ε)(δ) < ∞;
(b) λ(0)
c(0)
ϕ(0)(δ) = α(0)
∫ ∞
0
eδsḠ(0)(ds) ∈ (1,∞).
Let us also consider the following characteristic equation
α(ε)
∫ ∞
0
eρs(1 − Ḡ(ε)(s))ds = 1. (3)
Conditions A, B, and C guarantee that (a) there exist ε1 = ε1(δ) > 0
such that the characteristic equation (3) has, for every 0 ≤ ε ≤ ε1, a unique
non-negative root ρ(ε) < δ, and (b) ρ(ε) → ρ(0) as ε → 0.
Note also that the root ρ(0) > 0 if the limit value of the safety loading
coefficient α(0) < 1, and ρ(0) = 0 if α(0) = 1. These cases correspond, respec-
tively, to the models of Cramér-Lundberg and diffusion approximations.
Let us introduce, for n = 0, 1, . . ., the mixed power-exponential moment
generating functions
ϕ(ε)[ρ, n] =
∫ ∞
0
sneρs(1 − G(ε)(s))ds = μ(ε)
∫ ∞
0
sneρsḠ(ε)(ds), ρ ∈ R1.
186 DMITRII S. SILVESTROV
By the definition, ϕ(ε)[ρ, 0] = ϕ(ε)(ρ).
Let us choose an arbitrary ρ(0) < β < δ. Condition C implies that (c)
there exists ε2 = ε2(β) > 0 such that, for ε ≤ ε2 and n = 0, 1, . . .,
ϕ(ε)[β, n] ≤ cn
∫ ∞
0
eδs(1 − G(ε)(s))ds < ∞, (4)
where cn = cn(δ, β) = sups≥0 sne−(δ−β)s < ∞.
Note also that ϕ(ε)[ρ, n], for ρ ≤ β, is the derivative of order n of the
function ϕ(ε)(ρ).
Denote
π(ε)
x,y(ρ
(ε)) =
∫ ∞
0
eρ(ε)s(1 − Ḡ(ε)(s))ds∫ ∞
0
seρ(ε)sḠ(ε)(ds)
(5)
=
∫ ∞
0
eρ(ε)s(
∫ ∞
s
(1 − G(ε)(u))du)ds∫ ∞
0
seρ(ε)s(1 − G(ε)(s)ds
.
Relation (b) implies that (d) there exists ε3 = ε3(β) > 0 such that
ρ(ε) < β.
Define ε0 = min(ε1, ε2, ε3). Conditions A, B, and C imply, due to rela-
tions (b) and (d), that (e)
∫ ∞
0
eρ(ε)s(1 − Ḡ(ε)(s))ds < ∞ and
(f)
∫ ∞
0
seρ(ε)sḠ(ε)(ds) < ∞ for ε ≤ ε3. Therefore, the quantity π(ε)(ρ(ε))
is well defined for all ε ≤ ε0.
The following theorem describe asymptotic behaviour of the surplus
probabilities.
Theorem 1. Let conditions A, B, and C hold. Then, for any u(ε) → ∞
as ε → 0, the following asymptotic relation holds for every x, y ≥ 0,
P
(ε)
x,y(u(ε))
exp{−ρ(ε)u(ε)} → π(0)
x,y(ρ
(0)) as ε → 0. (6)
Let now assume that the following nonlinear perturbation conditions
hold for some integer k ≥ 1:
D
(k)
1 : c(ε) = c(0) + c1ε + · · ·+ ckε
k + o(εk), where |cl| < ∞, l = 1, . . . , k;
D
(k)
2 : λ(ε) = λ(0) + d1ε + · · · + dkε
k + o(εk), where |dl| < ∞, l = 1, . . . , k;
D
(k)
3 : ϕ(ε)[ρ(0), n] = ϕ(0)[ρ(0), n]+v1[ρ
(0), n]ε+· · ·+vk−n[ρ(0), n]εk−n+o(εk−n),
where |vl[ρ
(0), n]| < ∞, l = 1, . . . , k − n, n = 0, . . . , k.
Conditions D
(k)
1 , D
(k)
2 , and D
(k)
3 mean that the characteristic quantities
of the perturbed risk processes penetrating the above perturbation condi-
tions are nonlinear functions of ε. These conditions correspond to the model
NONLINEARLY PERTURBED RISK PROCESSES 187
of smooth perturbation where these functions have k derivatives at ε = 0,
i.e., they can be expanded in a power series with respect to ε up to and
including the order k.
The relationship between the rates at which the perturbation parameter
ε tends to zero and the initial capital u tends to infinity has an influence
upon the obtained results. Without loss of generality it can be assumed
that u = u(ε) is a function of the parameter ε. The relationship between
the rate of perturbation and the rate of growth of the initial capital is
characterized by the following balancing condition that is assumed to hold
for some integer 1 ≤ r ≤ k:
E(r): u(ε) → ∞ as ε → 0 such that εru(ε) → �r as ε → 0, where �r ∈ [0,∞).
The following theorem presents the asymptotic exponential expansions
for surplus probabilities P
(ε)
x,y(u) for nonlinearly perturbed risk processes.
Theorem 2. Let conditions A, B, C, D
(k)
1 , D
(k)
2 , D
(k)
3 , and E(r) hold.
Then, the following asymptotic relation holds for every x, y ≥ 0,
P
(ε)
x,y(u(ε))
exp{−(ρ(0) + a1ε + · · · + ar−1εr−1)u(ε)} → e−�rarπ(0)
x,y(ρ
(0)) as ε → 0, (7)
where the coefficients a1, . . . ak are given by explicit recurrence formulas as
functions of coefficients in the expansions penetrating the perturbation con-
ditions D
(k)
1 , D
(k)
2 , and D
(k)
3 .
3. Conclusion
In conclusion, I would like to note that this paper presents the result of
the author included in a new book [9] written in cooperation with Professor
Mats Gyllenberg. The algorithm for calculation of the coefficients in the
asymptotic expansions (7) and proofs of Theorems 1 and 2 are given in this
book.
The book mentioned above is devoted to studies of quasi-stationary phe-
nomena in nonlinearly perturbed stochastic systems. The methods based
on exponential asymptotics for nonlinearly perturbed renewal equation are
used. Mixed ergodic and large deviation theorems are presented for nonlin-
early perturbed regenerative processes, semi-Markov processes and Markov
chains. Applications to nonlinearly perturbed population dynamics and
epidemic models, queueing systems and risk processes are considered. The
book also includes an extended bibliography of works in the area.
References
1. Badescu, A.L., Breuer, L., Drekic, S., Latouche, G., Stanford, D.A., The
surplus prior to ruin and the deficit at ruin for a correlated risk process,
Scand. Actuar. J., No. 6, (2005), 433–445.
188 DMITRII S. SILVESTROV
2. Dickson, D.C.M., On the distribution of the surplus prior to ruin, Insur.
Math. Econom., 11, (1992), 191–207.
3. Dickson, D.C.M., Dos Reis, A.D.E., Waters, H.R., Some stable algorithms
in ruin theory and their applications, ASTIN Bull., 25, (1995), 153–175.
4. Dufresne, F., Gerber, H.U., The surpluses immediately before and at ruin,
and the amount of the claim causing ruin, Insur. Math. Econom., 7,
(1988), 193–199.
5. Gerber, H.U., Goovarerts, M.J., Kaas, R., On the probability of severity of
ruin, ASTIN Bull., 17, (1987), 151–163.
6. Gyllenberg, M., Silvestrov, D.S., Cramér-Lundberg and diffusion approxi-
mations for nonlinearly perturbed risk processes including numerical com-
putation of ruin probabilities, In: Silvestrov, D., Yadrenko, M., Borisenko,
O., Zinchenko, N. (eds) Proceedings of the Second International School on
Actuarial and Financial Mathematics, Kiev, 1999. Theory Stoch. Process.,
5(21), no. 1-2, (1999), 6–21.
7. Gyllenberg, M., Silvestrov, D.S., Cramér–Lundberg approximation for non-
linearly perturbed risk processes, Insur. Math. Econom., 26, (2000a), 75–
90.
8. Gyllenberg, M., Silvestrov, D.S., Nonlinearly perturbed regenerative pro-
cesses and pseudo-stationary phenomena for stochastic systems, Stoch. Pro-
cess. Appl., 86, (2000b), 1–27.
9. Gyllenberg, M., Silvestrov, D.S., Quasi-stationary Phenomena in Nonlin-
early Perturbed Stochastic Systems (submitted)
10. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., Stochastic Processes for
Insurance and Finance. Wiley Series in Probability and Statistics, Wiley,
New York, (1999).
11. Schmidli, H., On the distribution of the surplus prior and at ruin, Astin
Bull., 29, No. 2, (1999), 227–244.
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atn. Mat. Stat., 18, (1978), 144–161 (English translation in Theory
Probab. Math. Statist., 18, 155–172).
13. Silvestrov, D.S. The renewal theorem in a series scheme. 2, Teor. Veroyatn.
Mat. Stat., 20, (1979), 97–116 (English translation in Theory Probab.
Math. Statist., 20, 113–130).
14. Silvestrov, D.S., Exponential asymptotic for perturbed renewal equations,
Teor. Ǐmovirn. Mat. Stat., 52, (1995), 143–153 (English translation in
Theory Probab. Math. Statist., 52, 153–162).
15. Willmot, G.E., Lin, X.S., Exact and approximate properties of the distri-
bution of surplus before and after ruin, Insur. Math. Econom., 23, (1998),
91–110.
Department of Mathematics and Physics, Mälardalen University,
Box 883, SE-721 23 Väster̊as, Sweden.
E-mail address: dmitrii.silvestrov@mdh.se
|
| id | nasplib_isofts_kiev_ua-123456789-4522 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-30T13:55:10Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Silvestrov, D.S. 2009-11-24T15:35:27Z 2009-11-24T15:35:27Z 2007 Asymptotic expansions for distributions of the surplus prior and at the time of ruin / D.S. Silvestrov // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 183–188. — Бібліогр.: 15 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4522 Asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin are given for nonlinearly perturbed risk processes. en Інститут математики НАН України Asymptotic expansions for distributions of the surplus prior and at the time of ruin Article published earlier |
| spellingShingle | Asymptotic expansions for distributions of the surplus prior and at the time of ruin Silvestrov, D.S. |
| title | Asymptotic expansions for distributions of the surplus prior and at the time of ruin |
| title_full | Asymptotic expansions for distributions of the surplus prior and at the time of ruin |
| title_fullStr | Asymptotic expansions for distributions of the surplus prior and at the time of ruin |
| title_full_unstemmed | Asymptotic expansions for distributions of the surplus prior and at the time of ruin |
| title_short | Asymptotic expansions for distributions of the surplus prior and at the time of ruin |
| title_sort | asymptotic expansions for distributions of the surplus prior and at the time of ruin |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4522 |
| work_keys_str_mv | AT silvestrovds asymptoticexpansionsfordistributionsofthesurpluspriorandatthetimeofruin |