Another approach to the problem of the ruin probability estimate for risk process with investments
An exponential estimate of ruin probability for an insurance company which invests all its capital in risk assets is found. The process which describes the risky assets is assumed to follow a geometrical Brownian motion. Insurance premium flow depends on the value of reserves of the insurance company...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2007
|
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/4510 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Another approach to the problem of the ruin probability estimate for risk process with investments / M. Androshchuk, Y. Mishura // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 1–18. — Бібліогр.: 8 назв.— англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4510 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-45102009-11-25T12:00:26Z Another approach to the problem of the ruin probability estimate for risk process with investments Androshchuk, M. Mishura, Y. An exponential estimate of ruin probability for an insurance company which invests all its capital in risk assets is found. The process which describes the risky assets is assumed to follow a geometrical Brownian motion. Insurance premium flow depends on the value of reserves of the insurance company. The problem is solved by reduction of the generalized risk process to the classical risk process without investments. 2007 Article Another approach to the problem of the ruin probability estimate for risk process with investments / M. Androshchuk, Y. Mishura // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 1–18. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4510 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
An exponential estimate of ruin probability for an insurance company which invests all its capital in risk assets is found. The process which describes the risky assets is assumed to follow a geometrical Brownian motion. Insurance premium flow depends on the value of reserves of the insurance company. The problem is solved by reduction of the generalized risk process to the classical risk process without investments. |
| format |
Article |
| author |
Androshchuk, M. Mishura, Y. |
| spellingShingle |
Androshchuk, M. Mishura, Y. Another approach to the problem of the ruin probability estimate for risk process with investments |
| author_facet |
Androshchuk, M. Mishura, Y. |
| author_sort |
Androshchuk, M. |
| title |
Another approach to the problem of the ruin probability estimate for risk process with investments |
| title_short |
Another approach to the problem of the ruin probability estimate for risk process with investments |
| title_full |
Another approach to the problem of the ruin probability estimate for risk process with investments |
| title_fullStr |
Another approach to the problem of the ruin probability estimate for risk process with investments |
| title_full_unstemmed |
Another approach to the problem of the ruin probability estimate for risk process with investments |
| title_sort |
another approach to the problem of the ruin probability estimate for risk process with investments |
| publisher |
Інститут математики НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4510 |
| citation_txt |
Another approach to the problem of the ruin probability estimate for risk process with investments / M. Androshchuk, Y. Mishura // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 1–18. — Бібліогр.: 8 назв.— англ. |
| work_keys_str_mv |
AT androshchukm anotherapproachtotheproblemoftheruinprobabilityestimateforriskprocesswithinvestments AT mishuray anotherapproachtotheproblemoftheruinprobabilityestimateforriskprocesswithinvestments |
| first_indexed |
2025-07-02T07:44:16Z |
| last_indexed |
2025-07-02T07:44:16Z |
| _version_ |
1836520315801305088 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.1–18
MARYNA ANDROSHCHUK, YULIYA MISHURA
ANOTHER APPROACH TO THE PROBLEM OF
THE RUIN PROBABILITY ESTIMATE FOR RISK
PROCESS WITH INVESTMENTS
An exponential estimate of ruin probability for an insurance com-
pany which invests all its capital in risk assets is found. The process
which describes the risky assets is assumed to follow a geometrical
Brownian motion. Insurance premium flow depends on the value of
reserves of the insurance company. The problem is solved by re-
duction of the generalized risk process to the classical risk process
without investments.
1. Introduction
A generalized risk model for an insurance company which invests all its
reserves into risky assets is considered. We let the value of the insurance
premium flow depend on the current value of the insurer’s capital.
The risky asset is assumed to follow a geometrical Brownian motion
dSt = St(a dt + b dWt), S0 > 0, (1)
where S0 denotes the initial value of the risky asset, a > 0, b > 0 are some
fixed constants, and {Wt, t ≥ 0} is a standard Brownian motion.
Let σc := inf{t ≥ 0 : St ≤ c} denote a stopping time of the investing
activity, where c > 0 is some fixed constant. Thus we let the insurance
company terminate their risky asset investments if the price drops below c.
We assume further that S0 > c, otherwise investment problem has no sense.
Let us now consider a risk process described by the equation
Rt(u) = u −
Nt∑
k=1
Uk +
∫ t
0
p(Rs) ds +
∫ t
0
RsI{s ≤ σc}
Ss
dSs, t ≥ 0, (2)
where
2000 Mathematics Subject Classifications 91B30
Key words and phrases. Ruin process, ruin probability, geometrical Brownian mo-
tion, supermartingale approach.
1
2 MARYNA ANDROSHCHUK, YULIYA MISHURA
Rt is the value of insurer’s capital at a moment of time t ≥ 0;
u > 0 is the initial capital of the insurance company;
{Nt, t ≥ 0} is a Poisson process modeling the number of insurance
claims, and it is assumed to have a constant intensity β > 0;
{Tk, k ≥ 1} are jump times of the Poisson process;
{Uk, k ≥ 1} is a sequence of i.i.d. positive random variables which
models the claim sizes incurred by the insurer at times {Tk, k ≥ 1}. In this
paper we will only consider light tailed distributions, with existing moment
generating functions (see below);
p(Rs) is a premium rate process which depends on a value of current
insurer’s capital at time s.
We will assume that the sequence {Uk, k ≥ 1}, the processes {Nt, t ≥
0}, and the standard Brownian motion {Wt, t ≥ 0} are all independent.
Also, for Nt = 0 we put
∑Nt
k=1 Uk = 0.
Note that equation (2) can be rewritten as
Rt(u) = u−
Nt∑
k=1
Uk +
∫ t
0
p(Rs) ds+ a
∫ t∧σc
0
Rs ds+ b
∫ t∧σc
0
Rs dWs, t ≥ 0.
(3)
Thus, our model is similar to the model described in paper [1]. An
essential difference between the models is that the existence of the moment
generating function for claim distribution is not demanded in [1]. On the
other hand, independence of some processes is demanded in [1], but they
are not necessarily independent in our model (3) (look section 4).
Comparisons of the results obtained for model (3) with known results
for the classical risk model, and for the risk model with reinvestments in [1]
are presented in section 3 and section 4.
2. Modification of Risk Process
Denote by F = (Ft)t≥0 the filtration generated by a compound Poisson
process {∑Nt
k=1 Uk, t ≥ 0} and the process {Wt, t ≥ 0, }. Let Et(·) denote
the conditional expectation E(·|Ft).
Furthermore let us denote by τ(u) := inf{t ≥ 0 : Rt(u) ≤ 0} the ruin
time of the insurance company with an initial capital u. Put τ(u) = +∞ if
Rt > 0 for all t > 0.
Let h : R+ → R+ be the shifted moment generating function, defined
as
h(r) = E[erU1 ] − 1, h(0) = 0.
We will use the classical assumption, as in [2], about existence of r∞ ∈
(0, +∞] such that h(r) < ∞ on [0, r∞), and h(r) → ∞ as r ↑ r∞. The
function h(r) is an increasing, convex, and continuous function on [0, r∞). In
other words, we will consider light tailed distributions of random variables to
RISK PROCESS WITH INVESTMENTS 3
Distribution, Probability density Shifted moment generating
parameters function f(x) function h(r) r∞
Exponential
Exp(λ) λe−λx r
λ−r
λ
Gamma
Γ(α, λ) λα
Γ(α)
xα−1e−λx
(
λ
λ−r
)α − 1 λ
Uniform
U(a, b) 1
b−a
, 0 ≤ a < x < b ebr−ear
r(b−a)
− 1 +∞
Table 1: Examples of shifted moment generating functions
describe claims arriving to the insurance company. Some examples of such
distributions with corresponding moment generating functions are given in
table 1.
Lemma 1. Suppose that the following conditions (P1) and (P2) are satis-
fied:
(P1) p(·) : R → R+ is a measurable bounded from above on R nonneg-
ative function, i.e. ∃C > 0 : p(x) ≤ C, ∀x ≥ 0;
(P2) p(·) is a Lipschitz continuous function, i.e. ∃K > 0 such that for
all x, y ∈ R : |p(x) − p(y)| ≤ K|x − y|.
Then equation (3) has a unique, up to the stochastic equivalence, Ft-
adapted solution. It can be written in following form
Rt = St∧σc
(
u
S0
+
∫ t
0
S−1
t∧σc
(p(Rs) ds − Us dZs)
)
, (4)
where
∑Nt
k=1 Uk =
∫ t
0
Us dZs, and Zs is the jump measure of the Poisson
process with intensity β.
Proof. Let Z i
t , i = 1, 2 be semimartingales such that Z i
0 = 0 a.s., and Ht is
an adapted cádlág process, i.e. which is right-hand continuous process and
has left-hand limits. Then we use the vector form of the Theorem 14.6 [3],
p.183. Assume the following conditions are satisfied:
1) fi(s, ω, X·(ω)) are locally bounded predicted processes;
2)∃Mi > 0 : |fi(s, ω, X·(ω)) − fi(s, ω, Y·(ω))| ≤ Mi sup0≤v≤s |Xv(ω) −
Yv(ω)|, i = 1, 2 for any cádlág adapted processes X, Y .
Then the stochastic differential equation
Xt(ω) = Ht(ω)+
∫ t
0
f1(s, ω, X·(ω)) dZ1
s (ω)+
∫ t
0
f2(s, ω, X·(ω)) dZ2
s (ω) (5)
has a unique strong solution.
4 MARYNA ANDROSHCHUK, YULIYA MISHURA
Let us consider the following modification of equation (3):
Rt(u) = u−
Nt∑
k=1
Uk +
∫ t
0
p(Rs−) ds+a
∫ t∧σc
0
Rs− ds+b
∫ t∧σc
0
Rs− dWs. (6)
In terms of equation (5) we have that Xt := Rt; Z1
s := s, Z2
s := Ws;
f1(s, ω, Xs) := p(Xs−) + aXs−, f2(s, ω, Xs) := bXs−.
Obviously, condition 2) of Theorem 14.6 [3] holds for the process (6) as
a consequence of (P2). Besides, functions fi(s, ·, x), i = 1, 2 are measurable
for any fixed s and x; and also functions fi(·, ω, x), i = 1, 2 are constant
for any fixed ω and x. Then, using Lemma 14.14 [3], we obtain that the
processes fi(s, ω, Xs−(ω)), i = 1, 2 are predicted and locally bounded. This
yields the condition 1) of Theorem 14.6 [3] is fulfilled. Therefore, equation
(6) has a unique strong solution.
By construction, the process Rt has a finite number of jumps on any
trajectory, and the values of these jumps are also finite. Hence, a solution
of equation (6) may be written in the form (3).
Equation (3) is a semilinear stochastic differential equation.
Solving the following linear stochastic differential equation
dη(t) = [α(t) + γ(t)η(t)] dt + δ(t)η(t) dWt
we get
η(t) = e
t
0 γ(s)− δ2(s)
2
ds+ t
0 δ(s) dWs×
×
(
η(0) +
∫ t
0
e
− s
0
γ(v)− δ2(v)
2
dv− s
0
δ(v) dWv
α(s) ds
)
.
(See example 3 [4], p.37-38). Analogously, solution of equation (3) can be
given in a form
Rt = e
t
0 a− b2
2
I{s≤σc} ds+ t
0 bI{s≤σc} dWs×
×
(
u +
∫ t
0
e
− s
0
a− b2
2
I{v≤σc} dv− s
0
bI{v≤σc} dWv (p(Rs) ds − Us dZs)
)
,
or
Rt = e
a− b2
2
·(t∧σc)+b·Wt∧σc ×
×
(
u +
∫ t
0
e
− a− b2
2
·(s∧σc)−b·Ws∧σc (p(Rs) ds − Us dZs)
)
=
= St∧σc
(
u
S0
+
∫ t
0
S−1
t∧σc
(p(Rs) ds − Us dZs)
)
.
RISK PROCESS WITH INVESTMENTS 5
The last equality is obtained from the equation St
S0
= e
a− b2
2
t+bWt , which
follows from (1). Thus, we have obtained representation (4).
Validity of formula (4) can also be easily checked by simple substitution
of the process Rt from (4) into equation (3). �
Now by using the representation of risk process (4), we can define the
ruin time differently: τ(u) = inf{t ≥ 0|Gt(u) < 0}, where
Gt(u) =
u
S0
+
∫ t
0
S−1
s∧σc
p(Rs) ds −
∫ t
0
S−1
s∧σc
Us dZs. (7)
Thus we have reduced the original problem of ruin probability estimation for
the risk model with investments to the problem of ruin probability estima-
tion in the model without investments, but with another premium income
process and claims process.
3. Ruin Probability Estimation
Theorem 1. Let the risk model be described by equation (7), where Rt
is a solution of equation (3). Assume that a premium income function
p(x) satisfies conditions (P1), (P2) of Lemma 1, and also that there exist
r ∈ [0, r∞), such that max{ r
c
+ 1, 2r
c
} < r∞, that the following condition
(P3) is satisfied:
(P3) p(x) ≥ β
(
h
(
r
c
+ 1
)− h
(
r
c
))
, ∀x ≥ 0.
Then the process Xt(u, r) := e−rGt(u), t ≥ 0 is an Ft-supermartingale.
Proof. Note that the process Xt(u, r) = e−rGt(u) is a semimartingale because
the process Gt is semimartingale as a sum of a martingale and a process
of a finite variation. (The process
∫ t
0
S−1
s∧σc
Us d(Zs − EZs) is a martingale,
and the processes
∫ t
0
S−1
s∧σc
p(Rs) ds,
∫ t
0
S−1
s∧σc
Us d(EZs) = β
∫ t
0
S−1
s∧σc
Us ds are
increasing processes, that is why they have a finite variation).
By Ito formula for semimartingale processes we have
F (Yt) − F (Y0) =
∫ t
0+
F ′(Ys−) dYs +
1
2
∫ t
0+
F ′′(Ys−) d〈Y, Y 〉cs+
+
∑
0<s≤t
(F (Ys) − F (Ys−) − F ′(Ys−)(Ys − Ys−)) , (8)
where {Yt, t ≥ 0} is a semimartingale, and F ∈ C2(R) (see [5], p.78-79).
If Yt(u, r) := −rGt(u) = −r
(
u
S0
+
∫ t
0
S−1
s∧σc
p(Rs) ds − ∫ t
0
S−1
s∧σc
Us dZs
)
,
and F (Yt) = eYt = F ′(Yt) = F ′′(Yt), then the formula (8) can be rewritten
as
eYt = eY0 +
∫ t
0+
eYs− dYs +
1
2
∫ t
0+
eYs− d〈Y, Y 〉cs+
6 MARYNA ANDROSHCHUK, YULIYA MISHURA
+
∑
0<s≤t
(
eYs − eYs− − eYs−(Ys − Ys−)
)
, (9)
if all integrals on the right-hand side of the equality (9) exist.
Yt−(u, r) = −r
(
u
S0
+
∫ t
0
S−1
s∧σc
p(Rs) ds −
∫ t−
0
S−1
s∧σc
Us dZs
)
, (10)
dYs = −rS−1
s∧σc
p(Rs) ds + rS−1
s∧σc
Us dZs, (11)
Ys − Ys− = rS−1
s∧σc
UNsI{�Ns
= 0}, (12)
eYs − eYs− = eYs−
(
erS−1
s∧σc
UNsI{�Ns �=0} − 1
)
, (13)
d〈Y, Y 〉cs = 0. (14)
Thus, in view of (10)-(14), we can formally rewrite formula (9) as
eYt = e
−r u
S0 − r
∫ t
0+
eYs−S−1
s∧σc
p(Rs) ds + r
∫ t
0+
eYs−S−1
s∧σc
Us dZs+
+
∑
0<s≤t
eYs−
(
erS−1
s∧σc
UNsI{�Ns �=0} − 1 − rS−1
s∧σc
UNsI{�Ns
= 0}
)
. (15)
The summands r
∫ t
0+
eYs−S−1
s∧σc
Us dZs and
−r
∑
0<s≤t e
Ys−S−1
s∧σc
UNsI{�Ns
= 0} in (9) cancel each other.
Consequently, (15) can be rewritten as
eYt = e
−r u
S0 − r
∫ t
0+
eYs−S−1
s∧σc
p(Rs) ds +
∑
0<s≤t
eYs−
(
erS−1
s∧σcUNsI{�Ns �=0} − 1
)
.
The process Xt = eYt is a supermartingale if it is integrable, and the
inequality
Et(XT − Xt) ≤ 0, ∀T > t ≥ 0 (16)
holds true almost surely.
Let us prove the integrability:
E|Xt| ≤ e
−r u
S0 + rE
∣∣∣∣∫ t
0+
eYs−S−1
s∧σc
p(Rs) ds
∣∣∣∣+
+E
∣∣∣∣∣ ∑
0<s≤t
eYs−
(
erS−1
s∧σc
UNsI{�Ns �=0} − 1
)∣∣∣∣∣ ≤
≤ e
−r u
S0 +
r
c
E
∣∣∣∣∫ t
0+
e−rGs−p(Rs) ds
∣∣∣∣+ E
∣∣∣∣∣
Nt∑
k=1
e−rGTk− · e r
c
Uk
∣∣∣∣∣ .
RISK PROCESS WITH INVESTMENTS 7
Here we used the definition of the stopping moment σc. According to it
S−1
s∧σc
≤ 1/c. Further, we find an estimate from above for the process −Gt:
−Gt = − u
S0
−
∫ t
0
S−1
s∧σc
p(Rs) ds +
∫ t
0
S−1
s∧σc
Us dZs ≤
∫ t
0
Us
c
dZs =
1
c
Nt∑
k=1
Uk.
Thus,
E|Xt| ≤ e
−r u
S0 +
r
c
E
∣∣∣∣∫ t
0+
e
r
c
Ns−
k=1 Ukp(Rs) ds
∣∣∣∣+ E
∣∣∣∣∣
Nt∑
k=1
e
r
c
k
m=1 Um · e r
c
Uk
∣∣∣∣∣ ≤
≤ e
−r u
S0 +
r
c
E
∣∣∣∣∫ t
0+
e
r
c
Ns−
k=1 Ukp(Rs) ds
∣∣∣∣+ E
∣∣∣∣∣
Nt∑
k=1
e
2r
c
k
m=1 Um
∣∣∣∣∣ .
Here we estimate the second and the third summands.
E
∣∣∣∣∫ t
0+
e
r
c
Ns−
k=1 Ukp(Rs) ds
∣∣∣∣ ≤ C
∫ t
0+
E e
r
c
Ns−
k=1 Uk ds =
= C
∫ t
0+
(
+∞∑
K=0
e
r
c
K
k=1 Uk · P{Ns = K}
)
ds =
= C
∫ t
0+
(
+∞∑
K=0
(
h
(r
c
)
+ 1
)K
· (βs)K
K!
e−βs
)
ds =
= C
∫ t
0+
(
e(h( r
c)+1)·βs · e−βs
)
ds = C
∫ t
0+
eh( r
c )·βs ds < ∞.
Further,
E
∣∣∣∣∣
Nt∑
k=1
e
2r
c
k
m=1 Um
∣∣∣∣∣ ≤
≤
+∞∑
K=1
∣∣∣E (e 2r
c
U1 + e
2r
c
(U1+U2) + ... + e
2r
c
(U1+U2+...+UK)
)∣∣∣ · P{Nt = K} =
=
+∞∑
K=1
((
h
(
2r
c
)
+ 1
)
+
(
h
(
2r
c
)
+ 1
)2
+ ... +
(
h
(
2r
c
)
+ 1
)K
)
×
×(βt)K
K!
e−βt =
+∞∑
K=1
(
h
(
2r
c
)
+ 1
)
·
(
h
(
2r
c
)
+ 1
)K − 1
h
(
2r
c
) · (βt)K
K!
e−βt =
=
h
(
2r
c
)
+ 1
h
(
2r
c
) · e−βt ·
(
+∞∑
K=1
((
h
(
2r
c
)
+ 1
)
βt
)K
K!
−
+∞∑
K=1
(βt)K
K!
)
=
8 MARYNA ANDROSHCHUK, YULIYA MISHURA
=
h
(
2r
c
)
+ 1
h
(
2r
c
) · e−βt ·
(
e(h( 2r
c )+1)βt − eβt
)
=
h
(
2r
c
)
+ 1
h
(
2r
c
) ·
(
eh(2r
c )·βt − 1
)
< ∞.
Thus we have show that E|Xt| < ∞.
The left-hand side of inequality (16) can be written as
Et(XT − Xt) = Et
⎛⎝−r
T∫
t+
eYs− p(Rs)
Ss∧σc
ds+
+
∑
t<s≤T
eYs−
(
e
r
UNs
Ss∧σc
I{�Ns �=0} − 1
))
. (17)
According to the mean value theorem we have
e
r
UNs
Ss∧σc
I{�Ns �=0} − 1 ≤ e
r
UNs
Ss∧σc · r UNs
Ss∧σc
· I{�Ns
= 0}.
Hence inequality (16) holds true if
Et
( ∑
t<s≤T
eYs− · er
UNs
Ss∧σc · UNs
Ss∧σc
· I{�Ns
= 0}
)
−
−Et
(∫ T
t+
eYs− p(Rs)
Ss∧σc
ds
)
≤ 0.
Using the fact that S−1
s∧σc
≤ 1/c, we see that inequality (16) holds if the
following inequality holds true
Et
( ∑
t<s≤T
eYs−er
UNs
c
UNs
Ss∧σc
· I{�Ns
= 0}
)
≤ Et
(∫ T
t+
eYs− · p(Rs)
Ss∧σc
ds
)
.
Now we transfer the right-hand term of the last inequality to the left, and
estimate the result from above.
Et
⎛⎝ ∑
t<s≤T
eYs−
Ss∧σc
· er
UNs
c · UNs · I{�Ns
= 0}
⎞⎠ − Et
⎛⎜⎝ T∫
t+
eYs−
Ss∧σc
· p(Rs) ds
⎞⎟⎠ ≤
≤ Et
⎛⎜⎝−
T∫
t+
eYs−
Ss∧σc
p(Rs) ds +
∑
t<s≤T
eYs−
Ss∧σc
e
r
c
UNs
(
eUNs − 1
)
I{�Ns
= 0}
⎞⎟⎠ (18)
To get inequality (18) we used the fact that every x ∈ R satisfies inequality
x+1 ≤ ex. For the validity of (17) it is enough to prove that the right-hand
side of inequality (18) is not positive. The process
Pt :=
∑
0<s≤t
eYs−
Ss∧σc
(
e(
r
c
+1)UNs − e
r
c
UNs
)
· I{�Ns
= 0}
RISK PROCESS WITH INVESTMENTS 9
is nondecreasing process almost surely. Write it in the integral form:
Pt =
∫ t
0
eYs−
Ss∧σc
dOs,
Ot :=
∑
0<s≤t
(
e(
r
c
+1)UNs − e
r
c
UNs
)
· I{�Ns
= 0}.
According to Wald identity ([6], p. 32, we can use it as random variables
{Uk}k≥1 and process {Nt}t≥0 are independent), and as a consequence the
definition of function h(r) , we have
EOt = E
Nt∑
k=1
(
e(
r
c
+1)Uk − e
r
c
Uk
)
= βt
(
h
(r
c
+ 1
)
− h
(r
c
))
.
The process {Ot}t≥0 is a process with independent increments, therefore the
corresponding compensated process {Ot − EOt}t≥0 is a martingale.
We have proved above that the process Xt = eYt is integrable. Hence
the process eYt−
St∧σc
is integrable too, as E| eYt−
St∧σc
| ≤ E|Xt−
c
| < ∞.
Consequently, the process∫ t
0
eYs−
Ss∧σc
d(Os − EOs) = Pt − β
(
h
(r
c
+ 1
)
− h
(r
c
))∫ t
0
eYs−
Ss∧σc
ds
is a martingale.
Therefore, it is obvious that the right-hand side of inequality (18) is not
positive if
Et
(
−
∫ T
t+
eYs−
Ss∧σc
· p(Rs) ds + β
(
h
(r
c
+ 1
)
− h
(r
c
))∫ T
t+
eYs−
Ss∧σc
ds
)
≤ 0.
(19)
It follows from condition (P3) that inequality (19) holds true.
Thus the process {Xt}t≥0 is a nonnegative Ft-supermartingale. �
Theorem 2. Let conditions (P1) and (P2) hold true for the risk process
with investments described by equation (2). If the equation
inf
x≥0
p(x) = β
(
h
(
r̂
c
+ 1
)
− h
(
r̂
c
))
. (20)
has a solution r̂ which satisfies 0 < max{ r
c
+ 1, 2r
c
} < r∞, then the ruin
probability can be bounded from above by
P{τ(u) < ∞} ≤ e
−r u
S0 . (21)
for all u ∈ R+.
10 MARYNA ANDROSHCHUK, YULIYA MISHURA
Proof. Evidently, for the constant r̂ defined in (20) condition (P3) of The-
orem 1 holds true. Then, as all conditions of Theorem 1 hold true, the
process {Gt}t≥0 is a supermartingale with respect to the flow Ft.
The process {Gt∧τ(u)}t≥0 is a supermartingale as well (it can be shown
easily by replacing t by t ∧ τ(u) in the proof of Theorem 1).
Then a ruin probability estimate of an insurance company can be found
in ordinary way.
e
−r u
S0 = X0(u, r̂) ≥ EXt∧τ(u)(u, r̂) =
= Xτ(u)(u, r̂) · Iτ(u)<t + Xt(u, r̂) · Iτ(u)≥t ≥ Xτ(u)(u, r̂) · Iτ(u)<t.
lim
t→∞
EXτ(u)(u, r̂) · Iτ(u)<t = EXτ(u)(u, r̂) · Iτ(u)<∞.
Hence
e
−r u
S0 ≥ E(Xτ(u)(u, r̂) / τ(u) < ∞) · P{τ(u) < ∞}
⇒ P{τ(u) < ∞} ≤ e
−r u
S0
EXτ(u)(u, r̂)
≤ e
−r u
S0 .
The last inequality holds true as a result of the fact that Gτ(u) < 0, and
Xτ(u) > 1. �
Remark 1. Due to the estimations made by means of the mean value
theorem and the inequality x + 1 ≤ ex in the proof of Theorem 1, we got
equation (20) for an adjustment coefficient r̂, which is of a ”non-classical
type”.
Let us remind of the classical results in risk theory (see for example [2],
p. 11). If the risk process is described by equation
Rt(u) = u + c1t −
Nt∑
k=1
Uk, t ≥ 0,
where c1 > 0 is a uniform process of insurance premiums income, then in
the case of positive safety loading ρ := c1 − βEU1 > 0 we can estimate ruin
probability as
P{τ(u) < ∞} ≤ e−Ru, (22)
where R is a unique positive solution of the equation
βh(R) = c1R. (23)
If we set in the model (3) S0 := 1 and c1 := infx≥0 p(x) then Theorem 2 will
give us ruin probability estimation
P{τ(u) < ∞} ≤ e−ru, (24)
RISK PROCESS WITH INVESTMENTS 11
where r̂ is a solution of equation
c1 = β
(
h
(r
c
+ 1
)
− h
(r
c
))
. (25)
We are interested if the estimation obtained in (24) can be better than
the estimation (22), that is whether r̂ ≥ R.
According to (23), constant R is a solution of the equation c1
β
= h(R)
R
;
according to (25) constant r̂ is a solution of c1
β
= h
(
r
c
+ 1
)− h
(
r
c
)
.
Let us consider the functions f1(r) := h(r)
r
and f2(r) := h
(
r
c
+ 1
)−h
(
r
c
)
,
and study their interrelation.
Since f1(0) = EU1, f2(0) = EeU1 − 1, we have that f2(0) > f1(0).
Further, f ′
1(r) = h′(r)r−h(r)
r2 , f ′
2(r) = 1
c
(
h′ (r
c
+ 1
)− h′ ( r
c
))
.
Note that h′(r) = EU1e
rU1 , h′′(r) = EU2
1 erU1, and these derivatives
exist at some neighborhood of r for r < r∞.
With the help of Taylor formula we have
h(0) = h(r) − h′(r)r +
1
2
h′′(θ1)r
2. (26)
Substitution of h(0) = 0 in formula (26) gives us
f ′
1(r) =
h′(r)r − h(r)
r2
=
1
2
h′′(θ1),
where θ1 ∈ (0, r).
On the other hand, according to the mean value theorem
f ′
2(r) = h
(r
c
+ 1
)
− h
(r
c
)
= h′′(θ2),
where θ2 ∈
(
r
c
, r
c
+ 1
)
.
Evidently h′′(r) = EU2
1 erU1 is an increasing function of r. Since c <
R0 = 1 we have r
c
> r which means that θ1 < θ2. This way we proved that
f ′
1(r) =
1
2
h′′(θ1) ≤ h′′(θ2) = f ′
2(r).
Thus, the function f2(r) crosses Y-axis higher then f1(r), and has faster
growth than the function f1(r). This means that the function f2(r) will
reach level c1
β
earlier, and hence r̂ < R. Consequently, we can not get as
good ruin probability estimate by means of inequality (21) as the Cramer-
Lundberg estimate gives.
Remark 2. Let us consider a modification of the risk model with invest-
ments by changing the stopping time of the investment activity. Let an
12 MARYNA ANDROSHCHUK, YULIYA MISHURA
insurance company invests into risky assets only when the price of the as-
sets is lying in the range of c∗ < St < c∗. Assume that S0 belongs to this
range. The price of the risky assets is modelled by a geometrical Brownian
motion (1).
Denote by σ̃c := inf{t ≥ 0 : St /∈ (c∗, c∗)} the stopping time, where
0 < c∗ < c∗ are some constants. So an insurer will terminate his investments
into the risk asset if the asset price drops below c∗ or rises above c∗.
Then, by analogy to Lemma 1, if conditions (P1) and (P2) hold true the
equation
Rt(u) = u−
Nt∑
k=1
Uk +
∫ t
0
p(Rs) ds + a
∫ t∧σc
0
Rs ds + b
∫ t∧σc
0
Rs dWs, t ≥ 0
(27)
has a unique, accurate within stochastic equivalence, Ft-measured solution,
which can be represented as
Rt = St∧σc
(
u
S0
+
∫ t
0
S−1
t∧σc
(p(Rs) ds − Us dZs)
)
,
where
∑Nt
k=1 Uk =
∫ t
0
Us dZs, Zs is a measure of jumps of a Poisson process
with intensity β.
Then we can reformulate Theorem 1.
Theorem 1∗. Let the risk model be described by an equation
G̃t(u) =
u
S0
+
∫ t
0
S−1
s∧σc
p(Rs) ds −
∫ t
0
S−1
s∧σc
Us dZs,
where Rt is risk a process which is a solution of the equation (27). If for
some r ∈ [0, r∞) such that r
c∗ < r∞, and premium income function p(x)
conditions (P1), (P2) and (P3)∗ hold true,
(P3)∗ r
c∗p(x) ≥ βh
(
r
c∗
)
, ∀x ≥ 0,
then the process X̃t(u, r) := e−rGt(u), t ≥ 0 is Ft-supermartingale.
The proof of the Theorem is similar with the one of Theorem 1. In the
same way we show that E|X̃t| < ∞.
Denote Ỹt(u, r) := −rG̃t(u), and find an estimation for the expectation
of exponential process increment: Et(X̃T − X̃t).
Et(X̃T − X̃t) =
= Et
(
−r
∫ T
t+
eYs− p(Rs)
Ss∧σc
ds +
∑
t<s≤T
eYs−
(
e
r
UNs
Ss∧σc
I{�Ns �=0} − 1
))
≤
RISK PROCESS WITH INVESTMENTS 13
≤ Et
⎛⎝−r
T∫
t+
eYs− p(Rs)
c∗
ds +
∑
t<s≤T
eYs−
(
er
UNs
c∗ − 1
)
I{�Ns
= 0}
⎞⎠ . (28)
The process P̃t :=
∑
0<s≤t e
Ys−
(
er
UNs
c∗ − 1
)
I{�Ns
= 0} is almost sure-
ly nondecreasing. It can be presented in an integral form
P̃t =
∫ t
0
eYs− dÕs,
where Õt :=
∑
0<s≤t
(
er
UNs
c∗ − 1
)
I{�Ns
= 0}.
According to Wald identity, we have
EÕt = E
Nt∑
k=1
(
e
r
c∗ Uk − 1
)
= βt · h
(
r
c∗
)
.
The process {Õt}t≥0 is a process with independent increments. Thus the
corresponding compensated process {Õt − EÕt}t≥0 is a martingale. Then
the process ∫ t
0
eYs− d(Õs − EÕs) = P̃t − βh
(
r
c∗
)∫ t
0
eYs− ds
is a martingale too.
Hence the right-hand side of inequality (28) is not positive if
Et
(
− r
c∗
∫ T
t+
eYs−p(Rs) ds + βh
(
r
c∗
)∫ T
t+
eYs− ds
)
≤ 0. (29)
The inequality (29) holds true as a result of condition (P3)∗. This proves
that the process {X̃t}t≥0 is a nonnegative Ft-supermartingale. �
Theorem 2 can be reformulated then in the following way.
Theorem 2∗. Let conditions (P1) and (P2) hold true for the risk process
described by equation (27). If the equation
R̂
c∗
inf
x≥0
p(x) = βh
(
R̂
c∗
)
has a solution R̂ > 0, then for the risk model described, the ruin probability
can be bounded from above by
P{τ(u) < ∞} ≤ e
−R u
S0 .
14 MARYNA ANDROSHCHUK, YULIYA MISHURA
The proof of the Theorem repeats completely the proof of Theorem 2.
Thus we can easily see that by reducing risk model (27) to the classical
risk model putting p(x) := c1, ∀x ≥ 0; St = c∗ = c∗ := 1, ∀t ≥ 0, we will
receive classical ruin probability estimate in Theorem 2∗:
P{τ(u) < ∞} ≤ e−Ru,
where R̂ is a solution of the equation R̂c1 = βh(R̂).
4. Analysis of Results in the Case of Exponential Claims
Distribution
Suppose that the distribution of the random variables modeling claim
sizes is exponential, i.e. {Uk, k ≥ 1} ∼ Exp(λ), and h(r) = r
λ−r
, r∞ = λ.
Let us denote c̃1 := infx≥0 p(x). Then the equation (20) can be written
as
c̃1 = β
(
r
c
+ 1
λ − (r
c
+ 1
) − r
c
λ − r
c
)
=
βλ(
λ − r
c
− 1
) (
λ − r
c
) ,
whence
r̂2 + c(1 − 2λ)r̂ + λc2
(
λ − 1 − β
c̃1
)
= 0. (30)
Solving equation (30) gives us
D = c2
(
1 +
4λβ
c̃1
)
> 0
r̂1,2 =
c
2
(
−1 + 2λ ±
√
1 +
4λβ
c̃1
)
.
We are interested in solutions of equation (30), which satisfy the condition
0 < max
{
r̂
c
+ 1,
2r̂
c
}
< r∞. (31)
of the Theorem 2.
As r2
c
+ 1 = λ + 1
2
+ 1
2
√
1 + 4λβ
c1
> λ, then the larger solution does not
satisfy (31).
Let us find sufficient conditions for characteristic λ, c, c̃1 such that the
solution r̂1 = c
2
(
−1 + 2λ −
√
1 + 4λβ
c1
)
satisfies conditions (31).
Write a minimum capital income into insurance company per unit of
time c̃1 as a function of average payoffs
c̃1 := (1 + K) · βEU1 = (1 + K) · β
λ
,
RISK PROCESS WITH INVESTMENTS 15
where K is some constant. (The necessary condition of existence of non-
trivial ruin probability estimation in the classical risk model is positiveness
of a safety loading K > 0).
Inequalities (31) for r̂1 can be rewritten as
0 < max
{
1
2
+ λ − 1
2
√
1 +
4λ2
1 + K
, −1 + 2λ −
√
1 +
4λ2
1 + K
}
< λ
or as a system of inequalities⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
√
1 + 4λ2
1+K
< 2λ + 1,√
1 + 4λ2
1+K
< 2λ − 1,
1 <
√
1 + 4λ2
1+K
,
λ − 1 <
√
1 + 4λ2
1+K
.
(32)
The first and the third inequalities in the system (32) are trivial. From the
second inequality we get that λ > 1, and K > 1
λ−1
. The consequence of
the forth inequality is if λ > 2 then we need one more restriction for K:
K < 3λ+2
λ−2
. Note that the last superior bound for K is adjusted with inferior
one written above, as inequality 1
λ−1
< 3λ+2
λ−2
holds true for all λ > 2.
Thus Theorem 2 provides us by ruin probability estimate P{τ(u) <
∞} ≤ e
−r u
S0 , where r̂ = c
2
(
−1 + 2λ −
√
1 + 4λβ
c1
)
, if characteristic λ and
K are in bounds [
λ ∈ (1, 2], 1
λ−1
< K,
λ > 2, 1
λ−1
< K < 3λ+2
λ−2
.
(33)
1. Comparison with the classical results. Equation (23) for an
adjustment coefficient in the case of claims distributed exponentially, i.e.
when h(r) = r
λ−r
can be written as
β
λ − R
= c1. (34)
To compare ruin probability estimates (21) and (22) we will assume that
p(x) := c̃1 = c1 = (1 + K) · β
λ
, ∀x ≥ 0, where K is some positive safety
loading coeficient, St := c = 1, ∀t ≥ 0.
Then we get R as a solution of equation (34)
R =
λK
1 + K
. (35)
Recall that
r̂ = λ − 1
2
− 1
2
√
1 +
4λ2
1 + K
. (36)
16 MARYNA ANDROSHCHUK, YULIYA MISHURA
Example 1. For λ = 5 restrictions of system (33) for the value of safety
loading K can be written as K ∈ (0.25, 5.66667). Set K := 0.3. Then
according to formulas (35) and (36), R = 1.15385, r̂ = 0.0862976. In fact,
as we see, in the case of exponential distribution of claims, adjustment
coefficients do not depend on the frequency of claims arrivals β. However
in order to find the value of premiums per unit of time and ruin probability
estimates, we set β := 20, and u := 50.
Then we get we get c1 = 5.2, and ruin probability estimates: 8.80135 ·
10−26 using classical formula of Cramer and Lundberg (22), and 0.0133682
using formula (21) from Theorem 2.
2. Comparison with the results of paper [1]. A model similar
to (3) has been examined in the paper [1]
Rt = u −
Nt∑
k=1
Uk +
∫ t
0
ps ds + a
∫ t
0
Rs ds + b
∫ t
0
Rs dWs, t ≥ 0, (37)
where
a, b are some positive constants,
ps is a nonnegative integrable predicted process,
Nt is Poisson process with intensity β and moments of successive jumps
{Tn, n ≥ 1},
Uk, k ≥ 1 are i.i.d. random variables with probability distribution func-
tion F .
Processes Wt, Nt, and random variables {Uk, k ∈ N} are assumed to be
independent. Denote θn := Tn − Tn−1, T0 := 0.
An assumption used in the paper [1] is
(F) The sequence (λn, ηn) is a sequence of two-dimensional i.i.d. random
variables, where
λn := ebW n
θn
+kθn,
ηn :=
∫ θn
0
pv+Tn−1e
b(W n
θn
−W n
v )+k(θn−v) dv,
k = a − b2
2
, W n
t = Wt+Tn−1 − WTn−1 .
Remark 3. The model (3) does not require anything similar to the con-
dition (F). Moreover, such condition would not be true for the model (3).
To show this rewrite λn and ηn as
λn := eb(WTn−WTn−1
)+k(Tn−Tn−1),
ηn :=
∫ Tn
Tn−1
pve
b(WTn−Wv)+k(Tn−v) dv.
RISK PROCESS WITH INVESTMENTS 17
Whereas in the model (3) we have pv = p(Rv), pv depends on realization of
the process {Wt, t ∈ [Tn−1, v]}, as well as λn. That is why λn and ηn are
not necessarily independent.
The main result of the paper [1] is
Theorem 2.1 [1]: For the risk model (37) in the case of exponential
claim size distribution F (x) = 1 − e−λx, λ > 0 we can estimate the ruin
probability in such a way
(i) if
:= 2a
b2
> 1, then for some M > 0
P{τ(u) < ∞} = Mu1−
(1 + o(1)), u → ∞;
(ii) if
< 1 then P{τ(u) < ∞} = 1, ∀u.
According to the Theorem 2 of the present work, we got exponential ruin
probability estimate P{τ(u) < ∞} ≤ e
−r u
S0 . In the case of exponentially
distributed claims we had r̂ = c
2
(
−1 + 2λ −
√
1 + 4λβ
infx≥0 p(x)
)
. Thus as we
got exponential estimate, it improves result of [1] as u → ∞. Moreover, our
result does not depend on the volatility of the underlying asset price (on the
value of
). Hence for
> 1 we can get by means of Theorem 2 non-trivial
ruin probability estimate.
5. Conclusions
We have considered a risk model with investment of all insurer’s capital
into the risky asset. The price of the asset is assumed to follow a geometrical
Brownian motion. We let the insurance company invest all its capital into
one type of risky assets, but if the price of the risky assets drops below some
predetermined fixed level the company is assumed to stop their investing.
For such model we have found exponential ruin probability estimate. It is
turned out, that the estimate found do not improve the Cramer-Lundberg
estimate. Hence we can conclude that, from the point of view of riskiness,
investing all the capital into risky assets is worse than not investing at all
for the insurance company.
On the other hand, the estimate obtained can be used for risk models
which have variable premium income process depending on the current value
of the company’s capital. Classical estimates can not be used for such
processes.
Besides, method introduced in this paper can give better results than
estimates of paper [1], where similar problem have been considered. In
the case of highly volatile assets inequality (21) will give us exponential
estimate, while [1] can give only trivial estimate.
Note that by diversifying investments into risky and non-risky assets, we
can get improved ruin probability estimates, sharper than the classical one,
18 MARYNA ANDROSHCHUK, YULIYA MISHURA
see, for example, [7] and [8]. This yields to the conjecture that if minimiza-
tion of ruin probability is the criteria of insurer’s investment behavior, then
we should search for an optimal investment strategy in between diversified
portfolios.
References
1. Frolova, A. G., Kabanov, Yu. M., Pergamenshchikov, S. M., In the Insur-
ance Business Risky Investments are Dangerous, Finance and Stochastics
6, no. 2, (2002), 227–235.
2. Grandell, J., Aspects of Risk Theory, Springer-Verlag, New York, (1991),
175 p.
3. Elliott, R., J., Stochastic Calculus and Applications, Springer, New York,
(1982), 305 p.
4. Gikhman, I. I., Skorokhod, A. V., Stochastic Differential Equations, Nauko-
va Dumka, Kiev, (1968), 354 p. (Rus.)
5. Protter, P. E., Stochastic Integration and Differential Equations, Springer,
Berlin, (2004), 410 p.
6. Skorokhod, A. V., Random Processes with Independent Increments, Nauka,
Moscow, (1986), 320 p. (Rus.)
7. Gaier, J., Grandits, P., Schachermayer, W., Asymptotic Ruin Probabilities
and Optimal Investment, The Annals of Applied Probability, 13, (2003),
1054–1076.
8. Androshchuk, M. O., Mishura, Yu. S., An Estimate of Ruin Probability
for an Insurance Company which is Functioning on BS-market, Ukrainian
Mathematical Journal, 11, (2007) (Ukr.)
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail: andr m@univ.kiev.ua
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail: myus@univ.kiev.ua
|