Dividend payments in a perturbed compound Poisson model with stochastic investment and debit interest
UDC 519.21 We consider a compound Poisson insurance risk model perturbed by diffusion with stochastic return on investment and debit interest. If the initial surplus is nonnegative, then the insurance company can invest its surplus in a risky asset and risk-free asset based on a fixed proportion....
Gespeichert in:
| Datum: | 2019 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1462 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507240311554048 |
|---|---|
| author | Li, Y. F. Lu, Y. H. Лі, Я. Ф. Лю, Я. Х. |
| author_facet | Li, Y. F. Lu, Y. H. Лі, Я. Ф. Лю, Я. Х. |
| author_sort | Li, Y. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:56:08Z |
| description | UDC 519.21
We consider a compound Poisson insurance risk model perturbed by diffusion with stochastic return on investment and debit interest. If the initial surplus is nonnegative, then the insurance company can invest its surplus in a risky asset and risk-free asset based on a fixed proportion. Otherwise, the insurance company can get the business loan when the surplus is negative. The integrodifferential equations for the moment generating function of the cumulative dividends value are obtained under the barrier and threshold dividend strategies, respectively. The closed-form of the expected dividend value is obtained when the claim amount is exponentially distributed. |
| first_indexed | 2026-03-24T02:06:10Z |
| format | Article |
| fulltext |
UDC 519.21
Y. H. Lu, Y. F. Li (School Statist., Qufu Normal Univ., Shandong, China)
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL
WITH STOCHASTIC INVESTMENT AND DEBIT INTEREST*
ДИВIДЕНДНI ВИПЛАТИ У ЗБУРЕНIЙ СКЛАДНIЙ
ПУАССОНIВСЬКIЙ МОДЕЛI ЗI СТОХАСТИЧНИМИ IНВЕСТИЦIЯМИ
ТА ДЕБЕТОВИМИ ВIДСОТКАМИ
We consider a compound Poisson insurance risk model perturbed by diffusion with stochastic return on investment and
debit interest. If the initial surplus is nonnegative, then the insurance company can invest its surplus in a risky asset and
risk-free asset based on a fixed proportion. Otherwise, the insurance company can get the business loan when the surplus
is negative. The integrodifferential equations for the moment generating function of the cumulative dividends value are
obtained under the barrier and threshold dividend strategies, respectively. The closed-form of the expected dividend value
is obtained when the claim amount is exponentially distributed.
Розглядається складна пуассонiвська модель страхових ризикiв, збурена дифузiєю, зi стохастичним доходом по
iнвестицiях та дебетовим вiдсотком. Якщо початковий надлишок є невiд’ємним, то стрaхова компанiя може вкладати
цей надлишок у ризиковi або безризиковi активи в фiксованiй пропорцiї. В протилежному випадку, коли надлишок є
вiд’ємним, страхова компанiя може отримувати бiзнесовi кредити. Iнтегро-диференцiальнi рiвняння для функцiї, що
породжує моменти значень кумулятивних дивiдендiв, отримано для бар’єрних та порогових дивiдендних стратегiй
вiдповiдно. Очiкувану величину дивiдендiв отримано в замкненiй формi у випадку експоненцiального розподiлу
суми позову.
1. Introduction. The compound Poisson insurance risk model perturbed by diffusion was first
introduced by Gerber [8] and further studied by many authors, such as Dufresne and Gerber [7],
Yuen et al. [23], Asmussen [1], Chiu and Yin [5], Lu et al. [15]. If investment income is
introduced in the perturbed compound Poisson model, the security loading will be a variable. In the
field of actuarial mathematics, researchers paid lots of attention to the issue of stochastic investment
and debit, for example, Gerber [9], Zhu and Yang [24] considered the absolute ruin problem in the
compound Poisson model when the debt and credit interest rates were the same. Cai [4] studied the
Gerber – Shiu function in the classical insurance risk model with debit interest. Paulsen and Gjessing
[19] simplified the version of the model which was used by Paulsen [16] through incorporating a
stochastic rate of return on investments. Then they computed the probability of eventual ruin and the
infinitesimal generator of the risk process. Gerber and Yang [11] proposed the general risk model
with investment, in which the insurance company can invest its surplus in a risky asset and risk-free
asset. If the company invests money in the bank or borrows money from the bank, the rate of return
can be described as
dRt =
\left\{ rRtdt, if surplus is nonnegative,
\alpha Rtdt, if surplus is negative,
(1.1)
where r is the company’s lending rate, \alpha is the borrowing rate which satisfies \alpha \geq r > 0. The model
of the risky asset satisfies the stochastic differential equation
* This research was supported by National Science Foundation of China (Grant No. 11571198); the Program for
Shandong Postgradute Education Innovation Project (Grant No. SDYY16094).
c\bigcirc Y. H. LU, Y. F. LI, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5 631
632 Y. H. LU, Y. F. LI
dSt = \mu Stdt+ \sigma StdB0t, (1.2)
where \mu t + \sigma B0t is the return on investment St to the risky asset, \{ B0t\} is the standard Brownian
motion. Yin and Wen [22] extended the model of Paulsen [17] to the following form:
Ut = u+ Pt +
t\int
0
Us - dRs with P0 = R0 = 0,
where \{ Pt\} , \{ Rt\} were independent Lévy processes, u was the initial value, \{ Pt\} was a general
two-sided jump-diffusion risk model.
Dividend strategies were first introduced by De Finetti [6]. He formulated the problem and
solved it under the assumption that the surplus was a discrete process without investment. The study
of the dividend strategy was also be conducted by Bühlmann [3], Paulsen [16], Jeanblanc-Picqué and
Shiryaev [12], Asmussen and Taksar [2], Gerber and Shiu [10], Wan [20], Lu and Wu [14], Yin and
Yuen [21]. Many results were obtained using the property of stationary and independent increments
of the Poisson process and Brownian motion. Recently, Yin and Wen [22] extended the model raised
by Paulsen [17] and obtained the integrodifferential equations of the Gerber – Shiu functions and total
discounted dividends, respectively. For further references, see two survey papers Paulsen [17, 18].
In recent papers the model was extended to renewal risk model with stochastic return, see Yin and
Wen [22] and Li [13].
Motivated by the previously mentioned papers, we are going to study the moment generating
function of dividend value under barrier and threshold dividend strategies on a perturbed compound
Poisson model with stochastic investment and debit interest. The rest of the paper is structured as
follows. In Section 2, the perturbed compound Poisson model with stochastic investment and debit
interest is introduced. In Section 3, the second-order integrodifferential equations for the moment
generating function of aggregate dividends under the barrier dividend strategy are established. In the
case of the exponential claim size, the exact solution of the third-order differential equations and the
closed-form of the expected dividend value are obtained. In Section 4, the threshold dividend strategy
is discussed. The integrodifferential equations for the moment generating function of aggregate
dividends and the expected dividend value are obtained.
2. The insurance risk model with investment income. We consider a company with initial
surplus u. If no dividends are paid, the surplus at time t is
Pt = u+ ct -
Nt\sum
i=1
Xi + \sigma 1Wt,
where c > 0 is the premium rate, \sigma 1 is a constant. The aggregate claim process
\Bigl\{ \sum Nt
i=1
Xi
\Bigr\}
is
the compound Poisson process with parameter \lambda , \{ Xi\} is a sequence of independent and identically
distributed random variables, the density function is defined as p(x). \{ Wt\} is the standard Brownian
motion independent of the aggregate claim process. Now, suppose that the surplus is nonnegative,
then the company can invest its money in a risk-free asset and a risky asset. The risk-free asset price
is assumed to follow the stochastic differential equation (1.1), and the risky asset price is the same
as (1.2). Wt and B0t are correlated with dWtdB0t = \rho dt.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL . . . 633
Assume that if the surplus Ut > 0 then the insurance company invests the surplus by a fixed
proportion \tau to the risk-free asset and the rest proportion 1 - \tau of the surplus to the risky asset. If
the surplus Ut \leq 0, then the insurance company borrows money from the bank under the debt rate \alpha .
Hence, the modified surplus risk process \{ Ut\} satisfies the following equation. Let t > 0, for small
enough \Delta t,
\Delta Ut = Ut\Delta t\times
\left\{ \tau r, Ut > 0,
\alpha , Ut \leq 0,
+ (1 - \tau )Ut(\mu \Delta t+ \sigma \Delta B0t)\times
\left\{ 1, Ut > 0,
0, Ut \leq 0,
+\Delta Pt.
Let initial u > 0, for small enough t and \Delta t = dt, we have
dUs = \tau rUs - ds+ (1 - \tau )\mu Us - ds+ (1 - \tau )\sigma Us - dB0s + dPs =
= [\tau r + (1 - \tau )\mu ]Us - ds+ (1 - \tau )\sigma Us - dB0s + dPs =
= Us - dLs + dPs, 0 \leq s < t, (2.1)
where Lt = adt+ bdB0t. We use a and b that are defined below
a = \tau r + (1 - \tau )\mu , b = (1 - \tau )\sigma .
From (2.5) in Yin and Wen [22] as a special case we get the another form of the equation (2.1) listed
as follows:
Ut = u+
t\int
0
(c+ aUs) ds+
t\int
0
\sqrt{}
(\sigma 1 + \rho bUs - )2 + b2(1 - \rho 2)U2
s - dBs -
Nt\sum
i=1
Xi, u > 0,
where \{ Bt, t \geq 0\} is a standard Brownian motion independent of the compound Poisson processes
involved.
If u \leq 0, for small enough t and \Delta t = dt, then \{ Ut\} satisfies the equation
dUt = dPt + \alpha Utdt. (2.2)
To solve the stochastic differential equation (2.2) we consider the process Vt = Ute
- \alpha t. By using the
product rule, we have dVt = e - \alpha tdUt - \alpha e - \alpha tUtdt. Combining with (2.2) we obtain dVt = e - \alpha tdPt.
This gives Vt = V0 +
\int t
0
e - \alpha sdPs. The solution for stochastic differential equation (2.2) is
Ut = e\alpha t
\left( u+ c
t\int
0
e - \alpha sds
\right) - e\alpha t
t\int
0
e - \alpha sd
Ns\sum
i=1
Xi + \sigma 1e
\alpha t
t\int
0
e - \alpha sdW (s), u \leq 0.
According to the above results, for small enough t > 0, we get the model
Ut =
\left\{
u+
\int t
0
(c+ aUs) ds+
\int t
0
\sqrt{}
(\sigma 1 + \rho bUs - )
2 + b2 (1 - \rho 2)U2
s - dBs -
\sum Nt
i=1
Xi, u > 0,
e\alpha t
\biggl(
u+ c
\int t
0
e - \alpha sds
\biggr)
- e\alpha t
\int t
0
e - \alpha sd
\sum Ns
i=1
Xi + \sigma 1e
\alpha t
\int t
0
e - \alpha sdW (s), u \leq 0.
(2.3)
We assume that the company can still run as long as the surplus is above the level - c/\alpha , that is, if
the surplus is below - c/\alpha , then the company’s interest can no longer pay off the debt and absolute
ruin occurs immediately.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
634 Y. H. LU, Y. F. LI
3. The barrier dividend strategy of the model. In this section we focus on the barrier dividend
strategy. If the surplus Ut \geq \eta , \eta > 0, then the excess D(t) = Ut - \eta all will be paid to the
shareholders as dividends. However, if the surplus Ut < \eta , then no dividends are paid. If the surplus
is below - c
\alpha
, then absolute ruin happens and the surplus process stops. We define the dividend value
until time t:
Dt(\eta ) =
t\int
0
I(Us > \eta )dD(s).
Then the modified surplus is
U\eta (t) = Ut - Dt(\eta ),
where \{ Ut\} is modeled by (2.3).
For simplicity, we let
E[\cdot | U0 = u] = Eu(\cdot ).
We denote V (u, \eta ) as the dividend value function of the strategy, which is
V (u, \eta ) = Eu[DT1 ],
where DT1 =
\int T1
0
e - \delta tdDt(\eta ), \delta > 0 is the force of interest for valuation, and T1 is the absolute
ruin time defined by
T1 = \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
t \geq 0, Ut(\eta ) \leq - c
\alpha
\Bigr\}
.
Let M(u, y, \eta ) = Eu[e
yDT1 ] be the moment generating function of DT1 and
M(u, y, \eta ) =
\left\{
ey[(u - \eta )+V (\eta ,\eta )], u > \eta ,
M1(u, y, \eta ), 0 \leq u \leq \eta ,
M2(u, y, \eta ), - c
\alpha
< u < 0.
We assume that, for any y < \infty , M(u, y, \eta ) exists, M1(u, y, \eta ) and M2(u, y, \eta ) are twice continu-
ously differentiable for u \in
\Bigl(
- c
\alpha
, \eta
\Bigr)
.
Theorem 3.1. If 0 < u < \eta , then M1(u, y, \eta ) satisfies the integrodifferential equation
\lambda M1(u, y, \eta ) = (au+ c)
\partial M1(u, y, \eta )
\partial u
- \delta y
\partial M1(u, y, \eta )
\partial y
+
+
1
2
(\sigma 2
1 + 2\rho b\sigma 1u+ b2u2)
\partial 2M1(u, y, \eta )
\partial u2
+
+\lambda
u\int
0
M1(u - x, y, \eta )p(x) dx+
+\lambda
u+ c
\alpha \int
u
M2(u - x, y, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx. (3.1)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL . . . 635
If - c
\alpha
< u < 0, then M2(u, y, \eta ) satisfies the following equation:
\lambda M2(u, y, \eta ) =
1
2
\sigma 2
1
\partial 2M2(u, y, \eta )
\partial u2
+ (\alpha u+ c)
\partial M2(u, y, \eta )
\partial u
- \delta y
\partial M2(u, y, \eta )
\partial y
+
+\lambda
u+ c
\alpha \int
0
M2(u - x, y, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx (3.2)
with boundary conditions
M2
\Bigl(
- c
\alpha
, y, \eta
\Bigr)
= 1, M1(0+, y, \eta ) = M2(0 - , y, \eta ),
\partial M(u, y, \eta )
\partial u
\bigm| \bigm| \bigm| \bigm|
u=\eta
= yM1(\eta , y, \eta ).
(3.3)
Proof. If 0 \leq u < \eta , we consider an infinitesimal time interval (0, t). By the Markov property
of the process \{ Ut\} , we have
Mi(u, y, \eta ) = EuMi(Ut, ye
- \delta t, \eta ) + o(t), i = 1, 2, 0 \leq u < \eta . (3.4)
Let A
d
= B means that A and B have the same distribution. Set
Yt = u+
t\int
0
(c+ aUs)ds+
t\int
0
\sqrt{}
(\sigma 1 + \rho bUs - )2 + b2(1 - \rho 2)U2
s - dBs.
In the infinitesimal time interval (0, t), the insurance risk process (2.3) has three possible cases.
(i) If no jump happen in (0, t), then its probability is e\lambda t and Ut
d
= Yt.
(ii) If there is only one jump in (0, t) and the claim size is X1, then its probability is \lambda te - \lambda t
and Ut
d
= Yt - X1.
Further, according to the amount of the claim, there are the following situations:
(1) if 0 < X1 < Yt, then ruin do not occur,
(2) if Yt < X1 < Yt +
c
\alpha
, then the ruin occur,
(3) if X1 \geq Yt +
c
\alpha
, then the absolute ruin occur.
(iii) If there are two or more jumps, then its probability is o(t).
So, we have
M1(u, y, \eta ) = e - \lambda tEu
\bigl[
M1(Yt, ye
- \delta t, \eta )
\bigr]
+
+\lambda te - \lambda tEu
Yt\int
0
M1(Yt - x, ye - \delta t, \eta )p(x) dx+
+\lambda te - \lambda tEu
Yt+
c
\alpha \int
Yt
M2(Yt - x, ye - \delta t, \eta )p(x) dx+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
636 Y. H. LU, Y. F. LI
+\lambda te - \lambda tEu
\infty \int
Yt+
c
\alpha
p(x) dx+ o(t). (3.5)
By Itô’s formula, we obtain
dM1(Yt, ye
- \delta t, \eta ) =
\partial M1(Yt, ye
- \delta t, \eta )
\partial u
dYt + y
\partial M1(Yt, ye
- \delta t, \eta )
\partial y
de - \delta t+
+
1
2
\partial 2M1(Yt, ye
- \delta t, \eta )
\partial u2
(dYt)
2.
Then
EuM1(Yt, ye
- \delta t, \eta ) = M1(u, y, \eta )+
+Eu
\left[ t\int
0
(c+ aUs)
\partial M1(Ys, ye
- \delta s, \eta )
\partial u
ds - \delta y
t\int
0
\partial M1(Ys, ye
- \delta s, \eta )
\partial y
e - \delta sds
\right] +
+
1
2
Eu
t\int
0
\partial 2M1(Ys, ye
- \delta s, \eta )
\partial u2
\bigl[
(\sigma 1 + \rho bUs - )
2 + b2(1 - \rho 2)U2
s -
\bigr]
ds. (3.6)
Substituting (3.6) into (3.5) and dividing both sides of (3.4) by t, then letting t \rightarrow 0 and rearranging
it we get (3.1).
If - c
\alpha
< u < 0, we let that
h\alpha (t, u) = ue\alpha t +
c(e\alpha t - 1)
\alpha
+ \sigma 1e
\alpha t
t\int
0
e - \alpha s dWs.
By full probability formula we have
M2(u, y, \eta ) = e - \lambda tM2(h\alpha (t, u), ye
- \delta t, \eta )+
+\lambda te - \lambda tEu
h\alpha +
c
\alpha \int
0
M2(h\alpha (t, u) - x, ye - \delta t, \eta )p(x) dx+
+\lambda te - \lambda tEu
\infty \int
h\alpha +
c
\alpha
p(x) dx+ o(t). (3.7)
By using Itô’s formula, we obtain
dM2(h\alpha (t, u), ye
- \delta t, \eta ) =
\partial M2(h\alpha (t, u), ye
- \delta t, \eta )
\partial u
dh\alpha (t, u)+
+
\partial M2(h\alpha (t, u), ye
- \delta t, \eta )
\partial u
dye - \delta t +
1
2
\partial 2M2(h\alpha (t, u), ye
- \delta t, \eta )
\partial u2
(dh\alpha )
2. (3.8)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL . . . 637
By using (3.7) and (3.8), we get (3.2).
Noting that if u = - c
\alpha
, then T1 = 0 and DT1 = 0. So we have
M2
\Bigl(
- c
\alpha
, y, \eta
\Bigr)
= E - c
\alpha
\bigl(
eyDT1
\bigr)
= 1, M1(0+, y, \eta ) = M2(0 - , y, \eta ),
due to Mi(u, y, \eta ), i = 1, 2, are continuous at u = 0.
By the definition of M(u, y, \eta ), we get
\partial M(u, y, \eta
\partial u
| u=\eta = yM1(\eta , y, \eta ). The boundary condi-
tions (3.3) are obtained.
Theorem 3.1 is proved.
Let M(u, y, \eta ) = 1 + yV (u, \eta ) + o(y), for small enough y, where
V (u, \eta ) = Eu[DT1 ] =
\left\{ V1(u, \eta ), 0 < u < \eta ,
V2(u, \eta ), - c
\alpha
< u < 0.
(3.9)
Substituting (3.9) into (3.1) and (3.2), comparing the coefficients of y, we get the following results.
Theorem 3.2. If 0 < u < \eta , then Vi(u, \eta ), i = 1, 2, satisfy the equation
(\lambda + \delta )V1(u, \eta ) =
1
2
\bigl(
b2u2 + 2\rho b\sigma 1u+ \sigma 2
1
\bigr)
V \prime \prime
1 (u, \eta ) + (au+ c)V \prime
1(u, \eta )+
+\lambda
u\int
0
V1(u - x, \eta )p(x)dx+ \lambda
u+ c
\alpha \int
u
V2(u - x, \eta )p(x)dx. (3.10)
If - c
\alpha
< u < 0, then V2(u, \eta ) satisfies the equation
(\lambda + \delta )V2(u, \eta ) =
1
2
\sigma 2
1V
\prime \prime
2 (u, \eta ) + (\alpha u+ c)V \prime
2(u, \eta )+
+\lambda
u+ c
\alpha \int
0
V2(u - x, \eta )p(x) dx (3.11)
with boundary conditions
V \prime
1(\eta ) = 1, V \prime
1(0+) = V \prime
2(0 - ), V1(0+) = V2(0 - ). (3.12)
Example 3.1. Assume that the claim size has an exponential distribution with the density function
p(x) = e - x. Applying the operator
\biggl(
d
du
+ 1
\biggr)
on (3.10) and (3.11), we get the following results.
If 0 < u < \eta , then V1(u, \eta ) satisfies the equation
1
2
\bigl(
b2u2 + 2\rho b\sigma 1u+ \sigma 2
1
\bigr)
V \prime \prime \prime
1 (u, \eta )+
+
\biggl[
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1) + (a+ b2)u+ \rho b\sigma 1 + c
\biggr]
V \prime \prime
1 (u, \eta )+
+(au+ c+ a - \delta - \lambda )V \prime
1(u, \eta ) - \delta V1(u, \eta ) = 0. (3.13)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
638 Y. H. LU, Y. F. LI
If - c
\alpha
< u < 0, then V2(u, \eta ) satisfies the equation
1
2
\sigma 2
1V
\prime \prime \prime
2 (u, \eta ) +
\biggl(
\alpha u+ c+
1
2
\sigma 2
1
\biggr)
V \prime \prime
2 (u, \eta ) + (\alpha u - \lambda - \delta + \alpha )V \prime
2(u, \eta ) - \delta V2(u, \eta ) = 0. (3.14)
Letting \delta = b2 = 0, \lambda = a+ c, then (3.13) gives the equation
1
2
\sigma 2
1V
\prime \prime \prime
1 (u, \eta ) +
\biggl(
au+ c+
1
2
\sigma 2
1
\biggr)
V \prime \prime
1 (u, \eta ) + auV \prime
1(u, \eta ) = 0.
Using substitutions x = au, V \prime
1(u, \eta ) = e - ug(x); z = ax+ac - a\sigma 2
1
2
, g(x) =
2
a2\sigma 2
1
f(x); t = - z2
2
,
f(z) = h(t), (3.13) yields
th\prime \prime (t) +
\biggl(
1
2
- 2
a3\sigma 2
1
t
\biggr)
h\prime (t) + ch(t) = 0. (3.15)
For convenience, we let \sigma 2
1a
3 = 2, thus (3.15) changes into a confluent hypergeometric equation
th\prime \prime (t) +
\biggl(
1
2
- t
\biggr)
h\prime (t) + ch(t) = 0.
The general solution to the above equation is a linear combination of two independent solutions
h(t) = c5H
\biggl(
- c,
1
2
, t
\biggr)
+ c6L
\biggl(
- c,
1
2
, t
\biggr)
,
where H
\biggl(
- c,
1
2
, t
\biggr)
and L
\biggl(
- c,
1
2
, t
\biggr)
are the first and second kind of confluent hypergeometric
functions respectively, c5 and c6 are constants. Transforming back to the original variables, we get
the following solution:
V \prime
1(u, \eta ) =
2c5
a2\sigma 2
1
e - uH
\biggl(
- c,
1
2
,
- a2(au+ c - \sigma 2
1/2)
2
2
\biggr)
+
+
2c6
a2\sigma 2
1
e - uL
\biggl(
- c,
1
2
,
- a2(au+ c - \sigma 2
1/2)
2
2
\biggr)
.
So V1(u, \eta ) =
\int u
0
V \prime
1(x) dx+ V1(0+, \eta ). Next, we compute c5, c6. For (3.14), let \sigma 1 = 0, we have
(\alpha u+ c)V \prime \prime
2 (u, \eta ) + (\alpha u+ c - \lambda - \delta + \alpha )V \prime
2(u, \eta ) - \delta V2(u, \eta ) = 0. (3.16)
With change of variable y = - u - c
\alpha
, g(y) = V2(u, \eta ), (3.16) becomes
yg\prime \prime (y) +
\biggl(
\alpha - \lambda - \delta
\alpha
- y
\biggr)
g\prime (y) +
\delta
\alpha
g(y) = 0.
This is an confluent hypergeometric equation. The general solution is
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL . . . 639
g(y) = c7( - y)
\lambda +\delta
\alpha eyH
\biggl(
1 +
\delta
\lambda
, 1 +
\lambda + \delta
\alpha
, - y
\biggr)
+ c8e
yL
\biggl(
1 - \delta
\lambda
, 1 - \lambda + \delta
\alpha
, - y
\biggr)
.
Transforming back to the original variable, we have
V2(u, \eta ) = g
\Bigl(
- u - c
\alpha
\Bigr)
= c7
\Bigl(
u+
c
\alpha
\Bigr) \lambda +\delta
\alpha
e - u - c
\alpha H
\biggl(
1 +
\delta
\lambda
, 1 +
\lambda + \delta
\alpha
, u+
c
\alpha
\biggr)
+
+c8e
- u - c
\alpha L
\biggl(
1 - \delta
\lambda
, 1 - \lambda + \delta
\alpha
, u+
c
\alpha
\biggr)
. (3.17)
As
\mathrm{l}\mathrm{i}\mathrm{m}
u\downarrow - c
\alpha
V2(u, \eta ) = c8L
\biggl(
1 - \delta
\lambda
, 1 - \lambda + \delta
\alpha
, 0
\biggr)
= c8
\Gamma
\Bigl( \delta + \delta
\alpha
\Bigr)
\Gamma
\Bigl(
1 +
\delta
\alpha
\Bigr) = 0,
so c8 = 0.
Letting \delta = 0 in (3.17), then we obtain
V2(u, \eta ) = c7
\Bigl(
u+
c
\alpha
\Bigr) \lambda
\alpha
e - u - c
\alpha H
\biggl(
1, 1 +
\lambda
\alpha
, u+
c
\alpha
\biggr)
.
By using the boundary condition (3.12), we get
2c5
a2\sigma 2
1
e - \eta H
\biggl(
- c,
1
2
, - a2(a\eta - \sigma 2
1/2)
2
2
\biggr)
+
2c6
a2\sigma 2
1
e - \eta L
\biggl(
- c,
1
2
,
a2(a\eta + c+ \sigma 2
1/2)
2
\biggr)
= 1, (3.18)
V \prime
1(0+) =
2c5
a2\sigma 2
1
H
\biggl(
- c,
1
2
, - a2(c - \sigma 2
1/2)
2
2
\biggr)
+
2c6
a2\sigma 2
1
L
\biggl(
- c,
1
2
, - a2(c - \sigma 2
1/2)
2
8
\biggr)
=
= c7
\Bigl( c
\alpha
\Bigr) \lambda
\alpha
e -
c
\alpha
\biggl[
\lambda
c
H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
- H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
+
\alpha
\alpha + \lambda
H
\biggl(
2, 2 +
\lambda
\alpha
,
c
\alpha
\biggr) \biggr]
=
= V \prime
2(0 - ), (3.19)
and
V1(0+) =
2c5
a2\sigma 2
1
\eta \int
0
e - xH
\biggl(
- c,
1
2
, - a2(ax - \sigma 2
1/2)
2
2
\biggr)
dx+
+
2c6
a2\sigma 2
1
\eta \int
0
e - xL
\biggl(
- c,
1
2
,
a2(ax+ c+ \sigma 2
1/2)
2
\biggr)
dx =
= c7
\Bigl( c
\alpha
\Bigr) \lambda
\alpha
H
\biggl(
1, 1 +
\lambda
\alpha
;
c
\alpha
\biggr)
= V2(0 - ). (3.20)
Let
P =
\lambda
c
H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
- H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
+
\alpha
\alpha + \lambda
H
\biggl(
2, 2 +
\lambda
\alpha
,
c
\alpha
\biggr)
,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
640 Y. H. LU, Y. F. LI
A1 =
\eta \int
0
e - xH
\biggl(
- c,
1
2
, - a2(ax - \sigma 2
1/2)
2
2
\biggr)
dx,
B1 =
\eta \int
0
e - xL
\biggl(
- c,
1
2
,
a2(ax+ c+ \sigma 2
1/2)
2
\biggr)
dx,
A2 = H
\biggl(
- c,
1
2
, - a2(a\eta - \sigma 2
1/2)
2
2
\biggr)
,
B2 = L
\biggl(
- c,
1
2
, - a2(c - \sigma 2
1/2)
2
8
\biggr)
,
A3 = H
\biggl(
- c,
1
2
, - a2(c - \sigma 2
1/2)
2
2
\biggr)
,
B3 = L
\biggl(
- c,
1
2
, - a2(c - \sigma 2
1/2)
2
8
\biggr)
,
then (3.18), (3.19) and (3.20) can be rewritten by
2c5
a2\sigma 2
1
A2 +
2c6
a2\sigma 2
1
B2 = e\eta ,
2c5
a2\sigma 2
1
A3 +
2c6
a2\sigma 2
1
B3 = c7
\Bigl( c
\alpha
\Bigr) \lambda
\alpha
e -
c
\alpha P,
2c5
a2\sigma 2
1
A1 +
2c6
a2\sigma 2
1
B1 = c7
\Bigl( c
\alpha
\Bigr) \lambda
\alpha
e -
c
\alpha H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
.
The solutions to the above three equations are
c5 =
a2\sigma 2
1e
\eta
2
B1P - B3H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
(A2B1 - A1B2)P +H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
(B2A3 - A2B3)
,
c6 =
a2\sigma 2
1e
\eta
2
A3H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
- A1P
(A2B1 - A1B2)P +H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
(B2A3 - A2B3)
,
c7 = e
c
\alpha
+\eta
\Bigl( \alpha
c
\Bigr) \lambda
\alpha P (B1A3 - A1B3)
(A2B1 - A1B2)P +H
\biggl(
1, 1 +
\lambda
\alpha
,
c
\alpha
\biggr)
(B2A3 - A2B3)
.
4. The threshold dividend strategy. In this section, we consider the threshold dividend strategy
for surplus Ut, where Ut is defined by (2.3). The company will pay dividends to its shareholders
according to the following strategy governed by parameters \eta > 0 and \beta > 0. Whenever the
(modified) surplus is below the level \eta , no dividends are paid. However, when the modified surplus
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL . . . 641
is above \eta , dividends are paid continuously at a constant rate \beta . If the surplus is below the level
- c
\alpha
, the absolute ruin happens and the dividend process stops.
Let \~Dt(\eta ) denote the aggregate dividends paid by time t \geq 0. The modified surplus process
\~U\eta (t) is given by
\~U\eta (t) = Ut - \~Dt(\eta ).
Let \delta > 0 be the force of interest for valuation, and let DT2 denote the present value of all dividends
until the absolute ruin,
DT2 =
T2\int
0
e - \delta td \~Dt(\eta ),
where
T2 = \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
t > 0, \~Ut(\eta ) \leq - c
\alpha
\Bigr\}
.
We denote \~V (u, \eta ) as the expected dividend value function of the strategy, that is
\~V (u, \eta ) = Eu[DT2 ].
Let \~M(u, y, \eta ) = Eu[e
yDT2 ] be the moment generating function of DT2 . For any y < \infty , we have
0 < \~M(u, y, \eta ) \leq \mathrm{l}\mathrm{i}\mathrm{m}u\uparrow \infty \~M(u, y, \eta ) = ey
\beta
\delta < +\infty , then \~M(u, y, \eta ) exists.
We define
\~M(u, y, \eta ) =
\left\{
\~M1(u, y, \eta ), if u \geq 0,
\~M2(u, y, \eta ), if -
c
\alpha
< u < 0.
Theorem 4.1. Suppose that \~M(u, y, \eta ) is twice continuously differentiable u \in
\Bigl(
- c
\alpha
, \eta
\Bigr)
, then
\~M1(u, y, \eta ) and \~M2(u, y, \eta ) satisfy the following equations:
\lambda \~M2(u, y, \eta ) = (\alpha u+ c)
\partial \~M2(u, y, \eta )
\partial u
- \delta y
\partial \~M2(u, y, \eta )
\partial y
+
1
2
\sigma 2
1
\partial 2 \~M2(u, y, \eta )
\partial u2
+
+\lambda
u+ c
\alpha \int
0
\~M2(u - x, y, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx, - c
\alpha
< u < 0, (4.1)
\lambda \~M1(u, y, \eta ) = (au+ c)
\partial \~M1(u, y, \eta )
\partial u
- \delta y
\partial \~M1(u, y, \eta )
\partial y
+
+
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1)
\partial 2 \~M1(u, y, \eta )
\partial u2
+ \lambda
u\int
0
\~M1(u - x, y, \eta )p(x) dx+
+\lambda
u+ c
\alpha \int
u
\~M2(u - x, y, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx, 0 \leq u < \eta , (4.2)
(\lambda - y\beta ) \~M1(u, y, \eta ) =
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1)
\partial 2 \~M1(u, y, \eta )
\partial u2
+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
642 Y. H. LU, Y. F. LI
+
\bigl[
(c - \beta ) + au - \delta y
\bigr] \partial \~M1(u, y, \eta )
\partial y
+ \lambda
u\int
0
\~M1(u - x, y, \eta )p(x) dx+
+\lambda
u+ c
\alpha \int
u
\~M2(u - x, y, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx, u \geq \eta , (4.3)
with boundary conditions
\~M2
\Bigl(
- c
\alpha
, y, \eta
\Bigr)
= 1, \mathrm{l}\mathrm{i}\mathrm{m}
u\uparrow \infty
\~M1(u, y, \eta ) = ey
\beta
\delta , \~M1(0+, y, \eta ) = \~M2(0 - , y, \eta ).
Proof. The proof of (4.1), (4.2) are similar to (3.1), (3.2), and are omitted. Next, we prove (4.3).
By Markovian property, we get
\~M(u, y, \eta ) = Eu[ \~M(Ut, ye
- \delta t, \eta )] + o(t).
Hence,
\~M1(u, y, \eta ) = (1 - \lambda t)e - \lambda tey\beta tEu[ \~M1(u, ye
- \delta t, \eta )]+
+\lambda te - \lambda tey\beta tEu
Zt\int
0
\~M1(Zt - x, ye - \delta t, \eta )p(x) dx+
+\lambda te - \lambda tey\beta tEu
Zt+
c
\alpha \int
Zt
\~M2(Zt - x, ye - \delta t, \eta )p(x) dx+
+\lambda te - \lambda tey\beta tEu
\infty \int
Zt+
c
\alpha
p(x) dx+ o(t),
where
Zt = u+
t\int
0
(c - \beta + aUs)ds+
t\int
0
\sqrt{}
(\sigma 1 + \rho bUs - )2 + b2(1 - \rho 2)U2
s - dBs.
By using the same arguments as in the proof of Theorem 3.1, we get (4.3).
Noting that if u = - c
\alpha
, then T2 = 0 and DT2 = 0. So we have
\~M2
\Bigl(
- c
\alpha
, y, \eta
\Bigr)
= E - c
\alpha
\bigl(
eyDT2
\bigr)
= 1.
\~M1(0+, y, \eta ) = \~M2(0 - , y, \eta ) due to \~M(u, y, \eta ) is continuous at u = 0.
By the definition of \~M1(u, y, \eta ), we get \mathrm{l}\mathrm{i}\mathrm{m}u\uparrow \infty \~M1(u, y, \eta ) = ey
\beta
\delta . The boundary conditions
for \~Mi, i = 1, 2, are obtained.
Theorem 4.1 is proved.
Set \~M(u, y, \eta ) = 1 + y \~V (u, \eta ) + o(y), where
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
DIVIDEND PAYMENTS IN A PERTURBED COMPOUND POISSON MODEL . . . 643
\~V (u, \eta ) = Eu[DT2 ] =
\left\{
\~V1(u, \eta ), u \geq 0,
\~V2(u, \eta ), - c
\alpha
< u < 0.
(4.4)
Substituting (4.4) into (4.1), (4.2) and (4.3), comparing the coefficients of y, we get the following
results.
Theorem 4.2. If - c
\alpha
< u < 0, then \~V2(u, \eta ) in (4.4) satisfies the equation
(\delta + \lambda ) \~V2(u, \eta ) = (\alpha u+ c) \~V \prime
2(u, \eta ) +
1
2
\sigma 2
1
\~V \prime \prime
2 (u, \eta )+
+\lambda
u+ c
\alpha \int
0
\~V2(u - x, y, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx. (4.5)
If 0 \leq u < \eta , then \~Vi(u, \eta ), i = 1, 2, in (4.4) satisfy the equation
(\delta + \lambda ) \~V1(u, \eta ) = (au+ c) \~V \prime
1(u, \eta ) +
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1) \~V
\prime \prime
1 (u, \eta )+
+\lambda
u\int
0
\~V1(u - x, \eta )p(x) dx+ \lambda
u+ c
\alpha \int
u
\~V2(u - x, \eta )p(x) dx+ \lambda
\infty \int
u+ c
\alpha
p(x) dx. (4.6)
If u \geq \eta , then \~Vi(u, \eta ), i = 1, 2, in (4.4) satisfy the equation
(\delta + \lambda ) \~V1(u, \eta ) =
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1) \~V
\prime \prime
1 (u, \eta ) + (c - \beta + au) \~V \prime
1(u, \eta )+
+\lambda
u\int
0
\~V1(u - x, \eta )p(x) dx+ \lambda
u+ c
\alpha \int
u
\~V2(u - x, \eta )p(x) dx+ \beta (4.7)
with boundary conditions
\~V2
\Bigl(
- c
\alpha
+, \eta
\Bigr)
= 0, \mathrm{l}\mathrm{i}\mathrm{m}
u\uparrow \infty
\~V1(u, \eta ) =
\beta
\delta
,
\~V1(0+, \eta ) = \~V2(0 - , \eta ), \~V \prime
1(0+, \eta ) = \~V \prime
2(0 - , \eta ),
\~V \prime
1(\eta +, \eta ) = \~V \prime
1(\eta - , \eta ), \~V1(\eta +, \eta ) = \~V1(\eta - , \eta ).
Remark 4.1. Letting \lambda = 0 in (4.5), (4.6) and (4.7), then they can be changed to
\delta \~V2(u, \eta ) = (\alpha u+ c) \~V \prime
2(u, \eta ) +
1
2
\sigma 2
1
\~V \prime \prime
2 (u, \eta ), - c
\alpha
< u < 0, (4.8)
\delta \~V1(u, \eta ) = (au+ c) \~V \prime
1(u, \eta ) +
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1)
\~V \prime \prime
1 (u, \eta ), 0 \leq u < \eta , (4.9)
\delta \~V (u, \eta ) =
1
2
(b2u2 + 2\rho b\sigma 1u+ \sigma 2
1)
\~V \prime \prime
1 (u, \eta ) + (c - \beta + au) \~V \prime (u, \eta ) + \beta , u \geq \eta . (4.10)
In the case of u \geq 0, (4.9) and (4.10) are obtained by Yin and Wen [22], where they got the same
results by using different methods. We extend their results for the more general case.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
644 Y. H. LU, Y. F. LI
References
1. Asmussen S. Ruin probabilities. – Singapore: World Sci., 2000.
2. Asmussen S., Taksar M. Controlled diffusion models for optimal dividend pay-out // Insurance: Math. and Econom. –
1997. – 20. – P. 1 – 15.
3. Bühlmann H. Mathematical methods in risk theory. – Heidelberg: Springer, 1970.
4. Cai J. On the time value of absolute ruin with debit interest // Adv. Appl. Probab. – 2007. – 39. – P. 343 – 359.
5. Chiu S. N., Yin C. C. The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process
perturbed by diffusion // Insurance: Math. and Econom. – 2003. – 33, № 1. – P. 59 – 66.
6. De Finetti B. Su un’ impostazione alternativa dell teoria collettiva del rischio // Trans. XVth Intern. Congr. Actuar. –
1957. – 2. – P. 433 – 443.
7. Dufresne F., Gerber H. U. Risk theory for the compound Poisson process that perturbed by diffusion // Insurance:
Math. and Econom. – 1991. – 10. – P. 51 – 59.
8. Gerber H. U. An extension of the renewal equation and its application in the collective theory of risk // Scand.
Actuar. J. – 1970. – P. 205 – 210.
9. Gerber H. U. Der Einfluss von Zins auf die Ruinwahrscheinlichkeit // Mitt. Ver. schweiz. Versicherungsmath. – 1971. –
S. 63 – 70.
10. Gerber H. U., Shiu S. W. On optimal dividend strategies in the compound Poisson model // North-Amer. Actuar. J. –
2006. – 10. – P. 76 – 93.
11. Gerber H. U., Yang H. L. Absolute ruin probabilities in a jump-diffusion risk model with investment // North-Amer.
Actuar. J. – 2008. – 11, № 3. – P. 159 – 169.
12. Jeanblanc-Picqué M., Shiryaev A. N. Optimization of the flow of dividends // Russian Math. Surveys. – 1995. – 20. –
P. 257 – 277.
13. Li J. Z. Asymptotics in a time-dependent renewal risk model with stochastic return // J. Math. Anal. and Appl. –
2012. – 387. – P. 1009 – 1023.
14. Lu Y. H., Wu R. The differentiability of dividends function on jump-diffusion risk process with a barrier dividend
strategy // Front. Math. China. – 2014. – 9, № 5. – P. 1073 – 1088.
15. Lu Y. H., Wu R., Xu R. The joint distributions of some actuarial diagnostics for the jump-diffusion risk process //
Acta Math. Sci. Ser. B. – 2010. – 30, № 3. – P. 664 – 676.
16. Paulsen J. Risk theory in a stochastic economic environment // Stochast. Process. and Appl. – 1993. – 46. –
P. 327 – 361.
17. Paulsen J. Sharp conditions for certain ruin in a risk process with stochastic return on investments // Stochast. Process.
and Appl. – 1998. – 75. – P. 135 – 148.
18. Paulsen J. Ruin models with investment income // Probab. Surv. – 2008. – 5. – P. 416 – 434.
19. Paulsen J., Gjessing H. K. Optimal choice of dividend barriers for a risk process with stochastic return on investments //
Insurance: Math. and Econom. – 1997. – 20. – P. 215 – 223.
20. Wan N. Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion //
Insurance: Math. and Econom. – 2007. – 40. – P. 509 – 523.
21. Yin C. C., Yuen K. C. Optimal dividend problems for a jump-diffusion model with capital injections and proportional
transaction costs // J. Ind. and Manag. Optim. – 2015. – 11, № 4. – P. 1247 – 1262.
22. Yin C. C., Wen Y. Z. An extension of Paulsen – Gjessing’s risk model with stochastic return on investments // Insurance:
Math. and Econom. – 2013. – 52. – P. 496 – 476.
23. Yuen K. C., Lu Y. H., Wu R. The compound Poisson process perturbed by a diffusion with a threshold dividend
strategy // Appl. Stochast. Models Bus. and Ind. – 2009. – 25, № 1. – P. 73 – 93.
24. Zhu J., Yang H. L. Estimates for the absolute ruin probability in the compound Poisson risk model with credit and
debit interest // J. Appl. Probab. – 2008. – 45, № 3. – P. 818 – 830.
Received 06.11.15,
after revision — 25.02.19
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
|
| id | umjimathkievua-article-1462 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:10Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f2/b78189f93f541557b4ba55462302f5f2.pdf |
| spelling | umjimathkievua-article-14622019-12-05T08:56:08Z Dividend payments in a perturbed compound Poisson model with stochastic investment and debit interest Дивiденднi виплати у збуренiй складнiй пуассонiвськiй моделi зi стохастичними iнвестицiями та дебетовими вiдсотками Li, Y. F. Lu, Y. H. Лі, Я. Ф. Лю, Я. Х. UDC 519.21 We consider a compound Poisson insurance risk model perturbed by diffusion with stochastic return on investment and debit interest. If the initial surplus is nonnegative, then the insurance company can invest its surplus in a risky asset and risk-free asset based on a fixed proportion. Otherwise, the insurance company can get the business loan when the surplus is negative. The integrodifferential equations for the moment generating function of the cumulative dividends value are obtained under the barrier and threshold dividend strategies, respectively. The closed-form of the expected dividend value is obtained when the claim amount is exponentially distributed. УДК 519.21 Розглядається складна пуассонівська модель страхових ризиків, збурена дифузією, зі стохастичним доходом по інвестиціях та дебетовим відсотком. Якщо початковий надлишок є невід'ємним, то стрaхова компанія може вкладати цей надлишок у ризикові або безризикові активи в фіксованій пропорції. В протилежному випадку, коли надлишок є від'ємним, страхова компанія може отримувати бізнесові кредити. Інтегро-диференціальні рівняння для функції, що породжує моменти значень кумулятивних дивідендів, отримано для бар'єрних та порогових дивідендних стратегій відповідно. Очікувану величину дивідендів отримано в замкненій формі у випадку експоненціального розподілу суми позову. Institute of Mathematics, NAS of Ukraine 2019-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1462 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 5 (2019); 631-644 Український математичний журнал; Том 71 № 5 (2019); 631-644 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1462/446 Copyright (c) 2019 Li Y. F.; Lu Y. H. |
| spellingShingle | Li, Y. F. Lu, Y. H. Лі, Я. Ф. Лю, Я. Х. Dividend payments in a perturbed compound Poisson model with stochastic investment and debit interest |
| title | Dividend payments in a perturbed compound Poisson model with stochastic
investment and debit interest |
| title_alt | Дивiденднi виплати у збуренiй складнiй пуассонiвськiй моделi зi стохастичними iнвестицiями та дебетовими вiдсотками |
| title_full | Dividend payments in a perturbed compound Poisson model with stochastic
investment and debit interest |
| title_fullStr | Dividend payments in a perturbed compound Poisson model with stochastic
investment and debit interest |
| title_full_unstemmed | Dividend payments in a perturbed compound Poisson model with stochastic
investment and debit interest |
| title_short | Dividend payments in a perturbed compound Poisson model with stochastic
investment and debit interest |
| title_sort | dividend payments in a perturbed compound poisson model with stochastic
investment and debit interest |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1462 |
| work_keys_str_mv | AT liyf dividendpaymentsinaperturbedcompoundpoissonmodelwithstochasticinvestmentanddebitinterest AT luyh dividendpaymentsinaperturbedcompoundpoissonmodelwithstochasticinvestmentanddebitinterest AT líâf dividendpaymentsinaperturbedcompoundpoissonmodelwithstochasticinvestmentanddebitinterest AT lûâh dividendpaymentsinaperturbedcompoundpoissonmodelwithstochasticinvestmentanddebitinterest AT liyf dividendniviplatiuzburenijskladnijpuassonivsʹkijmodelizistohastičnimiinvesticiâmitadebetovimividsotkami AT luyh dividendniviplatiuzburenijskladnijpuassonivsʹkijmodelizistohastičnimiinvesticiâmitadebetovimividsotkami AT líâf dividendniviplatiuzburenijskladnijpuassonivsʹkijmodelizistohastičnimiinvesticiâmitadebetovimividsotkami AT lûâh dividendniviplatiuzburenijskladnijpuassonivsʹkijmodelizistohastičnimiinvesticiâmitadebetovimividsotkami |