=Paper=
{{Paper
|id=Vol-1726/paper-01
|storemode=property
|title=Survival Probability in the Life Annuity Insurance Model with Stochastic Return on Investments
|pdfUrl=https://ceur-ws.org/Vol-1726/paper-01.pdf
|volume=Vol-1726
|authors=Tatiana A. Belkina,Nadezhda B. Konyukhova,Bohdan V. Slavko
}}
==Survival Probability in the Life Annuity Insurance Model with Stochastic Return on Investments
==
https://ceur-ws.org/Vol-1726/paper-01.pdf
Survival Probability in the Life Annuity
Insurance Model with Stochastic Return on
Investments
Tatiana A. Belkina1 , Nadezhda B. Konyukhova2 , and Bohdan V. Slavko3
1
Central Economics and Mathematics Institute of RAS, Moscow, Russia,
2
Dorodnicyn Computing Center of RAS FRC CSC of RAS, Moscow, Russia
3
National Research University – Higher School of Economics, Moscow, Russia
slavkobogdan@gmail.com
Abstract. We investigate the survival probability in the life annuity,
or pension, insurance model when whole of the surplus (or a fixed its
part) is invested into a risky asset with the price following the geometric
Brownian motion. For the case of exponential distribution of revenue
sizes, we formulate a singular boundary value problem for linear integro-
differential equation and prove that the survival probability as a function
of the initial surplus is the unique solution of this problem. Moreover,
asymptotic representations for the survival probability both for small
and large values of initial surplus are obtained. The efficient algorithm
for the numerical calculation of the survival probability is described.
Using computational experiments, we show that in the pension insurance
business risky investments play a very important role in strengthening
of the insurers solvency in a zone of small sizes of the surplus.
Keywords: life annuity insurance, dual risk model, survival probability,
investment, risky asset, geometric Brownian motion, exponential distri-
bution of revenue sizes
1 Introduction and Statement of the Problem
We consider the life annuity insurance model [9] (so called “dual risk model”,
see, e. g., [2]), where the surplus or equity of a company (in the absence of
investments) is of the form
N (t)
X
Rt = u − ct + Zk , t ≥ 0. (1)
k=1
Here Rt is the surplus of a company at time t ≥ 0; u is the initial surplus, c is
the life annuity rate (or the pension payments per unit of time), assumed to be
deterministic and fixed. N (t) is a homogeneous Poisson process with intensity
λ > 0 that, for any t > 0, determines the number of random revenues up to
the time t; Zk (k = 1, 2, . . . ) are independent identically distributed random
variables with a distribution function F (z) (F (0) = 0, EZ1 = m < ∞) that
determine the revenue sizes and are assumed to be independent of N (t). These
revenues arise at the moments of the death of policyholders.
In comparison with the classical non-life collective risk model (so called
Cramér-Lundberg (CL) model, see [9]), the circumstances in (1) are reversed:
in the classical model second and third summands have opposite signs. Since
the “claims” in the dual model are negative (the jumps of the process (1) are
positive), this model is also called the insurance model with negative risk sums
or compound Poisson model with negative claims [3].
Let now the whole surplus be continuously invested into risky asset with
price St following the geometric Brownian motion
dSt = µSt dt + σSt dBt , t ≥ 0, (2)
where µ is the expected return rate, σ is the volatility, Bt is a standard Brownian
motion.
Then the resulting surplus process Xt is governed by the equation
dXt = µXt dt + σXt dBt + dRt , t ≥ 0, (3)
with the initial condition X0 = u, where Rt is defined by (1).
Remark 1. For the case when only fixed part of the surplus is invested into
risky asset while the rest part is invested into a risk free asset with fixed return
rate, the equation (3) is valued with modified parameters for the corresponding
surplus process (see, e. g., [4]).
Denote ϕ(u) = P (Xt ≥ 0, t ≥ 0) the survival probability (i. e., the probabil-
ity that bankruptcy will never happen); Ψ (u) = 1 − ϕ(u) is the ruin probability.
Let ϕ0 (u) = P (Rt ≥ 0, t ≥ 0) be the survival probability for the process (1)
and Ψ0 (u) = 1 − ϕ0 (u) be the corresponding ruin probability (in the absence of
investments).
For the case of exponential distribution of revenue sizes, the following theo-
rem is proved in [10].
Theorem 1. Let F (z) = 1 − e−z/m , m > 0, β := 2µ/σ 2 − 1. Then:
1) if β > 0 then Ψ (u) = Ku−β (1 + o(1)), u → ∞, for some K > 0;
2) if β ≤ 0 then Ψ (u) = 1 for any u.
The formulation of this theorem is exactly the same as in [8], [14] for the
non-life insurance model.
For the ruin probability Ψ0 (u), it is easy to obtain an integro-differential equa-
tion (IDE) using the obvious modifications of the “differential argument” (see,
e. g., [9]). In the case of exponential distribution of revenue sizes and if the safety
loading is positive, i. e., the inequality λm > c is valid, this IDE has an exact so-
lution satisfying boundary conditions limu→+0 Ψ0 (u) = 1, limu→∞ Ψ0 (u) = 0;
this solution has the form
Ψ0 (u) = exp {−(λ/c − 1/m)u} . (4)
Such slow decay as in Theorem 1 is in contrast with the exponential rep-
resentation (4) in corresponding model without investment. This fact leads to
the following conclusion, which has been made earlier for various models with
non-heavy-tailed distributions of claims: investment of the whole surplus into
risky assets in a zone of large values of the surplus impair the insurers solvency.
The same conclusion remains true when only fixed part of the surplus is invested
into risky assets (so called simple investment strategies), see [4] and references
therein.
At the same time, the studies of optimal investment strategies, which maxi-
mize the survival probabilities in various settings of a problem for the CL model,
show that the risky assets play a crucial role in strengthening of the insurer’s
solvency in a zone of small sizes of the surplus (see, e. g., [5] and references
therein). The same conclusions concern the simple investment strategies for CL
model and some its modifications, see [6].
The main goal of our paper is to identify the impact of simple investment
strategies on the solvency in the dual risk model not only in the case of large
surplus levels but also in the case of its small levels. For this purpose, we use the
approach based on so called sufficiency theorems [4], which state that the solu-
tions of singular problems for linear IDEs, generated by infinitesimal operators
of the resulting surplus processes, define the corresponding survival probabili-
ties. This approach is rather different from the one used in [10] and eliminates
the need a priory to prove the twice continuously differentiability of the survival
probability as well as the justification of boundary conditions. Solving the sin-
gular problem for IDE, we calculate the survival probability as a function of the
initial surplus on the whole nonnegative semi-axis by means of the proposed algo-
rithm; some results of numerical experiments are described. These results allow
us to make conclusions about impact of the simple risky investment strategies
on the solvency in the life annuity insurance model.
2 Some Preliminary Results
Recall at first that the infinitesimal generator A (see, e. g., [11]) of the process
Xt defined by (3) has the form
Z ∞
1
(Af )(u) = σ 2 u2 f 00 (u) + f 0 (u)(µu − c) − λf (u) + λ f (u + z) dF (z), (5)
2 0
for any function f from a certain subclass D of the space C 2 (IR+ ) of real-valued,
twice continuously differentiable on (0, ∞) functions.
In the assumption ϕ(u) ∈ D, where ϕ(u) is the survival probability of the
process Xt , some considerations based on the generalized Ito’s formula and the
complete probability formula allow us to write the following equation for u > 0:
(Aϕ)(u) = 0 (6)
(see [10]; for the corresponding equation in the CL model with stochastic return
on investment, see, e. g., [13]).
For the case of exponential distribution of revenue sizes, namely when
F (z) = 1 − exp (−z/m), m > 0, (7)
the equation (6) has the form
1 2 2 00
σ u ϕ (u) + (µu − c)ϕ0 (u) − λϕ(u) +
2
Z∞
λ
+ ϕ(z) exp(−(z − u)/m)dz = 0, u > 0. (8)
m
u
Here we use the following transformation of non-Volterra integral operator in (5)
with F of the form (7):
Z∞ Z∞
1 1
(Jm ϕ)(u) := ϕ(u+z) exp(−z/m)dz = ϕ(z) exp(−(z − u)/m)dz. (9)
m m
0 u
As noted in [10], the life annuity insurance case is rather different from the non-
life insurance one because the change of two signs to the opposite ones in the
equation defining the dynamics of the reserve leads to technical complications. In
contrast to the non-life insurance case, the risk process (3) may leave the positive
half-axis only in a continuous way. In [10] it was emphasized that the main
difficulty in deriving the IDE is to prove the smoothness of the ruin probability.
The other difficulty is to establish lower and upper asymptotic bounds for ruin
probability in order to identify boundary conditions at infinity for the survival
probability as the solution of the IDE. The smoothness of the ruin probability
is studied in [10] using a method based on integral representations; as a tool for
the proof of the asymptotic bounds for the ruin probability some theorems of
the renewal theory are used.
As mentioned above in the introduction, in this paper we apply the approach
based on sufficiency theorems (developed in [4] for the CL model and its mod-
ification with stochastic premiums; see also the earlier paper [13]) which allows
us to avoid the a-priori proof of the the smoothness of the ruin probability as
well as the justification of the boundary conditions at infinity.
For this purpose, we need a few preliminary propositions.
Lemma 1. For the survival probability ϕ(u) of the process (3) with the initial
condition X0 = u, the following relation is valid:
ϕ(0) = 0, (10)
i. e., the ruin occurs immediately at zero initial surplus.
The proof of this lemma is obvious due to the negativity of the deterministic
component and we omit it. We provide below several statements concerning the
properties of the solutions to IDE (8) satisfying the various conditions.
Lemma 2. Let in IDE (8) the parameters c, λ, σ, m be fixed positive numbers
and µ is arbitrary fixed number. Then if there exists a solution ϕ(u) to IDE (8)
with conditions
lim ϕ(u) = 0, lim ϕ(u) = 1, (11)
u→+0 u→∞
then this solution is unique.
This lemma can be easily proved by contradiction using the linearity of IDE.
To formulate some further auxiliary propositions, we will use also the follow-
ing limiting conditions:
lim |ϕ0 (u)| < ∞, lim uϕ00 (u) = 0. (12)
u→+0 u→+0
lim uϕ0 (u) = 0, lim u2 ϕ00 (u) = 0. (13)
u→∞ u→∞
Lemma 3. Let in IDE (8) the parameters c, λ, σ, m be fixed positive numbers
and µ is arbitrary fixed number. If there exists the solution ϕ(u) of IDE (8),
satisfying conditions (11), (12), then
0 ≤ ϕ(u) ≤ 1, u ∈ IR+ ; (14)
moreover,
0 < lim ϕ0 (u) < ∞. (15)
u→+0
The first part of this statement may be proved by contradiction (see the proof of
a similar assertion in [6]). Let us prove the second part of the statement. Indeed,
from the IDE (8) and conditions (11) and (12) we have the relation
λ ∞
Z
0
−cϕ (+0) + ϕ(z) exp (−z/m)dz = 0, (16)
m 0
wherefrom, taking into account the proved above relation (14), the conditions (11)
and positiveness of c and other parameters, we conclude that (15) is valid.
The following lemma is essential auxiliary statement for further study of
the initial problem.
Lemma 4. Let in IDE (8) all the parameters c, λ, µ σ, m be fixed positive num-
bers. Then the singular IDE problem (8), (11)–(13) is equivalent to the singular
problem for ODE
1 2 2 000 1 2 2
2
σ u ϕ (u) + µu + σ u − c − σ u ϕ00 (u) +
2 2m
µu − c
+ µ−λ− ϕ0 (u) = 0, (17)
m
defined on IR+ , with conditions (11)–(13).
Proof. Let ϕ(u) be satisfying IDE (8). Let us show that it satisfies also ODE (17).
It is easy to check that, for the operator (9), the following relation is valid:
d 1 1
(Jm ϕ)(u) = (Jm ϕ)(u) − ϕ(u). (18)
du m m
By differentiating IDE (8) and taking into account the relation (18) we obtain
1 2 2 000
σ u ϕ (u) + σ 2 uϕ00 (u) + (µu − c)ϕ00 (u) +
2
λ λ
+ (µ − λ)ϕ0 (u) + (Jm ϕ)(u) − ϕ(u) = 0. (19)
m m
The obvious linear combination of IDEs (8) and (19), which exclude the integral
term (Jm ϕ)(u), leads to ODE (17).
Conversely, let now ϕ̂(u) be satisfying ODE (17) and conditions (11) and (13).
Let us show that it satisfies also the IDE (8). Denote g(u) the left-hand side of
the equation (8) with function ϕ̂(u). Then
1
g(u) = −λϕ̂(u) + λ(Jm ϕ̂)(u) + (µu − c)ϕ̂0 (u) + σ 2 u2 ϕ̂00 (u),
2
λ
g 0 (u) = [(Jm ϕ̂)(u) − ϕ̂(u)] + (µ − λ)ϕ̂0 (u) +
m
1
+ (µu + σ 2 u − c)ϕ̂00 (u) + σ 2 u2 ϕ̂000 (u).
2
Hence,
0 g(u) µu − c
g (u) − = µ−λ− ϕ̂0 (u) +
m m
1 2 2 1
2
+ µu + σ u − c − σ u ϕ̂00 (u) + σ 2 u2 ϕ̂000 (u),
2m 2
and, in view of the fact that the function ϕ̂(u) is a solution of ODE (17), we
obtain
g(u)
g 0 (u) − = 0. (20)
m
The solution of ODE (20) has the form
g(u) = C exp (u/m), u > 0, (21)
where C is an arbitrary constant. It is easy to see that for the function ϕ̂(u),
which satisfies conditions (11) and (13), the following relation is valid:
1 ∞
Z
lim ϕ̂(s) exp (−(s − u)/m)ds = 1. (22)
u→+∞ m u
Then, taking into account the definition of g(u), equality (22) and conditions (11),
(13) we conclude that the equality limu→∞ g(u) = 0 holds. Consequently, in view
of positiveness m, the constant C in (21) should be equal to zero for this solution,
i. e., g(u) ≡ 0. Thus, ϕ̂(u) is the solution of IDE (8).
It remains to note that the whole set of conditions is the same for the two
considered problems for IDE and ODE, and lemma is proved.
To establish a connection between the original problem of the survival prob-
ability investigation and a singular problem for IDE, we need also the following
statement which we call the sufficiency theorem [4].
Theorem 2. Let in IDE (8) all the parameters be positive numbers and the
inequality
2µ > σ 2 (23)
be fulfilled. Suppose IDE (8) has a twice continuously differentiable on (0, ∞)
solution ϕ(u) subject to conditions
0 ≤ ϕ(u) ≤ 1, u ∈ IR+ , (24)
lim ϕ(u) = 1. (25)
u→+∞
Then ϕ(u) is the survival probability for the process (3) with initial state X0 = u.
The proof of this theorem is completely analogous to the proof of Theorem 3.1
in [4].
3 Main Theorem
For the considered case of the exponential distribution of revenue sizes, we es-
tablish the following statement.
Theorem 3. Let F (z) be of the form (7), all the parameters µ, σ 2 , m, c, λ be
fixed positive constants, and let the condition (23) be satisfied. Then the following
assertions hold:
(I) the survival probability ϕ(u) of the process (3) with initial condition
X0 = u is the solution to the singular boundary value IDE problem (8), (11);
(II) this solution is unique and satisfies the following relations:
0 ≤ ϕ(u) ≤ 1, u ∈ IR+ , (26)
0 < lim ϕ0 (u) < ∞; (27)
u→+0
(III) the survival probability ϕ(u) may be calculated by the formula
Z∞
ϕ(u) = 1 − ψ(s)ds, (28)
u
where ψ(u) = ϕ0 (u) is the solution of the following singular problem for ODE:
1 2 2 00 1 2 2
σ u ψ (u) + µu + σ 2 u − c − σ u ψ 0 (u) +
2 2m
µu − c
+ µ−λ− ψ(u) = 0, (29)
m
0 < u < ∞,
lim |ψ(u)| < ∞, lim uψ 0 (u) = 0, (30)
u→+0 u→+0
lim uψ(u) = 0, lim u2 ψ 0 (u) = 0; (31)
u→∞ u→∞
Z ∞
ψ(s)ds = 1; (32)
0
(IV) ϕ(u) has the asymptotic representations
∞
!
X
k
ϕ(u) ∼ D1 u + Dk u /k , u ∼ +0, (33)
k=2
where
D1 = ϕ0 (+0),
D2 = (µ − λ + c/m) /c, (34)
D3 = D2 (2µ + σ 2 − λ + c/m) − µ/m /(2c),
(35)
2
Dk+1 = [Dk (k(k − 1)σ /2 + µk − λ + c/m) −
− Dk−1 ((k − 2)σ 2 /(2m) + µ/m)]/(kc), k = 3, 4, . . . , (36)
and 2
ϕ(u) = 1 − Ku1−2µ/σ (1 + o(1)), u → ∞, (37)
where K > 0 is a constant;
(V) as u → +0, the behavior of the survival probability derivatives depends
on the relations between the parameters, in particular on a sign of the coefficient
ir = (λ − µ)m − c: (1) if ir ≥ 0, then limu→+0 ϕ00 (u) ≤ 0, moreover, the solution
ϕ is concave on IR+ ; (2) if ir < 0, then limu→+0 ϕ00 (u) > 0, the solution ϕ is
convex in a some neighborhood of zero and has an inflexion point.
Sketch of the proof. At first, we need to establish the existence and uniqueness
of the solution to the problem (29)–(32) and to study its asymptotic behaviors
for large and small values of u. For this purpose, we have to investigate the
singular problems (29), (30) and (29), (31) separately, taking into account that
ODE (28) has irregular singular points at zero and infinity (about singular points
for ODEs see, e. g. [15]).
By using methods of the investigation of ODEs with singular points [15],
[12] we obtain asymptotic representation for families of solutions to this sin-
gular problems (see also [6] and references therein for analogous investigation
with application of these methods for CL model with investment in details). As
result we have that ODE (29) for small u > 0 has a two-parameter family of
solutions ψ(u, D1 , C1 ) and for these solutions the following asymptotic represen-
tation holds:
ψ(u, D1 , C1 ) =
2
= D1 (1 + ψ1 (u)) + C1 exp −2c/(σ 2 u) u−2µ/σ (1 + ψ2 (u)),
u → 0. (38)
Here C1 is a parameter, ψ2 (u) = o(1), u → +0, the function ψ1 (u) can be
represented by asymptotic series
∞
X
ψ1 (u) ∼ Dk+1 uk , u ∼ +0, (39)
k=1
where the coefficients Dk , k = 2, 3, . . . , may be found from the recurrence rela-
tions (34)–(36).
It is obvious that the conditions (30) hold for all the solutions of the family
ψ(u, D1 , C1 ), i. e., for all the solutions to ODE (29).
Under condition (23), ODE (29) has a one-parameter family of solutions
ψ(u, C2 ) which are integrable at infinity. For these solutions, the following asymp-
totic representation holds as u → ∞:
2 2µ 2
ψ(u, C2 ) = C2 u−2µ/σ (1 + o(1)), ψ 0 (u, C2 ) = − C2 u−2µ/σ (1 + o(1)). (40)
σ2 u
It is obvious that the conditions (31) hold for all solutions of this family (i. e.,
for all integrable at infinity solutions of (29)) iff the condition (23) is fulfilled.
Thus, if the condition (23) is fulfilled, then there exists one-parameter fam-
ily of solutions to the problem (29)-(31). All the solutions of the equation (29)
are bounded at zero, and all bounded and integrable at infinity solutions be-
long to this family and have the asymptotic representations (38) and (40). The
condition (32) extracts the unique solution from this family. It is clear that
this solution satisfies the conditions (30) and (31). Then, taking into account
Lemma 4, it is easy to see that the function ϕ(u) defined by the formula (28)
is the solution to the IDE problem (8), (11)–(13) and has the asymptotic rep-
resentations (33) and (37). Therefore, according to Lemma 3, the relations (14)
also hold and, in view of Theorem 2, for any u ∈ IR+ , the value of ϕ(u) is the
survival probability for the process (3) with initial state X0 = u. In accordance
to Lemma 2 (about the uniqueness) this probability as a function of u satisfies
conditions (12) and (13) with necessity. In view of Lemma 3, we have also that
the inequalities (15) take place, and, in accordance to the asymptotic represen-
tation (33), we conclude that ϕ00 (+0) and the expression µ − λ + c/m are of the
same sign. The sketch of the proof is completed.
4 Numerical Examples
The studies given in previous sections allow us to suggest computationally simple
and theoretically justified algorithm for numerical calculation of the survival
probability in the considered model. This algorithm requires to solve the singular
Cauchy problem from infinity (29), (31) with the normalizing condition (32).
Then it remains to use the relation (28) (recall that all the solutions of ODE (29)
are bounded as u → +0 and all the integrable at infinity solutions form the
one-parameter family). To solve numerically the problem (29), (31) we realize
previously the equivalent transfer of the limit conditions (31) from infinity to
a large finite point using the results [7], [12]. For the first approximation, such
approach yields boundary condition at a finite point u = u∞ 1 as follows
2µ
ψ 0 (u∞ ) ≈ − ψ(u∞ )
σ 2 u∞
(the same relation follows also from (40)).
For general ODE systems with pole-type singular points, the theory of bound-
ary condition transfer from singular points is developed; such transfer can be re-
alized by construction of the stable initial manifolds, or the Lyapunov manifolds
of conventional stability, at the neighborhoods of singular points (see, e. g., [1]
and references therein). On the application of such approach in actuarial math-
ematics, see [6] and references therein.
Numerical experiments (see in particular Figs. 1, 2) show that risky invest-
ments improves survival probability at a zone of small values of initial surplus
in the case of positive safety loading (Fig. 2). Moreover, risky investments make
survival possible in the case of negative safety loading, see Fig. 1 (in this case
survival is impossible without investments).
Fig. 1. The graphs of the survival probability as a function of initial surplus in the
case of negative safety loading for two different scales (λ = 1, m = 2, c = 4, µ = 0.2,
σ 2 = 0.23).
5 Conclusions
To study the impact of investments with stochastic return on survival probabil-
ity in the life annuity insurance model we use the approach based on so called
Fig. 2. Survival probability as a function of initial surplus in the case of positive safety
loading (λ = 1, m = 2, c = 1.8) both with risky investments (solid curve, µ = 0.2,
σ 2 = 0.22) and without investments (dotted curve).
sufficiency theorem and the existence theorems for the corresponding singular
problems for IDEs (see [4]). This unified approach eliminates need to proof reg-
ularity of the survival probability as well as to use its upper and lower bounds.
Moreover, the solving of above singular problem for IDE leads to calculation of
the survival probability on all the non-negative semi-axis. We reduce the prob-
lem (8), (11) to a certain initial problem from infinity for some second order
ODE with respect to the derivative of the survival probability with normalizing
condition. As a result of calculations, we conclude in particular that if the value
of safety loading (λm − c) in the model (1) is negative or sufficiently small and
the surplus is small too, then the use of the risky investments allows to increase
the survival probability significantly.
References
1. Abramov, A.A., Konyukhova, N.B.: Transfer of admissible boundary conditions
from a singular point for systems of linear ordinary differential equations. Sov. J.
Numer. Anal. Math. Modelling (VNU Science Press, Utrecht, The Netherlands)
1(4), 245–265 (1986)
2. Albrecher, H., Badescu, A., Landriault, D.: On the dual risk model with tax pay-
ments. Insurance Math. Econom. 42, 1086–1094 (2008)
3. Asmussen, S., Albrecher, H.: Ruin Probabilities. Advanced Series on Statistical
Science and Applied Probability, V.14. World Scientific, Singapore (2010)
4. Belkina, T.A.: Risky investment for insurers and sufficiency theorems for the sur-
vival probability. Markov Processes Relat. Fields. 20, 505–525 (2014)
5. Belkina, T., Hipp, C., Luo, S., Taksar, M.: Optimal constrained investment in the
Cramer-Lundberg model. Scandinavian Actuarial Journal 5, 383–404 (2014)
6. Belkina, T.A., Konyukhova, N.B., Kurochkin, S.V.: Dynamical insurance models
with investment: Constraint singular problems for integro-differential equations.
Comput. Math. Math. Phys. 56(1), 47-98 (2016)
7. Birger, E.S., Lyalikova, N.B.: Discovery of the solutions of certain systems of dif-
ferential equations with a given condition at infinity, I. U.S.S.R. Comput. Maths.
Math. Phys. 5(6), 1–17 (1965); On finding the solutions for a given condition at
infinity of certain systems of ordinary differential equations, II. U.S.S.R. Comput.
Maths. Math. Phys. 6(3), 47–57 (1966)
8. Frolova, A., Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky
investments are dangerous. Finance Stochast. 6(2), 227-235 (2002)
9. Grandell, J.: Aspects of Risk Theory. Springer, Berlin–New York (1991)
10. Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments
are dangerous: the case of negative risk sums. Finance Stochast. 20(2), 355–379
(2016)
11. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer-
Verlag, NY (1988)
12. Konyukhova, N.B.: Singular Cauchy problems for systems of ordinary differential
equations. U.S.S.R. Comput. Maths Math. Phys. 23(3), 72–82 (1983)
13. Paulsen, J., Gjessing, H.K.: Ruin theory with stochastic return on investments.
Adv. Appl. Probab. 29(4), 965–985 (1997)
14. Pergamenshchikov, S. and Zeitouny, O.: Ruin probability in the presence of risky
investments. Stochastic Process. Appl. 116(2), 267–278 (2006)
15. Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Dover,
New York (1987).