VNU Journal of Mathematics – Physics, Vol. 29, No. 1 (2013) 9-17
Ruin Probabilities in Generalized Risk Process
with a Marko Chain Interest Model
Nguyen Thi Hien*
Viet Nam University of Comerce, Ho Tung Mau, Mai Dich, Ha Noi, Vietnam
Received 15 January 2013,
Revised 07 March 2013; accepted 28 March 2013
Abstract: This paper deals with ruin probabilities in generalized discrete time risk process with
Markov chain interest model. After, recursive and integral equations for the ruin probabilities are
given. When interest rates can be negative and loss distribution have regularly varying tails, the
paper built an asymptotic formula for the finite time ruin probability by an inductive approach on
the recursive equations.
Keywords: Discrete time risk model; Time of ruin; Ruin probability; Recursive eqution; Rate of
interest.
1. Introduction
Recently, estimates for the probability of ruin within finite time or infinite time for a discrete time
risk model as the initial capital tends to infinity, with emphasis on heavy-tailed insurance risk and
financial risk, have drawn a lot attention. In [1], authors consider the discrete time risk process
(1.1)
U k  U k 1 (1  I k )  Yk ,
k  1, 2,...
where U 0  u  0 is the initial capital, Yk , k  1, 2,... (the net loss in period k) is a sequence of
independent and identically distributed (i.i.d.) random variables and the interest rate {Ik, k =0,1,…}in
period k is a sequence of random variables and independent of Yk , k  1, 2,... . Thus Uk given by
(1.1) is the surplus of an insurer at the end of period k. The main results dealt with in [1] are to first
derive a recursive equation of the finite time ruin probabilities and then an integral equation for
infinite time ruin one. After, authors generalize Lundbergs upper bound for the infinite time ruin
probability. Later, X. Wei and Y. Hu consider a more general model
(1.2)
U k  (U k 1  X n )(1  I k )  Yk ,
k  1, 2,...
_______

Tel.: 84- 984032636
E-mail: hienvcu83@gmail.com
9
10
N.T. Hien / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 9-17
Where, Xn is the total amount of premiums;  X k , k  1, 2,... ; Yk , k  1, 2,... are two
sequences independent and identically distributed random variables and the interest rate
I k , k  0,1,... is a sequence of random variables and independent of  X k , Yk  , k  1, 2,... . They

claim
that
the
time
of
ruin,
denoted
by  u  inf n  1;U n  0 ,

has
an
estimate
P  u    u  as u   , where  is a specific positive parameter. See [2], [3].
In this paper, we continue to consider the model described by (1.2) by studying a recursive
equation for the finite ruin probabilities and an integral equation for infinite time ruin one. We also
investigate the asymptotic formula for ruin probability.
The organization of this paper is as follows: Section 2 derives recursive and integral equation for
ruin probability. In Section 3 we concern with asymptotic formulas for ruin probabilities.
2. Recursive and integral equation for ruin probpability
Consider the risk model
U k  (U k 1  X n )(1  I k )  Yk , k  1, 2,...
With the sequences of random variable
 X n  ; Yn  and  I n  satisfy
(2.1)
the assumptions in the
section 1. Denote by H(x) the common distribution function of the i.i.d sequence
X k ,
k  1, 2,... ,
that is
H ( x)  P  X1  x ,
And by F(x) the common loss distribution function of Yk , k  1, 2,... , i.e.,
F ( y)  P Y1  y , with F (0)  0.
It is easy to see that the solution of (2.1) has the expression
k
k
k 

U k  u (1  I k )    X j (1  I j )  Y j   (1  I t )  k  1, 2,...
j 1 
j 1
t  j 1

(2.2)
Assume that the interest rates I n , n  0,1,... is a homogeneous Markov chain with state space
I  i0 , i1 ,..., iN  and
the
transition
probabilities pij , i, j  I .
That
is
for
all
n  0,1,...and all states is , it , ito ,..., itn1 ,

P I n1  it | I n  is , I n 1  itn1 ,..., I1  it1 , I 0  it0

 P I n1  it | I n  is   pst  0; s, t  0,1, 2,..., N ,
where
N
p
t 0
st
 1, s  0,1, 2,..., N .
We define the finite and infinite time ruin probabilities in risk model (2.1) with the initial surplus
u and I 0  is given, respectively, by
N.T. Hien / VNU Journal of Mathematics-Physics Vol. 29, No. 1 (2013) 9-17

 n  u , is   P 
n
 k 1
U k

 0  | I 0  is 

k

n  k
 P   u  1  I j    X j 1  I j   Y j
j 1

 k 1  j 1
and


  u , is   P 

U k
11

  (1  I )  0  | I
k
t
t  j 1

0


 is  ,



 0  | I 0  is 

 k 1
k

  k
= P   u  1  I j    X j 1  I j   Y j
j 1
 k 1  j 1



  1  I   0  | I
k
t  j 1
t

0


 is  ,


where U k is given by (2.2). It is clear that
 1  u, is    2 (u, is )   3  u, is   ...,
and
lim n  u, is    (u, is ).
 2.3 
n 
Throughout this paper, if B is a distribution function then the function B  x   1  B  x  is
called tail of the distribution B  x  . We first give a recursive equation for  n (u, is ) and an integral
equation for  (u, is ) .
Lemma 2.1. For n  1, 2,...and any u  0,
N

t 0
0
 n 1  u , is    pst  F   u  x 1  it   dH  x 
N
  pst 

0
t 0

 u  x 1 it 
0
 n   u  x 1  it   y, it  dF ( y ) dH  x  , (2.4)
with
N
 1 (u, is )   pst  F   u  x 1  it   dH  x 
t 0

0
and
N
 (u , is )   pst 0 F   u  x 1  it   dH  x 

t 0
N
  pst 
t 0

0

 u  x 1 it 
0

  u  x 1  i   y, i  dF ( y )dH  x  .  2.5 
t
t
Proof. Given X1  x, Y1  y and I1  it , from  2.1 , we have
U1   u  X1 1  I1   Y1   u  x 1  it   y  h  y,
where h   u  x 1  it  . Thus, for y  h it yields
P U1  0 | Y1  y, X1  x, I1  it , I 0  is   1
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N.T. Hien / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 9-17
 n1
Which implies that for y  h, P 
 k 1

U k  0  | Y1  y, X1  x, It  it , I 0  is   1.

In case 0  y  h one has P U1  0 | Y1  y, X1  x, I1  it , I 0  is   0.
Let
 X , n  1, 2,... ;Y , n  1, 2,... and I , n  0,1, 2,...
be
n
 X n , n  1, 2,... , Yn , n  1, 2,... ,
and
I n , n  0,1, 2,... ,
independent
 n 1

P  U k  0  | Y1  y, X 1  x, I t  it , I 0  is 
 k 1

 n 1

 P  U k  0  | Y1  y, X 1  x, I t  it , I 0  is 
k 2

n

1
k
k
k

 

 P   u  1  I j    X j 1  I j   Y j  (1  I t )  0  | I1  it 
j 1
t  j 1

 k  2  j 1


k
k
 n 1 
 P    h  y   1  I j    X j 1  I j   Y j
j 2
j 2
 k  2 
  (1  I )  0  | I
k
k
 n 1 
 P    h  y   1  I j   X j 1  I j  Yj
j 1
j 1
 k 1 
  (1  I )  0  | I



 


k
t  j 1
t

1

k
t  j 1
t
  n (h  y, it )   n  u 1  it   x  y, it  .

0

 it 


 it 

Therefore, by conditioning on Y1, X1 and I1 , we get
 n 1
 n 1 (u, is )  P 

U k  0  | I 0  is 
 k 1

n

1
 


  pst   P  U k  0  | Y1  y , X 1  x, I t  it , I 0  is dF ( y )dH  x 
0 0
t 0
 k 1

N
N
  pst 

t 0
0
N

  pst 
t 0
0
N



u  x 1 it 
dF ( y )dH  x 
 u  x 1 it 
0
 n   u  x 1  it   y, it  dF ( y )dH  x 
  pst  F   u  x 1  it  dH  x 

t 0
0
N

  pst 
t 0
0

 u  x 1 it 
0
copies
of
respectively. From (2.6) and (2.2) it
follows that for 0  y  h

 2.6
 n   u  x 1  it   y, it  dF ( y )dH  x  .
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N.T. Hien / VNU Journal of Mathematics-Physics Vol. 29, No. 1 (2013) 9-17
Which yields the recursive equation (2.4) for the finite ruin probabilities  n (u, is ) in lemma 2.1.
By using the Lesbegue dominated convergence theorem and letting n   in (2.4) we obtain (2.5).
The proof is complete.
We remark that the techniques used in this proof are similar in [4]
3. Asymptotic formula for ruin probability
A distribution F on  ,   is said to have a regularly varying tail, or simply says F  R   , if
there exists some constant   0 such that for any y  0 ,
lim
x 
F ( xy )
 y .
F ( x)
We say that F  D is dominant – tailed if for any 0  y  1 one has
lim sup
x 
F ( xy )
 .
F ( x)
Similarly, we say F  L to have long – tailed if for any y  0
lim
x 
F ( x  y)
 1.
F ( x)
In case
lim e x F ( x)  ,   0,
x 
F is called heavy – tailed.
It is easy to see that
 R   D  L.
 All three classes of distributions R   , D and L are heavy - tailed.
 The set R   b is stable under convolution operator, that is if F1  R   and F2  R   then
F1 * F2  R   Moreover,
F1 * F2  x  ~ F 1  x   F 2  x  as x  .
(3.1)
See in [5] and [6].
 In addition, the class R   is closed under tail – equivalences, i.e., for two distributions F1 and
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N.T. Hien / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 9-17
F2, if F1  R   and F1  x  ~ cF2  x  for some c  0 , then F2  R  
 Further, it is easy to see that if F  D then for any constant c  0 , the function
F (cu ) / F (u ) is uniformly bounded in u   ,   .
We consider asymptotic formula for the finite time ruin probability  n  u, is  when the loss
distribution F is heavy – tailed. We now allow that the interest rates may be negative, so that we can
think of interest rates I n , n  0,1,... as the rates of return on a risky investment satisfying:
1  I n  0, n  0,1,..., or equivalently, 1  is  0, s  0,1,..., N .
(3.2)
We rewrite (2.1) under the form
U k  U k 1 1  I k   X k (1  X k )  Yk , k  1, 2,...


Define Qs ( z)  P Yk  X k 1  is   z for s=0, 1, 2 ,…,N. Since Yk and Xk are independent,

 x  dH  x 
Qs  z   P Yk  X k (1  is   z )  E I[Yk  X k (1is ) z ]


=  E I[Yk  X k (1is ) z ] | X k
0
=  P Yk  x 1  is   z  dH  x  ,

0

3.3
Which implies that
Qs  z   1  Qs  z    F  x 1  is   z  dH  x  .

0
By lemma 2.1 we obtain
N
N
1 (u, is )    F   u  x 1  it   dH  x    pst Qt  u 1  it   .
t 0

0
(3.4)
t 0
Denote
Bn,is  u   1   n  u, is  and B n,is  u    n  u, is  , u  0
For n=1, 2, … and s=1, 2, ….,N. All distributions Bn,is  u   1  n  u, is  , n  1, 2,... are
supported on [0, ).
By lemma 2.1 and equation (3.3) we have
N.T. Hien / VNU Journal of Mathematics-Physics Vol. 29, No. 1 (2013) 9-17
N
15
 n 1  u, is    pst  F   u  x 1  it   dH  x 

0
t 0
N
  pst 

0
t 0

 u  x 1 it 
0
 n   u  x 1  it   y, it  dF  y  dH  x  ,
N
N
t 0
t 0
  pst Qt  u 1  it     pst 
u (1 it )
0
 n  u 1  it   z, it  dQt  z  .
Further, for any distribution F1 on [0, ) and a distribution F2 on  ,   , the tail of convolution
F1*F2 satisfies
F1 * F2  x   F2  x    F 1  x  y  dF2  y  for    x  .
x

Therefore,
B n 1,is  u    n 1  u , is 
N

  pst Qt  u 1  it    
t 0
u 1 it 
0
 n  u 1  it   z, it  dQt  z 
N
  pst Bn ,is * Qt  u 1  it   .

3.5 
t 0
Theorem 3.1. Let F  R   for some   0 then, for any n=1,2,… and is  I
 n  u, is  ~ Dn  is  F  u  as u  ,
3.6 
where Dn  is  is given recursively by
Dn  is   E[1  Dn 1  I1   1  I1 

3.7 
| I 0  is ],
with D0  is   0.
Proof. Since F is increasing function, F  z  x   F  z  . Therefore, for any F  R   .
0
F  z  x
F  z
 1 and lim
z 
F  z  x
F z
 1.
 3.8 
Using Lebesgue dominated convergent theorem we obtain
lim
z 
Qt  z 
F  z
 lim 

z  0
F  x 1  it   z 
F z
dH  x   1.
 3.9 
This means that Qt  z  ~ F  z  as z   for t =1, 2, …, N. Further, with F  R   we have
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N.T. Hien / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 9-17
lim
F  u 1  it  
F u 
u 
 1  it  .

By (3.4) and (3.9) we get
lim
1  u , is 
u 
F u 
N
 lim  pst
u 
t 0
N
Qt  u 1  it   F  u 1  it  
.
F  u 1  it  
F u 
= pst 1  it 

t 0
 E[1  I1 

| I 0  is ]  D1  is  .
Hence, with any is  I we have
1  u, is   B1,is  u  ~ D1 (is ) F  u  as u  .
Assume in induction that for any is  I
n  u, is   B n,is  u  ~ Dn  is  F  u  as u  .
3.10 
We need to prove that
 n1  u, is   B n1,is  u  ~ Dn1  is  F  u  as u  .
From (3.10) it follows that Bn ,is and F are tail – equivalent. Thus, Bn ,is  R   . Hence, by (3.1) and
(3.9) we have for any is  I ,
Bn,is * Qt  u  ~ B n,is  Qt  u  ~ 1  Dn  is   F  u  u  .
3.11
Hence,
lim
Bn ,it * Qt  u 1  it  
u 
F u 
 lim
Bn ,it * Qt  u 1  it  
u 
F  u (1  i t ) 
.lim
u 
F  u 1  it  
F u 
= 1  Dn  it   1  it  .

Thus, by (3.5) we get
lim
u 
 n 1  u, is 
F u 
N
 lim  pst
u 
N
t 0
Bn,it * Qt  u 1  it  
F u 
=  pst 1  Dn  it   1  it 
t 0

=D n 1  is  .
Hence, (3.6) holds for all n=1, 2, … This completes the proof of theorem.
N.T. Hien / VNU Journal of Mathematics-Physics Vol. 29, No. 1 (2013) 9-17
17
4. Conclusion
The article gave an asymptotic formula for the finite time ruin probability in case of possibly
negative interest rates. It is seen that the rate of convergence of ruin probability does not depend on the
distribution of premiums Xn when the initial capital tends to infinity. This result confirmed a capital
injection at each period whether there is premium or without premium does not play a role in the ruin
probability of the model. This is similar to the results already known in game theory, when the player
has an infinite amount of capital, the ruin probability gradually to 0 wich does not depend on the
person’s capacity to play. We think this is an important discovery in the field of financial mathematics.
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