•
•
•
•
•
Introduction
A recursive approach
A Gerber Shiu function at claim instants
Numerical illustrations
Conclusions
• Chan et al. (2003) , Dang et al. (2009)
U i (t )  ui  ci t 
Ni (t )
i
X
 k,
t  0; i  1,2
(1)
k 1
• u i - the initial capital of the i-th class of business;
• ci - the premium rate of the i-th class of business;
i
• X k - the k-th claim amount in the i-th risk process, with common cdf Fi () and
pdf f i () ;
• N i (t ) - the counting process for the i-th risk process.
N1 (t ) and N 2(t ) are common shock correlated Poisson processes
occurring at rates 1 and 2 respectively.
N1 (t )  N11 (t )  N12 (t )
N2 (t )  N22 (t )  N12 (t )
where N11 (t ), N22 (t ) and N12 (t ) are independent Poisson processes with
rates 11 , 22 and 12 ;
•
Tor  inf{t  0 | min{U1 (t ),U 2 (t )}  0}  min(1, 2 )
•
Tsim  inf{t  0 | max{U1 (t ),U 2 (t )}  0}
•
Tand  max(1 , 2 )
•
Tsum  inf{t  0 | U1 (t )  U 2 (t )  0}
• Chan et al. (2003)
• Cai and Li (2005)
• Yuen et al. (2006)
• Li et al. (2007)
• Dang et al. (2009)
Chan et al. (2003) for 11  22  0
•
c1
•
u1 u2
(u1 , u2 )
(u1 , u2 )
 c2
 12 (u1 , u2 )  12   (u1  z1 , u2  z2 )dF2 ( z2 )dF1 ( z1 ) (2)
0 0
u1
u2
Dang et al. (2009)
u
12 1 2
 n 1 (u1 , u2 ) 
{
12  c11  c2  2 0
1




u1
u2
c2
( a1 u1 )  u 2
c1
0

c1
( a2 u 2 )  u1
c2
0


u2
0
 n (a1 , a2 )e 1 ( a1 u1 ) e  2 ( a2 u2 ) da2 da1
12
 n (a1 , a2 )e c1
( u1  a1 )   2 [
12
 n (a1 , a2 )e c2
wit h st art ingpoint 0 (u1 , u2 )  1
e
c2
( a1 u1 )  u 2  a2 ]
c1
( u 2  a2 )  1 [
e
da2 da1
c1
( a2 u 2 )  u1  a1 ]
c2
da1 da2 
(3)
 0 (u1 , u 2 )  1,
1 (u1 , u 2 )  

0


0

u 2  c2t
0


0
u1  c1t

0



0
0

u 2  c2t

u 2  c2t
0
0
f1 ( x1 )11e ( 11  22  12 ) t dx1dt  

0

u 2  c2t
0
f 2 ( x2 )22 e ( 11  22  12 ) t dx2 dt
f1 ( x1 ) f 2 ( x2 )12 e ( 11  22  12 ) t dx1dx2 dt,
0
 n 1 (u1 , u 2 )  

u1  c1t

u1  c1t
0
 n (u1  c1t  x1 , u 2  c2t ) f1 ( x1 )11e ( 11  22  12 ) t dx1dt
(4)
 n (u1  c1t , u 2  c2t  x2 ) f 2 ( x2 )22 e ( 11  22  12 ) t dx2 dt

u1  c1t
0
 n 1 (u1 , u 2 ) 
 n (u1  c1t  x1 , u 2  c2t  x2 ) f1 ( x1 ) f 2 ( x2 )12 e ( 11  22  12 ) t dx1dx2 dt
12 1 2
S  c11  c2  2

12 1 2
 
 
 22 2
c1
 S  c11  c2  2
u1
u2
0
0
 
 n ( a1 , a2 )e 1 ( a1 u1 ) e  2 ( a2 u2 ) da2 da1

s
( u1  a1 )   2 [ 2 ( a1  u1 )  u 2  a2 ]
  cc2 ( a1 u1 )  u 2
c1
c1
   1
 n ( a1 , a2 )e
e
da2 da1
u1 0


c
s
( u 2  a2 )  1 [ 1 ( a 2  u 2 )  u1  a1 ]

12 1 2
11 1   cc12 ( a2 u2 ) u1
c2
  
 

 n ( a1 , a2 )e
e c2
da1da2
u2 0


c


c

c
1 1
2 2
2

 S
where s  11  12  22
c
(5)
• Let {X k1}k 1 and{X k2}k 1 follow independent PH distributions with
parameters (α, T) and (β, Q).
 n 1 (u1 , u2 )  
u1 u 2
0

0
 n (a1 , a2 )12 (   )[12  (c1T )  (c2Q)]1
e[T (u1 a1 )][Q (u2 a2 )] (t  q)da2 da1


u1
c2
( a1 u1 )  u 2
c1
0

 n (a1 , a2 )12 (   )[12  (c1T )  (c2Q)]1 e
[ 12 (  c1T ) (  c2Q )](
a1 u1
)
c1
e[T (u1 a1 )][Q (u2 a2 )] (t  q)da2 da1


u2
c1
( a2 u 2 )  u1
c2
0

 n (a1 , a2 )12 (   )[12  (c1T )  (c2Q)]1 e
e[T (u1 a1 )][Q (u2 a2 )] (t  q)da1da2
[ 12 (  c1T ) (  c2Q )](
a2 u 2
)
c2
(6)
• m(u1, u2 )  E(u1 ,u2 ) [eTo r w(U1 (Tor ),U2 (Tor ))I (Tor  )] (7)
• w(,) is a penalty function that depends on the surplus levels at time
Tor in both processes.
w(U1 (Tor ),U 2 (Tor ))  w1 (U1 (Tor ),U 2 (Tor ))I ( 1   2 )  w2 (U1 (Tor ),U 2 (Tor ))I ( 2   1 )
 w12 (U1 (Tor ),U 2 (Tor ))I ( 1   2 )
(8)
• Here are few choices of the penalty functions
1. w1 (,)  w2 (,)  w12 (,)  1
2.   0, w1 (,)  w2 (,)  0 and w12 (,)  1
3. w1 ( y, z)   y, w2 ( y, z)   z and w12 ( y, z)   y  z
E(u1 ,u2 ) [e1 | U1 (1 ) | I (1   2 ;1  )]  E(u1 ,u2 ) [e 2 | U2 ( 2 ) | I ( 2  1; 2  )] (9)
4. w1 ( y, z)  y  z and w2 (,)  w12 (,)  0
E(u ,u ) [e (U1 (1 )  U2 ( 2 ))I (1   2 )] (10)
1
1
2
• mn (u1, u2 )  E(u1 ,u2 ) [e
• m(u1 , u2 ) 
To r
w(U1 (Tor ),U2 (Tor ))I (Tor  Sn )] (11)


n 1
mn (u1 , u2 ) (12)
• m1 (u1 , u2 )  m1 (u1 , u2 )  m1 (u1 , u2 )  m1
1
2
12
(u1 , u2 ) (13)
Where m1 (u1 , u2 ), m1 (u1 , u2 ) and m1 (u1 , u2 ) correspond to the
cases {τ1<τ2}, {τ2<τ1} and {τ1=τ2} respectively.
1
2
12
• Considering the first case when ruin occurs at the first claim instant in
{U1(t)} only and using a conditional argument gives
m (u1 , u2 )  
1
1

0


0

u 2  c2 t
0


0


0
w1 ( y, u2  c2t ) f1 (u1  c1t  y )11e ( s  )t dydt
(14)
w1 ( y, z ) f1 (u1  c1t  y ) f 2 (u2  c2t  z )12 e ( s  )t dzdydt
• By similar method, one immediately has m12 and m112. Hence by adding
m11 , m12 and m112, we obtain the starting point of recursion.
• If w1 ( y, z)   y, w2 ( y, z)   z , and w12   y  z , the three integrals reduce to
m11 (u1 , u2 )  

0
m12 (u1 , u2 )  
0

0
m (u1 , u2 )  
12
1



y f1 (u1  c1t  y )[11  12 F2 (u2  c2t )]e ( s  ) t dydt,

0

0


0
y f 2 (u2  c2t  y )[22  12 F1 (u1  c1t )]e ( s  ) t dydt,
y
u1  c1t  y
u1  c1t
(15)
f1 ( x1 ) f 2 (u2  c2t  u1  c1t  y  x1 )12 e ( s  ) t dx1dydt.

• mn1 (u1 , u2 )  
0




0
0

u 2  c2 t

u 2  c2 t
0
0

u1  c1t
0
mn (u1  c1t  x1 , u2  c2t ) f1 ( x1 )11e ( s  ) dx1dt
mn (u1  c1t , u2  c2t  x2 ) f 2 ( x2 )22 e ( s  ) dx2 dt

u1  c1t
0
(16)
mn (u1  c1t  x1 , u2  c2t  x2 ) f1 ( x1 ) f 2 ( x2 )12 e ( s  ) dx1dx2 dt
• The idea that we use to find a computational tractable solution of (16) is based
on mathematical induction.
11  12
12
• m11 (u1 , u2 ) 
e u 
e (  u   u ) ,
1 1
1 1
2
2
1 (s    c11 )
1 (s    c11  c2  2 )
22  12
12
m12 (u1 , u 2 ) 
e u 
e (  u   u ) ,
 2 (  s    c2  2 )
 2 (s    c11  c2  2 )
1
1
12
m112 (u1 , u2 )  (

)
e (  u   u ) .
1  2 s    c11  c2  2
2
2
1 1
1 1
2
2
(17)
2
2
• Therefore, the expected discounted deficit when ruin happens at the instant of
the first claim is given by
11  12
22  12
m1 (u1 , u2 ) 
e u 
e u
(18)
1 (s    c11 )
 2 (s    c2  2 )
1 1
2
2
mn 1 (u1 , u 2 ) 
n
n
n
 a[ n 1, j ]u1j e  1u1   b[ n 1, j ]u2j e   2u2  
j 0
j 0
k 0
n
e
j 0
u1j u 2k e  ( 1u1   2u 2 ) ,
[ n 1, j , k ]
for n  0,1,  , wit h
a[ n 1, j ] 
b[ n 1, j ] 
a[1, 0 ] 12 a[ n ,i ]i!c1i 1 j
n 1

i  max( j 1, 0 )
j!(s    c1 1 ) i 1 j
b[1, 0 ]  b[ n ,i ]i!c
n 1
i 1 j
2
i 1 j
2
2
2

i  max( j 1, 0 )
j!(s    c2  )
n 1

e[ n 1, j , k ]  I ( k  0)(
i  max( j 1, 0 )
 I ( j  0)(
n 1

i j
n 1

i j
22 a[ n ,i ]i!c1i  j
,
j!(s    c1 1 ) i 1 j
11b[ n ,i ]i!c2i  j
,
j!(s    c2  2 ) i 1 j
n 1
 1 a[ n ,i ]c1i 1 j i!
22 a[ n ,i ]c1i  j i!

)

i 1 j
j!(s    c11  c2  2 ) i 1 j
i  j j!(s    c1 1  c2  2 )
n 1

i  max(k 1, 0 )
n 1
  2 b[ n ,i ]c2i 1 k i!
11b[ n ,i ]c2i  k i!

)

i 1 k
k!(s    c1 1  c2  2 ) i 1 k
i  j k!(s    c1 1  c2  2 )
e[ n ,i , q ]22  2i! q!c1i  j c2q 1 k
i  q 1 j  k 


 
 j! k!(    c   c  ) i  q  2  j  k
i j
i  j q  max(k 1, 0 ) 

s
1 1
2
2
n 1
n 1
e[ n ,i , q ]11 1i! q!c1i 1 j c2q  k
i  q 1 j  k 



 
 j! k!(    c   c  ) i  q  2  j  k
qk
q  k i  max( j 1, 0 ) 

s
1 1
2
2
n 1
n 1

e[ n ,i , q ]12 1 2i! q!c1i 1 j c2q 1 k
i  q  2  j  k 



  q  1  k
 j! k!(    c   c  ) i  q  3 j  k ,
q  max(k 1, 0 ) i  max( j 1, 0 ) 

s
1 1
2
2
n 1
n 1
12
. T hest art ingpointis given by
s    c11  c2  2
22  12

,e
 0.
 2 (s    c2  2 ) [1, 0 , 0 ]
for n  1,2, ; j , k  0,1,  , n, where  
a[1, 0 ] 
11  12
,b
1 (s    c11 ) [1, 0 ]
Not e t hat weassum e i  j  0 for any j  k .
k
i i 1   x

1
• f1 ( x1 )   qi x1 e ,
(i  1)!
i 1
m
1 1
 2j x2j 1e  x
f 2 ( x2 )   p j
( j  1)!
i 1
m
2 2
• Using equation (4) for n=0 and λ11=λ22=0 along with the trivial condition
0 (u1, u2 )  1 , we obtain
m 1
1 (u1 , u 2 )  1   a
s 0
s
[1, s ] 1
u e
 1u1
m 1
  b[1, s ]u e
s 0
s
2
  2u 2
m 1 m 1
   e[1, s ,v ]u1s u 2v e ( 1u1   2u 2 ) ,
s 0 v 0
where
a[1, s ]
qi c1k  s 1k
,
 
k  s 1
i  s 1k  s s!( k  s )!(  c1 1 )
b[1, s ]
p j c2k  s  2k
,
  
k  s 1
)

c


(
)!
s

k
(
!
s
j  s 1 k  s
2 2
m
m
i 1
j 1
p j qi c1k  s c2l v 1k  2l
k  l  s  v

,
e[1, s ,v ]      
k  l  s  v 1
s

k
)

c


c


(
!
l
!
k
!
v
!
s
i  s 1k  s j  v 1 l  v 

2 2
1 1
for s, v  0,1,  , m  1.
m
i 1
m
j 1
 (u1 , u 2 )  lim n   n (u1 , u 2 )
 n 1 (u1 , u 2 )  1 
( n 1) m 1
 a[ n 1,w]u1we  1u1 
w 0
( n 1) m 1
 b[ n 1,w]u2we   2u2 
( n 1) m 1( n 1) m 1

w0
w0
e
y 0
u1wu 2y e  ( 1u1   2u 2 ) ,
[ n 1, w , y ]
for n  0,1,  , wit h
a[ n 1, w ]  a[1, w ] I (0  w  m  1)

m

nm 1
s
 qi a[ n, s ]

i  max(w  mn 1,1) s  max(w  i , 0 ) g  0
( 1) s  g 1i c1s  i  w ( s  i )!s!
,
(i  1)! g!( s  g )!w!( s  g  i )(  c11 ) s  i 1 w
b[ n 1, w ]  b[1, w ] I (0  w  m  1)

m

nm 1
s
 p j b[ n ,s ]

j  max(w  mn 1,1) s  max(w  j , 0 ) g  0
( 1) s  g  2j c2s  j  w ( s  j )!s!
,
( j  1)! g!( s  g )!w!( s  g  j )(  c2  2 ) s  j 1 w
e[ n 1, w, y ]  e[1, w, y ] I (0  w  m  1,0  y  m  1)
m
 I (0  y  m  1)

nm 1

s
m
i 1
  p
i  max(w  mn 1,1) s  max(w  i , 0 ) g  0 i  w 1 k  w
j
qi b[ n , s ]
 s  i  k  w  y  s 
( 1) s  g c1s  i  w c2l  y 1k  2j ( s  i )!



s  i  l  w  y 1




siw

 s  g  (i  1)!w! y!( s  g  i )(  c11  c2  2 )
 I (0  w  m  1)
m

nm 1

s
m
i 1
  p
i  max(w  mn 1,1) s  max(w  i , 0 ) g  0 i  w 1 k  w
j
qi b[ n , s ]
 s  j  k  w  y  s 
( 1) s  g c1k  w c2s  j  y 1k  2j ( s  j )!





 s  g  ( j  1)!w! y!( s  g  j )(  c   c  ) k  j  s  w y 1
s j y



1 1
2
2
m
nm 1
s
m
nm 1
v
p j qi e[ n , s ,v ]







i  max(w  mn 1,1) s  max(w  i , 0 ) g  0 j  max( y  mn 1,1) v  max( y  j , 0 ) z  0 ( s  i  g )(v  j  z )
 s  i  v  j  w  y  s  v 
( 1) s  v  g  z c1s  i  w c2v  j  y 1i  2j ( s  i )!(v  j )!







 s  g  v  z  (i  1)!( j  1)!w! y!(  c   c  ) s  v  i  j  w y 1 ,
siw




1 1
2
2
for m  1,2,  , y  0,1,  , ( n  1) m  1.
• We denote the survival probability associated to the time of ruin Tand,
by and (u1, u2 )  PTand   | u1, u2   P1   or 2   | u1, u2 .
and
1
2
or
• n (u1, u2 )  n (u1 )  n (u2 )  n (u1, u2 ) .

•  (u1 )  0
1
n1
u1 c1t

0
 (u1  c1t  x1 ) f1 ( x1 )(11  12 )e
1
n
s t

1
0
n
dx1dt    (u1  c1t )22est dt
T heunivariat esurvivalprobability 1n 1 (u1 ) up t o and including t he(n  1) - t h claim
event sadmit s t herepresent at ion

n
1
n 1
(u1 )  1   a[1n 1, j ]u1j e  1u1 ,
j 0
for n  0,1, , wit h
a[1n 1, j ]  a[11, 0 ] I ( j  0) 
n 1
a[11, 0 ] 1a[1n ,i ]c1i 1 j i!
i  max( j 1, 0 )
j!(s  c11 ) i  2 j

n 1
a[1n ,i ]22 c1i  j i!
i j
j!(s  c11 ) i 1 j

for n  1,2, ; j  0,1,  , n. T hest art ingpointis a[11, 0 ] 
11  12
.
s  c11
,
• u1=2 ,u2=10, c1=3.2, c2=30, 1/µ1=1 and 1/µ2=10.
• Case 1: Independent model — λ11=λ22=2; λ12=0.
Case 2: Three-states common shock model — λ11=λ22=1.5; λ12=0.5.
Case 3: Three-states common shock model — λ11=λ22=0.5; λ12=1.5.
Case 4: One-state common shock model — λ11 = λ22 = 0; λ12 = 2.
• Note that λ1 = λ2 = 2, and θ1 = 0.6 and θ2 = 0.5.
• In Case 1, after
100 iterations we
obtain a ruin
probability of
0.6306428 that is
very close to the
exact value of
0.6318894.
• max{1 (u1 ),  2 (u2 ) }
 or (u1 , u2 )
 1 (u1 )   2 (u2 )
 1 (u1 ) 2 (u2 )
• Cai and Li (2005,
2007) provided
simple bounds for
Ψand(u1, u2) given
by
 1 (u1 )  2 (u 2 )
 and (u1 , u 2 )
 min{ 1 (u1 ),  2 (u 2 )}
• δ = 0.05
• This quantity
is achieved by
letting w1(y,
z) = y+z and
w2(.,. ) =
w12(.,.) =0
• This quantity
is achieved
by letting
w2(y, z) =
y+z and
w1(.,. ) =
w12(.,.) =0
• Several extensions:
1. Correlated claims
2. Correlated inter-arrival times and the resulting
claims
3. Renewal type risk models