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2010� 9� ACTA MATHEMATICA SINICA, Chinese Series Sep., 2010
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E-mail: liangzhibin111@
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E-mail: jyguo@
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Poisson MN
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MR(2000) x!"# 62P05, 91B30
$%"# , 0232
Ruin Probabilities under Optimal Combining Quota-Share
and Excess of Loss Reinsurance
Zhi Bin LIANG
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, P. R. China
E-mail : liangzhibin 111@
Jun Yi GUO
School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China
E-mail : jyguo@
Abstract In this paper, we study the optimal retentions for an insurance company,
which intends to reinsure its risk by means of a pure proportional treaty, a pure excess
of loss treaty or any combination of the two. Under the criterion of maximizing the
adjustment coefficient, the closed form expressions of the optimal results are given not
only for the Brownian motion risk model but also for the compound Poisson risk model.
����: 2008-09-18; ����: 2010-03-16
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Moreover, a smallest exponential upper bound for the ruin probability is obtained. We
also conclude that, under some conditions, there exists a pure excess of loss reinsurance
strategy which is better than any combinational reinsurance strategy. Some numerical
examples are presented, which illustrate the results of this paper.
Keywords ruin probability; adjustment coefficient; compound Poisson process; quota-
share reinsurance and excess of loss reinsurance
MR(2000) Subject Classification 62P05, 91B30
Chinese Library Classification , 0232
1 )*
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�!��" ��#�� Poisson �� (!Æ$ C-P ��) "3#�#%��$4� (!Æ$
D-A ��). -����!&%', +� "3+( &!'"#), +5(�+�$)�6#)
��**. ++, � [1] .13��� D-A ��.+� ,-#)�+�%+/����, ��
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�� D-A ��.�+�17�+�/����, 234���+�6�56&'.. -+(
/()0 ("3+� 7*)0) �+�89', Schmidli [4] � Liang [2] ��� D-A ��.�
+�%+/����, +(��+�$)�+(/()0�18&'.. � [6] .139,-�
�9����.�+�,-:;/����, ./�+�$)�<:��.
;�<=, C-P ��./()0 ψ(u) �18*#02(��, =12 C-P ��./()
0�63>344�����.�.?5�. ���>6?@
A/()0�53>$, +(�
C-P ��./()0��6 [7, 8]. 78�6.�97B) R 4=14$����.8�!0?
C���:9;), <7,
�9��>-:;@�<D97B)+� �+�$) [9−15].
=�-+� 97B)�+�89', � C-P��� D-A��.�+�A�/����.
� [12] 4��� C-P ��'�+�A�/����, >#B�/������
AE=?. 7
Zhang @� [16] 9F�� D-A ��.�+�A�/��, +>���+�$)�+(/()0
��*&'.. =��G?A2@� [16] �BC, 4��� D-A��.+�6��2&'.. 2
@ C-P ��, H����%� [12] D18�+�$)�+�97B)�&'., +>./�8
��'/()0�+(C)�6. 1I, E:@� Lundberg A@.AA2 C-P ��./()0
4�, 2J42 D-A ��.�/()044�. +F�/=�: K�,- C-P ��E# D-A �
�, -��&%', D*-�EGH,-:;/��$)%BI�CA�/��$)CJ.
2 FG
-DK����., LM�� {Xt}t≥0 ENO4
Xt = u + ct− St, ()
F. u ≥ 0 #LÆLM, c #MPJG,H��N0, St &6�JI t $H���" -. H�
�! St =
∑N(t)
i=1 Yi #�� Poisson ��, 4J#Q, N(t) #�
K:;)$ λ �LO Poisson
5� MIPQ: JRRNKSTSTLUMVUVWOXYP 859
��, Yi, i ≥ 1 #Z�2
�, +>Z�@ N(t) �N6��[9WQ.
! Y #X Yi �
�2
�#)��\��[9, ./6 E(Y ) Æ$ μ. H��! Y �
�
#) F (y) RON'&%:
(1) F (0) = 0, >
� 0 < y < N J, 0 < F (y) < 1, � y ≥ N J, F (y) = 1, F.
N = sup{y : F (y) ≤ 1} ≤ +∞;
(2) Y �K:#) f(y) := dF (y)/dy *-7>SY;
(3) TU#) MY (r) RO Crame´r–Lundberg &% [17, 18], V*-&!�) 0 < r∞ ≤ ∞, D
� MY (r) - r ∈ (−∞, r∞) �*-, >RO
lim
r→r∞
MY (r) = +∞.
=��!��]ZEN��N'[E/��B.�P%�����:
^�E#GH%+/��;
^_E#GH,-:;/��;
^[E9#WXC/��B.�A�.
!�) a ∈ (0, 1] #%+/��.�Y$[9, �) m ∈ (0, N ] #,-:;/��.�Y$
[9. ��0XC/��B.�A�, ���-^ i O" ., 9B\Q`" -
Yi(a,m) = min{aYi,m} (i = 1, . . . , Nt).
�! (/) ���#aÆ./6R��4]�N�, Z[^>A�/��$) (a,m) F, ���
�MP�N,_[$
c(a,m) = (1 + θ)λμ− (1 + η)λE(Yi − aYi ∧m)
= (θ − η)λμ + (1 + η)λE(aYi ∧m)
= (θ − η)λμ + (1 + η)λaE(Yi ∧m/a),
F., θ = cλμ − 1 #����b?c\�N0, η #/����b?c\�N0. A;�\E, H
��! η > θ. $��:�N0 c(a,m) > 0 4�, H�EC;
η <
aE(Yi ∧m/a) + θμ
μ− aE(Yi ∧m/a) .
! Zi = Yi ∧m/a, Z[ {Zi}i≥1 4#Z�2
�, +>Z�@ N(t) �N6��[9WQ.
A2��
EZi =
∫ m/a
0
[1− F (y)]dy := μ(a,m),
EZ2i =
∫ m/a
0
2y[1− F (y)]dy := σ2(a,m),
MZ(r) = 1 + r
∫ m/a
0
[1− F (y)]erydy.
=1, LM��EN&6$
Xa,mt = u + c(a,m)t− aS1(t), ()
F.
S1(t) =
N(t)∑
i=1
Zi
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#8�`.���" ��.
']�a/(JG
τa,m = inf{t ≥ 0 : Xa,mt < 0}
�+5/()0
ψa,m(u) = P{τa,m < ∞|Xa,m0 = u}.
3 |S}TUVWXYZ[\
D-A ]^$_~`ab
0�7., �!�� () .���" ��#�#%��$4�, V
Sˆ1(t) = λμ(a,m)t−
√
λσ(a,m)Wt, ()
F. {Wt, t ≥ 0} #�"8�$4�. 0^, Sˆ1(t) EN_4#�� Poisson �� S1(t) �`bd
� (a� [19]).
e Sˆ1(t) cf�� () .� S1(t), �� !, ��N'd�G`bLM��:
Xˆa,mt = u + [(θ − η)λμ + aλημ(a,m)]t + a
√
λσ(a,m)Wt, ()
F. a ∈ (0, 1], m ∈ (0, N ].
�a�� () '�/(JG$
τa,mD = inf{t ≥ 0 : Xˆa,mt ≤ 0},
+5/()0$
ψa,mD (u) = P (τ
a,m
D < ∞| Xˆa,m0 = u).
��cBC, H���N'=�:
de �� {e−RD(a,m)Xˆa,mt , t ≥ 0} #c, +>
ψa,mD (u) = e
−RD(a,m)u, ()
F.
RD(a,m) =
2[(θ − η)λμ + aλημ(a,m)]
a2λσ2(a,m)
. ()
fg < () ., H�A2_/
E(e−r(Xˆ
a,m
t −u)) = e{−r[(θ−η)λμ+aλημ(a,m)]+
1
2a
2r2λσ2(a,m)}t.
b
g˜(r) = r[(θ − η)λμ + aλημ(a,m)]− 1
2
a2r2λσ2(a,m),
Z[B� g˜(r) = 0 �
Ng
RD(a,m) =
2[(θ − η)λμ + aλημ(a,m)]
a2λσ2(a,m)
,
<7
E(e−RD(a,m)(Xˆ
a,m
t −u)) = 1.
5� MIPQ: JRRNKSTSTLUMVUVWOXYP 861
e1H��/=�: {e−RD(a,m)Xˆa,mt , t ≥ 0} #c. 0^� RD(a,m) J#�� () �2��9
7B), 4J#�f� Lundberg C). ��"8�cBC [19], H�ENhc�/
ψa,mD (u) = e
−RD(a,m)u.
<7���:. :h.
']igd@<D/()0+( �+�$) (a∗,m∗), V
ψD(u) := ψ
a∗,m∗
D (u) = infa,m ψ
a,m
D (u).
< () .A2ih, +( /()0�+�$) (a∗,m∗) 2J4D97B)+� , jjie.
=1, H�h-BF�D97B)+� �+�$) (a∗,m∗), V
RD := RD(a∗,m∗) = sup
a,m
RD(a,m).
e@#) g˜(r) -k r = RD lHmN6, E= RD #B�
sup
a,m
{
r[(θ − η)λμ + aλημ(a,m)]− 1
2
a2r2λσ2(a,m)
}
= 0 ()
�+(Ng. 7#) g˜(r) �@ m �+�6k$
m1 =
η
r
,
=1, H���
m∗D = m1 ∧N.
� m∗D = N J, +�A�/����G?@f@+�GH%+/����. 7+
� 97B)�+�%+/����jD-� [14] .kg���. =1, -'�., H�A,-
m∗D = m1 < N �h`.
h-e m1 f_#) g˜(r), eF�@ a ;n, ��
∂g˜(r)
∂a
= rλ
∫ m1/a
0
[1− F (y)](η − 2ary)dy.
H�
:
le 2BI a ∈ (0, 1], A@.∫ m1/a
0
(1− F (y))(η − 2ary)dy > 0
4�.
fg b
g1(y) = η − 2ary, y ∈ [0,m1/a],
H�
g′1(y) = −2ar < 0, =1#) g1(y) #RO g1(0) = η > 0 � g1(m1/a) = −η < 0 �
kopiSY#). �N, *- y1 ∈ (0,m1/a), D� g1(y1) = 0. 7>, � 0 ≤ y < y1 J,
g1(y) > 0; � y1 < y ≤ m1/a J, g1(y) < 0. 1I, 1− F (y) #M9pi#), 9
, � y < y1 J,
1− F (y) > 1− F (y1); � y > y1 J, 1− F (y) < 1− F (y1).
862 � � � � y z { 53�
je
∫m1/a
0
g1(y)dy = 0, H�
∫ m1/a
0
(1− F (y))g1(y)dy
=
∫ y1
0
(1− F (y))g1(y)dy +
∫ m1/a
y1
(1− F (y))g1(y)dy
> (1− F (y1))
∫ m1/a
0
g1(y)dy = 0.
<7l��:. :h.
<l� , H�
∂g˜(r)∂a > 0, =1 a
∗ = 1. 0#m�
mn��!=?, 0moi�
m1 < N J, D#*-�!GH�,-:;/��$) m∗D %BI�CA�/��$) (a,m)
CJ, F. a < 1.
=1, H�h-B\,-GH�+�,-:;/��$). e (a∗,m∗) = (1,m1) f_ ()
., ��B�
[(θ − η)λμ + λημ(1,m1)]r − 12λσ
2(1,m1)r2 = 0.
=$ r = ηm1 , H����@ m1 �8�!d�B�
[(θ − η)μ + ημ(1,m1)]m1 − 12σ
2(1,m1)η = 0. ()
<']l�H�EN_/, 8B�*-p�Ng.
le B� () *-p�Ng m¯D.
fg !
f(m) = [(θ − η)μ + ημ(1,m)]m− 1
2
σ2(1,m)η.
H�
f(0) = 0,
f ′(0) = (θ − η)μ < 0,
f ′′(m) = η[1− F (m)] > 0,
lim
m→+∞ f(m) = +∞.
0moiB� f(m) = 0
p�Ng m¯D, l��:. :h.
=1, H�
m∗D = m¯D ∧N. ()
<�� , H�0kq��+'=�:
de ! m¯D < N #B� () �p�Ng. Z[�� () .D/()0+( �+
�/��$)# (1, m¯D), +(/()0$ ψD(u) = e−RDu, F. RD = ηm¯D .
�� .�=�X� [16] .=�G?�m.
C-P ]^$_~`ab
-0�7., H��� C-P ���� () .�+�A�/����. =$ C-P ��./
()0�18&'.Aq(�, -0^, H�hc,-+� 97B)�+�A�/��$).
5� MIPQ: JRRNKSTSTLUMVUVWOXYP 863
! RC(a,m) # C-P ��.�97B), Z[ RC(a,m) ROB�
c(a,m)r = λ[MZ(ar)− 1],
"3
[(θ − η)λμ + (1 + η)aλμ(a,m)]r − λ[MZ(ar)− 1] = 0. ()
H��'"#C@�+� RC(a,m) �+�$) (a∗,m∗), D�
RC := RC(a∗,m∗) = sup
a,m
RC(a,m).
=$ () .�nÆ-k r = RC lHmN6, =1, RC #N'B�
sup
a,m
{[(θ − η)λμ + (1 + η)aλμ(a,m)]r − λ[MZ(ar)− 1]} = 0
�*. 8B�EEN@fq&6$
sup
a,m
{
[(θ − η)λμ + (1 + η)aλ
∫ m/a
0
[1− F (y)]dy]r − aλr
∫ m/a
0
[1− F (y)]earydy
}
= 0. ()
h-2 () .�n�@ m ;n, ��r�+�6k
m2 =
ln(1 + η)
r
.
=1, H�
m∗C = m2 ∧N.
2 D-A ��, � m∗C = N J, +�A�/��$)���G?@f@+�GH%
+/����. =1, -'�., H�B�� m∗C = m2 < N �h`.
e m2 fl� () .., /�@ a ;n, ��
∂�
∂a
= λr
∫ m2/a
0
(1− F (y))[(1 + η)− eary − aryeary]dy,
F.
� =
{
(θ − η)λμ + (1 + η)aλ
∫ m2/a
0
[1− F (y)]dy
}
r − aλr
∫ m2/a
0
[1− F (y)]earydy.
H�
le 2Bm a ∈ (0, 1], A@.∫ m2/a
0
(1− F (y))[(1 + η)− eary − aryeary]dy > 0
4�.
fg !
g(y) = (1 + η)− eary − aryeary, y ∈ [0,m2/a].
H�
g′(y) = −areary − a2r2yeary < 0, y ∈ [0,m2/a].
=1, #) g(y) #RO g(0) = η > 0 � g(m2/a) = −(1 + η) ln(1 + η) < 0 �koM9pi#).
�N, *- y0 ∈ (0,m2/a), D� g(y0) = 0, 7>, � 0 ≤ y < y0 J, g(y) > 0, � y0 < y ≤ m2/a
J, g(y) < 0. 23=$ 1− F (y) #M9pi#), H�
, � y < y0 J, 1− F (y) > 1− F (y0),
� y > y0 J, 1− F (y) < 1− F (y0).
864 � � � � y z { 53�
s=$
∫m2/a
0
g0(y)dy = 0, H�
∫ m2/a
0
(1− F (y))g0(y)dy
=
∫ y0
0
(1− F (y))g0(y)dy +
∫ m2/a
y0
(1− F (y))g0(y)dy
> (1− F (y0))
∫ m2/a
0
g0(y)dy = 0,
<7l��:. :h.
<l� E=
∂�
∂a
> 0, a ∈ (0, 1],
=1,H�
a∗ = 1. 0!=?X D-A��.=?�3. 4J#Q,- C-P��.,�m2 < N J,
D*-�!GH,-:;/��$) m∗C %BI�CA�/��$) (a,m) CJ, F. a < 1.
']H�A,-GH+�,-:;/����. =$ r = ln(1+η)m2 , e (a
∗,m∗) = (1,m2) f
_ () ., H����@ m2 �B�
(θ − η)μ + (1 + η)
∫ m2
0
(1− F (y))dy −
∫ m2
0
(1− F (y))(1 + η) ym2 dy = 0. ()
<']l�, H�E=8B�*-p�Ng.
le B� ()
p�Ng m¯C .
fg !
g(m) = (θ − η)μ + (1 + η)
∫ m
0
(1− F (y))dy −
∫ m
0
(1− F (y))(1 + η) ym dy,
A2:@
g(0) = (θ − η)μ < 0,
g′(m) =
∫ m
0
(1− F (y)) ln(1 + η)(1 + η) ym · y
m2
dy > 0,
lim
m→∞ g(m) = θμ > 0.
�� g(m)�0�Eo, H�A2_/*-p�N�) m¯C , D�B� g(m¯C) = 0, =1l��:.
:h.
el�=�, H���+�$)
m∗C = m¯C ∧N, ()
<7
:
de ! m¯C < N #B� () �p�Ng, 9�� () .+� 97B)�+�A
�/��$)# (1, m¯C), F.+�97B)#
RC =
ln(1 + η)
m¯C
. ()
23e"8cBC, HENhc��N'=�:
de ! (1,m∗C) #+�A�/��$), RC #�� () �+�97B), Z[��
{e−RCX1,m
∗
C
t } #c, 7>
ψ1,m
∗
C (u) ≤ e−RCu ≤ e−RC(a,m)u ()
5� MIPQ: JRRNKSTSTLUMVUVWOXYP 865
4�, F. (a,m) #BmA�/��$).
<']��, H�A2ih�� .� Lundberg A@.2 ψD(u) 44�.
de ! RC #�� () . X
a,m
t �+�97B), ψD(u) #�� () .�� Xˆ
a,m
t
�+(/()0. H�
N'A@.
ψD(u) ≤ e−RCu ()
4�. F.�� Xˆa,mt #�� X
a,m
t �`bd�.
$�:@�� , H�t:@']l� .
le ! RC(a,m) � RD(a,m)
r#�� () ��� () �97B), (a,m) #
B�A�/��$). H�
N'=�4�:
RC(a,m) ≤ RD(a,m). ()
fg <�� �B� (), E=97B) RD(a,m) RON'B�:
(θ − η)μ + aημ(a,m) = a
∫ m/a
0
[1− F (y)]arydy, ()
RC(a,m) 9RO8�B�
(θ − η)μ + aημ(a,m) = a
∫ m/a
0
[1− F (y)](eary − 1)dy. ()
%m�]B� () � (), E=XB��nÆ&'.#�3�. !
f1(r) =
∫ m/a
0
[1− F (y)](eary − 1)dy,
f2(r) =
∫ m/a
0
[1− F (y)]arydy.
< EN_/, C:@ RC(a,m) ≤ RD(a,m), B\:@: 2Bm r ≥ 0 s
f1(r) ≥ f2(r) 4
�. =$
f1(r)− f2(r) =
∫ m/a
0
[1− F (y)](eary − 1− ary)dy,
A2:@: 2Bm r ≥ 0, y ≥ 0, s
eary − 1− ary ≥ 0
4�, =1 f1(r)− f2(r) ≥ 0, V f1(r) ≥ f2(r) 2Bm r ≥ 0 4�. <7l��:. :h.
l� .�:@no#u Dickson [20] �pi.
']H�q�l� �=�:@�� .
de _fg el� E=A@. RC(a,m) ≤ RD(a,m) 2BmE>$) (a,m) s
4�, =1H�
RC = RC(a∗,m∗C) ≤ RD(a∗,m∗C) ≤ RD ()
4�. F. (a∗,m∗C) #�� () .�+�A�/��$), 7 RD #�� () .+�97B
). < () .Nr�� =�, H�A2��
ψD(u) = e−RDu ≤ e−RCu,
=1���:. :h.
866 � � � � y z { 53�
0 1 2 3
0
1
r
Figure 1: Functions f1 and f2 giving RC(a,m)≤ RD(a,m)
f2(r)f1(r)
RC(a,m) RD(a,m)
< () � () ., H�E=+�/��$)AAX����/�����N4
]R�
�, EX" -�
�
�, 0�� [20] .��=�#G?A�3�. -� [20] ., +�
56./0��+�/��$)+Avs@" -�
�.
�� ./��� () .+�$)'/()0�34, VC)�6 ψ1,m
∗
C (u) ≤
e−RCu. =$ RC #+�97B), �NH����/()0�+(C)�6. e�� E=,
Lundberg A@.2 ψD(u) 44�. tt0moi ψD(u) Au�u34 ψ1,m
∗
C (u), >v
E<
v3 ψ1,m
∗
C (u), 70Ev3-w]^w.#m�x��.
4 pqrs
-0�7., �!" - Yi w<xG [0, 2] ��yt
�, Æ$ Yi ∼ U [0, 2], 1J μ = 1.
H���N'=�:
le �!" - Yi∼U [0, 2], 9�� () ��� () �+�,-:;/��$)$
m∗D = m¯D ∧ 2 ()
�
m∗C = m¯C ∧ 2, ()
F.
m¯D =
3η −
√
12ηθ − 3η2
η
,
m¯C=
η−(1+η) ln(1+η)+√[η−(1+η) ln(1+η)]2−(θ−η){2(1+η) ln(1+η)−2η−(1+η)[ln(1+η)]2}
1 + η − ηln(1+η) − 1+η2 ln(1 + η)
.
le �!" - Yi ∼ U [0, 2], 9
:
5� MIPQ: JRRNKSTSTLUMVUVWOXYP 867
(1) � m¯D ≤ 2 J, �� () �+�97B)$
RD =
η2
3η −
√
12ηθ − 3η2 . ()
(2) � m¯C ≤ 2 J, �� () �+�97B)$
RC=
(1 + η) ln(1 + η)− η − 1+η
2
[ln(1 + η)]2
η−(1+η) ln(1+η)+√[η−(1+η) ln(1+η)]2−(θ−η){2(1+η) ln(1+η)−2η−(1+η)[ln(1+η)]2} . ()
u 1 ! θ = , η = , H�,- D-A ����.�/()0� C-P ����.��
6. =?56- – ..
0 2 4 6 8 10
0
1
the initial surplus u
th
e
ru
in
pr
ob
ab
ilit
y ψ
Da,
m
(u)
Figure 2: the effect of u on the ψD
a,m(u) with m=m*D
a=1
a=
a=
a=
0 2 4 6 8 10
0
1
the initial surplus u
th
e
ru
in
pr
ob
ab
ilit
y ψ
D1,
m
(u)
Figure 3: the effect of u on the ruin probability ψD
1,m(u)
mD
*
=
m=
m=1
m=
868 � � � � y z { 53�
0 2 4 6 8 10
0
1
the initial surplus u
the
up
pe
r b
ou
nd
ex
p(−
R C
(a,m
)u)
Figure 4: the effect of u on the upper bound exp(−RC(a,m)u) with m=m*C
a=1
a=
a=
a=
0 2 4 6 8 10
0
1
the initial surplus u
the
up
pe
r b
ou
nd
ex
p(−
R C
(1,m
)u)
Figure 5: the effect of u on the upper bound exp(−RC(1,m)u)
m*C=
m=
m=1
m=
0 1 2
0
1
2
η
m
D*
1 2
0
1
2
η
m
C*
Figure 6: The effect of η on the optimal retention level mD* and mC*
θ=
θ=
θ=
θ=
θ=
θ=
5� MIPQ: JRRNKSTSTLUMVUVWOXYP 869
u 2 ! θ = , � , H�,-+�,-:;/��$) m∗C � m
∗
D. =?56-
..
0 1 2 3 4 5
0
1
initial surplus u
Figure 7: the effect of u on the ψD(u) and exp(−RCu)
exp(−RCu)
ψD(u)
< , H�EN_/, � m = m∗D = J, �� () �/()0�ivzy{ a �
w�7i(, - a = 1 J'�+(.
< , H�EN_/, � a = 1 J, /()02Bm m 7z, 4#- m = m∗D =
J'�+(. 4J#Q, �� () �/()0-/��$)$ (1,m∗D) J'�+(, 70!=?
NJX�� �=��x�.
< � , H�4<��X�] � �2�=�. V� m = m∗C =
J, �� () ./()0�C)�6�ivzy{ a �w�7i(, +>- a = 1 J'�+(.
� a = 1 J, �� () ./()0�C)�62Bm m 7z, 4#- m = m∗C = J'
�+(. 04NJx��� �=�.
< , H�EN_/+�$) m∗D ("3 m
∗
C ) #�i η �pw7pw�. 0moi�/
���4={uJ, ���|vzDy���.v}z{. �j�, � θ w\J, ���|P%
Dy���./���. H�EENih, � θ ≥ η J, ���|e�
���P%./���,
V|x$) m∗D = 0 ("3 m
∗
C = 0), D�2��/()0 ψD(u) ("3 ψ
1,m∗C (u))= 0. 0Eh
`', =��
�����J[�|Kma, -^}4AE</h. =1, H�B,-
η > θ �h`.
< EN_/, A@. ψD(u) ≤ e−RCu 2BmLÆLM u ≥ 0 s4�, 0NJx���
�=�.
yz 13{?}~~k2=��{/�~ma.
| } ~
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