Universidad Carlos III de Madrid
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INDEM - Working Paper Business Economic Series
2014-06-01
Optimal reinsurance under risk and uncertainty
Balbas de la Corte, Alejandro
http://hdl.handle.net/10016/19024
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UNIVERSIDAD CARLOS III DE MADRID
Working Paper
Business Economic Series
WP. 14-04
ISSN 1989-8843
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Instituto para el Desarrollo Empresarial.
Universidad Carlos III de Madrid
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Optimal reinsurance under risk and uncertainty
Alejandro Balbás1
Departamento de Economía e la Empresa
Universidad Carlos III de Madrid
Beatriz Balbás2
Departamento de Análisis Económico y Finanzas
Universidad de Castilla-La Mancha
Raquel Balbás3
Departamento Actuarial y Financiero
Universidad Complutense de Madrid
Antonio Heras4
Departamento Actuarial y Financiero
Universidad Complutense de Madrid
1
2
C/ Madrid, 126. 28903 Getafe (Madrid, Spain).
Avda. Real Fábrica de Seda, s/n. 45600 Talavera (Toledo, Spain).
3
Somosaguas-Campus. 28223 Pozuelo de Alarcón (Madrid, Spain).
4
Somosaguas-Campus. 28223 Pozuelo de Alarcón (Madrid, Spain).
Optimal reinsurance under risk and uncertainty
Alejandro Balbás, Beatriz Balbás,y Raquel Balbász and Antonio Herasx
Abstract. This paper deals with the optimal reinsurance problem if both insurer and
reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also
included. The insurer and reinsurer degrees of uncertainty do not have to be identical.
The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to
the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the
reinsurer moral hazard is totally eliminated.
Three main contributions seem to be reached. Firstly, necessary and su¢ cient optimality conditions are given. Secondly, the optimal contract is often a bang-bang solution,
i:e:, the sensitivity between the retained risk and the total claims saturates the imposed
constraints. For some special cases the optimal contract might not be bang-bang, but there
is always a bang-bang contract as close as desired to the optimal one. Thirdly, the optimal
reinsurance problem is equivalent to other linear programming problem, despite the fact
that risk, uncertainty, and many premium principles are not linear. This may be important because linear problems are easy to solve in practice, since there are very e¢ cient
algorithms.
Key Words. Risk and Uncertainty, Moral Hazard, Optimal Reinsurance and Optimality Conditions, Bang-Bang Solution, The Optimal Reinsurance Linear Problem.
A.M.S. Classification Subject. 91B30, 90C48.
JEL Classification. G22.
1.
Introduction
Since Borch (1960) and Arrow (1963) published their celebrated seminal papers, the
optimal reinsurance problem has been addressed by many authors and under many
di¤erent risk measurement methods and premium principles. Recent approaches are,
amongst many others, Kaluszka (2005), Cai and Tan (2007) and Chi and Tan (2013).
University Carlos III of Madrid. CL. Madrid 126. 28903 Getafe (Madrid, Spain). [email protected]
y
University of Castilla la Mancha. Avda. Real Fábrica de Seda, s/n. 45600 Talavera (Toledo,
Spain). [email protected]
z
University Complutense of Madrid. Department of Actuarial and Financial Economics.
Somosaguas-Campus. 28223 Pozuelo de Alarcón (Madrid, Spain). [email protected]
x
University Complutense of Madrid. Department of Actuarial and Financial Economics.
Somosaguas-Campus. 28223 Pozuelo de Alarcón (Madrid, Spain). [email protected]
1
Optimal reinsurance under risk and uncertainty
2
Usually, researchers consider the insurer point of view, though the reinsurer viewpoint
may be also incorporated (Cai et al., 2012, Cui et al., 2013, etc.). An interesting
survey about the State of the Art in 2009 may be found in Centeno and Simoes
(2009).
All the papers above assume that the statistical distribution of claims is known.
Nevertheless, measurement errors or lack of complete information may provoke discrepancies between the real and the estimated probabilities of the states of nature,
generating uncertain (also called ambiguous) frameworks. Actuarial and …nancial literature is recently paying signi…cant attention to those cases where the probabilities
of the scenarios are not totally known. Interesting examples are, among many others,
portfolio management (Zhu and M. Fukushima, 2009), equilibrium in asset markets
(Bossaerts et al., 2010) and optimal stopping (Riedel, 2009).
The …rst objective of this paper is to incorporate ambiguity in the optimal reinsurance problem, though many results will be also new in the uncertainty free setting.
Both insurer and reinsurer may be ambiguous, but their degrees of ambiguity do not
have to be identical. Since the reinsurer information about the reinsured set of policies
could be lower than the information of the insurer, it seems natural to assume that
the reinsurer ambiguity is higher, but we will not impose this hypothesis because
we will not need it. According to the empirical evidence and the famous Ellsberg
paradox, agents usually re‡ect ambiguity aversion, so we will accept this assumption
in our analysis. Though there are other recent approaches (Maccheroni et al., 2006),
the worst-case principle properly incorporates the ambiguity aversion (Gilboa and
Schmeidler, 1989), and therefore our analysis will deal with this principle when considering the insurer expected wealth, the insurer global risk (integrating uncertainty
too) and the reinsurer premium principle. Actually, all of the papers above deal with
ambiguity by means of a worst-case approach.
Stop-loss or closely related contracts frequently solve the optimal reinsurance
problem. These solutions have been often criticized by both theoretical researchers
and practitioners. In practice, reinsurers will rarely accept these solutions due to the
lack of incentives of the insurer to verify claims beyond some thresholds. Our second
objective will be to overcome this caveat. Consequently, the insurer decision variable
will be the (almost everywhere) mathematical derivative of the retained risk with respect to the global claims, rather than the retained risk itself. With this modi…cation
we can impose positive lower bounds to this decision variable, and therefore contracts
re‡ecting spreads with null derivative (‡at behavior of the retained risk with respect
to the global claims) become unfeasible. In other words, the usual reinsurer moral
hazard is eliminated with this approach.
The paper is organized as follows. Section 2 will present the general framework,
the set of priors, the properties of the insurer risk measure (integrating uncertainty),
the properties of the reinsurance premium principle (which may incorporate the rein-
Optimal reinsurance under risk and uncertainty
3
surer uncertainty) and the general optimal reinsurance problem we are going to deal
with. We will point out how our approach contains most of the usual cases and extends them all if ambiguity arises. Section 3 will be devoted to dealing with two dual
approaches. Theorems 4 and 5 will provide us with two alternative dual problems,
as well as two di¤erent families of necessary and su¢ cient optimality conditions. It
is worth to point out that one of the duals is linear.
The optimality conditions will generate two di¤erent ways permitting us to linearize the optimal reinsurance problem. The …rst one is the introduction of a linear
optimization problem generated by a dual solution. This method will allow us to
prove Theorem 7 in Section 4, which will show that the optimal contract is in the closure of a set composed of convex combinations of bang-bang contracts, i:e:, contracts
such that the derivative of the retained risk with respect to the total one saturates
the imposed constraints. A clear consequence is that in many classical approaches
one must …nd stop-loss or closely related optimal contracts. In our less restrictive
framework the optimal retention will be often a bang-bang solution.
Section 5 explores a second linearization procedure. In order to simplify the
exposition the focus is on the Robust Conditional Value at Risk (robust CV aR) as an
insurer risk/ambiguity measure and a reinsurer instrument to generate the premium
principle. The method applies for much more situations, but selecting one important
case we signi…cantly shorten the paper. Furthermore, the CV aR is becoming more
and more popular among researchers and practitioners due to its interesting properties
(Ogryczak and Ruszczynski, 2002).
Since one of the two duals of Section 3 is linear, we will construct the doubledual (dual of the dual) optimal reinsurance problem in Section 5, which is linear.
We will prove that the solution of the double-dual will directly lead to the optimal
reinsurance contract. This seems to be a very important property because there
are many e¢ cient algorithms solving linear problems in both …nite-dimensional and
in…nite-dimensional frameworks (Anderson and Nash, 1987). Besides, linear problems
often lead to extreme solutions, which explains why the non linear optimal reinsurance
problem may be solved by a bang-bang retention.
The last section of the paper summarizes the most important conclusions, emphasizing the two main novelties (uncertainty introduction and moral hazard elimination) and the three main contributions (necessary and su¢ cient optimality conditions,
bang-bang solutions and double-dual linear problems).
Throughout the paper we will need several mathematical notions about topological spaces, Banach and Hilbert spaces, strong and weak convergences, etc. Some
of them will be brie‡y summarized, but further discussions may be found in Luenberger (1969), Kelly (1975), Rudin (1973) and (1987), or Anderson and Nash (1987),
amongst others.
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H ≤ 1 A4?A4B4=C8=6 C74 ;>F4BC 0224?C01;4 B4=B8C8E8CH
#$ & %! 4C DB 5>;;>F C74 0??A>027 >5 ACI=4A et al 0=3
%>2:054;;0A et al 8= >A34A C> 8=CA>3D24 C74 =>C8>= >5 A8B: 5D=2C8>= '7DB
C7A>D67>DC C78B ?0?4A 0 A8B: <40BDA4 F8;; 14 0 5D=2C8>= ρ : L2 (0 ) −→ BD27 C70C
C74A4 4G8BCB 0 2>=E4G 0=3 F40:;H2><?02C BD1B4C Δρ >5 L2 (0 ) F8C7
ρ (y) = M ax {− IP (yz) ; z ∈ Δρ }
5>A 4E4AH y ∈ L2 (0 ) 5 7>;3B C74= 8C <0H 14 ?A>E43 C70C Δρ 8B D=8@D4 0=3
z ∈ Δρ 85 0=3 >=;H 85
−IP (yz) ≤ ρ (y)
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
5>A 4E4AH y ∈ L2 (0 ) DAC74A<>A4 7>;3B 85 0=3 >=;H 85 ρ 8B 2>=C8=D>DB
BD10338C8E4 ρ (y1 + y2 ) ≤ ρ (y1 ) + ρ (y2 ) 5>A yi ∈ L2 (0 ) i = 1, 2 0=3 7><>64
=4>DB ρ (λy) = λρ (y) 5>A λ ≥ 0 0=3 y ∈ L2 (0 ) *4 F8;; 0;B> 8<?>B4 ρ C> 14
Ẽρ −CA0=B;0C8>= 8=E0A80=C 5>A B><4 Ẽρ ∈ ρ (y + k) = ρ (y) − Ẽρ k 85 k ∈ 0=3
y ∈ L2 (0 ) C 8B 4@D8E0;4=C C> C74 5D;J;;<4=C >5
IP (z) = Ẽρ
5>A 4E4AH z ∈ Δρ &D<<0A8I8=6 F4 70E4
* %! -< Ẽρ ∈ *- ) :-)4 6=5*-: ρ : L2 (0 ) −→ 1; ;)1, <7 *- )
Ẽρ −<:)6;4)<176 16>):1)6< :1;3 5-);=:- 1. <0- <?7 -9=1>)4-6< +76,1<176; *-47? 074,
a) ρ : L2 (0 ) −→ 1; 67:5+76<16=7=; Ẽρ −<:)6;4)<176 16>):1)6< ;=*),,1<1>)6, 0757/-6-7=;
b) &0-:- -@1;<; ) +76>-@ )6, ?-)34A+758)+< ;-< Δρ ⊂ L2 (0 ) ;=+0 <0)< 074,; .7: ->-:A y ∈ L2 (0 ) )6, 074,; .7: ->-:A z ∈ Δρ .
*4 F8;; =>C ?A>E4 C74 4@D8E0;4=24 14CF44= >=38C8>=B a) 0=3 b) 01>E4 1420DB4
0=0;>6>DB ?A>>5B <0H 14 5>D=3 8= B4E4A0; ?0?4AB 5>A 8=BC0=24 B44 0;1RB et al.,
">C824 C70C 8<?;84B C70C ρ 8B F40:;H ;>F4A B4<82>=C8=D>DB
'74A4 0A4 <0=H A8B: <40BDA4B 8= ?A02C824 2CD0;;H 4E4AH 4G?42C0C8>= 1>D=343
A8B: <40BDA4 0=3 4E4AH 34E80C8>= <40BDA4 %>2:054;;0A et al B0C8B5H 4J=8C8>=
1 01>E4 F8C7 Ẽρ = 1 0=3 Ẽρ = 0 A4B?42C8E4;H '74 2>74A4=C A8B: <40BDA4B >5 ACI=4A
et al 0;B> B0C8B5H 4J=8C8>= 1 F8C7 Ẽρ = 1 B B083 8= C74 8=CA>3D2C8>=
C74 ;;B14A6 ?0A03>G 8=3820C4B C74 ?A4B4=24 >5 0<186D8CH 0E4AB8>= F7827 <0H 14
8=2>A?>A0C43 1H 340;8=6 F8C7 C74 F>ABC20B4 ?A8=28?;4 8;1>0 0=3 &27<483;4A 2>=B8BC4=C F0H C> 2>=B834A C78B ?A8=28?;4 8B C> 34J=4 MC74 A>1DBC 4GC4=B8>= >5 0
A8B: <40BDA4 A4;0C8E4 C> C74 B4C >5 ?A8>AB PU0 N >A 8=BC0=24 C74 A>1DBC CV aR F8C7
2>=J34=24 ;4E4; 0 ≤ μ < 1 A4;0C8E4 C> C74 B4C >5 ?A8>AB PU0 F8;; 14 68E4= 1H
RCV aR(
,μ) (y) := M ax
U
CV aR(p,μ) (y) ; p ∈ PU0
5>A 4E4AH y ∈ L2 (0 ) CV aR(p,μ) (y) 34=>C8=6 C74 DBD0; CV aR >5 y 85 p 8B C74 B4;42C43
?A>1018;8CH <40BDA4 0=3 μ 8B C74 ;4E4; >5 2>=J34=24 '74>A4< 2 14;>F B7>FB C70C
C78B 34J=8C8>= 8B 2>=B8BC4=C
!# %=887;- <0)< 0 ≤ μ < 1 &0-6 CV aR(p,μ) (y) -@1;<; .7: ->-:A p ∈ PU0
)6, ->-:A y ∈ L2 (0 ) <0- 5)@15=5 16 -@1;<; .7: ->-:A y ∈ L2 (0 ) )6,
RCV aR(
,μ) ;)<1;C-; -C61<176 1b) ?1<0 ẼRCV aR = 1 )6,
U
(PU ,μ)
ΔRCV aR =
(PU ,μ)
z ∈ L2 (0 ) ; IP (z) = 1 and ∃f ∈ PU with 0 ≤ z ≤
.
f
1μ
.
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
6 8):<1+=4): 4-),; <7
⎧
RCV aR(
,μ) (y) = M ax − IP (yz)
⎪
⎪
U
⎪
⎪
⎪
⎪
⎪ IP (f ) = 1
⎪
⎨
IP (z) = 1
⎪
⎪
⎪
f
⎪
0 ≤ z ≤ 1μ
, f ≤R
⎪
⎪
⎪
⎪
⎩
z ∈ L2 (0 ) and f ∈ L2 (0 ) are decision variables
.7: ->-:A y ∈ L2 (0 )
8ABC ;4C 0B ?A>E4 C70C C74 B4C ΔRCV aR 68E4= 1H 8B 2>=E4G =3443
(PU ,μ)
BD??>B4 C70C z1 0=3 z2 14;>=6 C> C78B B4C 0=3 C0:4 0 ≤ λ ≤ 1 5 f1 0=3 f2 0A4 C74
>1E8>DB <40BDA01;4 5D=2C8>=B 68E4= 1H C74= z = λz1 + (1 − λ) z2 0=3 f =
λf1 + (1 − λ) f2 CA8E80;;H B0C8B5H C74 2>=38C8>=B 6D0A0=C448=6 C70C z ∈ ΔRCV aR (PU ,μ)
R
ΔRCV aR 8B =>A<1>D=343 =3443 8<?;84B C70C z (2,IP ) ≤ 1μ
(PU ,μ)
(2,IP )
5>A 4E4AH z ∈ ΔRCV aR (PU ,μ)
ΔRCV aR 2;>B43 =3443 BD??>B4 C70C C74 B4@D4=24 (zn )
n=1 ⊂ ΔRCV aR P ,μ
(PU ,μ)
( U )
2
2>=E4A64B C> z 8= L (0 ) 0=3 ;4C DB ?A>E4 C70C z ∈ ΔRCV aR 4C (fn )n=1 ⊂
(PU ,μ)
2
L (0 ) C74 >1E8>DB B4@D4=24 &8=24 (fn )n=1 ⊂ PU 0=3 C78B B4C 8B F40:;H 2><?02C
C74A4 4G8BCB f ∈ PU F7827 8B 0 F40: 066;><4A0C8>= ?>8=C '7DB (z, f ) 8B 0 F40:
f
066;><4A0C8>= ?>8=C >5 (zn , fn )
n=1 0=3 C74A45>A4 0 ≤ z ≤ 1μ , f ≤ R CA8E80;;H 5>;;>F
fn
, fn ≤ R 5>A 4E4AH n ∈ 5A>< 0 ≤ zn ≤ 1μ
ΔRCV aR 8B F40:;H 2><?02C =3443 ΔRCV aR 8B F40:;H 2;>B43 1420DB4 8C
(PU ,μ)
(PU ,μ)
8B 2;>B43 0=3 0=3 2>=E4G 07=0=027 '74>A4< B> C74 ;0>6;DLB '74>A4< 8<?;84B
C70C ΔRCV aR 8B F40:;H 2><?02C 1420DB4 8C 8B =>A<1>D=343
(PU ,μ)
148=6 2>=E4G 0=3 F40:;H 2><?02C F4 70E4 C70C C74 <0G8<D< 8=
ΔRCV aR (PU ,μ)
C74 A867C 70=3 B834 >5 >1E8>DB;H 4G8BCB 4=>C4 1H ρ (y) C78B <0G8<D< C 8B
:=>F= %>2:054;;0A et al C70C 5>A p ∈ PU0
#!!
CV aR(p,μ) (y) = M ax
−p (yz) ; p (z) = 1, 0 ≤ z ≤
1
1−μ
7>;3B F74=4E4A y 8B 8=C46A01;4 F8C7 A4B?42C C> p 5 y ∈ L2 (0 ) C74= |y| R 8B 8=C46A01;4
F8C7 A4B?42C C> 0 0=3
M
p (|y|) =
0
M
|y| dp =
0
dp
|y|
d0 ≤
d0
M
0
|y| Rd0 < ∞
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
5>A 4E4AH p ∈ PU0 B> y 8B 8=C46A01;4 F8C7 A4B?42C C> p 0=3 CV aR(p,μ) (y) 4G8BCB
'74 C74>A4< F8;; 14 >1E8>DB;H ?A>E43 85 F4 B7>F C70C ρ (y) 4@D0;B C74 A867C 70=3
B834 >5 4=>C4 1H RCV aR(
,μ) (y) ≤ ∞ C74 BD?A4<D< 8= C74 A867C 70=3 B834
U
>5 5 p ∈ PU0 8<?;84B C74 4G8BC4=24 >5 0 ≤ z̃ ≤
1 = p (z̃) = IP z̃
1
1μ
F8C7
dp
d0
0=3
CV aR(p,μ) (y) = −p (yz̃) = −IP yz̃
dp
d0
.
dp
dp
z = z̃ dIP
∈ ΔRCV aR B0C8BJ4B C74 A4@D8A43 2>=38C8>=B 4=24
1420DB4 f = dIP
P
,μ
( U )
CV aR(p,μ) (y) = −IP (yz) ≤ ρ (y) 0=3 C74A45>A4 RCV aR(
,μ) (y) ≤ ρ (y)
U
'74 >??>B8C4 8=4@D0;8CH 0;B> 7>;3B =3443 85 ρ (y) = −IP (yz) F8C7 z ∈
ΔRCV aR C74= C0:4 C74 >1E8>DB f 0=3 C74 ?A>1018;8CH <40BDA4 dp = f d0 B0C
(PU ,μ)
1
0=3
8BJ4B p ∈ PU0 !>A4>E4A z̃ = fz C0:4 00 = 0 B0C8BJ4B 0 ≤ z̃ ≤ 1μ
M
p (z̃) =
M
z̃dp =
0
M
zd0 = IP (z) = 1,
z̃f d0 =
0
0
B>
RCV aR(
,μ) (y) ≥ CV aR(p,μ) (y) ≥ −p (yz̃) = −
U
−
M
0
yz̃f d0 = −
M
0
M
0
yz̃dp =
yzd0 = −IP (yz) = ρ (y) .
# &0- +76;<)6< :)6,75 >):1)*4- z = 1 *-476/; <7 ΔRCV aR *-+)=;(PU ,μ)
f = 1 ;)<1;C-; <0- +76,1<176; 7. -6+- ;07?; <0)< RCV aR(
,μ) (y) ≥
U
−P (y) .7: ->-:A y ∈ L2 (0 ) i.e. RCV aR(
,μ) 1; -@8-+<)<176 *7=6,-, 16 <0U
;-6;- 7. $7+3).-44): et al *-+)=;- f = 1 ;)<1;C-; <0# z ∈ ΔCV aRIP ,μ =⇒ z ∈ ΔRCV aR (PU ,μ)
:-9=1:-, +76,1<176; 7:-7>-: . R ≥ 1 1; +76;<)6< <0-6
0≤z≤
?1<0 μ̃ = 1 −
1μ
R
f
R
1
, f ≤ R =⇒ 0 ≤ z ≤
=
1−μ
1−μ
1 − μ̃
&0-:-.7:- )6, <:1>1)44A 4-), <7
CV aR(IP ,μ) (y) ≤ RCV aR(
,μ) (y) ≤ CV aR(IP ,μ̃) (y) .
U
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
< 1; -);A <7 ;-- <0)< 0 ≤ 1 − R1 ≤ μ̃ < 1 )6, μ̃ ≥ μ 6 7<0-: ?7:,; 1. R ≥ 1 1;
+76;<)6< <0-6 <0- :7*=;< CV aR 4)A; ?1<016 <0- ;8:-), 7. <?7 CV aRs
# %1514):4A 76- +)6 ,-C6- ) :7*=;< -@<-6;176 :-4)<1>- PU0 7. 5)6A +7
0-:-6< 7: -@8-+<)<176 *7=6,-, :1;3 5-);=:-; ;=+0 ); <0- ()6/ 5-);=:- <0- 87?-:
<:)6;.7:5 <0- ?-1/0<-, CV aR -<+ 44 7. <0-5 ?144 ;)<1;.A -C61<176 1 ?1<0 Ẽρ = 1
$7*=;< ,->1)<176; ;)<1;.A16/ Ẽρ = 0 5)A *- )4;7 ,-C6-, 7: 16;<)6+- <0- :7*=;<
)*;74=<- ,->1)<176 ?144 *- /1>-6 *A
RAD
U (y) := M ax
p (|y − p (y)|) ; p ∈ PU0 ,
)6, <0- :7*=;< ;<)6,):, ,->1)<176 ?144 *RSD
U (y) := M ax
p (y − p (y))2 ; p ∈ PU0
.
(- ?144 67< )6)4AB- <0- ,->1)<176; )*7>- *-+)=;- <0-A +)6 *- ;<=,1-, ?1<0 <0- 5-<0
7,; 7. &0-7:-5 2
"#& "# " 5 x ∈ L2 [0, M ] 8B C74 A4C08=43 A8B: 0=3 1 − x ∈
L2 [0, M ] 8B C74 24343 >=4 C74= A48=BDA0=24 ?A824 >5 F8;; 14 68E4= 1H
Re insurance/ Pr ice = Υ (J (1 − x)) ,
Υ : L2 (0 ) −→ 148=6 0 ẼΥ −CA0=B;0C8>= 8=E0A80=C A8B: <40BDA4 5>A B><4 ẼΥ ∈ 4J=8C8>= 1 ">C824 C70C 2>=C08=B C74 DBD0; ?A4<8D< ?A8=28?;4B =3443 5>A 0
;8=40A ?A8=28?;4
Υ (J (1 − x)) = IP (J (1 − x) c) ,
F8C7 c ∈ L2 (0 ) 0=3 c ≥ 0 8C 8B >1E8>DB C70C Υ B0C8BJ4B 4J=8C8>= 1b) F8C7 ΔΥ =
{−c} 0=3 ẼΥ = −IP (c) = ?0AC82D;0A C74 G?42C43 )0;D4 $A4<8D< $A8=28?;4
EV P P Re insurance/ Pr ice = (1 + α) IP (J (1 − x)) ,
α ≥ 0 148=6 C74 ;>038=6 A0C4 8B 0 ;8=40A ?A8=28?;4 F8C7 c = 1 + α 0=3 C74A45>A4 8C
8B 8=2;D343 8= B B083 01>E4 4E4AH 34E80C8>= <40BDA4 0=3 4E4AH 4G?42C0C8>=
1>D=343 >A 2>74A4=C A8B: <40BDA4 ρ B0C8BJ4B 4J=8C8>= 1 0=3 C74A45>A4 B> 3>4B
α, β ≥ 0 F8C7
Υ (J (1 − x)) = αIP (J (1 − x)) + βρ (−J (1 − x)) ,
ΔΥ = {−α − βz; z ∈ Δρ }
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
0=3 ẼΥ = −α − β Ẽρ B44 >=B4@D4=C;H <0=H 2;0BB820; =>=;8=40A ?A4<8D<
?A8=28?;4B 0A4 8=2;D343 8= *0=6 $A8=28?;4 &C0=30A3 4E80C8>= $A4<8D< $A8=
28?;4 4C2 !>A4 8<?>AC0=C;H 0;B> 8=2;D34B ?A4<8D< ?A8=28?;4B 8=E>;E8=6 C74
A48=BDA4A 0<186D8CH B8=24 5>A 8=BC0=24 A>1DBC 4GC4=B8>=B >5 C74 CV aR C74 F4867C43
CV aR C74 *0=6 <40BDA4 >A C74 BC0=30A3 34E80C8>= <0H ?;0H C74 A>;4 >5 ρ 8= !>A4>E4A Υ <0H 14 A4;0C43 C> C74 0<186D8CH ;4E4; >5 C74 A48=BDA4A F7827 3>4B =>C
=424BB0AH 4@D0; C74 8=BDA4A 0<186D8CH ;4E4; >A 8=BC0=24 8C <0H <0:4 B4=B4 C> 0B
BD<4 C70C C74 A48=BDA4A 0<186D8CH ;4E4; 8B 78674A 0=3 C74A45>A4 8C 8B 270A02C4A8I43 1H
C74 B4C >5 ?A8>AB f ∈ L2 (0 ) ; 0 ≤ f ≤ R̃, IP (f ) = 1 F8C7 R̃ ≥ R B44 ! "#! '74 ?A4B4=24 >5 0<186D8CH 0E4AB8>= 0=3 C74 F>ABC
20B4 ?A8=28?;4 8;1>0 0=3 &27<483;4A <0H 0;B> 9DBC85H C74 8=CA>3D2C8>= >5 MC74
A>1DBC 4G?42C0C8>= A4;0C8E4 C> C74 B4C >5 ?A8>AB PU N14;>F
U (W ) := M in
p (W ) ; p ∈ PU0 = M in {IP (W f ) ; f ∈ PU }
F74A4 W ∈ L2 (0 ) 8B C74 D=24AC08= 8=BDA4A F40;C7 0C T PU 8B C74 B4C >5 540B81;4
?A8>AB 0=3 p (W ) = IP (W f ) 8B C74 4G?42C0C8>= >5 W D=34A dp = f d0 ">C824
C70C C74 <8=8<D< 8= 4G8BCB 1420DB4
L2 (0 )
f → IP (W f ) ∈ 8B 0 F40:;H 2>=C8=D>DB 5D=2C8>= 5>A 4E4AH W ∈ L2 (0 ) 0=3 PU 8B 0 F40:;H 2><?02C
B4C *484ABCA0BB '74>A4< #1E8>DB;H ;403B C>
−
U (W ) = M ax {−p (W ) ; p ∈ P} = M ax {−IP (W f ) ; f ∈ PU } ,
5>A 4E4AH W ∈ L2 (0 ) = >C74A F>A3B −
U 0;B> B0C8BJ4B 4J=8C8>= 1 F8C7 ẼIEPU =
1 0=3 ΔIEPU = PU &D??>B4 C70C A 34=>C4B C74 8=2><4 64=4A0C43 1H C74 B>;3 8=BDA0=24 ?>;8284B
40A8=6 8= <8=3 C74 A48=BDA0=24 2>=CA02C 0=3 C74 8=BDA4A F40;C7 0C T F8;; 14
A − J (x) − Υ (J (1 − x)) .
'74 8=BDA4A <0H B4;42C 0 A8B: <40BDA4 ρ 4J=8C8>= 1 8= >A34A C> 2>=CA>; C74
A8B: >5 '74 >?C8<0; A48=BDA0=24 ?A>1;4< F8;; <0G8<8I4 C74 A>1DBC 4G?42C43
F40;C7 U F7827 8B 4@D8E0;4=C C> C74 <8=8<8I0C8>= >5 −
U 0=3 F8;; <8=8<8I4 ρ
'7DB 140A8=6 8= <8=3 C74 ?A>?4AC84B >5 −
U 0=3 ρ 4J=8C8>= 1 C74 >?C8<8I0C8>=
?A>1;4< 142><4B
⎧
⎨ M in ρ (A − J (x) − Υ (J (1 − x))) = ρ (−J (x)) + Ẽρ Υ (J (1 − x)) − Ẽρ A
M in − U (A − J (x) − Υ (J (1 − x))) = −
U (−J (x)) + Υ (J (1 − x)) − A ,
⎩
H≤x≤1
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
x ∈ L2 [0, M ] 148=6 C74 3428B8>= E0A801;4 &8=24 Ẽρ 0=3 A 0A4 2>=BC0=C F4 20= A4<>E4
Ẽρ A 0=3 A 8= C74 E42C>A >?C8<8I0C8>= ?A>1;4< 01>E4 F8C7>DC 0;C4A8=6 C74 $0A4C>
B>;DC8>=B !>A4>E4A B8=24 0=3 B7>F C70C −
0=3 ρ 0A4 2>=E4G 5D=2C8>=B
>= L2 (0 ) 4E4AH ?A>?4A $0A4C> >?C8<D< <0H 14 >1C08=43 1H <8=8<8I8=6 0 B20;0A
>1942C8E4
ρ (−J (x)) + Ẽρ Υ (J (1 − x)) + w (−
U (−J (x)) + Υ (J (1 − x))) ,
w > 0 148=6 0 MF4867CN5>A −
U "0:0H0<0 et al. 2CD0;;H w 8=3820C4B C74
A4;0C8E4 8<?>AC0=24 >5 C74 A>1DBC 4G?42C43 A4CDA= F8C7 A4B?42C C> C74 A8B: C74 78674A
C74 E0;D4 >5 w C74 78674A C74 8<?>AC0=24 >5 U 5>A C74 8=BDA4A !0=8?D;0C8=6 C74
>1942C8E4 5D=2C8>= 142><4B
ρ (−J (x)) + w (−
(−J (x))) + Ẽρ + w Υ (J (1 − x))
0=3 C74 >?C8<0; A48=BDA0=24 ?A>1;4< F8;; 14
M in ρ (−J (x)) + w (−
U (−J (x))) + Ẽρ + w Υ (J (1 − x))
,
H≤x≤1
x ∈ L2 [0, M ] 148=6 C74 3428B8>= E0A801;4
3.
D/& **,)"
4C DB 8=CA>3D24 0= 8=BCAD<4=C0; M!40= )0;D4 '74>A4<N F7>B4 ?A>>5 8B ><8CC43
1420DB4 0 B8<8;0A >=4 <0H 14 5>D=3 8= 0;1RB et al %=887;- <0)< Y 1; ) )6)+0 ;8)+- Z 1; 1<; ,=)4 )6, K ⊂ Z 1; ) +76>-@
)6, σ (Z, Y ) −+758)+< ;-< %=887;- <0)< ν 1; )6 166-: :-/=4): 8:7*)*141<A 5-);=:- 76
<0- 7:-4 σ−)4/-*:) 7. K -6,7?-, ?1<0 <0- σ (Z, Y ) −<78747/A &0-6 <0-:- -@1;<;
) =619=- kν ∈ K ;=+0 <0)<
K
074,; .7: ->-:A y ∈ Y.
≺ y, z
dν (z) =≺ y, kν
4C DB 8=CA>3D24 CF> 3D0; ?A>1;4<B 5>A 0B F4;; 0B =424BB0AH 0=3 BDP 284=C
0ADB7D7='D2:4A ;8:4 >?C8<0;8CH 2>=38C8>=B &C0=30A3 3D0;8CH C74>AH 3>4B =>C
6D0A0=C44 C74 01B4=24 >5 3D0;8CH 60? 85 8=J=8C438<4=B8>=0; ?A>1;4<B 0A4 8=E>;E43
D0;8CH 60?B <0H >22DA 4E4= 8= ;8=40A ?A>6A0<<8=6 =34AB>= 0=3 "0B7 =
>A34A C> >E4A2><4 C78B 20E40C JABC >5 0;; F4 F8;; 68E4 0 2>=20E4 3D0; B0C8B5H8=6 C74
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
&;0C4A @D0;8J20C8>= D4=14A64A F7827 6D0A0=C44B C74 01B4=24 >5 3D0;8CH 60?
'74= F4 F8;; 68E4 0 ;8=40A 3D0; =>C B0C8B5H8=6 C74 &;0C4A @D0;8J20C8>= "4E4AC74;4BB
C74 01B4=24 >5 3D0;8CH 60? F8C7 C74 2>=20E4 3D0; F8;; 0;;>F DB C> ?A>E4 C74 01B4=24
>5 3D0;8CH 60? F8C7 C74 ;8=40A >=4 C>>
!# 76;1,-: ":7*4-5
⎧
⎧
⎪
⎪
⎪
⎪
⎪ − Ẽρ + w ≺ 1, J (γ)
⎪
⎨
⎪
⎪
⎨
+ ≺ H, J z + wf + Ẽρ + w γ
M ax
⎪
⎪
⎪
⎪
⎩ − ≺ 1, J z + wf + Ẽ + w γ
⎪
⎪
ρ
⎪
⎪
⎩
f ∈ PU , z ∈ Δρ , γ ∈ ΔΥ
+
,
(f , z , γ ) ∈ PU × Δρ × ΔΥ *-16/ <0- ,-+1;176 >):1)*4-
&0-6
a) &0- 51615=5 7. )6, <0- 5)@15=5 7. ):- C61<- )<<)16)*4- )6,
1,-6<1+)4
b) %=887;- <0)< x 1; .-);1*4- )6, (f , z , γ ) 1; .-);1*4- &0-6 x ;74>-;
)6, (f , z , γ ) ;74>-; 1. )6, 764A 1. 7=< 7. -*-;/=- 6=44 ;-<; 7. [0, M ]
⎧
IP (J (x ) f ) ≥ IP (J (x ) f ) ,
∀f ∈ PU
⎪
⎪
⎪
⎪
(J
(x
)
z
)
≥
(J
(x
)
z)
,
∀z
∈ Δρ
⎪
IP
IP
⎪
⎨
IP (J (1 − x ) γ ) ≤ IP (J (1 − x ) γ) , ∀γ ∈ ΔΥ
⎪
x (t) = H (t) ,
J z + wf + Ẽρ + w γ (t) > 0
⎪
⎪
⎪
⎪
⎪
⎩ x (t) = 1,
J z + wf + Ẽρ + w γ (t) < 0
#!!
4C DB ?A>E4 C70C 8B B>;E01;4 =3443 C74 540B81;4 B4C H ≤ x ≤ 1 8B
F40:;H2><?02C 1420DB4 L2 [0, M ] 8B 0 8;14AC B?024 0=3 C74A45>A4 A4K4G8E4 ;0>6;DLB
'74>A4< 0=3 C74 >1942C8E4 5D=2C8>= >5 8B F40:;H ;>F4A B4<82>=C8=D>DB 1420DB4
8C 8B 0 2><?>B8C8>= >5 C74 σ (L2 [0, M ] , L2 [0, M ]) − σ (L2 (0 ) , L2 (0 ))2>=C8=D>DB
5D=2C8>= J 0=3 C74 σ (L2 (0 ) , L2 (0 ));>F4A B4<82>=C8=D>DB 5D=2C8>=B ρ −
0=3
Υ '74A45>A4 0CC08=B 0= >?C8<0; E0;D4 *484ABCA0BBL'74>A4<
B DBD0; φ+ (t) = M axt {φ (t) , 0} 0=3 φ− (t) = M axt {−φ (t) , 0} 5>A E4AH φ ∈ L2 [0, M ] 0=3
4E4AH 0 ≤ t ≤ M C 8B 40BH C> B44 C70C φ = φ+ − φ− 0=3 φ+ , φ− ∈ L2 [0, M ] &8<8;0A =>C0C8>=B <0H
14 DB43 8= B8<8;0A B8CD0C8>=B
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
G?A4BB8>=B F7827 0;B> 0??;84B 5>A Υ D=34A C74 >1E8>DB <>38J20C8>=B 0=3
B7>F C70C $A>1;4< 8B 4@D8E0;4=C C> $A>1;4<
⎧
⎪
M in θ1 + wθ2 + Ẽρ + w θ3
⎪
⎪
⎪
⎪
⎨ θ1 ≥ IP (J (x) z) ,
∀z ∈ Δρ
θ2 ≥ IP (J (x) f ) ,
∀f ∈ PU ,
⎪
⎪
⎪
∀γ ∈ ΔΥ
⎪ θ3 ≥ IP (J (x − 1) γ) ,
⎪
⎩
H ≤ x ≤ 1, θ1 , θ2 , θ3 ∈ (θ1 , θ1 , θ3 , x) ∈ × × × L2 [0, M ] 148=6 C74 3428B8>= E0A801;4 '74 JABC B42>=3
0=3 C78A3 2>=BCA08=CB >5 0A4 E0;D43 8= C74 B?024B >5 2>=C8=D>DB 5D=2C8>=B >= C74
F40:;H 2><?02C B?024B K = Δρ K = PU 0=3 K = ΔΥ A4B?42C8E4;H >;;>F8=6
D4=14A64A C74 0BB>280C43 3D0; E0A801;4B (ν 1 , ν 2 , ν 3 ) <DBC 14 8==4A A46D;0A
=>==460C8E4 <40BDA4B >= C74 >A4; σ−0;641A0B >5 C74 2><?02C B4CB 01>E4 0=3 C74
06A0=680= 5D=2C8>= 8B
L (θ1 , θ1 , θ3 , x, ν 1 , ν 2 , ν 3 ) = θ1 1 −
wθ2 1 −
Δρ
U
dν 1 +
dν 2 + Ẽρ + w θ3 1 −
IP (J (x) z) dν 1 (z) + w
Ẽρ + w
Δρ
Δ
U
Δ
dν 3 +
IP (J (x) f ) dν 2 (f ) +
IP (J (x) γ) dν 3 (γ) − Ẽρ + w
Δ
IP (J (1) γ) dν 3 (γ) .
&8=24 (ν 1 , ν 2 , ν 3 ) 8B 3D0;540B81;4 85 0=3 >=;H 85
Inf {L (θ1 , θ1 , θ3 , x, ν 1 , ν 2 , ν 3 ) ; H ≤ x ≤ 1, θ1 , θ1 θ3 , ∈ } > −∞
D4=14A64A ν 1 ν 2 0=3 ν 3 <DBC 14 ?A>1018;8CH <40BDA4B C>C0; <0BB 4@D0; C>
>=4 0=3 C74 3D0; >1942C8E4 142><4B B44 ⎧
⎧
(J (x) z) dν 1 (z) + w U IP (J (x) f ) dν 2 (f ) +
⎪
⎪
⎪
⎪
⎪ Δρ IP
⎪
⎪
⎪
⎨
⎪
⎪
⎨
Ẽρ + w Δ IP (J (x) γ) dν 3 (γ)
Inf
.
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
− Ẽρ + w Δ IP (J (1) γ) dν 3 (γ)
⎪
⎪
⎪
⎩
H≤x≤1
B 8= 0;1RB et al 4<<0 3 F8C7 Y = Z=L2 (0 ) 6D0A0=C44B C70C C74 3D0;
B>;DC8>= 8B 02784E43 8= 0 E42C>A >5 C7A44 8A02 4;C0B B> C74 3D0; E0A801;4 <0H B8<?;85H
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
C> (z, f, γ) ∈ Δρ × PU × ΔΥ 0=3 C74 3D0; >1942C8E4 142><4B B44 0=3 φ (z, f, γ) = − Ẽρ + w + IP (J (1) γ) +
InfHx1 IP (J (x) z) + wIP (J (x) f ) + Ẽρ + w IP (J (x) γ)
= − Ẽρ + w IP (J (1) γ) +
InfHx1 ≺ x, J (z)
+w ≺ x, J (f )
= − Ẽρ + w ≺ 1, J (γ)
+ Ẽρ + w ≺ x, J (γ)
+InfHx1 x, J z + wf + Ẽρ + w γ
.
'74 3D0; ?A>1;4< 8B
M ax φ (z, f, γ)
(z, f, γ) ∈ Δρ × PU × ΔΥ
'74 01B4=24 >5 3D0;8CH 60? 14CF44= >A 0=3 0=3 C74 B>;E018;8CH >5
7>;3 3D4 C> C74 &;0C4A 2>=38C8>= D4=14A64A F7827 8B CA8E80;;H 5D;J;;43
1H '74 8=J<D< >5 8B >1E8>DB;H 02784E43 0C
⎧
⎪
H (t) ,
if J z + wf + Ẽρ + w γ (t) > 0
⎪
⎪
⎨
H (t) ≤ x (t) ≤ 1, if J z + wf + Ẽρ + w γ (t) = 0
x (t) =
⎪
⎪
⎪
⎩ 1,
if J z + wf + Ẽρ + w γ (t) < 0
0=3 4@D0;B
HJ z + wf + Ẽρ + w γ
dt+
J ∗ (z+wf +(Ẽρ +w)γ )(t)>0
J z + wf + Ẽρ + w γ dt+
J ∗ (m+wp+(Ẽρ +w)π )(t)<0
=≺ H, M axt J z + wf + Ẽρ + w γ (t) , 0
− ≺ 1, M axt −J z + wf + Ẽρ + w γ (t) , 0
.
'7DB 0=3 ;403 C> C74 3D0; >1942C8E4 68E4= 8= 0=3 a) 8B ?A>E43
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
= >A34A C> ?A>E4 b) >F8=6 C> C74 =424BB0AH 0=3 BDP 284=C ?A8<0; 0=3 3D0;
>?C8<0;8CH 2>=38C8>=B 5>A 142><4 D4=14A64A ⎧ θ1 = IP (J (x ) z )
⎪
⎪
⎪
⎪
∀z ∈ Δρ
θ1 ≥ IP (J (x ) z) ,
⎪
⎪
⎪
⎪
θ2 = IP (J (x ) f )
⎪
⎪
⎪
⎪
∀f ∈ PU
⎨ θ2 ≥ IP (J (x ) f ) ,
,
θ3 = IP (J (x − 1) γ )
⎪
⎪
≥
(J
(x
−
1)
γ)
,
∀γ
∈
Δ
θ
⎪ 3
IP
Υ
⎪
⎪
⎪
⎪
x (t) = H (t) ,
if J z + wf + Ẽρ + w γ (t) > 0
⎪
⎪
⎪
⎪
⎪
⎩ x (t) = 1,
if J z + wf + Ẽρ + w γ (t) < 0
0=3 142><4B >1E8>DB
# -;81<- <0- .)+< <0)< 1; +76>-@ *=< 67< 416-): ?- ?144 ;-- <0)< 1< 1;
+47;-4A :-4)<-, <7 ;->-:)4 416-): 8:7*4-5; 1:;<4A 4-< =; ;-- <0)< 5)A *- -);14A
<:)6;.7:5-, 16<7 ) 416-): 8:7*4-5 )< 4-);< .7: )88:78:1)<- +071+-; 7. PU ρ )6, Υ <
1; 1587:<)6< .7: <?7 :-);76; 1:;<4A <0-:- ):- 5)6A )4/7:1<05; <7 ;74>- 16 8:)+<1+*7<0 C61<-,15-6;176)4 )6, 16C61<-,15-6;176)4 416-): 8:7*4-5; 6,-:;76 )6, );0
%-+76,4A 416-): 8:7*4-5; 7.<-6 8:-;-6< -@<:-5- 7: *)6/*)6/ ;74=<176; x
;=+0 <0)< x (t) = H (t) 7: x (t) = 1 5=+0 074, 7=< 7. -*-;/=- 6=44 ;-<; &01; +7=4,
-@84)16 ?0A <0- ;<78 47;; :-16;=:)6+- ;)<1;.A16/ x (t) = 0 7: x (t) = 1 .:-9=-6<4A
;74>-; 1. H = 0
!# 76;1,-: ":7*4-5
⎧
⎪
M ax ≺ H, λd − ≺ 1, λu − Ẽρ + w ≺ 1, J (γ)
⎪
⎪
⎪
⎨
J z + wf + Ẽρ + w γ − λd + λu = 0
⎪
⎪
λd ≥ 0, λu ≥ 0
⎪
⎪
⎩
f ∈ PU , z ∈ Δρ , γ ∈ ΔΥ , λd ∈ L2 [0, M ] , λu ∈ L2 [0, M ]
(f, z, γ, λd , λu ) ∈ PU × Δρ × ΔΥ × L2 [0, M ] × L2 [0, M ] *-16/ <0- ,-+1;176 >):1)*4-
a) &0- 51615=5 7. )6, <0- 5)@15=5 7. ):- C61<- )<<)16)*4- )6,
1,-6<1+)4
b) %=887;- <0)< x 1; .-);1*4- )6, (f , z , γ , λd , λu ) 1; .-);1*4- &0-6
<0-A ;74>- )6, :-;8-+<1>-4A 1. )6, 764A 1.
⎧
∀f ∈ PU
IP (J (x ) f ) ≥ IP (J (x ) f ) ,
⎪
⎪
⎪
⎪
∀z ∈ Δρ
⎨ IP (J (x ) z ) ≥ IP (J (x ) z) ,
IP (J (1 − x ) γ ) ≤ IP (J (1 − x ) γ) , ∀γ ∈ ΔΥ
⎪
⎪
λ
(x
−
H)
=
0
⎪
⎪
⎩ d
λu (1 − x ) = 0
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
#!! B 8= C74 ?A>>5 >5 '74>A4< 4 8B 4@D8E0;4=C C> '74 06A0=680= >5
<0H 0;B> 8=2>A?>A0C4 C74 <D;C8?;84AB λd λu ∈ L2 [0, M ] λd ≥ 0 λu ≥ 0 A4;0C43
C> C74 2>=BCA08=CB H ≤ x ≤ 1 = BD27 0 20B4 C74 06A0=680= 142><4B
L (θ1 , θ1 , θ3 , x, ν 1 , ν 2 , ν 3 , λd , λu ) =
L (θ1 , θ1 , θ3 , x, ν 1 , ν 2 , ν 3 ) + ≺ H − x, λd
+ ≺ x − 1, λu
,
F74A4 L 8B 68E4= 1H 4B834B C74 3D0; E0A801;4 <0H 14 B8<?;8J43 0608= 3D4 C>
4<<0 3 B44 0=3 C74 06A0=680= >5 142><4B B44 0=3 L (x, , z, f, γ, λd , λu ) = − Ẽρ + w IP (J (1) γ) + IP (J (x) z) +
wIP (J (x) , f ) + Ẽρ + w IP (J (x) , γ) + ≺ H − x, λd
= − Ẽρ + w ≺ 1, J (γ)
− ≺ 1, λu
≺ x, J z + wf + Ẽρ + w γ − λd + λu
+ ≺ x − 1, λu
+
+ ≺ H, λd
.
'74 8=J<D< >5 L (x, , z, f, γ, λd , λu ) 5>A x ∈ L2 [0, M ] 8B >1E8>DB;H J=8C4 85 0=3 >=;H
85 C74 JABC 2>=BCA08=C >5 7>;3B 8= F7827 20B4 8C 4@D0;B
− Ẽρ + w ≺ 1, J (γ)
− ≺ 1, λu
+ ≺ H, λd
.
'7DB 8B 0 3D0; ?A>1;4< >5 0=3 D4=14A64A '74 &;0C4A @D0;8
J20C8>= 3>4B =>C 7>;3 1420DB4 2>=BCA08=CB H ≤ x 0=3 x ≤ 1 0A4 E0;D43 8= L2 [0, M ]
0=3 C74 ?>B8C8E4 2>=4 >5 C78B B?024 70B E>83 8=C4A8>A '7DB 0 3D0;8CH 60? 14CF44=
0=3 <867C 4G8BC 0=3 C74 BD?A4<D< >5 <867C 14 BCA82C;H ;>F4A C70= C74
>?C8<0; E0;D4 "4E4AC74;4BB 85 F4 ?A>E4 C70C C74 >?C8<0; E0;D4B >5 0=3 0A4 A402701;4 0=3 2>8=2834 C74= C74 3D0;8CH 60? F8;; E0=8B7 0=3 C74 B>;DC8>=B >5 0=3 F8;; 14 270A02C4A8I43 1H C74 2><?;4<4=C0AH B;02:=4BB 2>=38C8>=B =34AB>=
0=3 "0B7 B 8= C74 ?A>>5 >5 '74>A4< 4 C74 2><?;4<4=C0AH B;02:=4BB 2>=38
C8>=B 14CF44= 0=3 2>8=2834 F8C7 B> 8C >=;H A4<08=B C> ?A>E4 C70C 02784E4B 8CB >?C8<0; E0;D4 0=3 8C 2>8=2834B F8C7 C74 >?C8<0; E0;D4 >=B834A 0
B>;DC8>= x >5 0=3 0 B>;DC8>= (f , z , γ ) >5 F7>B4 4G8BC4=24 8B 6D0A0=C443
1H '74>A4< 4a >=B834A
λd = J z + wf + Ẽρ + w γ 0=3
λu = J z + wf + Ẽρ + w γ +
.
19
Optimal reinsurance under risk and uncertainty
Obviously, (f ; z ;
d;
u)
is (33)-feasible, and the dual objective becomes
1; u
E~ + w
1; J ( )
o
n
;0
H; M ax J z + wf + E~ + w
n
o
~
1; M ax
J z + wf + E + w
; 0 + E~ + w J ( )
H;
=
;
d
;
which coincides with the (25)-optimal value because there is no duality gap between
(25) and (26) (Theorem 4). Hence, since the (25)-optimal value is an upper value
for the (33)-objective (Anderson and Nash, 1987), (f ; z ; ; d ; u ) solves (33) and
there is no duality gap between (25) and (33).
Remark 5. Theorem 5 implies that for appropriate PU , and the dual solution
(f ; z ; ; d ; u ) of (33) may be obtained in practice by linear programming methods,
which is a very good new because interesting algorithms to solve these problems are
available in both, …nite and in…nite dimensions (Anderson and Nash, 1987). Once
the dual solution was computed, the optimal retention J(x ) may be obtained by
(34), which is much easier to apply once we know (f ; z ; ; d ; u ). Besides, if (25)
has a linear dual, then this problem should be “almost linear” too. Sections 4 and
5 will show two di¤erent methods linearizing Problem (25) (see Problems (36) and
(40) below).
4.
On the existence of bang-bang solutions
The literature about the optimal reinsurance problem frequently obtains stop-loss or
closely related solutions such as stop-loss contracts with an upper bound, given by
J (1
x ) (t) = (t
a)+
(t
b)+ ;
0
t
M and 0
a < b
M .2 Obviously, the (Lebesgue almost everywhere)
derivative of J (x ) is given by
x (t) =
1; if t < a or t > b
0; if a < t < b
and Lemma 6 below will show that this is an extreme point of the (25)-feasible set
if H = 0. A natural question is how general this result is. Let us prove Theorem
7 below showing that some extreme points are “good approximations” of the (25)solution for general choices of H and general uncertainty levels for both the insurer
and the reinsurer.
2
Balbás et al. (2009) and (2011) obtain stop-loss optimal contracts even for many problems
without comonotonicity restrictions.
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
%=887;- <0)< x 1; .-);1*4- &0-6 <0- +76,1<176; *-47? ):- -9=1>)
4-6<
a) x 1; ) -@<:-5- 8716< 7. <0- .-);1*4- ;-< H ≤ x ≤ 1
b) &0-:- -@1;< ) 7:-45-);=:)*4- 8):<1<176 [0, M ] = A ∪ B ;=+0 <0)< x (t) = H (t)
1. t ∈ A )6, x (t) = 1 1. t ∈ B
-6+-.7:<0 <0- .-);1*4- ;74=<176 x )*7>- ?144 *- +)44-, *)6/*)6/ :-<-6<176
#!! b) =⇒ a) 8B >1E8>DB B> ;4C DB ?A>E4 C74 2>=E4AB4 8<?;820C8>= &D??>B4 C70C
a) 7>;3B >=B834A C = {0 ≤ t ≤ M ; H (t) < x (t) < 1} 0=3
Cn =
0 ≤ t ≤ M ; H (t) +
1
1
< x (t) < 1 −
n
n
414B6D4 <40BDA4 C74= L (C) =
5>A n ∈ #1E8>DB;H C = ∪
n=1 Cn 5 L 8B C74
Limn L (Cn ) 1420DB4 (Cn )nIN 8B 0= =>= 342A40B8=6 B4@D4=24 4=24 L (C) = 0 85
L (Cn ) = 0 5>A 4E4AH n 5 5>A B><4 n F4 70E4 L (Cn ) > 0 C74= H (t) < x (t) − n1 ≤ 1
0=3 H (t) ≤ x (t) + n1 < 1 5>A t ∈ Cn B> 5D=2C8>=B
x1 (t) = x (t) − n1 , x2 (t) = x (t) +
1
n
5>A t ∈ Cn 0=3 x1 (t) = x2 (t) = x (t) 5>A t ∈
/ Cn 0A4 1>C7 540B81;4 '7DB
1
1
x = 2 x1 + 2 x2 2>=CA0382CB a)
!# %=887;- <0)< Ẽρ + w ≥ 0 . x ;74>-; <0-6 .7: ->-:A n ∈ <0-:- -@1;<; ) *)6/*)6/ .-);1*4- :-<-6<176 xn ;=+0 <0)< <0- 7*2-+<1>- .=6+<176 7. ;)<1;C-;
ρ (−J (x )) + w (−
U (−J (x ))) + Ẽρ + w Υ (J (1 − x ))
≤ ρ (−J (xn )) + w (−
U (−J (xn ))) + Ẽρ + w Υ (J (1 − xn )) + n1 .
#!!
4C (f , z , γ ) B>;E4 '74= 0=3 B7>F C70C
ρ (−J (x )) = IP (J (x ) z )
−
U (−J (x )) = IP (J (x ) f )
Υ (J (1 − x )) = −IP (J (1 − x ) γ ) .
'74 540B81;4 B4C 8B F40:;H2><?02C 0=3 C74A45>A4 022>A38=6 C> C74 A48=!8;<0=
'74>A4< 85 n ∈ C74= C74 ;8=40A >?C8<8I0C8>= ?A>1;4<
M in IP (J (x) z ) + w (IP (J (x) f )) − Ẽρ + w IP (J (1 − x) γ )
H≤x≤1
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
B0C8BJ4B C74 4G8BC4=24 >5 0= 4GCA4<4 ?>8=C i.e. 0 10=610=6 A4C4=C8>= B44 4<<0 6
H ≤ xn ≤ 1 BD27 C70C
IP (J (x ) z ) + w (IP (J (x ) f )) − Ẽρ + w IP (J (1 − x ) γ ) ≤
IP (J (xn ) z ) + w (IP (J (xn ) f )) − Ẽρ + w IP (J (1 − xn ) γ ) +
1
n
=34AB>= 0=3 "0B7 4=24 0=3 CA8E80;;H ;403 C>
IP (J (xn ) z ) + w (IP (J (xn ) f )) + Ẽρ + w IP (J (1 − xn ) (−γ ))
≤ ρ (−J (xn )) + w (−
U (−J (xn ))) + Ẽρ + w Υ (J (1 − xn ))
0=3 142><4B >1E8>DB
# 76,1<176 Ẽρ + w ≥ 0 ?144 .:-9=-6<4A 074, 7: 16;<)6+- )++7:,16/ <7
$-5):3 3 1< 074,; .7: -@8-+<)<176 *7=6,-, :1;3 5-);=:-; ,->1)<176 5-);=:-; )6,
<0-1: :7*=;< -@<-6;176;
&0-7:-5 7 /=):)6<--; <0)< )88:78:1)<- *)6/*)6/ :-<-6<176; ?144 *- ); +47;- );
,-;1:-, <7 <0- 78<15)4 ;74=<176 &0- -@1;<-6+- 7. ) *)6/*)6/ 78<15)4 :-<-6<176 +)6 *8:7>-, 1. <0- 416-): 8:7*4-5 0); )6 -@<:-5- ;74=<176 6,-:;76 )6, );0 ;07? <0)< <01; -@<:-5- ;74=<176 +)6 *- /=):)6<--, =6,-: ),-9=)<- );;=58<176; 7:
-@)584- 1. 7: 1; ) C61<-,15-6;176)4 8:7*4-5 ?01+0 074,; .7: 16;<)6+-
1. <0- <7<)4 5);; 7. 0 1; +76+-6<:)<-, 76 ) C61<- ;=*;-< 7. [0, M ]
# =5-:1+)4 -@8-:15-6<; 76,1<176; 7. &0-7:-5 4 ?-:- =;-.=4 <7
8:7>- <0- -@1;<-6+- 7. *)6/*)6/ ;74=<176; )6, <0-A ?144 *- )4;7 =;-.=4 <7 16 7:,-: <7
;-- <0)< ?- +)667< /7 *-A76, &0-7:-5 7 )6, /1>- +47;-, -@8:-;;176; .7: ;-<;
A = {t; x (t) = H (t)} and B = {t; x (t) = 1} ,
J(x ) ,-67<16/ <0- 78<15)4 +76<:)+< '6,-: 57:- +4);;1+)4 )88:7)+0-; J(x ) 1; )
;<7847;; :-16;=:)6+- 7: ) +47;-4A :-4)<-, 76- 7: ) ;<7847;; +76<:)+< <0-:- -@1;<;
M0 ∈ [0, M ] ;=+0 <0)<
A = (M0 , M ) and B = (0, M0 ) .
->-:<0-4-;; 16 7=: ;-<<16/ A )6, B +:1<1+)44A ,-8-6, 76 <0- )5*1/=1<A ;-< PU )6,
<0- :1;3 .=6+<176; ρ )6, Υ )6, 5)6A ,1F-:-6< ;1<=)<176; 5)A ):1;- -< =; ;07? <01;
.)+< ?1<0 ;1584- 144=;<:)<1>- -@)584-; 7<1+- <0)< <0-;- -@)584-; )4;7 )884A 16 <0676)5*1/=7=; )6, 416-): +);-
%=887;- <0)< M = 5 <0- ;=887:< 7. 0 1; {1, 2, 3, 4, 5} )6, 0 (i) = 0.2 i =
1, 2, 3, 4, 5 %=887;- <0)< R = 1 676)5*1/=7=; +);- )6, <0-:-.7:- PU 764A +76<)16;
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
<0- B-:7>):1)6+- :)6,75 >):1)*4- f = 1 %=887;- C6)44A <0)< ρ )6, Υ ):- 416-): Δρ
764A +76<)16; <0- -4-5-6< z = 1 ;7 Ẽρ = 1 )6, ΔΥ 764A +76<)16; γ = −c ∈ L2 (0 )
?1<0 c ≥ 0 ;7 Υ 1; /1>-6 *A -67<- ci = c (i) ≥ 0 i = 1, 2, 3, 4, 5 )6, <)3w = 3 16 %16+- Ẽρ = 1 )6, 0 (z + wf = 4) = 1 @8:-;;176 15841-; <0)<
⎧
⎪
s<1⇒
J Ẽρ + w c = 0.6 5i=1 ci ,
⎪
J (z + wf ) = 12
⎪
⎪
⎪
⎪
⎪
1 < s < 2 ⇒ J Ẽρ + w c = 0.6 5i=2 ci .
⎪
J (z + wf ) = 9.6
⎪
⎨
J (z + wf ) = 7.2
s < 2 < 3 ⇒ J Ẽρ + w c = 0.6 5i=3 ci ,
⎪
⎪
⎪
⎪
J (z + wf ) = 4.8
s < 3 < 4 ⇒ J Ẽρ + w c = 0.6 (c4 + c5 ) ,
⎪
⎪
⎪
⎪
⎪
⎩ 4<s⇒
J (z + wf ) = 2.4
J Ẽρ + w c = 0.6c5 ,
&0=; 4-),; <7 <0- *)6/*)6/ 78<15)4 ;74=<176 x (t) +0):)+<-:1B-, *A
⎧
t < 1, 0.6 5i=1 ci < 12 ⇒
x = H
⎪
⎪
⎪
⎪
⎪
⎪
x = 1
t < 1, 0.6 5i=1 ci > 12 ⇒
⎪
⎪
⎪
⎪
⎪
⎪
1 < t < 2, 0.6 5i=2 ci < 9.6 ⇒ x = H
⎪
⎪
⎪
⎪
⎪
⎪
1 < t < 2, 0.6 5i=2 ci > 9.6 ⇒ x = 1
⎪
⎪
⎪
⎪
⎨ 2 < t < 3, 0.6 5 c < 7.2 ⇒ x = H
i=3 i
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
2 < t < 3, 0.6
5
i=3 ci
> 7.2 ⇒
x = 1
3 < t < 4, 0.6 (c4 + c5 ) < 4.8 ⇒ x = H
3 < t < 4, 0.6 (c4 + c5 ) > 4.8 ⇒ x = 1
4 < t, 0.6c5 < 2.4 ⇒
4 < t, 0.6c5 > 2.4 ⇒
x = H
x = 1
%=887;- <0)< ci = 6 i = 1, 2, 3 )6, ci = 2.5 i = 4, 5 &0-6 4-),; <7 x = 1 1.
t < 2 )6, x = H 1. t > 2 ?01+0 *-+75-; <0- ;<7847;; +76<:)+< J(1 − x ) = (t − 2)+
16 <0- +4);;1+ ;+-6):17 H = 0 ->-:<0-4-;; <0- 7887;1<- ;1<=)<176 5)A )4;7 ;74>
6 .)+< 1. <0- :-16;=:-: =;-; ) >):1)*4- 47),16/ :)<- α (t) 8-6)41B16/ 01/0 +4)15;
;=+0 ); ci = 1.05 i = 1, 2, 3 )6, ci = 5 i = 4, 5 <0-6 4-),; <7 x = H 1.
t < 3 )6, x = 1 1. t > 3 ?01+0 *-+75-; J(x ) = (t − 3)+ 16 <0- +4);;1+ +);- H = 0
=>C824 C70C J ∗
Ẽρ + w c (i) 0=3 J ∗ (z ∗ + wf ∗ ) (i) 0A4 =>C A4;4E0=C 5>A i = 1, 2, 3, 4, 5 B8=24
C74 414B6D4 <40BDA4 >5 C78B B4C E0=8B74B 0=3 J ∗
Ẽρ + w c , J ∗ (z ∗ + wf ∗ ) ∈ L2 [0, M ]
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
16)44A 57:- +7584-@ *)6/*)6/ ;74=<176; ):- )4;7 .-);1*4- 7: 16;<)6+- 1. c1 = 7
c2 = 1 c3 = 6 c4 = 1 )6, c5 = 6 <0-6 x = H .7: t ∈ (1, 2) ∪ (3, 4) )6, x = 1 .7:
t ∈ (0, 1) ∪ (2, 3) ∪ (3, 5)
5.
T" )*.#'& ,#(-/,( &#(, *,)&'
B 8=3820C43 8= %4<0A: 5'74>A4< 5 8<?;84B C70C 5>A 0??A>?A80C4 PU ρ 0=3 Υ C74 3D0;
B>;DC8>= (f , z , γ , λd , λu ) >5 <0H 14 >1C08=43 8= ?A02C824 1H ;8=40A ?A>6A0<<8=6
<4C7>3B #=24 (f , z , γ , λd , λu ) 8B :=>F= C74 ;8=40A ?A>1;4< <0H ;403 C> C74
>?C8<0; A4C4=C8>= J (x ) B44 C74 ?A>>5 >5 '74>A4< 7 "4E4AC74;4BB C74A4 8B B42>=3
<4C7>3 0;;>F8=6 DB C> ;8=40A8I4 0=3 C78B 8B C74 5>2DB >5 C78B B42C8>=
&D??>B4 C70C C74 BD16A0384=CB >5 ρ 0=3 Υ 0A4 68E4= 1H ;8=40A 2>=BCA08=CB A>1DBC
CV aR A>1DBC 01B>;DC4 34E80C8>= <0=H 20B4B >5 C74 A>1DBC F4867C43 CV aR 4C2 B44
'74>A4< 2 0=3 %4<0A: 3 '74= F4 20= 68E4 0 ;8=40A 3D0; >5 =34AB>= 0=3
"0B7 F7827 8B C74 3>D1;43D0; >5 0=3 2>D;3 ;403 C> C74 >?C8<0; A4C4=C8>=
C>> = >C74A F>A3B C74 >?C8<0; A4C4=C8>= 2>D;3 14 38A42C;H >1C08=43 1H B>;E8=6 0
;8=40A ?A>1;4< = >A34A C> B7>AC4= C74 4G?>B8C8>= ;4C DB 5>2DB >= C74 A>1DBC CV aR
1DC C74B4 8340B 0??;H 8= <D27 <>A4 5A0<4F>A:B
&D??>B4 C70C ρ = RCV aR(
,μ) 0=3 Υ (y) = αIP (y) + βRCV aR(
˜ ,μ̃) (−y)
U
U
α, β ≥ 0 B44 F74A4 C74 D=24AC08=CH ;4E4; >5 C74 A48=BDA4A P̃U0 8B 64=4A0C43 1H
C74 B4C >5 ?A8>AB
P̃U = h ∈ L2 (0 ) ; 0 ≤ h ≤ R̃, IP (h) = 1 ,
F74A4 R̃ ∈ L2 (0 ) 0=3 R̃ ≥ 1 B 0;A403H B083 85 R̃ ≥ R C74= C74 A48=BDA4A
D=24AC08=CH F8;; 14 78674A C70= C74 8=BDA4A >=4 '78B 8B 0 @D8C4 =0CDA0; 0BBD<?C8>=
1420DB4 C74 C74 A48=BDA4A 8=5>A<0C8>= 01>DC C74 8=E>;E43 ?>;8284B <0H 14 ;>F4A 1DC
F4 F8;; =>C 8<?>B4 8C = 64=4A0; F4 F8;; =>C =443 0=H A4;0C8>=B78? 14CF44= R 0=3
R̃ &8<8;0A;H F4 F8;; =>C 8<?>B4 0=H A4;0C8>=B78? 14CF44= C74 2>=J34=24 ;4E4;B μ 0=3
μ̃ DAC74A<>A4 5>A R = 1 R̃ = 1 F4 8=2;D34 C74 =>=0<186D>DB 8=BDA4A A48=BDA4A
8= C74 0=0;HB8B 0=3 0=0;>6>DB;H 5>A R̃ = 1 α ≥ 1 0=3 β = 0 F4 8=2;D34 C74 EV P P
0B 0 ?0AC82D;0A 20B4
'0:4 C = (1 + w) αIP (J (1)) $A>?>B8C8>= 1 G?A4BB8>= '74>A4< 2 G
">C824 C70C C74 B>;DC8>= >5 C78B 4G0<?;4 8B 8= C74 ;8=4 >5 C7>B4 B>;DC8>=B 5>D=3 1H D8 et al. 5>A A8B: 5D=2C8>=B 0=3 ?A4<8D< ?A8=28?;4B 68E4= 1H 38BC>AC8>=B '74 8=C4A?A4C0C8>= >5 '74>A4< 7
8B C70C C78B CH?4 >5 64=4A0; B>;DC8>= BC8;; 0??;84B 8= 0<186D>DB 5A0<4F>A:B F8C7 A8B: 5D=2C8>=B 0=3
?A4<8D< ?A8=28?;4B 8=2>A?>A0C8=6 1>C7 A8B: 0=3 0<186D8CH 0E4AB8>= 0B F4;; 0B 8<?>B8=6 0 ;>F4A
1>D=3 H F7827 C>C0;;H 4;8<8=0C4B C74 A48=BDA4A <>A0; 70I0A3
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
?A4BB8>=B 0=3 0=3 '74>A4< 5 8<?;H C70C 142><4B
⎧
⎪
M ax ≺ H, λd − ≺ 1, λu + (1 + w) β ≺ 1, J (γ) +C
⎪
⎧
⎪
⎪
⎪
J (z + wf − (1 + w) βγ) − λd + λu
⎪
⎪
⎧
⎪
⎪
⎪
⎪
⎪
⎪
= (1 + w) αJ (1)
IP (g) = 1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(f
)
=
1
g
IP
⎪
⎪
⎪
⎨ ≤R
⎨
⎪
⎪
⎪
⎪
f ≤R
⎨
IP (γ) = 1
⎪
⎪
⎪
⎪
h
⎪
⎪
γ ≤ 1μ̃
⎪
⎪
IP (z) = 1
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
IP (h) = 1
⎪
⎪
g
⎪
⎪
⎩ z ≤ 1μ
⎪
⎪
⎪
⎪
⎧
⎪
⎪
⎪
⎨ h ≤ R̃
⎪
⎪
⎪
⎪
λd , λu ∈ L2 [0, M ] , f, z, g, γ, h ∈ L2 (0 )
⎪
⎪
⎩
⎩
λ , λ , f, z, g, γ, h ≥ 0
d
u
&8=24 C78B ?A>1;4< 8B >1E8>DB;H ;8=40A F4 20= 2>=BCAD2C 8CB ;8=40A 3D0; D4=14A64A
>A =34AB>= 0=3 "0B7 F7827 F8;; 14 20;;43 3>D1;43D0; >5 0=3 8B
68E4= 1H
M in
s.t.
C − (1 + w) αIP (J (λ1 )) + λ2 + IP (Rλ3 ) + λ4
+λ6 + IP (Rλ7 ) + λ8 + λ10 + IP R̃λ11
⎧
⎧
(1 + w) βJ (1 − λ1 ) − λ8 − λ9 ≤ 0
H
−
λ
≤
0
⎪
⎪
1
⎪
⎪
⎪
⎪
1
⎪
⎪
−
1
≤
0
λ
⎨ 1
⎨ 1μ̃ λ9 − λ10 − λ11 ≤ 0
wJ (λ1 ) − λ2 − λ3 ≤ 0
λ1 ∈ L2 [0, M ] , λ2 , λ4 , λ6 , λ8 , λ10 ∈ ⎪
⎪
⎪
J (λ1 ) − λ4 − λ5 ≤ 0 ⎪
⎪
⎪
λ , λ , λ , λ , λ ∈ L2 (0 )
⎪
⎪
⎩ 3 5 7 9 11
⎩ 1
λ − λ 6 − λ7 ≤ 0
λ3 , λ5 , λ7 , λ9 , λ11 ≥ 0
1μ 5
'74 2><?;4<4=C0AH B;02:=4BB 2>=38C8>=B 14CF44= 0=3 0A4
⎧
⎧ ⎧ λd (λ1 − H) = 0
γ (λ8 + λ9 − (1 + w) βJ (1 − λ1 )) = 0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪ λ (1 − λ ) = 0
⎪
⎪
⎨ h λ10 + λ11 − 1μ̃
⎪
λ9 = 0
⎪ ⎨ u 1 ⎪
f (λ2 + λ3 − wJ (λ1 )) = 0
⎪
⎪
λ3 (f − R) = 0
⎪
⎪
⎪
⎪
(λ
+
λ
−
J
(λ
))
=
0
z
⎪
⎪
⎪
4
5
1
⎪
⎪
⎪
⎨ ⎪
⎪ λ z − g ∗ = 0
⎪
⎩ g λ + λ − 1 λ = 0 ⎪
⎩ 5
1μ
6
7
1μ 5
⎧
⎪
⎪
⎪
λ7 (g − R) = 0
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
h∗
⎪
λ9 γ − 1μ̃
=0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ ⎪
⎩ λ11 h − R̃ = 0
'74 2>=BCA08=CB >5 0=3 0;>=6 F8C7 270A02C4A8I4 C74 3D0; 0=3 C74
3>D1;43D0; B>;DC8>=B
D = (λd , λu , f , z , g , γ , h )
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
0=3
S = (λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 , λ8 , λ9 , λ10 , λ11 )
85 8B B>;E01;4 0=3 C74A4 8B => 3D0;8CH 60? 14CF44= 0=3 =34AB>= 0=3
"0B7 >=E4AB4;H 85 C74A4 4G8BC 540B81;4 0=3 540B81;4 B>;DC8>=B >5 C74= 8B B>;E01;4 0=3 C74A4 8B => 3D0;8CH 60? 14CF44= 0=3 = 64=4A0;
C74 B>;E018;8CH >5 0=3 C74 01B4=24 >5 3D0;8CH 60? 20= =>C 14 6D0A0=C443
"443;4BB C> B0H 8C 8B 8<?>AC0=C C> B44 C70C C74 B>;DC8>= >5 ;403B C> C74 >?C8<0;
B>;DC8>= x >5 0=3 C74 >?C8<0; A4C4=C8>= J (x ) $A>?>B8C8>= 8 14;>F B7>FB C70C
C74 01B4=24 >5 3D0;8CH 60? ?;0HB 0 2A8C820; A>;4
#!"!$%! %=887;- <0)< <0- ,7=*4-,=)4 8:7*4-5 1; ;74>)*4- )6, 0); 67
,=)41<A /)8 ?1<0 <0- ,=)4 8:7*4-5 %=887;- <0)< ;74>-; <0- ,7=*4-,=)4
&0-6 x = λ1 ;74>-; <0- 78<15)4 :-16;=:)6+- 8:7*4-5 #!! >=B834A C74 B>;DC8>= >5 F7>B4 4G8BC4=24 8B 6D0A0=C443 1H '74
>A4< 5 22>A38=6 C> '74>A4< 5 F4 <DBC ?A>E4 C70C 7>;3B 5>A λ1 0=3 '74 ?A>?>B8C8>= 0BBD<?C8>=B 8<?;H C70C 7>;3B 0=3 C74A45>A4 C74 5>DAC7 0=3 J5C7
2>=38C8>=B 8= 7>;3 C>> '7DB 8C >=;H A4<08=B C> B44 C70C C74 JABC B42>=3 0=3
C78A3 2>=38C8>=B 8= 7>;3 &8=24 C74 C7A44 8=4@D0;8C84B 70E4 0 B8<8;0A ?A>>5 ;4C DB
B7>F C70C
IP (J (λ1 ) z ) ≥ IP (J (λ1 ) z)
7>;3B 5>A 4E4AH z ∈ Δρ 0=3 ;4C DB ><8C C74 CF> A4<08=8=6 20B4B '74 5>DAC7 4@D0C8>=
>5 8<?;84B C70C J (λ1 ) z = z (λ4 + λ5 ) '74 =8=C7 4@D0C8>= 8<?;84B C70C
g∗
λ5 '74 J5C7 4@D0C8>= ;403B C> J (λ1 ) z = z λ4 + g λ6 + g λ7 J (λ1 ) z = z λ4 + 1μ
0=3 C74 C4=C7 >=4 J=0;;H ;403B C> J (λ1 ) z = z λ4 + g λ6 + Rλ7 4=24 B8=24
IP (z ) = IP (g ) = 1
IP (J (λ1 ) z ) = λ4 + λ6 + IP (Rλ7 ) .
&D??>B4 C70C z ∈ Δρ '74 2>=BCA08=CB >5 8<?;H C74 4G8BC4=24 >5 g ∈ PU B44
g
4=24 J (λ1 ) z ≤ z (λ4 + λ5 ) 3D4 C> z ≥ 0 0=3 C74 5>DAC7
BD27 C70C z ≤ 1μ
g
g
2>=BCA08=C >5 J (λ1 ) z ≤ zλ4 + 1μ
λ5 3D4 C> z ≤ 1μ
0=3 λ5 ≥ 0 J (λ1 ) z ≤
zλ4 + gλ6 + gλ7 3D4 C> g ≥ 0 0=3 C74 J5C7 2>=BCA08=C >5 8=0;;H J (λ1 ) z ≤
zλ4 + gλ6 + Rλ7 3D4 C> g ≤ R 0=3 λ7 ≥ 0 4=24 IP (z) = IP (g) = 1 ;403B C>
IP (J (λ1 ) z) ≤ λ4 + λ6 + IP (Rλ7 ) .
0=3 8<?;H $A>?>B8C8>= 8 8B DB45D; 8= ?A02C824 1420DB4 8C <0:4B C74 >?C8<0; A48=BDA0=24 ?A>1
;4< 142><4 ;8=40A 1DC C74 01B4=24 >5 3D0;8CH 60? 14CF44= 0=3 <DBC 14
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
6D0A0=C443 '74>A4< 10 14;>F F8;; B7>F C70C C78B 8B C74 20B4 85 H R 0=3 R̃ 0A4
B8<?;4 5D=2C8>=B %420;; C70C B8<?;4 5D=2C8>=B 0A4 68E4= 1H C74 64=4A0; 4G?A4BB8>=
n
i=1 ai χBi [0, M ] = B1 ∪ B2 ∪ ... ∪ Bn 148=6 0 <40BDA01;4 0=3 38B9>8=C ?0AC8C8>=
a1 , a2 , ....an 148=6 A40; =D<14AB 0=3 χBi 148=6 C74 8=3820C>A >5 Bi i = 1, 2, ..., n #1
E8>DB;H 2>=BC0=C 5D=2C8>=B BD27 0B C74 DBD0; >=4B H = 0 R = R̃ = 1 0A4 8=2;D343
!>A4>E4A A420;; C74 C74 B4C >5 B8<?;4 5D=2C8>=B 8B 34=B4 8= L2 [0, M ] 0=3 L2 (0 ) 0=3
5>A 64=4A0; H R 0=3 R̃ >=4 20= 0;F0HB 1D8;3 M6>>3 B8<?;4 0??A>G8<0C8>=BN
= >A34A C> ?A>E4 '74>A4< 10 F4 F8;; =443 B><4 2>=24?CB 0=3 8=BCAD<4=C0; A4BD;CB
F7827 F8;; 14 1A84KH BD<<0A8I43 5 B0 ⊂ B 8B 0 BD1σ−0;641A0 >5 C74 >A4; σ−0;641A0
B >5 [0, M ] C74= F8C7 >1E8>DB =>C0C8>=B L2 [0, M, B0 ] 8B 0 2;>B43 BD1B?024 >5 L2 [0, M ]
0=3 L2 (0 , B0 ) 8B 0 2;>B43 BD1B?024 >5 L2 (0 ) '74 >AC7>6>=0; ?A>942C8>= >5 L2 (0 )
>E4A L2 (0 , B0 ) 8B 68E4= 1H C74 2>=38C8>=0; 4G?42C0C8>=
L2 (0 ) → L2 (0 , B0 )
y
→ IP (y |B0 )
0=3 8C 8B :=>F= C70C
IP (y1 y2 ) = IP (IP (y1 |B0 ) y2 )
5>A 4E4AH y1 ∈ L2 (0 ) 0=3 4E4AH y2 ∈ L2 (0 , B0 ) 4B834B 85 L 8B C74
8B 0 ?A>1018;8CH C74 >AC7>6>=0; ?A>942C8>=
<40BDA4 C74= M
414B6D4
L2 [0, M ] → L2 [0, M, B0 ]
y
→ L (y |B0 )
M
8B 0;B> 68E4= 1H C74 1H 0 2>=38C8>=0; 4G?42C0C8>= 0=3 142><4B
L (y1 y2 ) = L
M
M
L (y1 |B0 ) y2 .
M
5 J0 8B C74 2><?>B8C8>= >5 J 0=3 C74 >AC7>6>=0; ?A>942C8>=B 01>E4
J0 := IP (− |B0 ) ◦ J ◦ L (− |B0 ) ,
M
C74= 140A8=6 8= <8=3 C70C 4E4AH >AC7>6>=0; ?A>942C8>= 8B B4;5039>8=C >=4 70B C70C
J0 = IP (− |B0 ) ◦ J ◦ L (− |B0 )
M
= L (− |B0 ) ◦ J ◦ IP (− |B0 )
M
= L (− |B0 ) ◦ J ◦ IP (− |B0 ) .
M
= >C74A F>A3B 85 J0 : L2 [0, M ] → L2 (0 ) 8B 68E4= 1H
J0 (y) := IP J L (y |B0 ) |B0
M
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
C74= J0 : L2 (0 ) → L2 [0, M ] 8B 68E4= 1H
J0 (y) = L (J (IP (y |B0 )) |B0 ) .
M
-< (Bn )
n=1 *- ) C4<:)<176 7. ;=*σ−)4/-*:); 7. B
2
⊂
L2 (0 ) :-;8-+<1>-4A (un )
a) -< (un )
n=1
n=1 ⊂ L [0, M ] *- ) 5):<16/)4
),)8<-, <7 <0- C4<:)<176 (Bn )n=1 . (un )n=1 1; 67:5 *7=6,-, <0-6 <0-:- -@1;<; u ∈
L2 (0 ) :-;8-+<1>-4A u ∈ L2 [0, M ] ;=+0 <0)< (un )
n=1 +76>-:/-; <7 u *7<0 16 67:5
)6, )457;< ->-:A?0-:- 7:-7>-: un = IP (u |Bn ) :-;8-+<1>-4A un = L (u |Bn )
M
.7: ->-:A n ∈ b) . <0-:- -@1;<; u ∈ L2 (0 ) :-;8-+<1>-4A u ∈ L2 [0, M ] ;=+0 <0)< un = IP (u |Bn )
:-;8-+<1>-4A un = L (u |Bn ) )6, B = σ (∪
n=1 Bn ) +716+1,-; ?1<0 <0- σ−)4/-*:)
M
/-6-:)<-, *A ∪n=1 Bn <0-6 (un )n=1 +76>-:/-; <7 u *7<0 16 67:5 )6, )457;< ->-:A
?0-:-
2
c) %=887;- <0)< B = σ (∪
n=1 Bn ) %=887;- <0)< u ∈ L (0 ) )6, +76;1,-: <0-
;-< A = {IP (u |Bn ) ; n ∈ } ∪ {u} &0-6 L (− |Bn ) ◦ J M
=61.7:54A 16 A
n=1
+76>-:/-; <7 J #!!
a) 0=3 b) 0A4 2>=B4@D4=24B >5 C74 >>1LB !0AC8=60;4 >=E4A64=24 '74>A4<
DAA4CC B> ;4C DB ?A>E4 c) BB4AC8>= b) 8<?;84B C70C A ⊂ L2 (0 ) 8B 0 2><?02C
B4C 4B834B
IP (− |Bn ) ◦ J ≤ IP (− |Bn ) J ≤ J
8<?;84B C70C L (− |Bn ) ◦ J M
8B 1>D=343 0=3 C74A45>A4 4@D82>=C8=D>DB '7DB
n=1
C74 B2>;8LB '74>A4< 8<?;84B C70C L (− |Bn ) ◦ J A 85 L (J (v) |Bn )
M
84D3>==S n=1
M
n=1
2>=E4A64B C> J D=85>A<;H 8=
2>=E4A64B C> J (v) 5>A 4E4AH v ∈ A F7827 7>;3B 3D4 C> b)
!# %=887;- <0)< μ > 0 )6, μ̃ > 0 . H R )6, R̃ ):- ;1584- .=6+<176;
<0-6 <0- ,7=*4-,=)4 8:7*4-5 1; ;74>)*4- )6, 0); 67 ,=)41<A /)8 ?1<0 <0- ,=)4
8:7*4-5 #!! C 8B :=>F= C70= C74 C>?>;>6H >5 [0, M ] 8B 64=4A0C43 1H 0 2>D=C01;4 10B8B >5
A4;0C8E4 >?4= B4CB 4;;H B> C74 >A4; σ−0;641A0 B 8B 64=4A0C43 1H 0 2>D=C01;4
B4@D4=24 (Bn )
n=1 ⊂ B >=B834A C74 σ−0;641A0 Bn = σ (B1 , B2 , ..., Bn ) 5>A 4E4AH
n ∈ C 8B >1E8>DB C70C 4E4AH Bn 70B 0 J=8C4 =D<14A >5 4;4<4=CB 0=3 C74 J;CA0C8>=
(Bn )
n=1 B0C8BJ4B
B = σ (∪
n=1 Bn ) .
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
&8=24 H R 0=3 R̃ 0A4 B8<?;4 5D=2C8>=B F4 70E4 C70C C74H 64=4A0C4 0 σ−0;641A0
σ H, R, R̃ F8C7 J=8C4;H <0=H <40BDA01;4 B4CB '7DB 1H A4?;028=6 Bn F8C7
σ Bn ∪ σ H, R, R̃
,
F4 >1C08= 0 =4F J;CA0C8>= BC8;; 34=>C43 1H (Bn )
n=1 F7827 0;B> B0C8BJ4B C70C 4E4AH
σ−0;641A0 8= C74 J;CA0C8>= 70B J=8C4;H <0=H 4;4<4=CB !>A4>E4A BC8;; 7>;3B 0=3
H R 0=3 R̃ 0A4 Bn −<40BDA01;4 5>A 4E4AH n ∈ >=B834A C74 ;8=40A >?C8<8I0C8>= ?A>1;4< −n A4B?42C8E4;H −n B8<8;0A
C> A4B?42C8E4;H 85 >=4 A4?;024B B44 0=3 J J L2 [0, M ] 0=3
L2 (0 ) F8C7 Jn = IP (− |Bn ) ◦ J ◦ L (− |Bn ) Jn = L (− |Bn ) ◦ J ◦ IP (− |Bn )
M
M
L2 [0, M, Bn ] 0=3 L2 (0 , Bn ) C 8B 40BH C> B44 C70C −n 8B C74 3D0; ?A>1;4< >5
−n DAC74A<>A4 B8=24 1>C7 ?A>1;4<B >1E8>DB;H 70E4 540B81;4 B>;DC8>=B C74H
0A4 1>D=343 8=0;;H B8=24 Bn >=;H 70B J=8C4;H <0=H 4;4<4=CB −n 0=3 −n
0A4 J=8C438<4=B8>=0; ?A>1;4<B L2 [0, M, Bn ] 0=3 L2 (0 , Bn ) 0A4 J=8C438<4=B8>=0;
B?024B B> C74H 0A4 B>;E01;4 0=3 C748A >?C8<0; E0;D4B 2>8=2834 5>A 4E4AH n ∈ 4=>C4 1H ϕn C74 >?C8<0; E0;D4 >5 −n 0=3 −n 1H
Dn = (λd,n , λu,n , fn , zn , gn , γ n , hn )
0 B>;DC8>= >5 −n 0=3 1H
Sn = (λ1,n , λ2,n , λ3,n , λ4,n , λ5,n , λ6,n , λ7,n , λ8,n , λ9,n , λ10,n , λ11,n )
0 B>;DC8>= >5 −n '74H B0C8B5H C74 2>=BCA08=CB >5 −n 0=3 −n F7827 0A4
C7>B4 >5 0=3 F8C7 C74 >1E8>DB <>38J20C8>=B 0=3 C74H 0;B> B0C8B5H −n
F7827 2>8=2834B F8C7 D=34A BCA0867C5>AF0A3 <>38J20C8>=B C>> >=B834A C74
B>;DC8>= D B44 >5 0=3 34=>C4 1H ϕ C74 >?C8<0; E0;D4 >5 C78B ?A>1;4< C74H
4G8BC >F8=6 C> '74>A4<B 2 0=3 5
(i) ϕn ≤ ϕn+1 ≤ ϕ 5>A 4E4AH n ∈ =3443 ;4C DB B44 C70C ϕn ≤ ϕ 8B 2;40A;H
540B81;4 85 (λd,n , λu,n ) 8B A4?;0243 1H
λd = (J (z + wf − (1 + w) (βγ − α)))+
λu = (J (z + wf − (1 + w) (βγ − α))) .
&8=24 H 8B Bn −<40BDA01;4 0=3 y → L (y |Bn ) 8B 8=2A40B8=6 ;403B C>
M
≺ H, λd
=≺ H, L (λd |Bn )
M
=
≺ H, L (J (zn + wfn − (1 + w) (βγ n − α)))+ |Bn
M
≥≺ H, L (J (zn + wfn − (1 + w) (βγ n − α)) |Bn )
M
+
=≺ H, λd,n
.
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
&8<8;0A;H − ≺ 1, λu
≥ − ≺ 1, λu,n
≺ 1, J (γ )
0=3 8<?;84B C70C
=≺ 1, L (J (γ ) |Bn )
M
.
4=24 ϕn ≤≺ H, λd − ≺ 1, λu + ≺ 1, J (γ ) ≤ ϕ '74 8=4@D0;8CH ϕn ≤ ϕn+1
<0H 14 ?A>E43 F8C7 0 B8<8;0A <4C7>3
(ii) (ϕn )
n=1 2>=E4A64B C> ϕ =3443 2>=B834A C74 B>;DC8>= >5 0=3 34=>C4
(fn , zn , gn , γ n , hn ) = IP ((f , z , g , γ , h ) |Bn ) .
5
+
λ
d,n = (Jn (zn + wfn − (1 + w) (βγ n − α)))
λ
u,n = (Jn (zn + wfn − (1 + w) (βγ n − α)))
C74=
(λ
d,n , λu,n , fn , zn , gn , γ n , hn )
8B −n540B81;4 B> C74 −n>1942C8E4 ϕ̃n 0C C78B ?>8=C B0C8BJ4B ϕ̃n ≤ ϕn '74=
4<<0 9 8<?;84B C70C
8C 8B BDP 284=C C> B44 C70C (ϕ̃n )
n=1 2>=E4A64B C> ϕ
(fn , zn , gn , γ n , hn )n=1
2>=E4A64B C> (f , z , g , γ , h ) 0=3 L (− |Bn ) ◦ J M
>= (fn , zn , gn , γ n , hn )n=1 4=24
n=1
2>=E4A64B C> J D=85>A<;H
+
Limn (λ
d,n ) = Limn (Jn (zn + wfn − (1 + w) (βγ n − α)))
= Limn L ((J (zn + wfn − (1 + w) (βγ n − α))) |Bn )
+
M
((J (z + wf − (1 + w) (βγ − α))))+ .
=0;>6>DB;H
Limn (λ
u,n ) = ((J (z + wf − (1 + w) (βγ − α)))) .
>=B4@D4=C;H 140A8=6 8= <8=3 C70C ≺ 1, Jn (γ n )
≺ 1, J (γ ) B44 Limn (ϕ̃n ) = Limn ≺ H, λ
d,n
− ≺ 1, λ
u,n
=≺ H, ((J (z + wf − (1 + w) (βγ − α))))
+
− ≺ 1, ((J (z + wf − (1 + w) (βγ − α))))
+ (1 + w) β ≺ 1, J (γ )
+C = ϕ.
=≺ 1, J (γ n )
2>=E4A64B C>
+ (1 + w) β ≺ 1, Jn (γ n )
+C
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
(iii) '74 B4@D4=24 (λ1,n )
n=1 8B =>A<1>D=343 1420DB4 C74 2>=BCA08=CB >5 −n
⊂
[H,
1]
8<?;H C70C (λ1,n )
n=1
(iv) λ2,n ≥ 0 5>A 4E4AH n ∈ =3443 85 λ2,n = −V < 0 C74 −n2>=BCA08=CB
F>D;3 ;403 C> 0 < V ≤ wJ λ1,n − λ2,n ≤ λ3,n 0=3 −n F>D;3 8<?;H fn = R
'7DB R ≥ 1 0=3 IP (R) = IP (fn ) = 1 F>D;3 8<?;H R = 1 5 C78B 4@D0;8CH 3>4B
=>C 7>;3 C74= F4 F8;; 70E4 0 2;40A 2>=CA0382C8>= 5 R = 1 7>;3B C74= 8C 8B 40BH C> B44
C70C
(λ1,n , 0, λ3,n − V, λ4,n , λ5,n , λ6,n , λ7,n , λ8,n , λ9,n , λ10,n , λ11,n )
8B −n540B81;4 0=3 0=3 C74 >1942C8E4 5D=2C8>= A4<08=B C74 B0<4 i.e. BC8;;
B>;E4B −n 0=3 8CB B42>=3 2><?>=4=C 8B =>==460C8E4
(v) λ4,n ≥ 0 0=3 λ8,n ≥ 0 5>A 4E4AH n ∈ =3443 1>C7 20B4B 70E4 0 B8<8;0A
?A>>5 B> ;4C DB 340; F8C7 λ4,n 5 λ4,n = −V < 0 C74 −n2>=BCA08=CB F>D;3 ;403
∗
gn
'74= 1 =
C> 0 < V ≤ J λ1,n − λ4,n ≤ λ5,n 0=3 −n F>D;3 8<?;H zn = 1μ
∗
gn
1
IP (zn ) = IP 1μ
F>D;3 64=4A0C4 C74 2>=CA0382C8>= μ = 0
= 1μ
(vi) λ6,n ≥ 0 0=3 λ10,n ≥ 0 5>A 4E4AH n ∈ =3443 1>C7 20B4B 70E4 0 B8<8;0A
?A>>5 B> ;4C DB 340; F8C7 λ6,n 5 λ6,n = −V < 0 C74 −n2>=BCA08=CB F>D;3 ;403
1
λ5,n − λ6,n ≤ λ7,n 0=3 −n F>D;3 8<?;H gn = R '74 A4BC >5 C74
C> 0 < V ≤ 1μ
?A>>5 8B B8<8;0A C> (iv)
(vii) '74 B4@D4=24 (λ2,n , λ4,n , λ6,n , λ8,n , λ10,n )
n=1 8B 1>D=343 =3443 8C 8B BDP 284=C
C> B44 C70C C74A4 8B 0= D??4A 1>D=3 (i) B7>FB C70C
5
2
Jn λ1,n
ϕ − C + (1 + w) αIP
4=24
2
i=1
−
Rλ2i+1,n
IP
i=1
− IP
R̃λ11,n
≥
λ2i,n .
i=1
IP Rλ2i+1,n + IP R̃λ11,n ≥ 0 0=3 λ1,n ≤ 1 ;403 C>
5
ϕ − C + (1 + w) αIP (Jn (1)) = ϕ − C + (1 + w) αIP (J (1)) ≥
λ2i,n .
i=1
(λ3,n , λ5,n , λ7,n , , λ9,n , λ11,n )
n=1
(viii) '74 B4@D4=24
8B =>A<1>D=343 =3443 C74
A4BCA82C8>=B 0=3 >1942C8E4 5D=2C8>= >5 −n 2;40A;H 8<?;H C70C
λ3,n = wJn λ1,n − λ2,n
λ5,n = Jn λ1,n − λ4,n
λ7,n =
1
λ
1μ 5,n
− λ6,n
+
+
+
≤ wJn λ1,n ≤ wJn (1)
≤ Jn λ1,n ≤ Jn (1)
≤
1
λ
1μ 5,n
≤
λ9,n = (1 + w) βJn 1 − λ1,n − λ8,n
λ11,n =
1
λ
1μ̃ 9,n
− λ10,n
+
≤
1
λ
1μ̃ 9,n
1
J
1μ n
+
≤
(1)
≤ (1 + w) βJn 1 − λ1,n ≤ (1 + w) βJn (1)
1
1μ̃
(1 + w) βJn (1)
O*.#'& ,#(-/,( /(, ,#-% ( /(,.#(.1
'74 B4@D4=24 (Jn (1))
n=1 2>=E4A64B C> J (1) 4<<0 9b) 0=3 C74A45>A4 8C 8B =>A<
1>D=343
(ix) '74 B4@D4=24 (Sn )
n=1 8B =>A<1>D=343 =3443 8C 8<<4380C4;H 5>;;>FB 5A><
(iii) − (viii)
(x) '74A4 4G8BCB 0 F40:;H 066;><4A0C8>= ?>8=C S >5 (Sn )
n=1 =3443 C78B 8B 0
CA8E80; 2>=B4@D4=24 >5 (ix) 0=3 C74 ;0>6;DLB '74>A4<
(xi) '74 066;><4A0C8>= ?>8=C S 8B 540B81;4 0=3 C74 >1942C8E4 >5 C78B ?A>1;4<
02784E4B C74 >?C8<0; E0;D4 ϕ >5 0C S =3443 JABC ;4C DB B44 C70C S 8B 540B81;4 >=BCA08=CB >5 =>C 8=E>;E8=6 J 0A4 >1E8>DB '74A4 0A4 C7A44 2>=BCA08=CB
8=E>;E8=6 J 1DC C74H 20= 14 0=0;HI43 F8C7 B8<8;0A <4C7>3B '7DB ;4C DB B7>F C70C
J (λ
1 ) − λ4 − λ5 ≤ 0 8G n0 ∈ 5 n0 ≤ n C74=
Jn λn,1 − λ4,n − λ5,n ≤ 0 ⇒ Jn λn,1 ≤ λ4,n + λ5,n .
&8=24 Jn = IP (− |Bn ) ◦ Jn F4 70E4 Jn λn,1 ≤ IP λ4,n |Bn + IP λ5,n |Bn 0=3 C74A45>A4 Jn (λ
1 ) ≤ IP (λ4 |Bn ) + IP (λ5 |Bn ) '7DB 4<<0 9b) ;403B C>
J (λ
1 ) = Limn Jn (λ1 ) ≤
Limn IP (λ
4 |Bn ) + Limn IP (λ5 |Bn )
= λ
4 + λ5 .
"4GC ;4C DB B7>F C70C C74 >1942C8E4 4@D0;B ϕ 0C S =3443 IP Jn λn,1
IP IP J
λn,1
|Bn
= IP J
λn,1
=
8<?;84B C70C
ϕn = C − (1 + w) αIP Jn λn,1
+ λn,2 + IP Rλn,3 + λn,4
+λn,6 + IP Rλn,7 + λn,8 + λn,10 + IP R̃λn,11 =
C − (1 + w) αIP J λn,1
+ λn,2 + IP Rλn,3 + λn,4
+λn,6 + IP Rλn,7 + λn,8 + λn,10 + IP R̃λn,11 .
'74 ;45C 70=3 B834 >5 C74 4@D0;8CH 01>E4 2>=E4A64B C> ϕ B44 (ii) B> 8C <DBC 4@D0;
4E4AH 066;><4A0C8>= ?>8=C >5 C74 A867C 70=3 B834 = >C74A F>A3B
ϕ = C − (1 + w) αIP (J (λ
1 )) + λ2 + IP (Rλ3 ) + λ4
+λ
6 + IP (Rλ7 ) + λ8 + λ10 + IP R̃λ11 .
(xii) 8=0;;H B8=24 C74 >1942C8E4 0C S 4@D0;B C74 >1942C8E4 0C D 1>C7
;8=40A >?C8<8I0C8>= ?A>1;4<B 0A4 B>;E01;4 0=3 C74A4 8B => 3D0;8CH 60? 14CF44= C74<
=34AB>= 0=3 "0B7 Optimal reinsurance under risk and uncertainty
6.
32
Conclusions
This paper has dealt with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also
included. The levels of uncertainty of insurer and reinsurer do not have to be identical. As a second novelty, the decision variable is not the retained (or ceded) risk, but
its sensitivity (mathematical derivative) with respect to the total claims. Thus, if one
imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is
totally eliminated. This may be an important property because stop-loss and closely
related (often optimal) contracts have been criticized. Indeed, the reinsurer would
not accept these solutions in practice due to the lack of incentives of the insurer to
verify claims beyond some thresholds.
Three main contributions seem to be reached. Firstly, necessary and su¢ cient
optimality conditions are given, and they apply in all the cases contained in the general setting above. Secondly, the optimal contract is often a bang-bang solution, i:e:,
the sensitivity between the retained risk and the total claims saturates the imposed
constraints. This …nding also explains why stop-loss or related contracts are often
optimal if appropriate constraints are not imposed. For some special cases the optimal contract might not be a bang-bang one, but there is always a bang-bang contract
as close as desired to the optimal one (the optimal reinsurance is in the closure of a
set described by bang-bang solutions). Thirdly, the optimal reinsurance problem is
equivalent to other linear programming problem (the double-dual problem), despite
the fact that risk and uncertainty (and many pricing principles) cannot be represented by linear expressions. This may be an important …nding for two reasons. On
the one hand, linear problems are easy to solve in practice, since there are very ef…cient algorithms in both …nite and in…nite dimensions. On the other hand, linear
problems often lead to extreme solutions, which explains why the non linear optimal
reinsurance problem may be solved by a bang-bang reinsurance.
Acknowledgments. Research partially supported by Ministerio de Economía,
Spain, Grant ECO2012 39031 C02 01. The usual caveat applies.
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