Connection between higher order measures of risk and stochastic dominance

Pichle , Alois
A icle — Published Ve sion
Connec ion be ween highe o de measu es o isk and
s ochas ic dominance
Compu a ional Managemen Science
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Pichle , Alois (2024) : Connec ion be ween highe o de measu es o isk and
s ochas ic dominance, Compu a ional Managemen Science, ISSN 1619-6988, Sp inge , Be lin,
Heidelbe g, Vol. 21, Iss. 2,
h ps://doi.o g/10.1007/s10287-024-00523-0
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ORIGINAL PAPER
Connec ion be weenhighe o de measu es o  isk
ands ochas ic dominance
AloisPichle 1
Recei ed: 8 Feb ua y 2024 / Accep ed: 21 Augus 2024 / Published online: 5 Sep embe 2024
© The Au ho (s) 2024
Abs ac
Highe o de isk measu es a e s ochas ic op imiza ion p oblems by design, and o
his eason hey enjoy aluable p ope ies in op imiza ion unde unce ain ies. They
nicely in eg a e wi h s ochas ic op imiza ion p oblems, as has been obse ed by he
in iguing concep o he isk quad angles, o example. S ochas ic dominance is
a bina y ela ion o andom a iables o compa e andom ou comes. I is demon-
s a ed ha he concep s o highe o de isk measu es and s ochas ic dominance
a e equi alen , hey can be employed o cha ac e ize he o he . The pape explo es
hese ela ions and connec s s ochas ic o de s, highe o de isk measu es and he
isk quad angle. Expec iles a e employed o exempli y he ela ions ob ained.
Keywo ds Highe o de isk measu e· Highe o de s ochas ic dominance· Risk
quad angle
Ma hema ics Subjec Classi ica ion 62G05· 62G08· 62G20
1 In oduc ion
Risk measu es a e conside ed in a ious disciplines o assess and quan i y isk.
Simila ly o assigning a p emium o an insu ance con ac wi h andom losses a e
app aising i s isk, isk measu es assign a numbe o a andom a iable, which i sel
has s ochas ic ou comes.
This pape ocuses on highe o de isk measu es, as hese isk measu es na u-
ally combine wi h s ochas ic op imiza ion p oblems o in ‘lea ning’ objec i es, as
hey a e he esul o op imiza ion p oblems. In addi ion, hese isk measu es ela e
o he isk quad angle.
The pape de i es explici ep esen a ions o highe o de isk measu es o gene al,
elemen a y isk measu es in a i s main esul . These cha ac e iza ions a e employed
* Alois Pichle
[email p o ec ed]hemni z.de
1 Technische Uni e si ä Chemni z, Facul y o Ma hema ics, 90126Chemni z, Ge many
A.Pichle
41 Page 2 o 28
o cha ac e ize s ochas ic dominance ela ions, which a e buil on gene al no ms. The
second main esul is a e i ica ion heo em. This is a cha ac e iza ion o highe o de
s ochas ic dominance ela ions, which is nume ically ac able.
Fo he no m in Lebesgue spaces, s ochas ic dominance ela ions ha e been con-
side ed o example in Dupačo á and Kopa (2014), Kopa e al. (2016, 2023), Pos and
Kopa (2017) and Consigli e al. (2023), in po olio op imiza ion in ol ing commodi-
ies (c . F ydenbe g e al. (2019)), and by Den che a and Ma inez (2012) and Mag-
gioni and P lug (2016, 2019) in a mul is age se ing. The pape employs he cha ac-
e iza ions ob ained o es ablish ela ions o gene al no ms. A compa ison o hese
me hods is gi en in Gu jah and Pichle (2013). The pape illus a es hese connec ions
o expec iles (Bellini e al. 2016; Bellini and Cape doni 2007) and adds a compa ison
wi h o he isk measu es.
Ou line o he pape The ollowing Sec . 2 ecalls he ma hema ical amewo k o
highe o de isk measu es. Sec ion3 add esses he highe o de isk measu e associ-
a ed wi h he spec al isks, as hese isk measu es cons i u e an elemen a y building
block o gene al isk measu es. This sec ion de elops he i s main esul , which is an
explici ep esen a ion o a spec al isk’s highe o de isk measu e. As a special case,
he subsequen Sec . 4 links and ela es s ochas ic dominance and highe o de isk
measu es. This sec ion p esen s he second main esul , which allows e i ying a s o-
chas ic dominance ela ion by in ol ing only ini ely many isk le els. The inal Sec .5
add esses he expec ile and es ablishes he ela ions o he p eceding sec ions o his
speci ic isk measu e. Sec ion6 concludes.
2 Ma hema ical amewo k
Highe o de isk measu es a e a special ins ance o isk measu es, o en also e med
isk unc ionals. To in oduce and ecall hei main p ope ies we conside a space
Y
o
ℝ
- alued andom a iables on a p obabili y space wi h measu eP con aining a leas all
bounded andom a iables, ha is,
L∞(P)⊆Y
. A isk measu e hen sa is ies he ol-
lowing axioms, o iginally in oduced by A zne e al. (1999).
De ini ion 2.1 (Risk unc ional) Le 
Y
be a space o
ℝ
- alued andom a iables on a
p obabili y space
(Ω,Σ,P)
. A mapping
R∶Y→ℝ
is
(i) mono one, i
R(X)≤R(Y)
, p o ided ha
X≤Y
almos e e ywhe e;
(ii) posi i ely homogeneous i
R(𝜆Y)=𝜆R(Y)
o all
𝜆>0
;
(iii) ansla ion equi a ian , i
R(c+Y)=c+R(Y)
o all
c∈ℝ
;
(i ) subaddi i e, i
R(X+Y)≤R(X)+R(Y)
o all X and
Y∈Y
.
A mapping sa is ying(i)–(i ) is called a isk unc ional, o a isk measu e.
The isk quad angle (c . Rocka ella and U yase (2013)) ela es isk measu es wi h
he measu e o eg e by
Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 3 o 28 41
whe e
V
is called eg e unc ion. Equa ion(2.1) was i s in oduced o he con-
di ional alue-a - isk in Rocka ella and U yase (2000). Fo he expec a ion
ype unc ion, i.e.,
V(X)=
𝔼
(X)
, he ela ionship(2.1) is s udied in Ben-Tal and
Teboulle (2007), whe e
V
was called op imized ce ain y equi alen ; also, K okhmal
(2007) s udy he ela ion(2.1).
I ollows om ela ion(2.1) ha
R
—i gi en as in(2.1)—is ansla ion equi -
a ian , i.e,
R
sa is ies
R(Y+c)=c+R(Y)
o any
c∈ℝ
(c .(iii) abo e). In an
economic in e p e a ion, he amoun c in(2.1) co esponds o an amoun o cash
spen oday, while he emaining quan i y
Y−c
is in es ed and consumed la e ,
hus subjec o
V
.
The isk unc ional
R
is posi i ely homogeneous, i he eg e unc ion
V
is
posi i ely homogeneous. I 
V
is no posi i ely homogeneous, hen one may con-
side he posi i ely homogeneous en elope
whe e

𝛽≥0
is a isk a e sion coe icien . The combined unc ional
is posi i ely homogeneous and ansla ion equi a ian (c . (ii) and (iii)). The
𝜑
-di e gence isk measu e is an explici example o a isk measu e, which is de ined
exac ly as(2.2), c . Dommel and Pichle (2021).
The pape sugges s a eg e o a highe -o de isk s a ing om a gi en isk
R
.
To his end conside a space
Y⊂L1(P)
endowed wi h no m
‖
⋅
‖
. We shall assume
he no m o be mono one, ha is,
‖X‖≤‖Y‖
p o ided ha
0≤X≤Y
almos e e-
ywhe e. We associa e he ollowing amily o isk measu e wi h a gi en no m.
De ini ion 2.2 (Highe o de isk measu e) Le
‖⋅‖
be a mono one no m on
Y⊂L1(P)
wi h
‖1‖=1
, whe e
1(⋅)=1
is he iden ically one unc ion on
Y
. The
highe o de isk measu e a isk le el
𝛽∈[0, 1)
associa ed wi h he no m
‖⋅‖
is
whe e
𝛽∈[0, 1)
is he isk a e sion coe icien and
x+∶= max(0, x)
.
We shall also omi he supe sc ip and w i e
R𝛽
ins ead o
R‖⋅‖
𝛽
in case he no m
is unambiguous gi en he con ex . We shall demons a e i s ha he highe o de
isk measu e is well-de ined o any
𝛽≥0
.
(2.1)
R(Y)=in
c∈ℝ
c+V(Y−c),
V

𝛽(Y)=in
>0
(

𝛽+V
(Y
)),
(2.2)
R
𝛽
(Y)=in
c∈ℝ
c+V
𝛽
(Y−c)
=in
>0
q∈
ℝ
(
𝛽+q+V(Y
−q
))
(2.3)
R‖
⋅
‖
𝛽(Y)=in
∈ℝ
+
1
1−𝛽‖
(Y− )+
‖,
A.Pichle
41 Page 4 o 28
P oposi ion 2.3 Le
(Y,‖⋅‖)
be a no med space o andom a iables. Fo he unc-
ional
R𝛽
de ined in(2.3) i holds ha
so ha
R𝛽(⋅)
is indeed well-de ined on
(Y,‖⋅‖)
o e e y
𝛽∈[0, 1)
.
P oo The uppe bound ollows i ially om he de ini ion by choosing
=0
in he
de ining equa ion(2.3).
Fo
≤0
, i holds ha
− =−Y+(Y− )≤−Y+(Y− )+
. I ollows om he
iangle inequali y ha
− ≤‖Y‖+‖(Y− )+‖
and hus
To es ablish he ela ion also o
≥0
, we s a by obse ing he ollowing mono o-
nici y p ope y o he objec i e in(2.3) in addi ion: o
Δ ≥0
, i ollows om he
e e se iangle inequali y ha
whe e we ha e used ha
0≤Y+−(Y−Δ )+≤Δ
oge he wi h mono onici y o
he no m. ReplacingY by
Y−
in he la e exp ession gi es
ha is, he unc ion
↦ +‖(Y− )+‖
is non-dec easing, which inally es ablishes
ha
The lowe bound in(2.4) hus ollows om he la e inequali y, as
R0(Y)≤R𝛽(Y)
o any
𝛽≥0
.
◻
Example 2.4 Fo Lebesgue spaces
Lp(P)
and no m
‖
Y
‖p
∶=( 𝔼
�
Y
�p
)
1∕p
,
p≥1
, he
highe o de isk measu e has been in oduced in K okhmal (2007) and s udied in
Den che a e al. (2010). Fo he no m
‖⋅‖∞
, he highe o de isk measu e is
indeed, i ollows om(2.3) ha
he subg adien o he con ex unc ion in he la e exp ession a
= ess sup Y
. The
in imum in(2.3) is a ained a
= ess sup Y
, and hus(2.5).
(2.4)
−‖
Y
‖≤
R𝛽(Y)
≤1
1−𝛽‖
Y
‖,
−‖Y‖≤ +‖(Y− )+‖ o all ≤0.
‖Y+‖−‖(Y−Δ )+‖≤‖Y+−(Y−Δ )+‖≤‖Δ 1‖=Δ ,
+‖(Y− )+‖≤ +Δ +‖(Y−( +Δ ))+‖;
−‖Y‖≤ +‖(Y− )+‖ o all ∈ℝ.
(2.5)
R‖⋅‖
∞
𝛽
(Y) = ess sup Y,𝛽>
0;
0
∈
�
1−1
1−𝛽,1
�
=𝜕
�
+1
1−𝛽
‖
(Y− )+
‖
∞
������ =ess sup Y
,

Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 5 o 28 41
Lemma 2.5
R𝛽(⋅)
is a isk unc ional, p o ided ha he no m is mono one. Fu he ,
R𝛽
is Lipschi z con inuous wi h espec o he no m, he Lipschi z cons an is
1
1−𝛽
.
P oo The asse ions(ii)–(i ) in De ini ion2.1 a e s aigh o wa d o e i y; o e -
i y(i) i is indispensable o assume ha he no m is mono one.
As o con inui y, i ollows om subaddi i i y oge he wi h (2.4) ha
R
𝛽(Y)−R𝛽(Z)
≤
R𝛽(Y−Z)
≤1
1−𝛽‖
Y−Z
‖
, and
�
R𝛽(Y)−R𝛽(Z)
�≤1
1−𝛽‖
Y−Z
‖
a e in e changing he oles o Y andZ. Hence, he asse ion.
◻
No e ha he highe o de isk measu e as de ined in(2.3) de ines a isk unc ional
based on a no m. In con as o his cons uc ion, a isk unc ional
R
de ines a no m ia
and a Banach space wi h
Y
=
{
Y∈L
1
∶R(
|
Y
|
)<∞
}
(c . Pichle (2013)). I s na u-
al dual no m o
Z∈Z∶=Y∗
is
The ollowing ela ionship allows de ining a eg e unc ional o connec a isk unc-
ional
R
wi h he highe -o de isk quad angle.
P oposi ion 2.6 (Duali y) Le
R
be a isk unc ional wi h associa ed no m
‖⋅‖
and
dual no m
‖⋅‖∗
. Fo he highe o de isk unc ional i holds ha
whe e
𝛽∈[0, 1)
.
Rema k 2.7 By he in e connec ing o mula(2.1), he highe o de isk unc ional
R‖⋅‖
𝛽
associa ed wi h he no m
‖⋅‖
is he eg e unc ion
V‖⋅‖
𝛽(⋅)∶=
1
1−𝛽‖
(⋅)+
‖
.
P oo I holds by he Hahn–Banach heo em and as
(Y− )+≥0
ha
This es ablishes he i s inequali y ‘
≤
’ in(2.9) wi h
+(Y− )+≥Y
, as
(2.6)
‖Y‖∶=R(�Y�)
(2.7)
‖
Z
‖∗
∶= sup {𝔼YZ ∶
‖
Y
‖≤
1}
=sup {
𝔼
YZ ∶
R
(�Y�)≤1}.
(2.8)
R
𝛽(Y)=sup
�
𝔼YZ ∶Z
≥
0, 𝔼Z=1 and
‖
Z
‖
∗
≤
1
1−𝛽
�
(2.9)
=
in
∈ℝ
+
1
1−𝛽‖
(Y− )+
‖,
1
1
−𝛽
⋅‖(Y− )+‖=sup
‖
Z
‖
∗≤1
1−𝛽
𝔼Z(Y− )+
≥
sup
𝔼Z=1, Z≥0,
‖
Z
‖
∗≤1
1−𝛽
𝔼Z(Y− )+
.
A.Pichle
41 Page 6 o 28
As o he con e se inequali y assume i s ha Y is bounded. No e, ha
so ha i ollows ha
Fu he , i holds ha
𝔼YZ = ∗+
𝔼
Z(Y− ∗)+
o
∗≤Y
a.s. and hus
hus he desi ed con e se inequali y, p o ided ha Y is bounded; i Y is no bounded,
hen he e is a bounded
Y𝜀
wi h
Y≤Y𝜀
(
𝜀>0
) and
‖Y𝜀
−Y‖<𝜀
, so ha
so ha we may conclude ha (2.9) holds o e e y
Y∈Y
.
◻
Example 2.8 (Lebesgue spaces) The dual no m o he genuine no m
‖
X
‖p
∶=( 𝔼
�
X
�p
)
1∕p
in he Lebesgue space
Lp(P)
is
‖Z‖∗=(
𝔼
�Z�q)1∕q
o he Hölde
conjuga e exponen q wi h
1
p
+
1
q
=
1
. Wi h P oposi ion2.6 i ollows ha
c . also Pichle and Shapi o (2015) and Pichle (2017).
In wha ollows, we shall elabo a e he highe o de isk measu e and he associ-
a ed eg e unc ion o speci ic isk measu es, speci ically he spec al isk measu e.
+
1
1−𝛽
⋅‖(Y− )+‖
≥
sup
𝔼Z=1
Z≥0, ‖Z‖∗≤1
1−𝛽
𝔼
�
+(Y− )+
�Z
≥sup
𝔼Z=1
Z≥0,
‖
Z
‖
∗≤1
1−𝛽
𝔼YZ.
in
∈ℝ
+𝔼(Y− )Z=𝔼YZ +in
∈ℝ
⋅(1−𝔼Z)=
{
𝔼YZ i 𝔼Z=
1,
−∞ else,
sup
𝔼Z=1
Z
≥0,
‖
Z
‖
∗≤1
1−𝛽
𝔼YZ =sup
Z
≥
0,
‖
Z
‖
∗≤1
1−𝛽
in
∈ℝ
+𝔼(Y− )Z.
sup
𝔼Z=1, Z≥0,
‖
Z
‖
∗≤1
1−𝛽
𝔼YZ =
sup
Z≥0,
‖
Z
‖
∗≤1
1−𝛽
∗+𝔼Z(Y− ∗)+= ∗+
1
1−𝛽‖(Y− ∗)‖
≥in
∈ℝ
+
1
1−𝛽‖(Y− )+‖
,
𝔼Z(Y𝜀− )+−𝜀
𝔼
Z≤
𝔼
Z(Y− )+≤
𝔼
Z(Y𝜀− )+,
R‖
⋅
‖
p
𝛽(Y)=in
∈ℝ +
1
1−𝛽‖(Y− )+‖p
=sup
�
𝔼YZ ∶
‖
Z
‖
q≤1
1−𝛽,Z≥0 and 𝔼Z=1
�,
Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 7 o 28 41
3 Highe o de spec al isk
By Kusuoka’s heo em (c . Kusuoka (2001)), e e y law in a ian isk unc ional
can be assembled by elemen a y isk unc ionals, each in ol ing he a e age
alue-a - isk.
The ollowing sec ion de elops he explici ep esen a ions o he highe o de
isk measu es associa ed wi h spec al isk measu es i s . The explici ep esen-
a ion hen is ex ended o gene al isk unc ionals.
De ini ion 3.1 (Spec al isk measu es) The unc ion
𝜎∶[0, 1)→ℝ
is called a spec-
al unc ion, i
(i)
𝜎(⋅)≥0
,
(ii)
∫1
0
𝜎(u)du=
1
and
(iii)
𝜎(⋅)
is non-dec easing.
The spec al isk measu e wi h spec al unc ion
𝜎
is
whe e
is he alue-a - isk, he gene alized in e se o quan ile unc ion.
The highe o de isk measu e o he spec al isk measu e is a spec al isk
measu e i sel . The ollowing heo em p esen s he co esponding spec al unc-
ion explici ly and gene alizes (P lug 2000). The esul is cen al owa ds he main
cha ac e iza ion p esen ed in he nex sec ions.
Theo em3.2 (Highe o de spec al isk) Le
𝛽∈[0, 1)
be a isk le el. The highe
o de isk unc ional o he isk unc ional
R𝜎
wi h spec al unc ion
𝜎(⋅)
has he
ep esen a ion
whe e
𝜎𝛽
is he spec al unc ion
he e,
u𝛽∈ℝ
is he
𝛽
-quan ile wi h espec o he densi y
𝜎
, ha is, he solu ion o
R
𝜎(Y)∶=
∫1
0
𝜎(u)F−1
Y(u)du
,
F−1
Y
(u)∶= 𝖵@𝖱
u
(Y)∶= in {x∈ℝ∶P(Y
≤
x)
≥
u
}
(3.1)
in
∈ℝ
+
1
1−𝛽
R𝜎
(
(Y− )+
)
=R𝜎𝛽(Y)
,
(3.2)
𝜎
𝛽(u)∶=
{
0 i u<u𝛽
,
𝜎(u)
1−𝛽else;
A.Pichle
41 Page 8 o 28
which is unique o
𝛽>0
.
P oo We ema k i s ha
𝜎𝛽
indeed is a spec al unc ion, as
∫1
0𝜎𝛽(u)du=1
1−𝛽∫1
u
𝛽
𝜎(u)du=1
−𝛽
1−𝛽
=
1
by he de ining p ope y (3.3) and (ii) in
De ini ion3.1. The quan ile
u𝛽
is uniquely de ined o
𝛽>0
, as he unc ion
𝜎
is
non-dec easing by (iii). In wha ollows we shall demons a e ha he in imum
in(3.1) is a ained a
∗
∶=F
−1
Y
(u
𝛽)
. No e i s ha
so ha
and
Assume i s ha
≤ ∗
. The inequali y
u≤FY( )
is equi alen o
F−1
Y
(u)
≤
(c .
ande Vaa (1998); his ela ion o unc ions
FY
and
F−1
Y
is occasionally called a
Galois connec ion), and hus
o equi alen ly
Assume nex ha
u𝛽≤FY( ∗)
, hen
∫1
FY(
∗
)
𝜎(u)du
≤
1−𝛽 so ha
Combining he inequali ies in he la e displays gi es
(3.3)
∫u
𝛽
0
𝜎(u)du=𝛽
,
F
−1
(Y− )+
(u)=
{
0 i u<FY( )
,
F−1
Y
(u)− else,
R
𝜎
(
(Y− )+
)
=∫
1
0
𝜎(u)F−1
(Y− )+
(u)du=∫
1
F
Y
( )
𝜎(u)
(
F−1
Y(u)−
)
d
u
(3.4)
(
R𝜎)𝛽(Y)=in
∈ℝ
+1
1−𝛽∫
1
F
Y
( )
𝜎(u)
(
F−1
Y(u)−
)
du
.
�F
Y
( ∗)
F
Y
( )
𝜎(u)
(
F−1
Y(u)−
)
du
≤0,
�1
F
Y
( )
𝜎(u)
(
F−1
Y(u)−
)
du
≤
�
1
F
Y
( ∗)
𝜎(u)
(
F−1
Y(u)−
)
du
.
− ∗
1
−𝛽�
1
FY(
∗
)
𝜎(u)du
≤
− ∗
.
(3.5)
∗+1
1−𝛽�
1
FY(
∗
)
𝜎(u)
(
F−1
Y(u)− ∗
)
du
≤
+1
1−𝛽�
1
FY( )
𝜎(u)
(
F−1
Y(u)−
)
d
u
Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 15 o 28 41
De ini ion 4.1 (S ochas ic dominance) Le X,
Y∈Y
be
ℝ
- alued andom a iables
in a Banach space
(Y,‖⋅‖)
. The andom a iableX is domina ed by Y, deno ed
i
I he no m is unambiguous om he con ex , we shall also simply w i e
≼
ins ead
o
≼‖⋅‖
.
The cone o andom a iables igge ed by a single a iable is con ex.
Lemma 4.2 (Con exi y o he s ochas ic dominance cone) Fo
X∈Y
gi en, he se
is con ex.
P oo The map
y↦( −y)+
is con ex, as ollows om e lec ing and ansla ing
he con ex unc ion
x↦x+
. Suppose ha
X≼Y0
and
X≼Y1
. Then i ollows o
Y𝜆∶=(1−𝜆)Y0+𝜆Y1
, oge he wi h mono onici y o he no m and(4.1), ha
Tha is, i holds ha
X≼Y𝜆
and hus he asse ion.
◻
4.1 Cha ac e iza ion o s ochas ic dominance ela ions
S ochas ic dominance ela ions can be ully cha ac e ized by highe o de isk
measu es. The ollowing heo em p esen s his main esul , which in eg a es
he de ails de eloped abo e o hese isk unc ionals and s ochas ic dominance
ela ions.
Theo em4.3 (Cha ac e iza ion o s ochas ic dominance, c . Gómez e  al. (2022))
The ollowing a e equi alen :
(i)
X≼‖⋅‖Y
,
(ii)
R𝛽(−X)≥R𝛽(−Y)
o all
𝛽∈[0, 1)
, and
X≼‖⋅‖Y,
(4.1)
‖( −X)+‖≥‖( −Y)+‖ o all ∈
ℝ
.
{Y∈Y∶X≼Y}
‖
( −Y𝜆)+‖
≤�
�
�
�
(1−𝜆)( −Y0)+𝜆( −Y1)
�
+
�
�
�
≤(1−𝜆)‖( −Y0)+‖+𝜆‖( −Y1)+
‖
≤(1−𝜆)
‖
( −X)+
‖
+𝜆
‖
( −X)+
‖
=
‖
( −X)
+‖
.

A.Pichle
41 Page 16 o 28
(iii)
in
Z∈Z
𝛽
𝔼ZX ≤in
Z∈Z
𝛽
𝔼ZY
o e e y
𝛽∈(0, 1)
, whe e
is he posi i e cone (
Z≥0
) in he dual ball wi h adius
1
1−𝛽
(
‖
Z
‖
∗
≤1
1−𝛽
),
in e sec ed wi h he simplex (
𝔼Z=1
).
P oo Suppose ha
X≼‖⋅‖Y
, hen, by de ini ion,
‖( −X)+‖≥‖( −Y)+‖
o e e y
∈ℝ
. I ollows ha
+
1
1−𝛽‖
(−X− )+
‖≥
+
1
1−𝛽‖
(−Y− )+
‖
o all
∈ℝ
, and
hus asse ion(ii) a e passing o he in imum.
As o he con a y, assume ha  (ii) holds. To demons a e (i) no e i s
ha
q↦‖(q−X)+‖
is con ex; indeed, wi h
q𝜆∶=(1−𝜆)q0+𝜆q1
and
(a+b)+≤a++b+
i holds ha
and hus
by he iangle inequali y o he no m.
Fo
q∈ℝ
ixed, choose
ha is, he subdi e en ial (o he con ex unc ion
𝜂↦‖(𝜂−Y)+‖
) e alua ed a
𝜂=q
, and no e ha
𝛼∈[0, 1]
. Se
𝛽∶=1−𝛼
, and obse e ha
so ha
by(2.3). Employing he de ini ion(2.3) again and assump ion(ii), i ollows ha
o equi alen ly
Z
𝛽∶=
�
Y∈Y∗∶
‖
Z
‖
∗
≤
1
1−𝛽,𝔼Z=1, Z
≥
0
�
(
q𝜆−X)+=
(
(1−𝜆)(q0−X)+𝜆(q1−X)
)+≤
(1−𝜆)(q0−X)+𝜆(q1−X)
+
‖(q𝜆−X)‖≤(1−𝜆)⋅‖(q0−X)+‖+𝜆⋅‖(q1−X)+‖
𝛼
∈𝜕𝜂
‖
(𝜂−Y)+
‖�
�
�𝜂=q
,
0
∈𝜕q−q+
1
1−𝛽‖
(q−Y)+
‖
R
𝛽(−Y)=−q+
1
1−𝛽‖
(q−Y)+
‖
−
q+
1
1−𝛽‖(−X+q)+‖
≥
R𝛽(−X)
≥R𝛽(−Y)
=−q+1
1−𝛽‖
(q−Y)+
‖,
‖(q−X)+‖≥‖(q−Y)+‖.
Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 17 o 28 41
The asse ion(i) ollows, as
q∈ℝ
was a bi a y; his es ablishes equi alence o (i)
and(ii).
Finally, le
𝛽∈(0, 1)
. Wi h(ii) and P oposi ion2.6 we ha e ha
whe e he in imum in bo h exp essions is among
Z
∈Z𝛽=
�
Z∈Z∶
‖
Z
‖
∗
≤
1
1−𝛽�
,
as he se
Z𝛽
collec s he cons ain s in (2.8). This es ablishes equi alence
be ween(ii) and(iii).
◻
Rema k 4.4 The quan i y
−R(−
Y
) =∶ A(
Y
)
a ising na u ally in Theo em 4.3 (ii)
abo e is o en called an accep abili y unc ional, c . P lug and Römisch (2007).
Co olla y 4.5 Suppose ha
hen X is domina ed by Y,
X≼‖⋅‖Y
. Fu he , he asse ion(4.2) is equi alen o
P oo Fix
𝛽∈(0, 1)
, hen
in
Z∈Z
𝛽
𝔼ZX ≤in
Z∈Z
𝛽
𝔼ZY
by (4.2). Wi h (iii) in he
p eceding Theo em4.3 i ollows ha
X≼Y
.
Wi h(2.7), he s a emen (4.3) is equi alen wi h
𝔼Z(X−Y)≤0
o
Z∈Z
and
hence he asse ion.
◻
Rema k 4.6 The asse ion (4.3), howe e , is s ic ly s onge han (ii) in Theo-
em4.3. Indeed, i ollows wi h con exi y and(4.3) ha
and hence(ii), he asse ion, al hough he e e se implica ion does no hold ue.
Example 4.7 (Uni o m no m) Fo he uni o m no m
‖⋅‖∞
, he de ining ela ion(4.1)
is equi alen o
his ela ion de i es om he cha ac e iza ion(i) in Theo em4.3 as well.
in
Z
∈Z
𝛽
𝔼ZX ≤in
Z∈Z
𝛽
𝔼ZY,
(4.2)
𝔼
ZX
≤
𝔼ZY o all Z∈Z∶=
⋃
𝛽∈(0,1)
Z𝛽
,
(4.3)
R𝛽(X−Y)≤0 o all 𝛽∈(0, 1).
R(−Y)≤R(X−Y)+R(−X)≤R(−X),
X≼‖⋅‖
∞
Y
⟺
ess in X≤ess in Y;
A.Pichle
41 Page 18 o 28
4.2 Highe o de s ochas ic dominance
A adi ional way o in oducing s ochas ic dominance ela ions is by i e a ing in e-
g als o he cumula i e dis ibu ion unc ion. This is a special case o he Lebesgue
no m
‖⋅‖p
,
p∈[1, ∞)
, wi h
p∈ℕ
.
De ini ion 4.8 (Highe o de s ochas ic dominance, c . Mülle and S oyan (2002))
The andom a iableX is domina ed byY in i s o de s ochas ic dominance, i
whe e
FX(x)∶=P(X≤x)
is he cumula i e dis ibu ion unc ion. We shall w i e
X≼(1)Y
. Fo
p∈[1, ∞]
, he andom a iableX is s ochas ically domina ed by Y in
p h-s ochas ic o de , i
we w i e
X≼(p)Y
.
By(4.1) in De ini ion4.1,
whe e
‖⋅‖p
is he usual no m in he Lebesgue space
Lp
. I is o his o ical—al hough
un o una e— easons ha he p-indici in he p eceding display do no ma ch. The
highe o de s ochas ic dominance o in eg al o de s has been ind oduced and con-
side ed in ea lie publica ions.
Lemma 4.9 (C . Og yczak and Ruszczyński (1999, 2001)) Wi h
F(1)
X
(⋅)∶=F
X
(⋅
)
,
he k h (
k=2, 3, …
) epea ed in eg al is
F(k)
X
(x)∶=
∫x
−∞
F
(k−1)
X
(y)d
y
. The ollowing
wo poin s a e equi alen , hey cha ac e ize s ochas ic dominance o in ege o de s
(
k=1, 2, …
) by epea ed in eg als:
(i)
X≼(k)Y
,
(ii)
F(k)
Y
(x)
≥
F
(k)
X
(x
)
o all
x∈ℝ
.
P oo I holds wi h Cauchy’s o mula o epea ed in eg a ion ha
By in eg a ion by pa s, he la e is
so ha
FX(x)≥FY(x) o all x∈
ℝ
,
(4.4)
𝔼
(x−X)
p−1
+≥
𝔼(x−Y)
p−1
+
o all x∈ℝ
;
X≼(p+1)Yis equi alen o X≼‖⋅‖
p
Y,p≥1,
F
(k)
X(x)= 1
(k−2)! ∫x
−∞
(x−y)k−2FX(y)dy
.
F
(k)
X(x)= 1
(k−1)! ∫x
−∞
(x−y)k−1dFX(y)
,
Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 19 o 28 41
om which he asse ion ollows om he de ining condi ion(4.1) in De ini ion4.1.
◻
Rema k 4.10 I ollows om he i e a ed in eg al and (ii) in Lemma 4.9 ha
X≼(k)Y⟹X≼(k+1)Y
o all na u al numbe s
k=1, 2, …
. We no ice nex ha
To his end no e i s ha he cha ac e iza ion(4.4) is equi alen o
Wi h
∫x
z
(x−y)𝛼
−
1(y−z)𝛽
−
1dy=B(𝛼,𝛽)(x−z)𝛽
+
𝛼
−1
(B is Eule ’s in eg al o he
i s kind) and in eg a ion by pa s i ollows ha
whe e we ha e used he cha ac e iza ion (4.6) in (4.7), as
x−y≥0
and ha
B(p,p�−p)
is well-de ined and posi i e o
p′>p
. The asse ion again ollows
wi h(4.6).
4.3 Cha ac e iza ion o s ochas ic dominance o spec al isk measu es
The ollowing builds on he spec al isk measu e
R𝜎(⋅)
in oduced in De ini ion3.1
and conside s he no m
o he spec al unc ion
𝜎
. Theo em 4.3 and he cha ac e iza ion o highe o de
spec al isk measu es (Theo em3.2) gi e ise o he ollowing esul .
Theo em4.11 The s ochas ic dominance ela ion
F
(k)
X(x)= 1
(k−1)! ∫∞
−∞
(x−y)k−1
+dFX(y)= 1
(k−1)!
𝔼(x−X)k−1
+
,
(4.5)
X≼(
p
)Y
⟹
X≼(
p
�)Y o all eal numbe s 1 ≤p≤p�∈
ℝ
.
(4.6)
�x
−∞
(x−z)p−1dFX(z)
≥�x
−∞
(x−z)p−1dFY(z) o all x∈ℝ
.
(4.7)
�x
−∞
(x−z)p�−1dFX(z)= 1
B(p,p�−p)
�x
−∞
�x
z
(x−y)p�−p−1(y−z)p−1dydFX(z)
=1
B(p,p�−p)�x
−∞
(x−y)p�−1−p�y
−∞
(y−z)p−1dFX(z)d
y
≥1
B(p,p�−p)�x
−∞
(x−y)p�−1−p�x
−∞
(y−z)p−1dFY(z)d
y
=
�
x
−∞
(x−z)p�−1dFY(z),
‖⋅‖𝜎∶=R𝜎(�⋅�)
A.Pichle
41 Page 20 o 28
wi h espec o he no m associa ed wi h he spec al isk measu e
R𝜎
is equi alen
o
whe e
𝜎
p∶=
∫1
1−p
𝜎(u)d
u
and
SX(x)∶=1−FX(x)=P(X>x)
is he su i al unc ion
o he andom a iableX.
P oo We a gue wi h he no m
‖Y‖𝜎∶=R𝜎(�Y�)
. No e, ha
(Y− )+≥0
, hence he
de ining equa ion(2.3) is
whe e we ha e used Theo em3.2 in(4.8).
F om(3.8) we ha e ha
whe e we ha e used ha
F−Y(y)=P(−Y≤y)=P(Y≥−y)=1−FY(−y)=SY(−y)
and
𝖵@𝖱𝛼(−Y)=−𝖵@𝖱1−𝛼(Y)
a poin s o con inui y o
FY(⋅)
.
Now se
1−u𝛽=∶p
. Then, by employing he cha ac e izing ela ion(3.3) o he
𝛽
-quan ile o 
𝜎
, i holds ha
so ha
X≼‖⋅‖
𝜎
Y
−
𝜎p⋅𝖵@𝖱p(Y)+�
𝖵@𝖱
p
(Y)
−∞
Σ
(
SY(y)
)
dy
≤
−𝜎p⋅𝖵@𝖱p(X)+
�
𝖵@𝖱p(X)
−∞
Σ
(
SX(x)
)
dx o all p∈(0, 1)
,
(4.8)
R‖
⋅
‖
𝜎
𝛽(Y)=in
∈ℝ
+
1
1−𝛽‖(Y− )+‖𝜎
=in
∈ℝ
+1
1−𝛽
R𝜎�(Y− )+
�
=R𝜎
𝛽
(Y),
R
𝛽(−Y)= 𝖵@𝖱u𝛽(−Y)+ 1
1−𝛽∫
∞
𝖵@𝖱u𝛽(−Y)
Σ
(
F−Y(y)
)
dy
=−𝖵@𝖱1−u𝛽(Y)+ 1
1−𝛽∫∞
−𝖵@𝖱1−u𝛽(Y)
Σ(SY(−y))d
y
=−𝖵@𝖱1−u𝛽(Y)+ 1
1−𝛽∫
𝖵@𝖱1−u𝛽(Y)
−∞
Σ
(
SY(y)
)
dy,
1
−𝛽=∫
1
u
𝛽
𝜎(u)du=∫
1
1−p
𝜎(u)du=𝜎p
,
R
𝛽(−Y)=−𝖵@𝖱p(Y)+ 1
𝜎
p
∫
𝖵@𝖱
p
(Y)
−∞
Σ
(
SY(y)
)
dy
.

Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 21 o 28 41
By Theo em4.3, he ela ion
X≼‖⋅‖
𝜎
Y
is equi alen o
R‖⋅‖
𝜎
𝛽
(−Y)
≤
R
‖⋅‖
𝜎
𝛽
(−X
)
o
all
𝛽∈(0, 1)
. Wi h ha , he asse ion ollows.
◻
4.4 Compa ison o s ochas ic o de ela ions
Di e en s ochas ic dominance ela ions may a y in s eng h ( he implica ion(4.5)
in he p eceding Rema k 4.10 is an example). In wha ollows, we p o ide an
explici ela ion o compa e s ochas ic dominance ela ions, which a e buil on di -
e en spec al unc ions.
P oposi ion 4.12 (Compa ison o spec al s ochas ic o de s) Suppose ha
o some p obabili y measu e
𝜇
, whe e
u𝛽
is as de ined in(3.3). Then he s ochas-
ic o de associa ed wi h
𝜎𝜇
is weake han he genuine s ochas ic o de associa ed
wi h
𝜎
. Speci ically, o di e en spec al unc ions
𝜎
and
𝜎𝜇
, i holds ha
Rema k 4.13 The unc ion
𝜎𝜇
in(4.9) is indeed a spec al unc ion. I is posi i e, as
𝜇
is a posi i e measu e ( hus(i) in De ini ion3.1). The unc ion is non-dec easing, as
u𝛽
is non-dec easing o
𝛽
inc easing. Finally, he unc ion
𝜎𝜇
is a densi y: indeed, i
holds ha
by in eg a ion by pa s, whe e we ha e used he de ini ion o
u𝛽
in(3.3).
P oo o P oposi ion 4.12 Since
x≼‖⋅‖
𝜎
Y
, i holds wi h Theo em 4.3 ha
R
𝜎
𝛽(−X)≥R
𝜎
𝛽(−Y)
o all
𝛽∈(0, 1)
, whe e
𝜎𝛽
is de ined in(3.2). By he cha ac-
e iza ion(3.1), his is
In eg a ing he la e exp ession wi h espec o
𝜇(d𝛽)
es ablishes he inequali y
In e changing he o de o in eg a ion oge he wi h(3.17) gi es ha
(4.9)
𝜎
𝜇(u)=𝜎(u)⋅
∫u
𝛽
0
𝜇(
d
𝛽)
1−𝛽
X≼‖
⋅
‖
𝜎
Y
⟹
X≼‖⋅‖
𝜎𝜇
Y.
∫1
0
𝜎𝜇(u)du=∫
1
0
𝜎(u)⋅∫
u
𝛽
0
𝜇(d𝛽)
1−𝛽
du=∫
1
0
∫
1
𝛽u
𝜎(u)du
𝜇(d𝛽)
1−𝛽=∫
1
0
𝜇(d𝛽)=
1
�1
u
𝛽
𝜎(u)
1−𝛽
F−1
−X(u)du
≥
�
1
u
𝛽
𝜎(u)
1−𝛽
F−1
−Y(u)du,𝛽∈(0, 1)
.
�1
𝛽�
1
u
𝛽
�
𝜎(u)
1−𝛽�F−1
−X(u)du𝜇(d𝛽�)
≥
�
1
𝛽�
1
u
𝛽
�
𝜎(u)
1−𝛽�F−1
−Y(u)du𝜇(d𝛽�),𝛽∈(0, 1)
.
A.Pichle
41 Page 22 o 28
which in u n is
This is he asse ion.
◻
5 Example: heexpec ile
The expec ile isk measu e, o iginally in oduced by Newey and Powell (1987), has
ecen ly gained addi ional in e es (c . Malandii e al. (2024), Balbás e al. (2023)
o Fa ooq and S einwa (2018) o condi ional eg essions). A main eason o he
addi ional in e es in his isk measu e is because i is he only elici able isk unc-
ional (c . Ziegel (2014)).
As P oposi ion2.6 indica es, he highe o de isk measu e can be based on he
dual no m. Fo his eason, he ollowing sec ion es ablishes he dual no m o expec-
iles i s , as i is c ucial in unde s anding i s eg e unc ion in he isk quad angle.
Nex , we p o ide an explici cha ac e iza ion o he highe o de expec iles, ha is,
he highe o de isk measu e based on he expec ile isk measu e.
The expec ile is de ined as a minimize . I s Kusuoka ep esen a ion is cen al in
elabo a ing he co esponding highe o de isk unc ional.
De ini ion 5.1 Fo
𝛼∈(0, 1)
, he expec ile is
whe e
is he asymme ic loss, o quad a ic e o unc ion.
The expec ile sa is ies he i s o de condi ion
and
e𝛼(⋅)
is a isk measu e o
𝛼∈[1∕2, 1]
. We men ion ha condi ion(5.2) p o ides
a de ini ion o
Y∈L1
, i is hus mo e gene al han(5.1), which equi es
Y∈L2
. The
Kusuoka ep esen a ion o he expec ile (c . Bellini e al. (2014,P oposi ion9)) is
gi en by
�1
u
𝛽
�
𝛽
u
𝛽
𝜎(u)
1−𝛽�𝜇(d𝛽�)F−1
−X(u)du
≥
�
1
u
𝛽
�
𝛽
u
𝛽
𝜎(u)
1−𝛽�𝜇(d𝛽)F−1
−Y(u)du,𝛽∈(0, 1)
,
�1
u
𝛽
𝜎𝜇(u)F−1
−X(u)du
≥
�
1
u
𝛽
𝜎𝜇(u)F−1
−Y(u)du,𝛽∈(0, 1)
.
(5.1)
e
𝛼
(Y) = a g min
x∈ℝ
𝔼
𝓁
𝛼
(Y−x),
𝓁
𝛼(x)=
{
𝛼x
2
i x
≥0,
(1−𝛼)x2else
(5.2)
(1−𝛼)𝔼(x−Y)+=𝛼𝔼(Y−x)+,
Connec ion be weenhighe o de measu es o  isk ands ochas ic… Page 23 o 28 41
whe e
𝜂
=
1−𝛼
𝛼
, so ha he isk le el in(5.3) is
1
−
𝛾
1−𝛾
𝜂
1−𝜂
=
𝛼(2−𝛾)−1
(2𝛼−1)(1−𝛾)
. In ol ing
spec al isk measu es, he expec ile can be ecas as
whe e
S
=
{
𝜎
𝛾
∶𝛾∈[0, 1 −𝜂]
}
collec s he spec al unc ions
The highe o de expec ile can be desc ibed by in ol ing i s dual no m (c .(2.9)), as
well as i s Kusuoka ep esen a ion (c . Co olla y3.6). The ollowing wo (sub)sec-
ions elabo a e hese possibili ies o he expec ile.
5.1 The dual no m o expec iles
The highe o de expec ile can be desc ibed wi h he dual ep esen a ion(2.8), o
which he dual no m o he expec ile is necessa y.
By he cha ac e iza ion o he loss unc ion (5.2) i holds ha
e𝛼(Y)
is well-
de ined o
Y∈L1(P)
. This is enough o conclude ha
𝔼|Y|≤C𝛼
⋅e𝛼(|Y|)
o some
cons an
C𝛼>0
(Lakshmanan and Pichle 2023,Co olla y2.16) elabo a e he igh
bound
C
𝛼
=
𝛼
1−𝛼
). I ollows ha
Y∗=L∞
, so ha
‖Z‖∞
is well-de ined o
Z∈Y∗
.
The ollowing esul p o ides he dual no m o he expec ile explici ly.
P oposi ion 5.2 (Dual no m o he expec ile) Fo
𝛼≥1∕2
, he dual no m is
(c .(2.7)) . I holds ha
No ably, he no m
‖⋅‖∗
𝛼
is no a isk measu e i sel , and(5.5) is no a Kusuoka
ep esen a ion; indeed, he o al weigh in he ep esen a ion(5.5) is
o
𝛼∈(1∕2, 1]
.
P oo o P oposi ion 5.2 We may assume ha
Z≥0
, as o he wise we may conside
sign (Z)⋅Y
ins ead o Y. Fo a bi a y se s B and G wi h
B⊂G
and
P(G)<1
de ine
he andom a iable
(5.3)
e
𝛼
(Y)= max
𝛾∈[0,1−𝜂](1−𝛾)⋅𝔼Y+𝛾⋅𝖠𝖵@𝖱
1−𝛾
1−𝛾
𝜂
1−𝜂
(Y),
e
𝛼(Y)=sup
{
R𝜎𝛾
(Y)∶𝜎𝛾∈S
},
s
𝛾(u)=
{
1−𝛾i u
≤
1−
𝛾
1−𝛾
𝜂
1−𝜂
,
1−𝛾
𝜂
else.
(5.4)
‖
Z
‖∗
𝛼
∶= sup
�
𝔼YZ ∶e
𝛼
(
�
Y
�
)
≤
1
�
(5.5)
‖
Z
‖
∗
𝛼=sup
𝛽∈(0,1)
(1−𝛽)⋅𝖠𝖵@𝖱𝛽(
�
Z
�
)+𝛽
1−𝛼
𝛼
‖
Z
‖
∞
.
(
1−𝛽)+𝛽
1−𝛼
𝛼
<
1
A.Pichle
41 Page 24 o 28
No e, ha
and hence
e𝛼
(

Y
B,G
)=
1
by he de ining equa ion(5.2). I ollows wi h(5.4) ha
As
B⊂G
a e a bi a y, we conclude in pa icula ha
because he andom a iables
sa is y all condi ions om abo e o any uni o m a iable U. Now le
P(G)→1
and
by deno ing
𝛽=P(B)
i ollows ha
as
𝖠𝖵@𝖱𝛾(Z)→ess sup Z
o
𝛾→1
.
As o he con e se obse e ha we may assume
e𝛼(Y)=1
o he op imal an-
dom a iable in(5.4). Conside he Lag angian
whe e he Lag angian mul iplie
𝜆∈ℝ
is associa ed wi h he equali y cons ain
e𝛼(Y)=1
, i.e.,(5.2), and he measu able a iable
𝜇∈L1
,
𝜇≥0
, is associa ed wi h
he inequali y cons ain
Y≥0
. P o ided Tha he de i a i e exis s, he i s o de
condi ions a e
o
Now no e ha he le -hand side o (5.8) in ol es he a iableZ, while he igh -
hand side only in ol es cons an s, excep on
{Y=0}
, whe e
𝜇
is no necessa ily
cons an . The i s o de condi ions (5.8) hus hold ue on pla eaus o Z, i hey
(5.6)

Y
B,G(𝜔)∶=
⎧
⎪
⎨
⎪
⎩
0 i 𝜔∈B,
1 i 𝜔∈G⧵B
, and
1−𝛼
𝛼
⋅
P(B)
1−P(G)+1 else.
(
1−𝛼)⋅P(B)(1−0)=𝛼⋅
(
1−P(G)
)(
(1−𝛼)P(B)
𝛼(1−P(G)) +1−1
),
‖
Z
‖∗
𝛼≥
𝔼ZY
B,G.
‖
Z
‖
∗
𝛼
≥�
(1−P(B)
�
⋅𝖠𝖵@𝖱P(B)(Z)+P(B)
1−𝛼
𝛼
⋅𝖠𝖵@𝖱P(G)(Z)
,

Y
B,G=
(
1−P(B)
)
⋅
1
1−P(B)
1[P(B),1](U)+P(B)
1−𝛼
𝛼
⋅
1
1−P(G)
1[P(G),1](U
)
‖
Z
‖
∗
𝛼
≥
sup
𝛽∈(0,1)
(1−𝛽)⋅𝖠𝖵@𝖱𝛽(Z)+𝛽
1−𝛼
𝛼
ess sup Z
,
(5.7)
L
(Y;𝜆,𝜇)∶= 𝔼ZY −𝜆
(
(1−𝛼)𝔼(1−Y)
+
−𝛼𝔼(Y−1)
+)
+𝔼𝜇Y
,
0
=
𝜕
𝜕Y
L(Y;𝜆,𝜇)
,
(5.8)
Z
=𝜆
(
−(1−𝛼)1
{Y<1}
−𝛼1
{Y>1})
−𝜇⋅1
{Y=0}.