Robust asymptotic insurance-finance arbitrage

Obe p ille , Ka ha ina; Ri e , Mo i z; Schmid , Tho s en
A icle — Published Ve sion
Robus asymp o ic insu ance- inance a bi age
Eu opean Ac ua ial Jou nal
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Obe p ille , Ka ha ina; Ri e , Mo i z; Schmid , Tho s en (2024) : Robus
asymp o ic insu ance- inance a bi age, Eu opean Ac ua ial Jou nal, ISSN 2190-9741, Sp inge ,
Be lin, Heidelbe g, Vol. 14, Iss. 3, pp. 929-963,
h ps://doi.o g/10.1007/s13385-024-00389-1
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315858
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h p://c ea i ecommons.o g/licenses/by/4.0/
Eu opean Ac ua ial Jou nal (2024) 14:929–963
h ps://doi.o g/10.1007/s13385-024-00389-1
ORIGINAL RESEARCH PAPER
Robus asymp o ic insu ance- inance a bi age
Ka ha ina Obe p ille 1·Mo i z Ri e 2·Tho s en Schmid 2
Recei ed: 9 Decembe 2022 / Re ised: 19 Feb ua y 2024 / Accep ed: 10 June 2024 /
Published online: 8 Augus 2024
© The Au ho (s) 2024
Abs ac
This pape s udies he alua ion o insu ance con ac s linked o inancial ma ke s, o
example h ough in e es a es o in equi y-linked insu ance p oduc s. We build upon
he concep o insu ance- inance a bi age as in oduced by A zne e al. (Ma h Financ,
2024), ex ending hei wo k by inco po a ing model unce ain y. This is achie ed by
in oducing s a is ical unce ain y in he unde lying dynamics o be ep esen ed by a
se o p io s P. Wi hin his amewo k we p opose he no ion o obus asymp o ic
insu ance- inance a bi age (RIFA) and cha ac e ize he absence o such s a egies
in e ms o he new concep o QP-e alua ions. This nonlinea wo-s ep e alua ion
ensu es absence o RIFA. Mo eo e , i domina es all wo-s ep e alua ions, as long as
we ag ee on he se o p io s P. Ou analysis highligh s he ole o QP-e alua ions
in e ms o showing ha all wo-s ep e alua ions a e ee o RIFA. Fu he mo e, we
in oduce a doubly s ochas ic model o add ess unce ain y o su ende and su i al,
u ilizing copulas o de ine condi ional dependence. This se ing illus a es how he
QP-e alua ion can be applied o he p icing o hyb id insu ance p oduc s, high-
ligh ing he lexibili y and po en ial o he p oposed app oach.
Keywo ds Insu ance- inance a bi age unde unce ain y ·Robus QP- ule ·
Enla gemen o il a ion ·Absence o obus insu ance- inance a bi age
The au ho s g a e ully acknowledge he suppo o he Ge man Resea ch Founda ion (DFG) h ough g an
SCHM 2160/15-1, he CRC SFB 1597/1 (Small Da a) and o he F eibu g Cen e o Da a Analysis and
Modeling (FDM).
BTho s en Schmid
[email p o ec ed]g.de
Ka ha ina Obe p ille
[email p o ec ed]
Mo i z Ri e
[email p o ec ed]g.de
1Uni e si y o Munich, The esiens . 39, 80333 München, Ge many
2Uni e si y o F eibu g, E ns -Ze melo-S . 1, 79104 F eibu g, Ge many
123
930 K. Obe p ille e al.
1 In oduc ion
This pape de elops and cha ac e izes he absence o insu ance- inance a bi age unde
model unce ain y. Ou s a ing poin is he obse a ion ha mos insu ance con ac s
a e linked o inancial ma ke s, o example h ough in e es a es o ia di ec links
o he con ac ual bene i s o s ocks o indices. Howe e , he modeling and he al-
ua ion o insu ance con ac s and p oduc s on inancial ma ke s a e undamen ally
di e en due o hei dis inc cha ac e is ics: insu ance con ac s a e s a ic and pe son-
alized p oduc s, whe eas p oduc s on inancial ma ke s a e s anda dized and aded
equen ly. Va ious app oaches ha e been p oposed in he li e a u e as o how insu -
ance and inancial ma ke s can be ea ed in a cohe en manne , see e.g., [10,11,21,
23] and e e ences he ein. Mo e ecen ly, 2-s ep and 3-s ep app oaches ha e been
p oposed as o example in [2,7] and in [19].
In his pape we aim o a undamen al analysis o a bi age in hese ma ke s and ely
on he no ion o insu ance- inance a bi age (IFA) in oduced by [1]. In his app oach
he insu ance company may issue con ac s o a la ge numbe o clien s on he one
hand. On he o he hand, i can simul aneously hedge i s posi ions by ading on he
inancial ma ke . In o de o model he wo in o ma ion lows o which he insu ance
company has access, we wo k wi h wo il a ions. The smalle il a ion ep esen s
he publicly a ailable in o ma ion on he inancial ma ke , deno ed by F=(F ) ≤T,
while he la ge il a ion G=(G ) ≤Taddi ionally con ains he insu e ’s in o ma ion.
Gi en a p icing measu e Qon he inancial ma ke (, F)and a s a is ical measu e P
on (, G)a cha ac e iza ion o he absence o IFA in e ms o he QP- ule is de i ed
in [1].
E en i a la ge se o homogeneous da a is a ailable, s a is ical unce ain ies in
p edic ing he u u e e olu ion o insu ance losses in he conside ed po olio emain
a p oblem ha needs o be add essed. In his pape we he e o e ake his unce ain y
in o accoun .
To do so, we ix he nullse s Non he inancial ma ke (, F)which oge he wi h
he aded asses s Sde e mine he se o equi alen ma ingale measu es Q. Second,
we conside a class o p obabilis ic models Pon (, G), such ha he measu es in P
es ic ed o Fha e exac ly he nullse s Nand s udy he associa ed QP- ule. Mos
no ably, his amewo k allows us o model unce ain y on he insu ance ma ke unde
he assump ion ha we do no ace any model isk on he inancial ma ke . Wo king
wi h a class o po en ial models Pis in line wi h he g owing li e a u e on model isk
and unce ain y, see e.g., [5,6,8,9,24,28].
To he bes o ou knowledge, his is he i s s udy o insu ance- inance a bi age
unde model unce ain y. Mo e speci ically, we p o e a cha ac e iza ion o obus
insu ance- inance a bi age (RIFA) by using he QP-e alua ion in Theo em 2.12.
Fu he mo e, we show ha his esul p o ides a heo e ical ounda ion o a class
o wo-s ep e alua ions in oduced in [23], which is applied o he p icing o hyb id
p oduc s depending on he inancial ma ke , as well as on o he andom sou ces. In
pa icula , we p o e ha e e y wo-s ep e alua ion, which is he combina ion o a
isk- ee measu e Qand a cohe en F-condi ional isk measu e, being con inuous
om below, equals a QP-e alua ion o a sui able subse Pon (, G). Thus, e e y
123
Robus insu ance- inance a bi age... 931
wo-s ep e alua ion o his kind leads o a obus a bi age- ee p ice in he asymp o ic
insu ance- inance se ing.
We conclude he pape by sugges ing possible applica ions in Sec . 4. Fi s , we
conside he case o wo condi ional independen andom imes such ha he condi-
ional dis ibu ion unc ions ace a ce ain deg ee o unce ain y. He e, he andom
imes ep esen he su ende ime and he ime o dea h o an insu ance seeke . In his
se ing, we in oduce a inancial ma ke ia a Cox-Ross-Rubins ein model and con-
side inance-linked insu ance bene i s wi h su ende op ions. We compu e he obus
a bi age- ee p ice o hese p oduc s nume ically and ind ha he unce ain y in his
example jus i ies ha he obus a bi age- ee p ice is highe han he sup emum o
all a bi age- ee p ices unde each possible model. This unde lines he impo ance
o ea ing unce ain y in a sys ema ic manne . Fu he mo e, we show ha he well-
known Cox model unde unce ain y is con ained in he ou lined se ing. Mo eo e , we
gene alize he se ing by allowing dependence be ween he andom imes desc ibed
by a copula.
The pape is s uc u ed as ollows. In Sec .2we in oduce he de ini ion o a obus
insu ance- inance a bi age and p o ide a cha ac e iza ion in ou main esul . A e
ha , Sec .3s udies he ela ion o he QP- ule o wo-s ep e alua ions. Then, in
Sec .4we s udy an insu ance- inance ma ke wi h wo condi ionally independen an-
dom imes and nume ically p o ide he obus a bi age- ee p ices o ce ain hyb id
p oduc s. In addi ion, we conside a copula amewo k o he wo andom imes unde
unce ain y.
2 Robus asymp o ic insu ance- inance a bi age
Le (, G)be a measu able space and deno e by P(, G) he se o all p obabili y
measu es on his space. We conside a disc e e ime model wi h imes =0,...,Tand
in oduce wo di e en kinds o in o ma ion lows desc ibed by he il a ions Fand G
on (, G). The il a ion F=(F ) ≤T ep esen s publicly a ailable in o ma ion and
con ains all in o ma ion a ailable on he inancial ma ke . The il a ion G=(G ) ≤T
con ains addi ional p i a e in o ma ion o he conside ed insu ance company, which
includes, o example, se e al da ase s on i s clien s. In pa icula , F⊆G. Mo eo e ,
le F0=G0={∅,}and Fbe a σ- ield such ha FT⊆F⊆G.
Gi en P⊆P(, G),ase A⊆is called P-pola i A⊆N o some N∈G
sa is ies P(N)=0 o all P∈P. Mo eo e , a p ope y holds P-quasi su ely
(P-q.s.) i i holds ou side a P-pola se .
Fo a ixed σ-ideal o F-nullse s N, i.e., he e exis s a measu e P0on (, F)such
ha Na e he nullse s o P0, we de ine he ollowing se s o p io s
PN(, F):=P∈P(, F)|Na e he nullse s o P(2.1)
and
PN(, G):=P∈P(, G)|Na e he nullse s o P|F.(2.2)
123
932 K. Obe p ille e al.
He ea e , we ix a p obabili y measu e P0∈P(, F), which de e mines he nullse
Non (, F), and a subse o p io s P⊆PN(, G), which speci ies he unce ain y
abou he model.
In he ollowing, we in oduce he concep o insu ance- inance a bi age. Fo sim-
plici y, we conside only a single insu e . We also assume ha he insu e can con ac
wi h an a bi a ily high numbe o clien s o educe i s isks, as speci ied in Assump-
ion 2.1. To es ablish his, we conside a ini e numbe o insu ance seeke s and s udy
he limi s o po olio alloca ions - a echnique inspi ed by la ge inancial ma ke s, see
e.g., [15,17,18]. Such s a egies may lead o insu ance a bi ages and i is pa ly ou
aim o cha ac e ize when and how such a bi ages can be achie ed and unde which
condi ions hey can be a oided.
A he same ime, he insu ance company also ades on he inancial ma ke , po en-
ially leading o a inancial a bi age. Thus, he combina ion o hese concep s esul s
in an insu ance- inance a bi age.
2.1 The insu ance con ac s
Insu ance con ac s o e a a ie y o bene i s a u u e imes in exchange o a single
p emium o a p emium s eam. We wo k wi h discoun ed quan i ies and, wi hou loss
o gene ali y, we conside a single p emium paid a ime 0 and an agg ega ed bene i
ecei ed a u u e ime T. Mo e p ecisely, we deno e by p∈R he p emium o be
paid a ime 0 and a GT-measu able (discoun ed) bene i Xi o be ecei ed by he i h
clien a ime T. This allows o co e a wide ange o con ac s, pa icula ly con ac s
depending on inancial ma ke s, such as a iable annui ies.
We assume ha all insu ance seeke s unde conside a ion can be ea ed as homoge-
neous (unde each model P∈P) and each insu ance seeke pays he same p emium p
in o de o ecei e his o he pe sonal bene i Xi. This idea is o malized in he ollow-
ing assump ion, which is a gene aliza ion o he amewo k o ac ua ial ma hema ics
o s ochas ic asse s, as discussed in [1].
Assump ion 2.1 Fo all P∈P, he ollowing holds:
(i) X1,X2,...∈L2
+(, G,P)a e F-condi ionally independen .
(ii) EP[Xi|F]=EP[X1|F] o all i∈N.
(iii) Va P[Xi|F]=Va P[X1|F] o all i∈N.
Rema k 2.2 (The implica ions o Assump ion 2.1) Assump ion 2.1 is a ai ly weak
assump ion and includes a wide ange o exis ing models:
(i) App oaches whe e insu ance bene i s a e i.i.d. and independen o he publicly
a ailable in o ma ion.
(ii) Any kind o a iable annui ies including F-independen and i.i.d. su i al
(o su ende ) imes τi. This co e s o example su i al bene i s o he
o m Xi=1{τi>T}e− T F(˜
ST)o paymen s a su ende imes like Xi=
1{τi=T}e− τiF(˜
Sτi), whe e ˜
Sis he F-adap ed (undiscoun ed) s ock p ice p ocess
and e− is he de e minis ic discoun ing ac o a ime . Simila ly, s ochas ic
in e es a es o pa h-dependen payo s can be included.
123

Robus insu ance- inance a bi age... 933
(iii) Doubly-s ochas ic andom imes a e also pa o he amewo k, i.e., andom
imes which a e d i en by a haza d a e λwhich is F-p og essi ely measu able
such ha he p ocess 1{τi≤ }−1[0, ∧τi]λsds is an F-ma ingale. This allows
o model impo an aspec s like sys emic isk o longe i y isk, o example by
inco po a ing ac o s which a ec λand hence all insu ance seeke s. I seems
impo an o poin ou , ha Fcon ains all publicly a ailable in o ma ion, and
hence migh be signi ican ly la ge hen he il a ion c ea ed by s ock p ices
only. In pa icula , publicly a ailable mo ali y ables would be included in F.
In ui i ely, λallows o co e isks which a e a e obse able by he public, such
as a gene al c isis gi ing ise o sys emic isk o inc easing longe i y e lec ed
in publicly a ailable li e ables. The indi idual isk, e e ing o he isk beyond
sys ema ic isk ha aligns wi h he indi idual cha ac e is ics o he policyholde ,
is subsequen ly cap u ed by he model’s addi ional s ochas ic componen . We
e e o Rema k 4.3 which shows how o include inc easing li e expec a ions in
he Gompe z model unde Assump ion 2.1.
Howe e , ou cu en amewo k does no include di e en coho s. To inco po a e o
example di e en ages o he insu ance seeke s, a sepa a e model o each age mus
be conside ed.
The insu ance po olio is ob ained as a limi o alloca ions o con ac s wi h a ini e
numbe o clien s. An alloca ion a ime 0, ψ=(ψi)i∈N,is c00- alued, de e minis ic
and non-nega i e, whe e c00 deno es he space o sequences wi h a ini e numbe o
non-ze o elemen s. Fo i∈N,ψi∈R+deno es he size o he con ac wi h he i h
policyholde . The accumula ed bene i s and p emiums associa ed wi h he alloca ion
ψa e gi en by i∈NψiXiand i∈Nψip, while he associa ed p o i s and losses a e
deno ed by
VT(ψ) := 
i∈N
ψi(p−Xi).
An insu ance po olio s a egy := (ψn)n∈N∈cN
00 is modeled as a sequence o
alloca ions ψn=(ψn,i)i∈N. Mo eo e , he p o i and loss o an insu ance po olio
s a egy is gi en (i i exis s) by
VT() := lim
n→∞ VT(ψn)=lim
n→∞ 
i∈N
ψn,i(p−Xi).
We in oduce he ollowing admissibili y condi ions o an insu ance po olio s a egy
=(ψn)n∈N.
Assump ion 2.3 (i) Uni o m boundedness: The e exis s C>0,such ha
ψn:=
i∈N
ψn,i≤C o all n≥1.
123
934 K. Obe p ille e al.
(ii) Con e gence o he o al mass: The e exis s γ≥0 such ha
γ=lim
n→∞ ψn.
(iii) Con e gence o he o al weal h: The e exis s a R- alued andom a iable
VT(), such ha
VT() =lim
n→∞ VT(ψn)P-q.s.
An insu ance po olio s a egy =(ψn)n∈Nwhich sa is ies Assump ion 2.3 is called
P-admissible.
2.2 The inancial ma ke
We in oduce a inancial ma ke model in disc e e ime consis ing o d isky asse s
S=(S1
,...,Sd
) =0,...,Ton (, F). Fo i=1,...,d, he discoun ed p ice o he
i h isky asse a ime is gi en by he F -measu able andom a iable Si
and he
bank accoun is gi en by S0≡1.We assume ha he insu ance company ades
wi h F- ading s a egies on he inancial ma ke , whe e a F- ading s a egy is a d-
dimensional F-p edic able p ocess ξ=(ξ ) =1,...,Twi h ξ =(ξ1
,...,ξd
). No e
ha each s a egy ξcan be ex ended by ξ0=(ξ0
) =1,...,T o a sel - inancing ading
s a egy ¯
ξ=(ξ0
,...,ξd
) =1,...,T,c .[14, Rema k 5.8]. The associa ed gain p ocess
a ime =1,...,Tis hen gi en by he disc e e s ochas ic in eg al
(ξ ·S) :=

s=1
ξs(Ss−Ss−1)=

s=1
d

i=1
ξi
sSi
s−Si
s−1.
The absence o a bi age in his ma ke wi h espec o one and hus any es ic ed
measu e P|F o P∈P⊆PN(, G)can be cha ac e ized by he exis ence o
an equi alen ma ingale measu e, c . [14, Theo em 5.16]. The se o all equi alen
ma ingale measu es is deno ed by Me(F)and de ined by
Me(F):= {Q∈PN(, F)|Sis a (Q,F)-ma ingale}.(2.3)
Rema k 2.4 By aking in o accoun a se o p io s P⊆PN(, G)and acco ding o
he de ini ion o PN(, G)in (2.2), all nullse s on (, F)a e al eady de e mined
by N.Thus, he unce ain y o ou model e e s only o he addi ional insu ance pa
de ined on (, G).
2.3 Robus insu ance- inance a bi age
Finally, we in oduce he insu ance- inance ma ke (S,X,p)on (, G)consis ing
o he bene i s X=(Xi)i∈N, he p emium p, and he discoun ed asse p ices S=
(S ) =0,...,T.
123
Robus insu ance- inance a bi age... 935
In o de o de ine a obus a bi age in ou se ing, we use he concep o a obus
a bi age s a egy in oduced in [6]. No e, howe e , ha we do no need o equi e
con exi y o P o he cha ac e iza ion o he absence o a obus a bi age s a egy.
De ini ion 2.5 AP- obus asymp o ic insu ance- inance a bi age RIFA(P)on he
insu ance- inance ma ke (S,X,p)is a pai (ξ, ) consis ing o an F-p edic able
ading s a egy ξand a P-admissible insu ance po olio s a egy such ha
(ξ ·S)T+VT() ≥0P-q.s. and EP[(ξ ·S)T+VT()]>0 o someP∈P.
(2.4)
I he e exis s no such pai (ξ, ) sa is ying (2.4), hen he e is no P- obus asymp o ic
insu ance- inance a bi age, which we deno e by NRIFA(P).
Rema k 2.6 I o all P∈Pi holds NRIFA({P}) hen NRIFA(P)is also sa is ied.
Howe e , he con e se s a emen does no hold in gene al.
In he case o no model unce ain y, i.e., when P={P} o a measu e P∈
PN(, G), i is shown in Co olla y 5.2 in [1] ha i he e exis s Q∈Me(F)such
ha
p≤EQP[X1]:=EQEP[X1|F],(2.5)
hen he e is no asymp o ic insu ance- inance a bi age. Thus, acco ding o Rema k
2.6, an insu ance- inance ma ke (S,X,p) ul ills NRIFA(P)i (2.5) holds o each
P∈P. Howe e , i should be no ed ha hese condi ions a e no necessa y.
2.4 Uni o m essen ial sup emum
In o de o cha ac e ize he absence o a P- obus asymp o ic insu ance- inance a bi-
age, we aim o iden i y assump ions ha allow us o ake in o accoun a obus e sion
o he condi ional expec a ions EP[X|F] o P∈Pin (2.5). Gi en a se o p io s
P⊆P(, G), i is in gene al no possible o conside supP∈PEP[X|F],as he
condi ional expec a ion is only de ined P-a.s. and he p io s in Pmay ha e di e en
nullse s. Fu he mo e, he sup emum no longe needs o be measu able.
The na u al app oach o sol e he measu abili y issue is o wo k wi h he essen ial
sup emum ins ead o he sup emum. Fo a ixed p obabili y measu e P∈P(, G)
and a se o andom a iables on (, G) he e is a andom a iable Y=: ess supP
such ha
(i) Y≥ϕP-a.s. o all ϕ∈, and
(ii) Y≤ψP-a.s. o e e y andom a iable ψsa is ying ψ≥ϕP-a.s. o all
ϕ∈.
We e e o [14, Theo em A.37] o a p oo o he exis ence o he essen ial sup emum
and o [3], in whose wo k a gene al cons uc ion o such a nonlinea condi ional
expec a ion is s udied.
123
936 K. Obe p ille e al.
Howe e , o a gene al se o p io s P⊆P(, G)and a se o andom a iables
on (, G) he e may be no uni o m essen ial sup emum, i.e., a andom a iable Y
such ha
Y=ess supPP-a.s. o all P∈P.(2.6)
The de ini ion o he se PN(, G)in (2.2) allows us o conside o P⊆
PN(, G) he se o andom a iables =(ϕP)P∈Psuch ha
ϕP=EP[X|F]P-a.s. o all P∈P.(2.7)
Indeed, we obse e ha by F-measu abili y oge he wi h he ac ha P|F∼P|F
o all P,P∈P, he condi ional expec a ion EP[X|F]is no only P-a.s. uniquely
de e mined bu also P-q.s.. Mo eo e , he e exis s an uni o m essen ial sup emum
which ul ills (2.6), as he ollowing esul demons a es.
Lemma 2.7 Le Nbe he nullse s gene a ed by he p obabili y measu e P0∈
P(, F),P⊆PN(, G)and a se o F-measu able andom a iables. Then
ess supP0=ess supPP-a.s. o all P ∈P.(2.8)
P oo We show ha ess supP0 ul ills condi ion (i) and (ii) om he de ini ion o
he essen ial sup emum o all P∈P. Fo he i s pa , using he de ini ion o
ess supP0, we ob ain ha
ess supP0≥ϕP0-a.s. o all ϕ∈. (2.9)
Since {ess supP0≥ϕ}is F-measu able and P|F∼P0 o all P∈P,(2.9)also
holds P-a.s. o all P∈P. Relying on he cons uc ion o he essen ial sup emum,
see, o example [14, Theo em A.37], he e exis s a coun able subse ∗⊆such
ha
ess supP0(ω) =sup ∗(ω) o all ω∈.
Fix any P∈P, hen o each andom a iable ψon (, G)such ha
ψ≥ϕP-a.s. o all ϕ∈,
we ge ψ≥sup ∗=ess supPP-a.s.. This shows he second pa and he esul is
p o en. 
In he ollowing, o he ixed measu e P0, which gene a es he F-nullse s Nand
use he no a ion
ess sup := ess supP0.
123
Robus insu ance- inance a bi age... 943
whe e we use [1, P oposi ion B.1] o he i h equali y and (2.20) o he inequali y.
We now de ine he p ocess Z=(Z ) =0,...,Tby
Z =ER[VT()|F ], o all =0,...T.
Zis a (F,R)-ma ingale wi h Z0<0by(2.21). Le ξbe some F-p edic able s a egy.
Then, as in [14, Rema k 9.5] he alue p ocess (ξ ·S)is a local (F,R)-ma ingale
and hus (ξ ·S)+Zis a local (F,R)-ma ingale. Fo he sake o con adic ion we
assume ha (ξ, ) is a P- obus asymp o ic insu ance- inance a bi age such ha 
has posi i e o al mass γ>0. Thus, i holds ha
(ξ ·S)T+ZT≥0P-a.s. o all P∈P.(2.22)
Gi en ha R|F∼P|F1
N
n

i=1
Pi|F, equa ion (2.22)isalso ueR-a.s., i.e.,
(ξ ·S)T+ZT≥0R-a.s.. (2.23)
Thus, acco ding o [14, P oposi ion 9.6] he p ocess (ξ ·S)+Zis a (F,R)-
supe ma ingale and we ob ain
ER[(ξ ·S)T+ZT]≤(ξ ·S)0+Z0<0.
This con adic s (2.23) and he e canno exis a P- obus asymp o ic insu ance- inance
a bi age (ξ, ) such ha has posi i e o al mass γ>0. Howe e , gi en ha he
pu e inancial ma ke is a bi age- ee wi h espec o F-p edic able ading s a egies
ξ, he e could also no be an a bi age (ξ, ) such ha has o al mass γ=0.
O e all, his leads o a con adic ion and hus he esul is p o en. 
Rema k 2.13 Le us compa e Theo em 2.12 in he case o P={P}wi h Theo-
em 1 in [26]. Acco ding o Assump ions 2.1 and 2.3, each condi ion (i)and (ii)in
Theo em 2.12 implies he absence o a bi age in he sense o De ini ion 2.5 and he e is
no need o an addi ional boundedness assump ion on he densi y o he co esponding
ma ingale measu es. By con as , in [26] he boundedness assump ion on he densi y
o he equi alen ma ingale measu e is essen ial and canno be subs i u ed by means
o Lemma 2.10, c . Example 2 in [26].
Mo i a ed by Theo em 2.12 and he p e ious discussion (see e.g., Equa ion (2.5)),
we de ine he ollowing obus e sion o he QP- ule.
De ini ion 2.14 Le P⊆PN(, G)and Q∈Me(F). Then, o X≥0P-q.s., we
de ine he QP-e alua ion o X by
EQP[X]:=EQess sup
P∈P
EP[X|F].(2.24)
No e ha QPdoes no de ine a p obabili y measu e as i is he case o QP.
123

944 K. Obe p ille e al.
Rema k 2.15 (On he choice o Q) I he inancial ma ke is comple e, Me(F)={Q}
and he e is only one possibly choice o Q. I he ma ke is incomple e, he choice o
Qbecomes mo e di icul . When su icien ly many aded de i a i es a e a ailable, a
ma ke -consis en Qcan be ob ained by calib a ing he model o hose de i a i es.
In he insu ance con ex , he mos challenging calib a ion p oblems occu when
long ma u i ies (like 10 o 30 yea s) a e conside ed. He e, he e a e ew o e en no
adeable ins umen s and a good calib a ion equi es much mo e e o , in pa icula
o exclude model isk.
Rema k 2.16 I he se {EP[X|F]|P∈P}is di ec ed upwa d, i.e., o all P,P∈P
he e exis s ˜
P∈Psuch ha
max{EP[X|F],EP[X|F]} ≤ E˜
P[X|F]P0-a.s.,(2.25)
hen, as demons a ed by [14, Theo em A.37], he e exis s a sequence o measu es
(Pn)n∈N⊂Psuch ha
EPn[X|F]ess sup
P∈P
EP[X|F]P0-a.s. o n→∞.
Using mono one con e gence we ind ha
EQess sup
P∈P
EP[X|F]=lim
n→∞ EQEPn[X|F]
≤sup
P∈P
EQ[EP[X|F]]
≤EQess sup
P∈P
EP[X|F]
and ha
EQP[X]= sup
P∈P
EQ[EP[X|F]].
Consequen ly, o a se o p io s P ha is di ec ed upwa ds in he sense o
Equa ion (2.25), he e is no P- obus asymp o ic insu ance- inance a bi age i and
only i he e is no asymp o ic insu ance- inance a bi age wi h espec o P o all
P∈P.
Rema k 2.17 We now b ie ly conside he case o G- ading s a egies on he inancial
ma ke in oduced in Sec .2.2, i.e., ξis a d-dimensional G-p edic able p ocess. This
e lec s he ac ha he insu e has access o in o ma ion on he inancial ma ke as
well as on he insu ance ma ke . In o de o de ine a no-a bi age condi ion o he
inancial ma ke , we in oduce some mo e no a ion. Le P⊆PN(, G). We say
ha a measu e Q∈P(, G)is domina ed by Pi he e exis s P∈Psuch ha
QP, and in his case we w i e Q≪P. Nex , we de ine he se
M(G):= {Q∈P(, G)|Q≪Pand Sis a (Q,G)-ma ingale}.(2.26)
123
Robus insu ance- inance a bi age... 945
In his case we assume ha o all P∈P he e exis s Q∈M(G)such ha PQ.
Then, acco ding o [6, Theo em 4.5], he e is no P- obus a bi age, deno ed by
NA(P,G), on he inancial ma ke , which means ha o all G- ading s a egies
(ξ ·S)T≥0P-q.s. implies (ξ ·S)T=0P-q.s. (2.27)
Gi en ha e e y F- ading s a egy is also a G- ading s a egy, i is clea ha
NA(P,G)implies NA(P,F), whe e Gand F e e s he e o he G- ading s a e-
gies and F- ading s a egies, espec i ely. I hus ollows ha NRIFA(P,G)implies
NRIFA(P,F)and hus (i) o (ii) in Theo em 2.12 is sa is ied. He e, NRIFA(P,G)
is de ined as in De ini ion 2.5, bu wi h a G- ading s a egy ξ. Howe e , he co e-
sponding i di ec ion o Theo em 2.12 is mo e delica e and could he e o e se e as a
opic o u u e esea ch.
3 Robus wo-s ep e alua ion
In his sec ion we show ha Theo em 2.12 p o ides a heo e ical ounda ion o he so-
called wo-s ep e alua ion,c .[11,23]. This kind o e alua ion is used o he p icing
o hyb id p oduc s depending on he inancial ma ke , as well as on o he andom
sou ces, e.g., indi idual isks depending on he policy holde o an insu ance con ac .
The idea o a wo-s ep e alua ion is o combine ac ua ial echniques wi h concep s om
inancial ma hema ics. In he ollowing, we ecap he idea and he concep . Howe e ,
no e ha in con as o he exis ing li e a u e we do no ix any p obabili y measu e on
(, G), bu only a p io P0on he measu able space (, F).
Le Xbe a G-measu able andom a iable, ep esen ing he discoun ed payo o an
insu ance p oduc . In a i s s ep we conside he F-condi ional isk o X, i.e. ρF(X),
whe e ρFis a sui able F-condi ional isk measu e de ined on he space o bounded
andom a iables on (, G), which is deno ed by Lb(, G). This co esponds o an
ac ua ial e alua ion esul ing in a F-measu able andom a iable ρF(X)de ined on
he inancial ma ke (, F,P0). In he second s ep we p ice ρF(X)on he inancial
ma ke unde an equi alen isk-neu al measu e Q∈Me(F). Combining hese, we
a i e a he ollowing wo-s ep e alua ion:
π:Lb(, G)→R,π(X)=EQ[ρF(−X)].(3.1)
As al eady men ioned in [1], he QP-e alua ion in (2.5) is a wo-s ep e alua ion wi h
he F-condi ional isk measu e ρF(X)=EP[X|F]as well as he new concep
o he obus QP-e alua ion om De ini ion 2.14. In he ollowing we ecap some
well-known ac s o condi ional isk measu es in o de o highligh ha o speci ic
cohe en F-condi ional isk measu es ρF he wo-s ep e alua ion in (3.1) can be
ew i en by a QP-e alua ion o a sui able choice o P.
123
946 K. Obe p ille e al.
3.1 Robus ep esen a ion o condi ional isk measu es
Fo he eade ’s con enience we ecall he de ini ion o condi ional isk measu es (see
e.g., [14]).
De ini ion 3.1 AmapρF:Lb(, G)→L∞(, F,P0)is called a con ex F-
condi ional isk measu e, i o all X,Y∈Lb(, G) he ollowing holds P0-a.s.:
(i) Condi ional cash in a iance: ρF(X+¯
X)=ρF(X)−¯
X o any ¯
X∈Lb(, F).
(ii) Mono onici y: X ≤Yimplies ρF(X)≥ρF(Y).
(iii) No maliza ion: ρF(0)=0.
(i ) Condi ional con exi y: ρF(λX+(1−λ)Y)≤λρF(X)+(1−λ)ρF(Y) o
λ∈Lb(, F)wi h 0 ≤λ≤1.
A con ex F-condi ional isk measu e ρFis called cohe en i i also sa is ies he
ollowing condi ion:
( ) Condi ional posi i e homogenei y: ρF(λX)=λρF(X) o λ∈Lb(, F)wi h
λ≥0.
We say ha ρFis con inuous om below i
( i) Con inui y om below: XnXpoin wise on implies ρF(Xn)ρF(X).
Mo eo e , we say ha ρ:Lb(, G)→Ris a (cohe en ) con ex isk measu e i
F={,∅}.
De ini ion 3.2 Le Sbe a inancial ma ke on (, F,F,P0)and deno e by Me(F) he
se o all ma ingale measu es which a e equi alen o P0.Amapπ:Lb(, G)→R
is called wo-s ep e alua ion i he e is a F-condi ional isk measu e ρFand an
equi alen ma ingale measu e Q∈Me(F)such ha π(X)=EQ[ρF(−X)] o all
X∈Lb(, G).
We now show ha Theo em 2.12 p o ides an economic ounda ion o he p icing o
inance-linked insu ance p oduc s ia wo-s ep e alua ions. Indeed, by using esul s
om [14], we o mula e su icien condi ion o a condi ional isk measu e ρFin
a wo-s ep e alua ion π(X)=EQ[ρF(−X)]such ha πis a QP-e alua ion. In
his way, he wo-s ep e alua ion πcha ac e izes he P- obus asymp o ic insu ance-
inance a bi age- ee p ice as p esen ed in Theo em 2.12.
Lemma 3.3 Le ρF:Lb(, G)→L∞(, F,P0)be a con ex F-condi ional isk
measu e which is con inuous om below. Then ρFis ep esen ed by
ρF(X)=ess sup
P∈PP0(,G)EP[−X|F]−αmin
F(P),(3.2)
whe e he accep ance se AF, he penal y unc ion αmin
Fand he se o p io s
PP0(, G)a e de ined by
AF:= {X∈Lb(, G)|ρF(X)≤0},
123
Robus insu ance- inance a bi age... 947
αmin
F(P):= ess sup
X∈AF
EP[−X|F],
and
PP0(, G):= {P∈P(, G)|P|F=P0}⊆PN(, G). (3.3)
I , in addi ion, ρFis a cohe en F-condi ional isk measu e, hen he e is a subse
P⊆PP0(, G)such ha
ρF(X)=ess sup
P∈P
EP[−X|F].(3.4)
P oo In he uncondi ional case he i s s a emen ollows om Theo em 4.16 and
Theo em 4.22 as p o en in [14]. Using he same idea as in he p oo o Theo em 11.2
in [14] yields he condi ional s a emen . Fo he second s a emen we e e o Co olla y
4.19 in [14] o he uncondi ional case. Using simila a gumen s o hose in Co olla y
11.6 in [14], he condi ional s a emen ollows. 
Thus, Lemma 3.3 shows ha e e y wo-s ep e alua ion π(X)=EQ[ρF(−X)]
gi en by an equi alen ma ingale measu e Q∈Me(F)and a cohe en F-condi ional
isk measu e ρFwhich is con inuous om below can be w i en as QP-e alua ion
π(X)=EQ[ess sup
P∈P
EP[X|F]] o a sui able subse P⊆PP0(, G).
Rema k 3.4 I we a p io i ix he nullse s on (, G)by a p obabili y measu e Pon
(, G)such ha P|F=P0, hen we can de ine he condi ional isk measu e ρFon
L∞(, G,P), ins ead o wo king wi h Lb(, G). In his case ρFonly needs o be
con inuous om abo e, as opposed o sa is ying he s onge assump ion o con inui y
om below, in o de o ha e a ep esen a ion as in Lemma 3.3, c . Theo em 4.33 and
Theo em 11.2 in [14]. The d awback o his app oach is ha we conside unce ain y in
a na ow sense because we ix all ele an nullse s on (, G)using a single p obabili y
measu e P. Ne e heless, his also leads o a obus p icing p oblem in he spi i o
Sec .2, which will be discussed in mo e de ail in Rema k 3.8.
Nex , we p o ide some examples o he se P⊆PP0(, G)in Lemma 3.3 and he
associa ed F-condi ional isk measu e in Equa ion (3.4).
Example 3.5 Le Pbe a p obabili y measu e on (, G)such ha P|F=P0.We
conside he se o p io s Pλ⊆PP0(, G)gi en by
Pλ:=˜
P∈PP0(, G)|˜
PPwi h d˜
P/dP ≤λ−1P-a.s. o λ∈(0,1).
(3.5)
In his case, he associa ed isk measu e ρF(X)is he condi ional a e age alue a
isk, deno ed by AVRλ(X|F), see also De ini ion 11.8 in [14]. No e ha he se Pλ
is domina ed by he p obabili y measu e P.
123
948 K. Obe p ille e al.
Example 3.6 Le Pbe a p obabili y measu e on (, G)such ha P|F=P0.We
conside he se o p io s Pc⊆PP0(, G) o c>0 gi en by
P
c:=˜
P∈PP0(, G)|H(˜
P|P)≤c,(3.6)
whe e H(˜
P|P)deno es he ela i e en opy o ˜
P wi h espec o P and is de ined by
H(˜
P|P):= E˜
Plog d˜
P
dP,i ˜
PP
+∞,o he wise.
He e, he associa ed isk measu e ρF(X)is he cohe en en opic isk measu e,in o-
duced in [13]. As in Example 3.5, hese P
cis domina ed by he p obabili y measu eP.
In he ollowing p oposi ion we show ha p icing wi h wo-s ep e alua ions leads o
a bi age- ee p emiums in he sense o Sec .2. This ema kable esul is a consequence
o Lemma 3.3 and Theo em 2.12.
P oposi ion 3.7 Le S be a inancial ma ke on (, F)and le πbea wo-s ep
e alua ion wi h F-condi ional con ex isk measu e o he o m (3.4) o he se
P={P∈PN(, G)|αmin
F(P)<∞}. Assume ha X=(Xi)i∈Nis a sequence o
insu ance bene i s ul illing Assump ion 2.1 and assume ha p <π(X1). Then he e
is NRIFA(P) wi h espec o he insu ance- inance ma ke (S,X,p).
P oo The esul is a di ec consequence o Theo em 2.12, since he QP-e alua ion
is an uppe bound o he wo-s ep e alua ion πand he chosen p emium pis e en
smalle by assump ion. 
3.2 Cons uc ion o condi ional iid copies
Ou nex goal is o apply Theo em 2.12 in he con ex o a obus wo-s ep e alua ion.
Mo e speci ically, gi en a andom a iable ˜
Xdesc ibing an insu ance bene i , we
de e mine a obus a bi age- ee p emium p o ˜
Xby using Theo em 2.12 and by
aking in o accoun some ac ua ial cons ain s, which a e e lec ed by he se o p io s
P⊆PN(, G)(see e.g., he se s Pλand Pcin Example 3.5 and 3.6, espec i ely).
To do so, wo ac o s mus o be conside ed. Fi s , he assump ions o Theo em 2.12
mus be sa is ied. To his end, we cons uc a sequence o bene i s (Xj)j∈Nwhich a e
copies o ˜
Xand which sa is y Assump ion 2.1. Second, he se o p io s Pmus be
shi ed o he p oduc space whe e we model he bene i s (Xj)j∈N. We obse e ha
hese s eps con ain some sub le ies which we discuss in mo e de ail in Rema k 3.8
a e o mally in oducing he se ing.
Le (F,FF)and (I,FI)be wo measu able spaces on which we model
pu ely inancial and pu ely insu ance e en s, espec i ely. On he p oduc space
(F×I,FF⊗FI)we in oduce he s ochas ic p ocess ˜
S=(˜
S ) =0,...,Tand
he andom a iable ˜
Xdesc ibing he inancial ma ke and a single insu ance bene i ,
espec i ely. Mo eo e , le P0be a measu e on (F×I,FF⊗{∅,I})which
123

Robus insu ance- inance a bi age... 949
de e mines he nullse s in FF⊗{∅,}and Pbe a se o p obabili y measu es on
(F×I,FF⊗FI)such ha
P|FF⊗{∅,I}∼P0 o all P∈P.
We now shi P o a se o p io s μPon (F×(I)N,FF⊗(FI)⊗N). Fu he mo e,
on his space we copy he inancial ma ke ˜
S o Sand cons uc insu ance bene i s
(Xj)j∈Nwhich a e iid condi ionally on Ssuch ha o all P∈P he law o (S,Xj)
o j∈Nunde μPcoincides wi h he law o (˜
S,˜
X)unde P.
Rema k 3.8 We also emphasize ha i he se Pis domina ed by a measu e P∈P,
as is he case in Example 3.5 and 3.6, his will no longe hold o he shi ed se
μP. The eason o his is ha absolu e con inui y o measu es is no s able unde
coun able p oduc s. The e o e, he seemingly no obus p oblem in he domina ed
case on (F×I,FF⊗FI,P)is indeed a obus p icing p oblem on (F×
(I)N,FF⊗(FI)⊗N,μ
P).
To be p ecise, we de ine
˜
:= F×I,:= F×(I)N,
˜
F:= FF⊗{I,∅},F:= FF⊗{I,∅}⊗N,
˜
G:= FF⊗FI,G:= FF⊗(FI)⊗N.
We deno e by ˜ω=(˜ωF,˜ωI)an elemen in ˜
and by ω=(ωF,(ωI
j)j∈N)an elemen
in . Fu he mo e, we in oduce he ollowing p ojec ions on ˜
:
˜πF:˜
→F
,˜πF(˜ω) =˜ωF
,
˜πI:˜
→I,˜πI
j(˜ω) =˜ωI,
as well as he ollowing he p ojec ions on :
πF:→F
,π
F(ω) =ωF
,
πI
j:→I,π
I
j(ω) =ωI
j.
Gi en a measu e Pon (˜
, ˜
G), he aim is o de ine a p obabili y measu e μPon (, G)
which ul ills he ollowing p ope ies:
The law o (πF,π
I
j)unde μPequals P o all j∈N,(3.7)
and
(πI
j)j∈Na e F-condi ionally independen unde μP.(3.8)
123
950 K. Obe p ille e al.
I Pis a p oduc measu e gi en by P=PF⊗PI o measu es PFon (F,FF)
and PIon (I,FI), hen he measu e μPcan be de ined by μP=PF⊗(PI)⊗N.
O he wise, we cons uc μP ia disin eg a ion as ollows. Fo some measu e Pon
(˜
, ˜
G) he measu e μpis de ined as
μP(A×B):= 1˜π−1
F(A)(P˜πI|˜
F)⊗N(B)dP o A∈FFand B∈(FI)⊗N,
(3.9)
whe e P˜πI|˜
Fdeno es he egula e sion o he condi ional p obabili y o ˜πIgi en
˜
F(see e.g., [16, Chap e 8]). No e ha we implici ly assume i s exis ence. This is no
es ic ion, howe e , because F, ep esen ing a inancial ma ke wi h d+1 asse s
and T ime s eps, can always be assumed o ha e he o m F=R(d+1)×(T+1)and,
hus, is a Bo el space. Mo eo e , using a mono one class a gumen , i ollows ha
(P˜πI|˜
F)⊗N:(FI)⊗Nט
→[0,1]
(B,˜ω) → (P˜πI|˜
F(·,˜ω))⊗N(B)
is a p obabili y ke nel om (˜
, ˜
G) o (, G)and, hus, he measu e μPis well de ined.
I Pis a measu e on (˜
, ˜
F), hen (3.9) de ines a measu e on (, F). In his case we
ha e B=N. No e ha by using (3.9) we ge he ollowing:
μP(A×(I)N)=P(A×I) o all A∈FF
and hus P|˜
F∼P0implies μP|F∼μP0 o all P∈P.
Le ˜
F=(˜
F ) ≤Twi h ˜
F :=FF
∨{I,∅}, be a il a ion on (˜
, ˜
F). He ea e , we
assume ha ˜
S=(˜
S ) ≤Tis a ˜
F-adap ed s ochas ic p ocess on (˜
, ˜
F)desc ibing he
p ices in a inancial ma ke . Mo eo e , le Me(˜
F)be he se o all ma ingale measu es
on (˜
, ˜
F)which a e equi alen some ixed measu e P0on (˜
, ˜
F). The insu ance and
inancial il a ion on (˜
, ˜
G)is deno ed by ˜
G=(˜
G ) ≤Tand he insu ance bene i , a
andom a iable on (˜
, ˜
G), is deno ed by ˜
X. In o de o shi all quan i ies o (, G)
we de ine
F=(F ) ≤Twi h F :=FF
∨{I,∅}⊗N,
G=(G ) ≤Twi h G :=σ((πF,π
I
j):→(˜
, ˜
G )|j∈N),
and
S := ˜
S ◦(πF,π
I
1)=˜
S ◦(πF,π
I
j) o all j∈N, =0,...,T,
Xj:= ˜
X◦(πF,π
I
j) o all j∈N.
123
Robus insu ance- inance a bi age... 951
No e ha ˜
S is assumed o be measu able wi h espec o ˜
F ⊆˜
Fand, hus, i does
no depend on he second coo dina e ˜ωI.
We show ha he measu e μPde ined in (3.9) sa is ies he desi ed p ope ies in (3.7)
and (3.8) and is he unique measu e wi h his p ope y. Fo he eade ’s con enience
we p o ide he p oo s o hese esul s in de ail in he Appendix.
P oposi ion 3.9 The measu e μP, as de ined by (3.9), is he unique measu e on (, G)
which ul ills (3.7)and (3.8).
Nex , we cha ac e ize he se o all equi alen ma ingales measu es on (, F).
P oposi ion 3.10 The se Me(F)o all measu es on (, F)such ha S is a F-
ma ingale and which a e equi alen o μP0is gi en by
Me(F)={μQ|Q∈Me(˜
F)}.
P oposi ion 3.11 Fo any P ∈Pand Q ∈Me(˜
F)we ha e he ollowing:
EQ[EP[˜
X|˜
F]] = EμQ[EμP[X1|F]],
EQ[ess sup
P∈P
EP[˜
X|˜
F]] = EμQess sup
P∈P
EμP[X1|F].
We emphasize ha Theo em 2.12 and P oposi ion 3.11 build a ounda ion o
wo-s ep e alua ions om a new pe spec i e. We shi he insu ance bene i o an
insu ance- inance ma ke such ha he assump ions o Theo em 2.12 a e ul illed and
cha ac e ize he obus insu ance- inance a bi age- ee p ices he ein. Then, P oposi-
ion 3.11 shows ha he p ices in he shi ed insu ance- inance ma ke coincide wi h
he wo-s ep e alua ion o he ini ial bene i .
Summa izing, he QP-e alua ion p o ides a alua ion me hodology which
excludes a obus insu ance- inance a bi age in he abo e sense. While he a gumen
in ol es he condi ional s ong law o la ge numbe s, and hus in ini ely many insu -
ance con ac s, he QP-e alua ion also p o ides a highly easonable alua ion ule
when only ini ely many con ac o s a e a ailable. Howe e , wi h only ini ely many
con ac s, he no-a bi age concep becomes less powe ul: his phenomenon a ises,
because he heo e ical limi , he condi ional expec a ion, is no ully eached in his
case. Consequen ly, his would pe mi highe p ices wi hou necessa ily c ea ing an
insu ance- inance a bi age since always a small isk emains.
4 Modeling o insu ance- inance ma ke s
In his sec ion we p o ide models o insu ance- inance ma ke s and calcula e he
obus insu ance- inance a bi age- ee p emium by means o he QP-e alua ion, c .
Theo em 2.12 and De ini ion 2.14.
As in Sec .2.2,le S0be he bank accoun and deno e by S1=(S1
) =0,...,T he
non-discoun ed p ice p ocess o a isky asse on (, F). We ix he F-nullse s N
123
952 K. Obe p ille e al.
gene a ed by a p obabili y measu e P0∈P(, F). We assume ha he il a ion Fis
gene a ed by S0and S1. Nex , we in oduce he N0- alued andom a iables τ1and τ2
ep esen ing he ime o dea h and he ime o su ende o a policy holde , espec i ely.
Le he σ-algeb a Gbe gi en by G=F∨σ(τ1)∨σ(τ2)and he il a ion Ggi en
by G=F∨H, whe e His he il a ion gene a ed by he p ocesses (1{τ1≤ }) =0,...,T
and (1{τ2≤ }) =0,...,T. No e ha τ1and τ2a e G-s opping imes bu , in gene al, hey
a e no F-s opping imes.
Gi en a pa ame e se , we in oduce he law o he s opping imes (τ1,τ2)unde
he pa ame e ized se o p io s P=(Pθ)θ∈⊆PN(, G). In pa icula , we
assume ha o each Pθ∈P he condi ional laws o τ1and τ2a e gi en by
Pθ(τ1≤ |F):=FF
1(θ, )and Pθ(τ 2≤ |F):=FF
2(θ, ) o ∈N0,(4.1)
whe e o ixed θ∈ he mappings FF
1(θ, ·)and FF
2(θ, ·)a e F-condi ional dis-
ibu ion unc ions.
4.1 Modeling unde condi ional independence
In his subsec ion we assume ha unde e e y Pθ∈P he andom a iables τ1and
τ2a e F-condi ionally independen , i.e.,
Pθ(τ1≤s,τ2≤ |F):=FF
1(θ, s)FF
2(θ, ) o s, ∈N0.(4.2)
In o de o de e mine he law o (S,τ1,τ2)unde Pθi now emains o in oduce he
es ic ed measu es Pθ|F∼P0and use disin eg a ion. Howe e , we could assume
ha Pθ|F=P0 o all θ∈since he QP-e alua ion is in a ian unde he speci ic
choice o he measu es {Pθ|F|θ∈}as he se o equi alen ma ingale measu es
Me(F)only depends on he nullse s Ngene a ed by P0.
We in oduce he discoun ed su i al bene i Xsu i al and he discoun ed su ende
bene i Xsu ende by
Xsu i al:=1{τ1>T,τ2>T}Y1(S0
T)−1,(4.3)
Xsu ende :=
T−1

=1
1{τ1> ,τ2= }Y2
(S0
)−1,(4.4)
whe e Y1is a F-measu able andom a iable and Y2:= (Y2
) =0,...,Tis a F-adap ed
p ocess. The insu ance bene i Xis hen gi en by
X:=Xsu i al +Xsu ende .(4.5)
Policy holde wi h such a policy ecei e he paymen Y1a ma u i y Ti hey su i es
un il Tand do no su ende be o e ime T. I hey su ende a ime <T hey ecei e
he paymen Y2
.
123
Robus insu ance- inance a bi age... 959
whe e we use he de ini ion o he condi ional p obabili y and he owe p ope y in
he hi d equali y. The second s a emen ollows by
μ(S,Xj)
P=μ
(πF,πI
j
)
P(˜
S,˜
X)
=P(˜
S,˜
X) o all j∈N.

P oposi ion A.2 Le j ∈Nand νPbe a measu e on (, G)which ul ills (3.7), i.e.,
ν
(πF,πI
j
)
P=P o all j ∈N.
Then i holds o all A ∈FI ha
EνP[1{πI
j
∈A}|F]
=EP[1{˜πI∈A}|˜
F]◦(πF,π
I
)ν
P-a.s. o all A ∈FIand ∈N.(A.1)
P oo Le B∈Fand A∈FI. Then we ha e
1B1{πI
j∈A}dνP=(1πF(B)◦πF)(1A◦πI
j)dνP
=(1πF(B)◦˜πF)(1A◦˜πI)dP
=(1πF(B)◦˜πF)EP[(1A◦˜πI)|˜
F]dP
=1BEP[(1A◦˜πI)|˜
F]◦((πF,π
I
))dνP,
whe e we use (3.7) in he second equali y and ˜π−1
F(πF(F)) ∈˜
Fin he hi d equali y.

P oposi ion A.3 The p ojec ions (πI
j)j∈Na e F-condi ionally iid unde μPand
consequen ly also he insu ance bene i s (Xj)j∈Na e F-condi ionally iid unde μP.
P oo We mus show ha o e e y ini e subse J⊂Nand Aj∈FI o all j∈J
ha
EμP⎡
⎣!
j∈J
1{πI
j
∈Aj}
F⎤
⎦=!
j∈J
EμP1{πI
j
∈Aj}
F.
123

960 K. Obe p ille e al.
This ollows because o e e y B∈F, i holds ha
EμP⎡
⎣1B!
j∈J
1{πI
j
∈Aj}⎤
⎦=EP⎡
⎣1˜π−1
F(πF(B)) !
j∈J
P˜πI|˜
F(Aj)⎤
⎦
=EP⎡
⎣1˜π−1
F(πF(B)) !
j∈J
EP1{˜πI∈Aj}|˜
F⎤
⎦
=EμP⎡
⎣1B!
j∈J
EP1{˜πI∈Aj}|˜
F]◦(πF,π
I
1)⎤
⎦
=1B!
j∈J
Eμp[1{πI
j
∈Aj}|F]dμP,
whe e we use he de ini ion o μPgi enin(3.9) in he i s equali y and P oposi ion A.2
in he ou h equali y. 
P oposi ion A.4 The measu e μPde ined by (3.9)is he unique measu e on (, G)
which ul ills (3.7)and (3.8).
P oo Using P oposi ion A.1 and P oposi ion A.3 he measu e μP ul ills (3.7) and
(3.8). Mo eo e , le νPbe a measu e on (, G)which also sa is ies (3.7) and (3.8)
(whe e μPis eplaced by νP). Le J⊂Nbe a ini e subse and A∈FFand Cj∈FI
o all j∈J. Then o Dgi en by
D=π−1
F(A)∩$
j∈J
π−1
I
j
(Cj)(A.2)
we ge he ollowing
νP(D)=EνP1π−1
F(A)EνP!
j∈J
1πI
j
∈Cj|F
=EνP1π−1
F(A)!
j∈J
EνP1πI
j
∈Cj|F
=EνP1π−1
F(A)!
j∈J
EP1˜πI∈Cj|˜
F◦(πF,π
I
1)
=EP1˜π−1
F(A)!
j∈J
EP1˜πI∈Cj|˜
F,
whe eweuse(3.8) in he hi d equali y and P oposi ion A.2 in he ou h equali y. The
same calcula ions can be done o μP. The se s in (A.2) o ma∩-s able gene a o o
Gand hus we ob ain μP=νP. This demons a es he uniqueness. 
123
Robus insu ance- inance a bi age... 961
P oposi ion A.5 The se Me(F)o all measu es on (, F)such ha S is a F-
ma ingale and which a e equi alen o μP0is gi en by
Me(F)={μQ|Q∈Me(˜
F)}.
P oo We show ha o all ,s∈{0,...,T}i holds ha
EμQ[S |Fs]=EQ[˜
S |˜
Fs]◦(πF,π
I
1). (A.3)
Le Bs∈Fs, hen we ha e
EμQ[1BsS ]=EμQ1Bs˜
S ◦(πF,π
I
1)
=EQ1˜π−1
F(πF(Bs)) ˜
S 
=EQ1˜π−1
F(πF(Bs)) EQ[˜
S |˜
Fs]
=EμQ1BsEQ[˜
S |˜
Fs]◦(πF,π
I
1)
(A.4)
Thus, i ollows (A.3) and so we can conclude ha Q∈Me(˜
F)implies μQ∈Me(F).
The equi alence o μP0and μQ ollows om he equi alence o P0and Q.Fo he
o he inclusion, le R∈Me(F). We now show ha he e exis s Q∈Me(˜
F)such ha
R=μQ. De ine Qby
Q:= RπF(˜πF(·)).
Then, by changing he oles o πFand ˜πF he esul ollows using (A.4). 
P oposi ion A.6 Fo any P ∈Pand Q ∈Me(˜
F)we ha e he ollowing:
EQEP[˜
X|˜
F]=EμQ[EμP[X1|F]],
EQess sup
P∈P
EP[˜
X|˜
F]=EμQess sup
P∈P
EμP[X1|F].
P oo By using he same a gumen s as in he p oo o P oposi ion 3.10, we ind ha
EμP[X1|F]=EP[˜
X|˜
F]◦(πF,π
I
1).
This implies ha
EμQEμP[X1|F]=EμQEP[˜
X|˜
F]◦(πF,π
I
1)=EQEP[˜
X|˜
F].
The second s a emen ollows analogously. 
123
962 K. Obe p ille e al.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence,
and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included
in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I
ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed
by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he
copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/.
Re e ences
1. A zne P, Eisele K-T, Schmid T (2024) Insu ance- inance a bi age. Ma h Financ. 34(3):739–773
2. Ba igou K, Linde s D, Yang F (2023) Ac ua ial-consis ency and wo-s ep ac ua ial alua ions: a new
pa adigm o insu ance alua ion. Scand Ac ua J 2:191–217
3. Ba l D (2020) Condi ional nonlinea expec a ions. S och P ocess Appl 130(2):785–805
4. Biagini F, Zhang Y (2019) Reduced- o m amewo k unde model unce ain y. Ann Appl P obab
29(4):2481–2522
5. Biagini S, Boucha d B, Ka da as C, Nu z M (2017) Robus undamen al heo em o con inuous
p ocesses. Ma h Financ 27(4):963–987
6. Boucha d B, Nu z M (2015) A bi age and duali y in nondomina ed disc e e- ime models. Ann Appl
P obab 25(2):823–859
7. Deels a G, De olde P, Gnameho K, Hiebe P (2020) Valua ion o hyb id inancial and ac ua ial
p oduc s in li e insu ance by a no el h ee-s ep me hod. ASTIN Bull 50(3):709–742
8. Denis L, Ma ini C (2006) A heo e ical amewo k o he p icing o con ingen claims in he p esence
o model unce ain y. Ann Appl P obab 16(2):827–852
9. Denis L, Hu M, Peng S (2010) Func ion spaces and capaci y ela ed o a sublinea expec a-
ion:applica ion o G-B ownian mo ion pa hs. Po en Analy 34:139–161
10. Dhaene J, Kukush A, Luciano E, Schou ens W, S assen B (2013) A no e on he (in-)dependence
be ween inancial and ac ua ial isks. Insu Ma h Econ 52(3):522–531
11. Dhaene J, S assen B, Ba igou K, Linde s D, Chen Z (2017) Fai alua ion o insu ance liabili ies:
me ging ac ua ial judgemen and ma ke -consis ency. Insu Ma h Econ 76:14–27
12. Dyb ig P (1992) Hedging non- aded weal h: when is he e sepa a ion o hedging and in es men . Op
Recen Ad Theo y P ac 2:13–24
13. Föllme H, Knispel T (2011) En opic isk measu es: cohe ence s con exi y, model ambigui y and
obus la ge de ia ions. S och Dyn 11(02n23):333–351
14. Föllme H, Schied A (2016) S ochas ic inance. de G uy e
15. Kabano YM, K amko DO (1995) La ge inancial ma ke s: asymp o ic a bi age and con igui y.
Theo y P obab Appl 39(1):182–187
16. Kallenbe g O (1997) Founda ions o mode n p obabili y, ol 2. Sp inge , Be lin
17. Klein I (2000) A undamen al heo em o asse p icing o la ge inancial ma ke s. Ma h Financ
10(4):443–458
18. Klein I, Schmid T, Teichmann J (2016) No a bi age heo y o bond ma ke s. Ad anced modelling
in ma hema ical inance. Sp inge , Be lin, pp 381–421
19. Linde s D (2023) The 3-s ep hedge-based alua ion: ai alua ion in he p esence o sys ema ic isks.
ASTIN Bull 53(2):418–442
20. Maje ek D, Nowak W, Zieba W (2005) Condi ional s ong law o la ge numbe . In J Pu e Appl Ma h
20(2):143–156
21. Malamud S, T ubowi z E, Wü h ich MV (2008) Ma ke consis en p icing o insu ance p oduc s.
ASTIN Bull 38(2):483–526
22. Nelsen RB (2007) An in oduc ion o copulas. Sp inge , Be lin
23. Pelsse A, S adje M (2014) Time-consis en and ma ke -consis en e alua ions. Ma h Financ 24(1):25–
65
123
Robus insu ance- inance a bi age... 963
24. Peng S (2019) Nonlinea expec a ions and s ochas ic calculus unde unce ain y: wi h obus CLT and
G-B ownian mo ion, ol 95. Sp inge , Be lin
25. Plachky D, Rüschendo L (1984) Con e sa ion o he ump- esp. maxmin-p ope y o s a is ical es s
unde ex ensions o p obabili y measu es. In: Colloquium on Godness o Fi , pp 439–457
26. Rásonyi M (2003) Equi alen ma ingale measu es o la ge inancial ma ke s in disc e e ime. Ma h
Me hods Ope Res 58:401–415
27. Schmid T (2007) Coping wi h copulas. Copulas: om heo y o applica ions in inance. In: Rank J
(eds) Risk Books
28. Sone HM, Touzi N, Zhang J (2011) Ma ingale ep esen a ion heo em o he G-expec a ion. S och
P ocess Appl 121(2):265–287
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
123