Djehiche, Boualem; Helgesson, Pe e
A icle
A isk based app oach o he p incipal-agen p oblem
Asian Jou nal o Economics and Banking (AJEB)
P o ided in Coope a ion wi h:
Ho Chi Minh Uni e si y o Banking (HUB), Ho Chi Minh Ci y
Sugges ed Ci a ion: Djehiche, Boualem; Helgesson, Pe e (2024) : A isk based app oach o he
p incipal-agen p oblem, Asian Jou nal o Economics and Banking (AJEB), ISSN 2633-7991, Eme ald,
Leeds, Vol. 8, Iss. 3, pp. 310-334,
h ps://doi.o g/10.1108/AJEB-05-2024-0065
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/334126
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A isk based app oach o he
p incipal–agen p oblem
Boualem Djehiche
Depa men o Ma hema ics, KTH Royal Ins i u e o Technology,
S ockholm, Sweden, and
Pe e Helgesson
Depa men o Ma hema ics, Chalme s Uni e si y o Technology,
Go henbu g, Sweden
Abs ac
Pu pose –We aim o gene alize he con inuous- ime p incipal–agen p oblem o inco po a e ime-inconsis en
u ili y unc ions, such as hose o mean- a iance ype, which a e p e alen in isk managemen and inance.
Design/me hodology/app oach –We use ecen ad ancemen s o he Pon yagin maximum p inciple o
o wa d-backwa d s ochas ic di e en ial equa ions (FBSDEs) o de elop a me hod o cha ac e izing op imal
con ac s in such models. This app oach add esses he challenges posed by he non-applicabili y o he
classical Hamil on–Jacobi–Bellman equa ion due o ime inconsis ency.
Findings –We p o ide a amewo k o de i ing op imal con ac s in he p incipal–agen p oblem unde
hidden ac ion, speci ically ailo ed o ime-inconsis en u ili ies. This is illus a ed h ough a ully sol ed
example in he linea -quad a ic se ing, demons a ing he p ac ical applicabili y o he me hod.
O iginali y/ alue –The wo k con ibu es o he exis ing li e a u e by p esen ing a no el ma hema ical
app oach o a class o con inuous ime p incipal–agen p oblems, pa icula ly unde hidden ac ion wi h ime-
inconsis en u ili ies, a scena io no p e iously add essed. The esul s o e po en ial insigh s o bo h
heo e ical de elopmen and p ac ical applica ions in inance and economics.
Keywo ds P incipal–agen p oblem, S ochas ic maximum p inciple, Pon yagin’s maximum p inciple,
Mean- a iance, Time inconsis en u ili y unc ions
Pape ype Resea ch pape
1. In oduc ion
Risk managemen o he p oblem o inding an op imal balance be ween expec ed e u ns
and isk aking is a cen al opic o esea ch wi hin banking, economics and inance.
Applica ions such as po olio op imiza ion, op imal s opping and liquida ion p oblems ha e
been o pa icula in e es in he li e a u e. In such applica ions i is common o conside
u ili y unc ions o mean- a iance ype. Mean- a iance u ili y unc ions cons i u e an
impo an subclass o he so called ime inconsis en u ili y unc ions o which he Bellman
p inciple o dynamic p og amming does no hold. P oblems in ol ing such u ili ies can
he e o e no be app oached by he classical Hamil on–Jacobi–Bellman equa ion. In his
pape we de elop a me hod o s udying a mean- a iance se ing o he celeb a ed p incipal–
agen p oblem by means o he s ochas ic gene aliza ion o Pon yagin’s maximum p inciple.
The p ecise s uc u e o he p incipal–agen p oblem goes as ollows. The p incipal
employs an agen o manage a ce ain well-de ined noisy asse o e a ixed pe iod o ime.
AJEB
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310
JEL Classi ica ion — B41, C00, C61, C70, C72
2010 Ma hema ics Subjec Classi ica ion — 93E20, 49N70, 49N90
© Boualem Djehiche and Pe e Helgesson. Published in Asian Jou nal o Economics and Banking.
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Recei ed 27 May 2024
Re ised 9 Augus 2024
Accep ed 27 Augus 2024
Asian Jou nal o Economics and
Banking
Vol. 8 No. 3, 2024
pp. 310-334
Eme ald Publishing Limi ed
e-ISSN: 2633-7991
p-ISSN: 2615-9821
DOI 10.1108/AJEB-05-2024-0065
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In e u n o his/he e o he agen ecei es a compensa ion acco ding o some ag eemen ,
se be o e he pe iod s a s. I could o ins ance in ol e a lump-sum paymen a he end o he
pe iod, a con inuously paying cash- low du ing he pe iod o bo h. Depending on wha
in o ma ion he p inciple has a hand o o m an ag eemen , one dis inguishes be ween wo
cases; he ull in o ma ion and he hidden ac ion-p oblem. The ull in o ma ion case di e s
om he hidden ac ion case in ha he p incipal can obse e he ac ions o he agen in
addi ion o he e olu ion o he asse . The e o e, unde ull in o ma ion he p incipal is
allowed o ailo a con ac based on bo h ou come and e o , no only ou come as o hidden
ac ions. In bo h cases he con ac is cons ained by he agen ia a so called pa icipa ion
cons ain , cla i ying he minimum equi emen s o he agen o engage in he p ojec . Unde
hidden ac ion he con ac is u he cons ained by he incen i e compa ibili y condi ion,
meaning ha as soon as a con ac is assigned he agen will ac as o maximize his/he own
u ili y and no necessa ily ha o he p incipal.
The pionee ing pape in which he p incipal–agen p oblem i s appea s is Holms €
om
and Milg om (1987). They s udy a con inuous ime model o e a ini e pe iod in which he
p inciple and he agen bo h op imize exponen ial u ili y unc ions. The p incipal ewa ds
he agen a he end o he pe iod by a lump-sum paymen . As a esul hey ind ha he
op imal con ac is linea wi h espec o ou pu . The pape Holms €
om and Milg om (1987) is
gene alized in Sch€
a le and Sung (1993) o a ma hema ical amewo k ha uses me hods
om dynamic p og amming and ma ingale heo y o cha ac e ize con ac op imali y.
The in e es in con inuous ime models o he p incipal–agen p oblem has g own
subs an ially since he i s s udies appea ed. In C i ani�
ce al. (2009),Sanniko (2008),
Wes e ield (2006) and Williams (2013) (only o men ion a ew) he au ho s analyze
con inuous ime models in a classical se ing, i.e. ha ing one p incipal and one agen . Such
models a e also co e ed in he ecen book C i ani�
c and Zhang (2013). O he models such as
a ious mul iplaye e sions ha e been s udied o ins ance in Kang (2013) and Koo
e al. (2008).
Ou goal is o cha ac e ize op imal con ac s in he classical se ing o p incipal–agen
p oblem unde hidden ac ion o ime inconsis en u ili y unc ions. We conside wo
di e en modeling possibili ies; hidden ac ion in he weak o mula ion and hidden con ac in
he s ong o mula ion. In he i s model he agen has ull in o ma ion o he mechanisms
behind he cash- low and he p incipal wishes o minimize his/he mean- a iance u ili y. In
he la e model he agen does no know he s uc u e o he cash- low and has o p o ec
him-/he -sel om high le els o isk by an addi ional pa icipa ion cons ain o a iance
ype. To he bes o ou knowledge his has no p e iously been add essed in he li e a u e. In
o de o ca y he p og am h ough we use ecen gene aliza ions o Pon yagin’s s ochas ic
maximum p inciple. The idea is o conside he p incipal–agen p oblem as a sequen ial
op imiza ion p oblem. We i s conside he agen ’s p oblem o cha ac e izing op imal choice
o e o . Then we p oceed o he P incipal’s p oblem which, by incen i e compa ibili y,
becomes a cons ained op imal con ol p oblem o a o wa d-backwa d s ochas ic
di e en ial equa ion ( om now on FBSDE). A simila scheme was conside ed in Djehiche
and Helgesson (2014) bu wi hou he non-s anda d mean- a iance conside a ion. Op imal
con ol wi h espec o mean- a iance u ili y unc ions has p e iously been s udied in o
ins ance Li (2000), and Ande sson and Djehiche (2011).
Op imal po olios based on ime inconsis en u ili ies ha e been add essed in Bj€
o k and
Mu goci (2010),Bj€
o k e al. (2014),Djehiche and Huang (2014),Ekeland and Laz ak (2006)
and Ekeland and Pi u (2008). See also he ecen book Bj€
o k e al. (2021). A disc e e e sion
o his class o p oblems boils down o s udy a sys em o o wa d-backwa d ime se ies
whose analysis ollows he same lines o easoning as he ime-con inuous e sion bu he
o mulas a e a bi clumsy.
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In he p esen li e a u e o he p incipal–agen p oblem he pape closes o ou s is
Williams (2013), in which a simila maximum p inciple app oach is used. The se ing is
classical (wi hou ime inconsis en u ili y unc ions) and he au ho inds a cha ac e iza ion
o op imal choice o e o in he agen ’s p oblem. The ull model in ol ing he cons ained
p incipal’s p oblem, howe e , is no conside ed. The main esul s o ou s udy a e p esen ed
in Theo em 4.3 and Theo em 4.4 in which a ull cha ac e iza ion o op imal con ac s is s a ed
o wo di e en models.
In p ac ical e ms, he assump ions made in ou s udy e lec ypical scena ios in inancial
decision-making. Fo example, he mean- a iance u ili y unc ion is a s anda d way o
ep esen an in es o ’s desi e o balance expec ed e u ns wi h isk. This assump ion helps
simpli y he complex eali y o inancial ma ke s in o a manageable model. Howe e , i migh
b eak down in si ua ions whe e in es o p e e ences a e no s able o e ime o when he e
a e ab up changes in ma ke condi ions. Ou esul s imply ha , unde hese assump ions, i
is possible o design con ac s ha align he in e es s o bo h he p incipal (e.g. an employe )
and he agen (e.g. an employee) e en when he agen ’s ac ions a e no di ec ly obse able.
The pape is o ganized as ollows. In Sec ion 2 we in oduce he ma hema ical machine y
om s ochas ic op imal con ol heo y ha is necessa y o ou pu poses. Mean- a iance
maximum p inciples a e hen de i ed in Sec ion 3 by esul s om Sec ion 2 in wo di e en
bu ela ed cases. Sec ion 4 is de o ed o i he me hods om he p e ious sec ions in o a
p inciple–agen amewo k. We conside wo di e en models unde hidden ac ion and ind
necessa y condi ions o op imali y. Finally in Sec ion 5 we make he gene al scheme o
Sec ion 4 conc e e by a simple and ully sol ed example in he linea -quad a ic (LQ)-se ing.
2. P elimina ies
Le T> 0 be a ixed ime ho izon and ðΩ;F;F;PÞbe a il e ed p obabili y space sa is ying
he usual condi ions on which a 1-dimensional B ownian mo ion W¼ W g ≥0is de ined.
We le Fbe he na u al il a ion gene a ed by Waugmen ed by all P-null se s NP, i.e.
F¼ F ∨NPwhe e F d
σ
ð Wsg:0≤s≤ Þ.
Conside he ollowing con ol sys em o o wa d s ochas ic di e en ial equa ions (SDEs)
o mean- ield ype:
dxð Þ ¼ bð ;xð Þ;E½xð Þ�;sð ÞÞd þ
σ
ð ;xð Þ;E½xð Þ�ÞdW ; ∈ð0;T�
xð0Þ ¼ x0
�
wi h a cos unc ional o he o m
Jðsð$ÞÞdEZT
0
ð ;xð Þ;E½xð Þ�;sð ÞÞd þhðxðTÞ;E½xðTÞ�Þ
� �;(2.1)
whe e b:½0;T�3R3R3S→R,
σ
:½0;T�3R3R→R, :½0;T�3R3R3S→Rand
h:R3R→Rand S⊂Ris a non-emp y subse . The con ol s($) is admissible i i is an
F-adap ed and squa e-in eg able p ocess aking alues in S. We deno e he se o all such
admissible con ols by S½0;T�. In o de o a oid echnicali ies in egula i y ha a e i ele an
o ou pu poses we s a e he ollowing assump ion.
Assump ion 1. The unc ions b,
σ
, and ha e C
1
wi h espec o xand ~
x, whe e ~
xdeno es
he explici dependence o E½xð$Þ�. Mo eo e , b,
σ
, and hand hei i s
o de de i a i es wi h espec o xand ~
xa e bounded and con inuous in x,
~
xand s.
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We a e in e es ed in he ollowing op imal con ol p oblem:
P oblem (S). Minimize (2.1) o e S½0;T�.
Any sð$Þ∈S½0;T�sa is ying
J�sð$Þ�¼in
sð$Þ∈S½0;T�Jðsð$ÞÞ
is called an op imal con ol and he co esponding xð$Þis called he op imal s a e p ocess. We
will e e o ðxð$Þ;sð$ÞÞas an op imal pai .
The ollowing s ochas ic maximum p inciple o cha ac e izing op imal pai s in p oblem (S)
was ound in Buckdahn e al. (2011).
Theo em 2.1. The s ochas ic maximum p inciple. Le he condi ions in Assump ion 1
hold and conside an op imal pai ðxð$Þ;sð$ÞÞo p oblem (S). Then he e
exis s a pai o p ocesses ðpð$Þ;qð$ÞÞ∈L2
Fð0;T;RÞ3ðL2
Fð0;T;RÞÞ
sa is ying he adjoin equa ion
dpð Þ ¼ −bx� ;xð Þ;Ehxð Þi;sð Þ�pð ÞþEhb~
x� ;xð Þ;Ehxð Þi;sð Þ�pð Þin
þ
σ
x� ;xð Þ;Ehxð Þi�qð ÞþEh
σ
~
x� ;xð Þ;Ehxð Þi�qð Þi
� x� ;xð Þ;Ehxð Þi�;sð Þ��E ~
x� ;xð Þ;Ehxð Þi�;sð ÞÞ
h iod
þqð ÞdW ;
pðTÞ ¼ −hx�xðTÞ;EhxðTÞi��Ehh~
x�xðTÞ;EhxðTÞi�i;
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
(2.2)
such ha
sð Þ ¼ a gmax
s∈SH� ;xð Þ;s;pð Þ;qð Þ�;a:e: ∈½0;T�;P�a:s:(2.3)
whe e he Hamil onian unc ion His gi en by
Hð ;x;s;p;qÞdbð ;x;E½x�;sÞ$pþ
σ
ð ;x;E½x�Þ$q� ð ;x;E½x�;sÞ(2.4)
o ð ;x;s;p;qÞ∈½0;T�3R3S3R3R.
Rema k 2.2. I is impo an o emembe ha Theo em 2.1 me ely s a es a se o
necessa y condi ions o op imali y in (S). I does no claim he exis ence o
an op imal con ol. Exis ence heo y o s ochas ic op imal con ols (bo h in
he s ong and he weak sense) has been a subjec o s udy since he six ies
(see e.g. Kushne (1965)) and, a leas in he case o s ong solu ions, he
esul s seem o depend a lo upon he s a emen o he p oblem. In he weak
sense an accoun o exis ence esul s is o be ound in Yong and Zhou (1999)
(Theo em 5.3, p. 71).
Rema k 2.3. Res ic ing he space U o be con ex allows o a di usion coe icien o he
o m
σ
ð ;x;E½x�;sÞ, wi hou changing he conclusion o Theo em 2.1. In he
case o a non-con ex con ol space he s ochas ic maximum p inciple wi h
con olled di usion was p o en in Peng (1990) and equi es he solu ion o
an addi ional adjoin BSDE. We choose o lea e his mos gene al maximum
p inciple as e e ence in o de o keep he p esen a ion clea .
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As poin ed ou in Rema k 2.2 i is a non- i ial ask o p o e he exis ence o an op imal pai
ðxð$Þ;sð$ÞÞin a gene al s ochas ic con ol model unde he addi ional assump ions.
Assump ion 2. The con ol domain Sis a con ex body in R. The maps b,
σ
and a e
locally Lipschi z in uand hei de i a i es in xand ~
xa e con inuous in x,~
x
and s, he ollowing heo em p o ides su icien condi ions o op imali y
in (S).
Theo em 2.4. Su icien condi ions o op imali y. Unde Assump ions 1 and 2 le
ðxð$Þ;sð$Þ;pð$Þ;qð$ÞÞbe an admissible 4- uple. Suppose ha h is con ex
and u he ha Hð ;$;$;$;pð Þ;qð ÞÞis conca e o all ∈½0;T�P-a.s.
and
sð Þ ¼ a gmax
s∈SH� ;xð Þ;Ehxð Þi;s;pð Þ;qð Þ�;a:e: ∈½0;T�;P�a:s:
Then ðxð$Þ;sð$ÞÞis an op imal pai o p oblem (S).
The s ochas ic maximum p inciple has since he ea ly days o he subjec (in pionee ing
pape s by, e.g. Bismu (1978) and Bensoussan (1982)) de eloped a lo and does by now apply
o a wide ange o p oblems mo e gene al han (S) (see o ins ance Peng (1990),Ande sson
and Djehiche (2011),Buckdahn e al. (2011),Djehiche e al. (2014)). Fo ou pu poses we need a
e ined e sion o Theo em 2.1, cha ac e izing op imal con ols in a FBSDE-dynamical
se ing unde s a e cons ain s. Mo e p ecisely we wish o conside a s ochas ic con ol
sys em o he o m
dxð Þ ¼ bð ;Θð Þ;sð ÞÞd þ
σ
ð ;Θð ÞÞdW
dyð Þ ¼ −cð ;Θð Þ;sð ÞÞd þzð ÞdW
xð0Þ ¼ x0;yðTÞ ¼ φðxðTÞÞ;
8
<
:(2.5)
whe e b;
σ
;c:½0;T�3R63S→Rand φ:R→R, wi h espec o a cos - unc ional o
he o m
Jðsð$ÞÞdEZT
0
ð ;Θð Þ;sð ÞÞd þhðxðTÞ;E½xðTÞ�Þþgðyð0ÞÞ
� �;(2.6)
and a se o s a e cons ain s
EZT
0
Fð ;Θð Þ;sð ÞÞd þHðxðTÞ;E½xðTÞ�ÞþGðyð0ÞÞ
� �d
EZT
0
1ð ;Θð Þ;sð ÞÞd þh1ðxðTÞ;E½xðTÞ�Þþg1ðyð0ÞÞ
� �
.
.
.
EZT
0
lð ;Θð Þ;sð ÞÞd þhlðxðTÞ;E½xðTÞ�Þþglðyð0ÞÞ
� �
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
∈Λ;
(2.7)
o some closed and con ex se Λ ⊆ Rl. In he abo e exp essions we ha e in oduced
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Θð Þdðxð Þ;yð Þ;zð Þ;E½xð Þ�;E½yð Þ�;E½zð Þ�Þ;
in o de o a oid unnecessa ily hea y no a ion. The op imal con ol p oblem is:
P oblem (SC). Minimize (2.6) subjec o he s a e cons ain s (2.7) o e he se S½0;T�.
To ge a good maximum p inciple o (SC) we equi e some u he egula i y condi ions
ensu ing sol abili y o (2.5). These condi ions a e lis ed in he ollowing assump ions and can
be ound in Li and Liu (2014).
Assump ion 3. The unc ions b,
σ
,ca e con inuously di e en iable and Lipschi z
con inuous in Θ, he unc ions h,g,h
i
,g
i
a e con inuously di e en iable
in xand y espec i ely, and hey a e bounded by Cð1þjxjþjyj
þjzjþj~
xjþj~
yjþj~
zjþjsjÞ,C(1 þ jxj) and C(1 þ jyj), espec i ely.
Assump ion 4. All de i a i es in Assump ion 4 a e Lipschi z con inuous and bounded.
Assump ion 5. Fo all Θ∈R6,s∈S,Að$;Θ;sÞ∈L2
Fð0;T;R3Þ, whe e we ha e A( ,Θ,s)
d(c( ,Θ,s), b( ,Θ,s),
σ
( ,Θ)) and
L2
F�0;T;Rk�d
ψ
:½0;T�3Ω→Rk��
ψ
is F�adap ed and EZT
0j
ψ
j2d
� �<∞
� �;
and o each x∈R,φðxÞ∈L2
FðΩ;RÞ. Fu he mo e, he e exis s a cons an C> 0 such ha
jAð ;Θ1;sÞ�Að ;Θ2;sÞj≤CjΘ1�Θ2j;P�a:s:and o a:e: ∈½0;T�;
jφðx1Þ�φðx2Þj≤Cjx1�x2j;P�a:s;
o all Θ1;Θ2∈R6:
8
<
:
Assump ion 6. The unc ions Aand φsa is y he ollowing mono onici y condi ions:
EDAð ;Θ1;sÞ�Að ;Θ2;sÞ;Θ1�Θ2E≤βEjΘ1�Θ2j2;P�a:s
Dφðx1Þ�φðx2Þ;x1�x2E≥
μ
jx1�x2j2
8
<
:
o all Θ1;Θ2∈R6,x1;x2∈R
In he spi i o Li and Liu (2014) we a e now eady o o mula e he s a e cons ained
s ochas ic maximum p inciple o ully coupled FBSDEs o mean- ield ype.
Theo em 2.5. The s a e cons ained maximum p inciple. Le Assump ions 3–6hold and
assume Λ ⊆ Rl o be a closed and con ex se . I ðxð$Þ;yð$Þ;zð$Þ;sð$ÞÞis an
op imal 4- uple o p oblem (SC), hen he e exis s a ec o ðλ0;λÞ∈R1þl
such ha
λ0≥0;jλ0j2þjλj2¼1;(2.8)
sa is ying he ans e sali y condi ion
�λ; �EZT
0
F� ;xð Þ;yð Þ;zð Þ;sð Þ�d þH�xðTÞ�þG�yð0Þ�
� ��≥0;∀ ∈Λ(2.9)
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and a 3- uple ð ð$Þ;pð$Þ;qð$ÞÞ∈L2
FðΩ;Cð½0;T�;RÞÞ3L2
FðΩ;Cð½0;T�;RÞÞ3L2
Fð0;T;RÞ
o solu ions o he adjoin FBSDE
d ð Þ ¼ cyð Þ ð Þ�byð Þpð Þ�
σ
yð Þqð ÞþX
l
i¼0
λi i
yð Þ
(
þEc~
yð Þ ð Þ�b~
yð Þpð Þ�
σ
~
yð Þqð ÞþX
l
i¼0
λi i
~
yð Þ
" #)d
þczð Þ ð Þ�bzð Þpð Þ�
σ
zð Þqð ÞþX
l
i¼0
λi i
zð Þ
(
þEc~
zð Þ ð Þ�b~
zð Þpð Þ�
σ
~
zð Þqð ÞþX
l
i¼0
λi i
~
zð Þ
" #)dW ;
dpð Þ ¼ −�cxð Þ ð Þþbxð Þpð Þþ
σ
xð Þqð Þ�X
l
i¼0
λi i
xð Þ
(
þE�c~
xð Þ ð Þþb~
xð Þpð Þþ
σ
~
xð Þqð Þ�X
l
i¼0
λi i
~
xð Þ
" #)d
þqð ÞdW ;
ð0Þ ¼ X
l
i¼0
λiEhgi�yð0Þ�i;
pðTÞ ¼ −φx�xðTÞ� ðTÞ�X
l
i¼0
λi�hi
x�xðTÞ;EhxðTÞi�þEhhi
~
x�xðTÞ;EhxðTÞi�i�;
(2.10)
such ha
sð Þ ¼ a gmax
s∈SH� ;Θð Þ;s; ð Þ;pð Þ;qð Þ;λ0;λ�a:e: ∈½0;T�;P�a:s:
whe e he Hamil onian unc ion His gi en by
Hð ;Θ;s; ;p;q;λ0;λÞd
� $cð ;Θ;sÞþp$bð ;Θ;sÞþq$
σ
ð ;ΘÞ�X
l
i¼0
λi ið ;Θ;sÞ:
Rema k 2.6. As in Rema k 2.3, analogue p inciples also hold in Theo em 2.5.
Rema k 2.7. The maximum p inciple in Theo em 2.5 wi hou s a e cons ain s is an easy
ex ension o he same esul in Li and Liu (2014) and ollows he p oo mu a is
mu andis. Ex ending he esul o allow o s a e cons ain s is a s anda d
p ocedu e and can be ound o ins ance in Djehiche and Helgesson (2014).
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3. U ili ies o mean- a iance ype
The mean- a iance u ili y unc ion we use models he ade-o be ween isk and ewa d ha
is cen al o many inancial decisions, such as in po olio managemen . By assuming mean-
a iance p e e ences, we a e cap u ing a ealis ic scena io whe e in es o s aim o maximize
e u ns while minimizing isk.
Fo ins ance, conside a po olio manage ( he agen ) who is employed by a und ( he
p incipal). The und wan s he manage o in es in a way ha maximizes e u ns while
con olling o isk. The manage ’s u ili y unc ion includes bo h he expec ed e u n and he
isk ( a iance) o he in es men po olio. The op imal con ac , de i ed using ou me hods,
ensu es ha he manage is incen i ized o in es in a way ha aligns wi h he und’s
objec i es, despi e he manage ha ing p i a e in o ma ion abou in es men oppo uni ies
and isks.
We a e now going o i he me hods p esen ed in Sec ion 2 o a mean- a iance amewo k,
i.e. we wan o con ol he o wa d-backwa d dynamics o mean- ield ype (2.5) wi h espec
o ei he o he ollowing wo cases:
(i) Minimize
IðuÞd�EZT
0
Uð ;Θð Þ;sð ÞÞd þVðxðTÞÞ
� �
þ
2Va ZT
0
Φð ;Θð Þ;sð ÞÞd þΨðxðTÞÞ
� �;
(3.1)
o e S½0;T� o some isk a e sion > 0.
(ii) Minimize
JðuÞdEZT
0
Uð ;Θð Þ;sð ÞÞd þVðxðTÞÞ
� � (3.2)
o e S½0;T�subjec o a se o s a e cons ain s (compa e (2.7)), including s a emen s o
he o m
Va ZT
0
Φð ;Θð Þ;sð ÞÞd þΨðxðTÞÞ
� �≤R0:(3.3)
In o de o ca y his h ough we in oduce he auxilia y p ocess
η
ð ÞdZ
0
Φð
τ
;Θð
τ
Þ;sð
τ
ÞÞd
τ
þΨðxð ÞÞ;
which by I ^
o’s Lemma sol es he SDE
d
η
ð Þ ¼ Φð Þþbð Þ$Ψ0ðxð ÞÞþ
σ
2ð Þ
2$Ψ00ðxð ÞÞ
� �d þ
σ
ð Þ$Ψ0ðxð ÞÞdW ;
η
ð0Þ ¼ 0:
8
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>
<
>
>
:
(3.4)
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The p incipal’s p oblem (s ong o mula ion): Gi en ha he agen ’s p oblem has
an op imal solu ion eð$Þin he weak o mula ion he P incipal’s p oblem is o ind a p ocess
sð$Þ∈S½0;T�, such ha he cos unc ional
JP�sð$Þ�d�EZT
0
Uð ;xð Þ;sð ÞÞd þVðxðTÞÞ
� �
þ
2Va ZT
0
Φð ;xð Þ;sð ÞÞd þΨðxðTÞÞ
� �;
is minimized and
JA�eð$Þ;s�¼EZT
0
uð ;xð Þ;eð Þ;sð ÞÞd þ ðxðTÞÞ
� �≤C0;
subjec o he dynamics
dxð Þ ¼
σ
ð ;xð ÞÞdW ; ∈ð0;T�;
xð0Þ ¼ 0:
�
Rema k 4.1. He e we ha e chosen o o mula e he p incipal’s p oblem in he s ong o m
a he han in he weak o m, which seems o be mos common in he
li e a u e. Howe e , as poin ed ou in C i ani�
c and Zhang (2013), because o
adap i eness his app oach can be p oblema ic in ce ain models. This is a
ac ha one should be awa e o .
In his con ex he ollowing de ini ion is na u al.
De ini ion 4.2. An op imal con ac is a pai ðeð$Þ;sð$ÞÞ∈E½0;T�3S½0;T�ob ained by
sequen ially sol ing i s he agen ’s and hen he p incipal’s p oblem.
In game heo e ic e minology an op imal con ac can hus be hough o
as a S ackelbe g equilib ium in a wo-playe non-ze o-sum game.
I is impo an o no e ha e en hough he p incipal canno obse e he
agen ’s e o , he/she can s ill o e he agen a con ac by sugges ing a
choice o e o e($) and a compensa ion s($). By incen i e compa ibili y,
howe e , he p incipal knows ha he agen only will ollow such a
con ac i he sugges ed e o sol es he agen ’s p oblem. To ind he
op imal e o , eð$Þ, he p incipal mus ha e in o ma ion o he agen ’s
p e e ences, i.e. he unc ions uand . The ealism o such an assump ion
is indeed ques ionable bu ne e heless necessa y in ou o mula ion due
o he pa icipa ion cons ain . In o de o make he in ui ion clea and o
a oid any con usion we adop he con en ion ha he p incipal has ull
in o ma ion o he agen ’s p e e ences uand . This gi es a ac able way
o hinking o how ac ual con ac ing is ealized.
Thus, he p incipal is able o p edic he op imal e o eð$Þo he agen ’s
p oblem and he eby sugges an op imal con ac ðeð$Þ;sð$ÞÞ, i i exis s.
The idea is o apply he me hods om Sec ion 2 o cha ac e ize op imal
con ac s in he gene al p incipal–agen model p esen ed abo e.
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Howe e , since he con ol a iable e igu es in he di usion o (4.5) we
equi e he ollowing con exi y assump ion in o de o a oid a second
o de adjoin p ocess in he maximum p inciple:
Assum ion 7. The se E⊂Ris con ex.
The agen ’s Hamil onian in he weak o mula ion is
HAð ;x;Γe;e;p;q;sÞdq$Γe$ ð ;x;eÞ
σ
ð ;xÞ�Γe$uð ;x;e;sÞ;(4.10)
and by Theo em 2.1 any op imal con ol eð Þsol ing he agen ’s p oblem mus maximize HA
poin wise. The pai (p($), q($)) sol es he agen ’s adjoin BSDE:
dpð Þ ¼ −qð Þ$
� ;xð Þ;eð Þ�
σ
ð ;xð ÞÞ �u� ;xð Þ;eð Þ;sð Þ�
8
<
:9
=
;d þqð ÞdW ;
pðTÞ ¼ − xðxðTÞÞ
8
>
>
>
>
<
>
>
>
>
:
(4.11)
I and ubo h a e di e en iable in he e a iable and we assume ha eð$Þ∈in ðEÞ,
maximizing HA ansla es in o he i s o de condi ion
qð Þ ¼
σ
ð ;xð ÞÞ$
ue� ;xð Þ;eð Þ;sð Þ�
e� ;xð Þ;eð Þ�;(4.12)
which is in ag eemen wi h Williams (2013). Be o e p oceeding o he P incipal’s p oblem we
assume sol abili y o ein (4.12) and we w i e
eð Þ ¼ e*� ;xð Þ;qð Þ;sð Þ�;
whe e e*:Rþ3R4→Ris a unc ion ha ing su icien egula i y o allow o he exis ence
o a unique solu ion o he FBSDE (4.13) below. Based on he in o ma ion gi en by e* he
p incipal wishes o minimize he cos JPby selec ing a p ocess s($) espec ing (4.4). The
dynamics o he co esponding con ol p oblem is, in con as o he SDE o he agen ’s
p oblem, a FBSDE buil up by he ou pu SDE coupled o he agen ’s adjoin BSDE. Mo e
p ecisely:
dxð Þ ¼
σ
ð ;xð ÞÞdW ;
dpð Þ ¼ −qð Þ$
� ;xð Þ;e*ð ;xð Þ;qð Þ;sð ÞÞ�
σ
ð ;xð ÞÞ �u� ;xð Þ;e*ð ;xð Þ;qð Þ;sð ÞÞ;sð Þ�
8
<
:9
=
;d
þqð ÞdW ;
xð0Þ ¼ 0;pðTÞ ¼ − xðxðTÞÞ:
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:(4.13)
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In o de o cha ac e ize cash- low op imali y in he P incipal’s p oblem we apply Theo em
3.1. The Hamil onian eads
HPð ;x;q;s;R;P1;P2;Q1;Q2;λP;λAÞd
R$�q$
� ;x;e*ð ;x;q;sÞ�
σ
ð ;xÞþu� ;x;e*ð ;x;q;sÞ;s�
8
<
:9
=
;þP2$Φð ;x;sÞþ
σ
2ð ;xÞ
2Ψ00ðxÞ
� �
þQ1$
σ
ð ;xÞþQ2$
σ
ð ;xÞΨ0ðxÞ�λA$u� ;x;e*ð ;x;q;sÞ;s�þλP$Uð ;x;sÞ;
(4.14)
and o any op imal 4- uple ðxð$Þ;pð$Þ;qð$Þ;sð$ÞÞ we ha e he exis ence o Lag ange
mul iplie s λA;λP∈Rsa is ying he condi ions in Theo em (3.1). The adjoin p ocesses (R($),
P
1
($), Q
1
($), P
2
($), Q
2
($)) sol e he FBSDE (3.7), in which case
sð Þ ¼ a gmax
s∈SHP� ;xð Þ;pð Þ;qð Þ;s;Rð Þ;P1ð Þ;Q1ð Þ;P2ð Þ;Q2ð Þ;λP;λA�:
Be o e s a ing he ull cha ac e iza ion o op imal con ac s in he Mean-Va iance P incipal-
Agen p oblem unde Hidden Ac ion we in oduce he ollowing echnical assump ion:
(PA1). All unc ions in ol ed in he Agen ’s p oblem sa is y Assump ion 1 om Sec ion 2
and he densi y o ou pu is a ma ingale. The unc ions de ining he P incipal’s p oblem
(including composi ion wi h he map e*) sa is y Assump ions 2-6, also om Sec ion 2, and Ψ
is h ee imes di e en iable.
Theo em 4.3. Le he s a emen s in (PA1) and Assump ion 7 hold and conside he
Mean-Va iance P incipal-Agen p oblem unde Hidden Ac ions wi h isk
a e sion > 0 and pa icipa ion cons ain de ined by C
0
< 0. Then, i
ðeð$Þ;sð$ÞÞ is an op imal con ac he e exis numbe s λA;λP∈R
such ha
λP≥0;λ2
Aþλ2
P¼1;
a pai ðpð$Þ;qð$ÞÞ∈L2
Fð0;T;RÞ3ðL2
Fð0;T;RÞÞsol ing he SDE in (4.11)
and a quin uple ðRð$Þ;P1ð$Þ;P2ð$Þ;Q1ð$Þ;Q2ð$ÞÞ∈L2
FðΩ;Cð½0;T�;RÞÞ
3L2
FðΩ;Cð½0;T�;RÞÞ3L2
Fð0;T;RÞsol ing he adjoin FBSDE (3.7)
de ined by (4.13) such ha , sequen ially,
eð Þ ¼ a gmax
e∈EHA� ;xð Þ;Γeð Þ;e;qð Þ;sð Þ�;
and
sð Þ ¼ a gmax
s∈SHP� ;xð Þ;qð Þ;s;Rð Þ;P1ð Þ;Q1ð Þ;P2ð Þ;Q2ð Þ;λP;λA�;
wi h Hamil onians HAand HPas in (4.10) and (4.14) espec i ely.
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4.2 Hidden Con ac in he s ong o mula ion
We a e now going o s udy a di e en ype o mean- a iance p incipal–agen p oblems
called hidden con ac models (in oduced in Djehiche and Helgesson (2014)). Compa ing o
he hidden ac ion model in Sec ion 4.1 he hidden con ac s di e in wo key aspec s. Fi s we
elax he in o ma ion se o he p incipal om Fx o he ull il a ion gene a ed by he
B ownian mo ion. Secondly we ea he p ocess s($) as hidden, meaning ha he Agen
eac s o he p o ided cash- low gi en as an F-adap ed p ocess, wi hou being awa e o he
unde lying dependence o he ou pu . This explains he name Hidden Con ac .
The ac ha he unde lying ma hema ical s uc u e o s($) is unknown o he Agen in he
Hidden Con ac model mo i a es he ele ance o a Mean-Va iance amewo k by an
ex ended pa icipa ion cons ain (compa ed o (4.4)). By equi ing an uppe bound o he
a iance o o ins ance he expec ed accumula ed weal h p o ided by s($) he Agen can
p o ec him/he -sel om undesi able high le els o isk. The se up goes as ollows.
Conside a P incipal-Agen model in which ou pu x( ) is modeled as a isky asse sol ing
he SDE
dxð Þ ¼ ð ;xð Þ;eð ÞÞd þ
σ
ð ;xð ÞÞdW ; ∈ð0;T�;
xð0Þ ¼ 0:
�(4.15)
He e T> 0 and W
is a 1-dimensional s anda d B ownian mo ion de ined on he il e ed
p obabili y space ðΩ;F;F;PÞ. The unc ions and
σ
ep esen p oduc ion a e and ola ili y
espec i ely, and we assume bo h o hem o sa is y Assump ion 1 om Sec ion 2. Jus as o
he Hidden Ac ion case we equi e any admissible e o p ocess e($) o be in E½0;T�. Fo he
admissible cash- lows, howe e , we enla ge S½0;T�(due o he ex ended low o in o ma ion
o he P incipal) o
S½0;T�d s:½0;T�3Ω→S;sis F-adap edg:
We conside he cos unc ionals
JAðeð$Þ;sÞdEZT
0
uð ;xð Þ;eð Þ;sð ÞÞd þ ðxðTÞÞ
� �;(4.16)
and
JPðsð$ÞÞdEZT
0Uð ;xð Þ;sð ÞÞd þVðxðTÞÞ
� �;(4.17)
and he pa icipa ion cons ain :
JA�eð$Þ;s�dEZT
0
u� ;xð Þ;eð Þ;sð Þ�d þ ðxðTÞÞ
� �≤C0;
IA�eð$Þ;s�dVa ZT
0
� ;xð Þ;eð Þ;sð Þ�d þ
ψ
ðxðTÞÞ
� �≤R0:
8
>
>
>
<
>
>
>
:
(4.18)
Jus as o he Hidden Ac ion case in Sec ion 4.1 we conside he Agen ’s- and he P incipal’s
p oblem sequen ially. The p ecise s a emen s a e:
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The Agen ’s P oblem. Gi en any sð$Þ∈S½0;T�( ul illing he pa icipa ion cons ain )
he Agen ’s p oblem is o ind a p ocess eð$Þ∈E½0;T�minimizing (4.16).
The P incipal’s P oblem. Gi en ha he Agen ’s p oblem has an op imal solu ion eð$Þ
he P incipal’s p oblem is o ind a p ocess sð$Þ∈S½0;T�minimizing he cos unc ional
(4.17) subjec o he pa icipa ion cons ain (4.18).
The ma hema ical i ue o Hidden Con ac s is he possibili y o wo king solely in he
s ong o mula ion. Fo he Agen ’s p oblem we a e acing he Hamil onian
HAð ;x;e;p;q;sÞdp$ ð ;x;eÞþq$
σ
ð ;xÞ�uð ;x;e;sÞ:(4.19)
The e o e, by Theo em 2.1 we ha e o any op imal pai ðxð$Þ;eð$ÞÞ he exis ence o adjoin
p ocesses (p($), q($)) sol ing he backwa d s ochas ic di e en ial equa ion (BSDE):
dpð Þ ¼ − x� ;xð Þ;eð Þ�pð Þþ
σ
x� ;xð Þ�qð Þ�ux� ;xð Þ;eð Þ�n od þqð ÞdW ;
pðTÞ ¼ − x�xðTÞ�;
8
<
:(4.20)
and he cha ac e iza ion
eð Þ ¼ a gmax
e∈EHA� ;xð Þ;e;pð Þ;qð Þ;sð Þ�;(4.21)
o a.e. ∈[0, T] and P-a.s.
As in he hidden con ac case we p oceed in o he p incipal’s p oblem by assuming he
exis ence o a unc ion e* such ha e ¼e*ð ;xð Þ;pð Þ;qð Þ;sð ÞÞ (ha ing su icien
egula i y o allow o exis ence and uniqueness o a solu ion o (4.22)). The p incipal is acing
he p oblem o minimizing JPsubjec o (4.18) by con olling he ollowing FBSDE:
dx ¼ � ;xð Þ;e*� ;xð Þ;pð Þ;qð Þ;sð Þ��d þ
σ
� ;xð Þ�dW ;
dpð Þ ¼ − x� ;xð Þ;e*� ;xð Þ;pð Þ;qð Þ;sð Þ��pð Þþ
σ
x� ;xð Þ�qð Þ
n
�ux� ;xð Þ;e*� ;xð Þ;pð Þ;qð Þ;sð Þ��od þqð ÞdW ;
xð0Þ ¼ 0;pðTÞ ¼ − x�xðTÞ�:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(4.22)
We now apply Theo em 3.2 in o de o cha ac e ize op imal cash- lows in he p incipal’s
p oblem. The associa ed Hamil onian is
HPð ;x;p;q;s;R;P1;P2;Q1;Q2;λE;λV;λPÞ ¼
R$ð xð ;x;eð ;x;p;q;sÞÞpþ
σ
xð ;xÞq�uxð ;x;eð ;x;p;q;sÞ;sÞÞ
þP1$ ð ;x;eð ;x;p;q;sÞÞþP2$
ð ;x;eð ;x;p;q;sÞ;sÞ
þ ð ;x;eð ;x;p;q;sÞÞ
ψ
0ðxÞþ
σ
2ð ;xÞ
2
ψ
00ðxÞ�þQ1$
σ
ð ;xÞþQ2$
σ
ð ;xÞ
ψ
0ðxÞ
�λE$uð ;x;eð ;x;p;q;sÞ;sÞ�λP$Uð ;x;sÞ:
(4.23)
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Fo any op imal 4- uple ðxð$Þ;pð$Þ;qð$Þ;sð$ÞÞ o he p incipal’s p oblem we ha e he
exis ence o Lag ange mul iplie s λE;λV;λP∈Rsa is ying ei he o he condi ions (i)-( ) in
Sec ion 2, wi h λ
P
≥0 and
λ2
Eþλ2
Vþλ2
P¼1;
and a iple o adjoin p ocesses (R($), P($), Q($)) sol ing he FBSDE (3.10) so ha
sð Þ ¼ a gmax
s∈SHPð ;xð Þ;pð Þ;qð Þ;s;Rð Þ;P1ð Þ;P2ð Þ;Q1ð Þ;Q2ð Þ;λE;λPÞ:
Fo he ull cha ac e iza ion o op imali y we equi e he ollowing echnical assump ion:
(PA2). All unc ions in ol ed in he agen ’s p oblem sa is y he Assump ion 1 om
Sec ion 2. The unc ions de ining he p incipal’s p oblem (including composi ion wi h he
map e*) sa is y he Assump ions 2–6, also om Sec ion 2, and
ψ
is h ee imes di e en iable.
Theo em 4.4. Le he s a emen s in (PA2) hold and conside he mean- a iance p incipal–
agen p oblem unde hidden con ac wi h pa icipa ion cons ain s de ined
by he gi en pa ame e s C
0
< 0 and R
0
> 0. Then, i ðeð$Þ;sð$ÞÞis an op imal
con ac he e exis numbe s λE;λV;λP∈Rsuch ha
λP≥0;λ2
Eþλ2
Vþλ2
P¼1;
a pai ðpð$Þ;qð$ÞÞ∈L2
Fð0;T;RÞ3ðL2
Fð0;T;RÞÞsol ing he BSDE in (4.20) and a quin uple
ðRð$Þ;P1ð$Þ;P2ð$Þ;Q1ð$Þ;Q2ð$ÞÞ∈L2
FðΩ;Cð½0;T�;RÞÞ3L2
FðΩ;Cð½0;T�;RÞÞ3L2
Fð0;T;
RÞsol ing he adjoin FBSDE (3.10) de ined by (4.22) such ha , sequen ially,
eð Þ ¼ a gmax
e∈EHA� ;xð Þ;e;pð Þ;qð Þ;sð Þ�;
and
sð Þ ¼ a gmax
s∈SHP� ;xð Þ;pð Þ;qð Þ;s;Rð Þ;P1ð Þ;P2ð Þ;Q1ð Þ;Q2ð Þ;λE;λV;λP�:
wi h Hamil onians HAand HPas in (4.19) and (4.23), espec i ely.
5. A sol ed example in he case o hidden con ac s
We now illus a e he me hod o Sec ion 4 by conside ing a conc e e example o hidden
con ac ype. In o de o ind explici solu ions we choose a linea -quad a ic se up. As a
esul we ge op imal con ac s adap ed o he il a ion gene a ed by ou pu .
Conside he ollowing dynamics o p oduc ion,
dxð Þ ¼ ðaxð Þþbeð ÞÞd þ
σ
dW ; ∈ð0;T�;
xð0Þ ¼ 0;a;b∈Rand
σ
>0;
�
and le he p e e ences o he agen and he p incipal be desc ibed by quad a ic u ili y unc ions:
JAðeð$Þ;sÞdEZT
0
ðs �e Þ2
2d �
α
$xðTÞ2
2
" #;(5.1)
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JPðsð$ÞÞdEZT
0
s2
2d �β$xðTÞ2
2
" #:(5.2)
No e ha we a e ollowing he con en ion o Sec ion 4 o conside cos - a he han payo -
unc ionals. Thus, he agen ’s u ili y unc ion should be in e p e ed as a desi e o main ain a
le el o e o close o he compensa ion gi en by he cash- low. We hink o he pa ame e s
α
> 0 and β> 0 as bonus ac o s o o al p oduc ion a ime T. Fo he pa icipa ion cons ain
we equi e any admissible cash- low s( ) o sa is y he ollowing:
JA�eð$Þ;s�≤C0;
Va xðTÞð Þ <R0;
((5.3)
whe e C
0
< 0, R
0
> 0 and eð$Þdeno es he op imal e o policy o he agen gi en s($).
Assume ha he p incipal o e s he agen s($) o e he pe iod 0 ≤ ≤T. The Hamil onian
unc ion o he agen is
HAðx;e;p;q;sÞdp$ðax þbeÞþq$
σ
�ðs�eÞ2
2;
so
HA
e¼bp þs�e¼0 and eð Þ ¼ bpð Þþsð Þ;(5.4)
whe e he pai (p,q) sol es he adjoin equa ion
dpð Þ ¼ −apð Þd þqð ÞdW ;
pðTÞ ¼
α
xðTÞ:
�
Tu ning o he p incipal’s p oblem we wan o con ol he FBSDE
dxð Þ ¼ �axð Þþb2pð Þþbsð Þ�d þ
σ
dW ;
dpð Þ ¼ −apð Þd þqð ÞdW ;
xð0Þ ¼ 0;pðTÞ ¼
α
xðTÞ;
8
>
<
>
:(5.5)
op imally wi h espec o he cos unc ion (5.2) and he pa icipa ion cons ain (5.3). The
p incipal’s Hamil onian is
HPðx;p;s;R;P1;P2;Q1;Q2;λE;λPÞd
�ap$Rþ�ax þb2pþbs�$P1þ�sþax þb2pþbs�$P2þ
σ
$ðQ1þQ2Þ
�λE$b2p2
2�λP$s2
2
(5.6)
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so
HP
s¼bP1þð1þbÞP2�λPsand sð Þ ¼ bP1ð Þþð1þbÞP2ð Þ
λP
;
whe e he quin uple (R( ), P
1
( ), P
2
( ), Q
1
( ), Q
2
( )) sol es he adjoin FBSDE:
dRð Þ ¼ �aRð Þ�b2ðP1ð ÞþP2ð ÞÞþλEb2pð Þ�d ;
dP1ð Þ ¼ −aðP1ð ÞþP2ð ÞÞd þQ1ð ÞdW ;
dP2ð Þ ¼ Q2ð ÞdW ;
Rð0Þ¼0;
P1ðTÞ ¼ −
α
RðTÞþð
α
λEþβλPÞxðTÞ;P2ðTÞ ¼ 2λVðE½
η
ðTÞ��
η
ðTÞÞ:
8
>
>
>
>
>
<
>
>
>
>
>
:
(5.7)
In his case, howe e , he auxilia y p ocess
η
( ) is he same as he ou pu x( ) in which case
P2ðTÞ ¼ 2λVðE½xðTÞ�−xðTÞÞ. To sol e he BSDE in (5.7) we can make a gene al linea
ansa z:
pð Þ ¼ A11ð Þxð ÞþB11ð ÞRð ÞþA21ð ÞE½xð Þ�þB21ð ÞE½Rð Þ�;
P1ð Þ ¼ A12ð Þxð ÞþB12ð ÞRð ÞþA22ð ÞE½xð Þ�þB22ð ÞE½Rð Þ�;
P2ð Þ ¼ A13ð Þxð ÞþB13ð ÞRð ÞþA23ð ÞE½xð Þ�þB23ð ÞE½Rð Þ�:
8
<
:(5.8)
Using he s anda d p ocedu e wi h I ^
o’s lemma i is elemen a y (bu edious) o de i e a se o
wel e coupled Ricca i equa ions o he coe icien s in (5.8). A nume ical example is
p esen ed in Figu e 2 below. We ge he unique semi-explici solu ion o he op imal con ac
eð Þ;sð Þg, d i en by he op imal dynamics ðxð Þ;Rð ÞÞ. Wha emains is o ind a easible
iple (λ
E
,λ
V
,λ
P
) so ha he op imal con ac ul ills he pa icipa ion cons ain in (5.3). One
way o inding such a iple is o ins ance by s ochas ic simula ion o ðxð Þ;Rð ÞÞ (e.g. a
simple Eule –Ma uyama scheme) and hen es ima e he payo and he a iance in (5.3) by
Mon e-Ca lo echniques o di e en alues o λ
P
. In Figu e 3 we ha e included he esul s o
such a scheme co esponding o case (i ) o he ans e sali y condi ion in Co olla y 3.2. No e
ha Rð Þsa is ies he linea ODE.
dR
d þ�b2B12 þb2B13 �λEb2B11 �a�Rð Þ ¼ �λEb2A11 �b2A12 �b2A13�xð Þ;
Rð0Þ ¼ 0;
8
>
<
>
:(5.9)
so
R0
@ 1
A¼Z
0
exp Zs
0
b2B12 þb2B13 �λEb2B11 �a du
� �$λEb2A11 �b2A12 �b2A13
� �xds
exp Z
0
b2B12 þb2B13 �λEb2B11 �a ds
� � ;
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and is by ha Fx-adap ed. The e o e, in his model he op imal con ac eð Þ;sð Þg is
Fx-adap ed and coincides wi h he co esponding s ong solu ion o he hidden ac ion
p oblem, i.e. when he in o ma ion se o he p incipal is gene a ed by ou pu .
6. Conclusion
In his pape , we ha e ex ended he con inuous- ime p incipal–agen p oblem by
inco po a ing ime-inconsis en u ili y unc ions, speci ically mean- a iance u ili ies. This
app oach o e comes he limi a ions o adi ional me hods ha ely on he Bellman p inciple
Sou ce(s): Figu e by he au ho s
θθθ
λλ
λPP P
No e(s): Pa ame e alues: a = b = σ = 1, α = 0.2, β = 1, T = 0.03
Sou ce(s): Figu e by he au ho s
Figu e 2.
Solu ion cu es o (5.7)
wi h pa ame e alues
chosen as:
a5b5
σ
51,
α
50.2,
β51, λ
P
50.1,
θ5
π
/2, T50.03
Figu e 3.
Mon e-Ca lo
simula ions o
JAðeð$Þ;sÞ,JPðsð$ÞÞ
and Va (x(T)) as
unc ions o λ
P
and θ
( ela ing o λ
E
and λ
V
ia case (i )) based on
10
6
sample pa hs a
each poin
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and he Hamil on–Jacobi–Bellman equa ion, which a e no sui able o ime-inconsis en
scena ios. By applying he Pon yagin maximum p inciple o FBSDEs, we ha e de eloped a
no el me hod o cha ac e izing op imal con ac s unde ime-inconsis en p e e ences. This
con ibu ion ills a signi ican gap in he li e a u e. Ou amewo k is pa icula ly ele an
o isk managemen applica ions in inance and economics, such as po olio op imiza ion
and pe o mance-based compensa ion. By illus a ing he me hod h ough a ully sol ed
linea -quad a ic example, we demons a ed how ou app oach can be applied o eal-wo ld
scena ios in ol ing ime-inconsis en p e e ences. This example unde sco es he p ac ical
alue o ou esul s o designing op imal con ac s and managing isk. Ou s udy p o ides a
igo ous amewo k o designing op imal con ac s in p incipal–agen p oblems wi h ime-
inconsis en p e e ences, which a e common in inancial se ings. By modeling hese
scena ios, we o e insigh s in o how p incipals can e ec i ely manage agen s who ha e
p i a e in o ma ion and di e ing isk p e e ences. Fu u e esea ch could build on ou
indings by explo ing addi ional o ms o ime inconsis ency, including hype bolic- ype
discoun ing ke nels, and ex ending he amewo k o mo e complex u ili y unc ions.
In es iga ing he applica ion o ou me hods in mul i-agen se ings o di e en economic
en i onmen s may p o ide u he insigh s. Empi ical alida ion h ough case s udies o
simula ions would also be bene icial, o e ing a deepe unde s anding o he p ac ical
pe o mance and obus ness o he p oposed me hods.
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