A mean-field model of optimal investment

Cal ia, Alessand o; Fede ico, Sal a o e; Fe a i, Gio gio; Gozzi, Faus o
Wo king Pape
A mean- ield model o op imal in es men
Cen e o Ma hema ical Economics Wo king Pape s, No. 690
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Sugges ed Ci a ion: Cal ia, Alessand o; Fede ico, Sal a o e; Fe a i, Gio gio; Gozzi, Faus o (2024) : A
mean- ield model o op imal in es men , Cen e o Ma hema ical Economics Wo king Pape s, No.
690, Biele eld Uni e si y, Cen e o Ma hema ical Economics (IMW), Biele eld,
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690
Ap il 2024
A Mean-Field Model o Op imal In es men
Alessand o Cal ia, Sal a o e Fede ico, Gio gio Fe a i, and Faus o Gozzi
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
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e-mail: [email p o ec ed]
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ISSN: 0931-6558
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A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT
ALESSANDRO CALVIAa,1, SALVATORE FEDERICOb,2, GIORGIO FERRARIc,3, AND FAUSTO GOZZId,4
Abs ac . We es ablish he exis ence and uniqueness o he equilib ium o a s ochas ic mean- ield
game o op imal in es men . The analysis co e s bo h ini e and in ini e ime ho izons, and he
mean- ield in e ac ion o he ep esen a i e company wi h a mass o iden ical and indis inguish-
able i ms is modeled h ough he ime-dependen p ice a which he p oduced good is sold. A
equilib ium, his p ice is gi en in e ms o a nonlinea unc ion o he expec ed (op imally con-
olled) p oduc ion capaci y o he ep esen a i e company a each ime. The p oo o he exis ence
and uniqueness o he mean- ield equilib ium elies on a p io i es ima es and he s udy o nonlin-
ea in eg al equa ions, bu employs di e en echniques o he ini e and in ini e ho izon cases.
Addi ionally, we in es iga e he de e minis ic coun e pa o he mean- ield game unde s udy.
Keywo ds: mean- ield games; mean- ield equilib ium; o wa d-backwa d ODEs; op imal in es -
men ; p ice o ma ion.
AMS 2020: 35Q89; 47H10; 49N10; 49N80; 91A07; 91B38; 91B70.
JEL classi ica ion: C02; C61; C62; C72; D25; D41.
1. In oduc ion
In his pape , we conside a mean- ield model o op imal in es men wi h compe i ion à la
Cou no , whe e he p ice o he good p oduced by a ep esen a i e company depends on he agg e-
ga e p oduc ion o he en i e economy h ough a nonlinea in e se demand unc ion. In he absence
o in e en ions, he p oduc ion capaci y o he ep esen a i e company e ol es s ochas ically o e
ime as a geome ic B ownian mo ion, and i s le el can be inc eased h ough in es men , subjec o
quad a ic cos s. The ep esen a i e company discoun s p o i s and cos s a a cons an a e and aims
o maximize o al expec ed p o i s om p oduc ion, ne o in es men cos s. Ins an aneous p o i s
depend linea ly on he company’s p oduc ion capaci y ( hus, p oduc ion occu s a ull capaci y)
and on he ime-dependen p ice o he p oduced good. The mean- ield equilib ium in es men and
a e age p oduc ion p ocesses (b
u,bq)a e such ha expec ed o al ne p o i s a e maximized and, as-
suming an isoelas ic in e se demand unc ion, he p ice is gi en in e ms o a nonlinea unc ion o
he a e age op imally con olled p oduc ion a each ime (see also Achdou e al. [2] and Rema k 2.4
below). We a e able o p o e he exis ence and uniqueness o he equilib ium pai (b
u,bq)when he
p oblem’s ime ho izon is ini e o in ini e. Fu he mo e, we demons a e ha he exis ence and
aUni e si y o Pa ma, Depa men o Economics and Managemen , Via J. F. Kennedy 6, 43125 Pa ma (I aly).
bUni e si y o Bologna, Depa men o Ma hema ics, Piazza di Po a San Dona o 5, 40126 Bologna (I aly).
cBiele eld Uni e si y, Cen e o Ma hema ical Economics (IMW), Uni e si ä ss asse 25, 33615 Biele eld (Ge many).
dLUISS Uni e si y, Depa men o Economics and Finance, Viale Romania 32, 00197 Roma (I aly).
1E-mail: [email p o ec ed].
2E-mail: [email p o ec ed].
3E-mail: [email p o ec ed].
4E-mail: [email p o ec ed].
Gio gio Fe a i g a e ully acknowledges inancial suppo om Deu sche Fo schungsgemeinscha (DFG, Ge man
Resea ch Founda ion)– P ojec -ID 317210226– SFB 1283. This wo k s a ed du ing he isi o Alessand o Cal ia,
Sal a o e Fede ico and Faus o Gozzi a he Cen e o Ma hema ical Economics (IMW) a Biele eld Uni e si y. Those
au ho s hank IMW and he SFB 1283 o he suppo and hospi ali y.
Alessand o Cal ia, Sal a o e Fede ico and Faus o Gozzi a e suppo ed by he I alian Minis y o Uni e si y and
Resea ch (MUR), in he amewo k o PRIN p ojec 2017FKHBA8 001 (The Time-Space E olu ion o Economic
Ac i i ies: Ma hema ical Models and Empi ical Applica ions).
Alessand o Cal ia and Sal a o e Fede ico a e membe s o he G uppo Nazionale pe l’Analisi Ma ema ica, la P oba-
bili à e le lo o Applicazioni (GNAMPA) o he Is i u o Nazionale di Al a Ma ema ica "F ancesco Se e i" (INdAM).
.
1
2 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
uniqueness ca y o e o he de e minis ic coun e pa o ou model, essen ially using he same
echniques as in he s ochas ic case.
The in es men p oblem unde s udy alls unde he ca ego y o mean- ield games wi h scala
in e ac ion. Mean- ield games, independen ly in oduced by Las y and Lions [17], and Huang,
Caines, and Malhamé [15], ep esen limi models o non-coope a i e symme ic N-playe games
wi h mean- ield in e ac ion as he numbe o playe s N ends o in ini y. An exhaus i e e iew o
mean- ield models can be ound in he wo- olume book by Ca mona and Dela ue [8].
In ou con ex , he consis ency condi ion ha he equilib ium p ice aligns a each ime wi h a
dec easing nonlinea unc ion o he expec ed (op imally con olled) p oduc ion capaci y can be
iewed as he limi , as he numbe No iden ical and indis inguishable companies ope a ing in he
ma ke di e ges, o he equi emen ha p ice in e sely depends on he agg ega e p oduc ion o
he en i e economy, scaled by a ac o o 1/N. As discussed in Huang e al. [15], his scaling can
be jus i ied by conside ing si ua ions whe e "an inc easing numbe o i ms join oge he o se e
an inc easing numbe o consume s" (see he discussion in Huang e al. [15], a e Equa ion (2.4)
he ein).
The equilib ium cons uc ion o a gi en ime ho izon T∈(0,+∞] ollows a h ee-s ep ap-
p oach. Fi s ly, gi en a de e minis ic pa h o he a e age p oduc ion q:= (qs)s∈[0,T ], we sol e he
ep esen a i e company’s op imal in es men p oblem. Because he p oduc ion capaci y e ol es as
a geome ic B ownian mo ion and is linea ly dependen on he in es men p ocess and since he
pe o mance c i e ion has linea dependence on he p oduc ion capaci y and quad a ic dependence
on he in es men cos s, he esul ing op imal con ol p oblem o he ep esen a i e company is
o linea -quad a ic ype and has an explici solu ion. Mo eo e , he op imal con ol b
u(T,q) o ixed
a e age p oduc ion ajec o y qis de e minis ic. In he second s ep, we calcula e he expec a ion o
he op imally con olled p oduc ion capaci y p ocess, which is easily compu able gi en he explici
ep esen a ion o he s a e p ocess. Finally, in he hi d s ep, we impose he consis ency condi ion,
equi ing ha his expec a ion mus ma ch qs o each ime s∈[0, T]. This condi ion leads o a
nonlinea in eg al equa ion o he equilib ium a e age p oduc ion ajec o y bq:= (bqs)s∈[0,T]. By
cons uc ion he con ol b
u:=b
u(T,bq)and he unc ion bq o m a mean- ield equilib ium.
I is impo an o no e ha al hough he company’s op imal con ol p oblem o a ixed a e age
p oduc ion ajec o y qis linea -quad a ic, due o he nonlinea dependence o he ne p o i unc-
ional on q, he o e all mean- ield p oblem is no o linea -quad a ic ype. In his ega d, ou esul s
di e om hose p esen ed in Bensoussan e al. [3], Dela ue and Tchuendom [12], Tchuendom [19],
among o he s, which ocus on linea -quad a ic mean- ield games.
While he wo-s ep app oach p e iously ou lined wo ks o bo h he case whe e T < +∞and he
case whe e T= +∞, di e en echnical a gumen s a e used o p o e he exis ence and uniqueness
o he equilib ium in he ini e and in ini e ho izon cases. As a ma e o ac , he in eg al equa ion
uniquely cha ac e izing he equilib ium a e age p oduc ion bqdoes no all in o he s anda d heo y
o in eg o-di e en ial equa ions, as we ha e an ini ial- alue p oblem wi h he in eg al being back-
wa d in ime, and hus equi es a ca e ul ad hoc analysis. Speci ically, he p oo o he exis ence
o a solu ion o i elies on a p io i es ima es and Schaude ’s ixed poin heo em when T < +∞,
whe eas i u ilizes a p io i es ima es and he F eche -Kolmo ogo o Lpcompac ness heo em in
he case o T= +∞. Uniqueness is hen es ablished in bo h cases h ough sui able con adic ion
a gumen s ha exploi he p ope ies o any mean- ield equilib ium. No ably, when T= +∞, we
can also p o e ha he ime-dependen equilib ium bqmono onically con e ges o he s a iona y
a e age p oduc ion le el. The la e is ob ained as he unique cons an solu ion o he in eg al
equa ion ha uniquely de e mines bq.
Gi en he geome ic dynamics o he p oduc ion capaci y, he linea dependence o he p o i
unc ional on he con olled s a e a iable, he quad a ic cos s o in es men , and he ac ha he
mean- ield in e ac ion is o scala ype and only in ol es he expec ed alue o he la e , i is no
su p ising ha he mean- ield equilib ium is de e minis ic and does no depend on he ola ili y
coe icien σappea ing in he p oduc ion capaci y’s dynamics. Guided by his obse a ion, we also
conside he de e minis ic coun e pa o he p e iously discussed mean- ield game and show ha
a unique equilib ium exis s in his se ing as well. This equilib ium can indeed be cons uc ed by
ollowing exac ly he same a gumen s employed in he s ochas ic case.
A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 3
Games o op imal in es men in bo h s ochas ic and de e minis ic se ings a e ex ensi ely co e ed
in he Economics li e a u e, and a comp ehensi e o e iew o models and esul s can be ound
in Vi es [20]. Speci ically, mean- ield p oblems wi h Cou no compe i ion ha e ga ne ed in e es
in ecen li e a u e; see Chan and Si ca [10] o an insigh ul o e iew. G abe and Bensoussan
[14] s udy he exis ence (and, unde ce ain condi ions, uniqueness) o a solu ion o a sys em o
pa ial di e en ial equa ions (PDEs) (namely, he Hamil on-Jacobi-Bellman (HJB) and Fokke -
Planck equa ions) associa ed wi h a mean- ield game in ol ing Be and and Cou no compe i ion
among a con inuum o playe s. In G abe and Si ca [13], exis ence and uniqueness o he mas e
equa ion associa ed wi h a mean- ield game o con ols wi h abso p ion a e p o en, while Chan and
Si ca [9] del e in o dynamic mean- ield games wi h exhaus ible capaci ies and in e ac ions akin o
Cou no and Be and compe i ions. An op imal anspo pe spec i e on Cou no -Nash equilib ia
is explo ed in Acciaio e al. [1]. Finally, Cao e al. [6] conside s s a iona y discoun ed and e godic
mean- ield games o singula con ols mo i a ed by i e e sible in es men and p o ide exis ence
and uniqueness esul s, as well as ela ions ac oss he wo classes o conside ed p oblems.
To he bes o ou knowledge, he exis ence and uniqueness o he (nons a iona y) mean- ield
equilib ium o a mean- ield model o op imal in es men wi h an isoelas ic demand unc ion, as
discussed in his pape , is p esen ed he e o he i s ime.
The es o he pape is o ganized as ollows. The s ochas ic mean- ield game is in oduced in
Sec ion 2. In Sec ion 3 he company’s op imal in es men p oblem is sol ed o bo h he cases
T < +∞and T= +∞, while exis ence and uniqueness o he equilib ium is shown in Sec ion 4,
again o he ini e and in ini e ime ho izon cases. The de e minis ic e sion o he mean- ield
p oblem is inally conside ed in Sec ion 5.
1.1. No a ion. In his sec ion we collec he main no a ion used in his wo k.
•Th oughou he pape he se Ndeno es he se o na u al in ege s wi hou he ze o elemen ,
i.e., N={1,2, . . . }, while Rdeno es he se o eal numbe s. Whene e T= +∞, he
no a ion [0, T]indica es he in e al [0,+∞).
•Fo any p≥1, any measu e space (E, E, µ), and any in e al I⊆R, we indica e by
Lp((E, E, µ); I) he se o all unc ions wi h alues in I ha a e p-in eg able wi h espec
o µ. I E⊆R, we ake L(E)as he Lebesgue σ-algeb a and as µ he Lebesgue measu e,
which is deno ed by Leb, and we simply w i e Lp(E;I). When compu ing in eg als wi h
espec o his measu e, we simply w i e dxins ead o Leb(dx).
•Fo any η > 0and E, I ⊆R, he no a ion Lp
η(E;I)indica es he se o all unc ions
:E→Isuch ha 7→ e−η ( )is p-in eg able wi h espec o he Lebesgue measu e.
•Gi en wo in e als I, J ⊆R, we deno e by C(I;J) he se o all con inuous unc ions om
I o J, endowed wi h he usual sup-no m. The no a ion Cp(I;J),p∈N∪ {+∞}, deno es
he se o unc ions om I o J ha a e con inuously di e en iable p imes.
2. The s ochas ic model
Le (Ω,F,F:= (Fs)s≥0,P)be a comple e il e ed p obabili y space, wi h Fsa is ying he
usual assump ions, suppo ing an F-adap ed s anda d B ownian mo ion Band an independen
F0-measu able andom a iable ξ. Th oughou he pape , T∈(0,+∞)∪ {+∞} deno es a ini e o
in ini e ime ho izon. In wha ollows, whene e T= +∞ he no a ion [0, T]indica es he in e al
[0,+∞).
We conside a eal- alued F-adap ed p ocess X= (Xs)s∈[0,T], sa is ying he s ochas ic di e en ial
equa ion (SDE) dXs=−δXsds+σXsdBs+usds, s ∈(0, T],
X0=ξ , (2.1)
whe e δ, σ > 0a e gi en coe icien s, and he con ol p ocess u:= (us)s∈[0,T]is chosen in ei he o
he ollowing wo classes o admissible con ols: i T < +∞,
UT:=u= (us)s∈[0,T ]s. . u: Ω ×[0, T]→[0,+∞)is (Fs)s∈[0,T ]-p og essi ely measu able and

4 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
EZT
0
u2
sds<+∞;(2.2)
i , ins ead, T= +∞,
U∞:=u= (us)s≥0s. . u: Ω ×[0,∞)→[0,+∞)is F-p og essi ely measu able,
EZ
0
usds<+∞,P-a.s., ∀ ≥0,and EZ+∞
0
e−ρsu2
sds<+∞.(2.3)
Whene e necessa y, o s ess he dependence o he solu ion o (2.1) on he ini ial condi ion ξand
on he con ol u, we deno e i by Xξ,u.
We obse e ha he solu ion o SDE (2.1) can be w i en explici ly (c . [16, P oblem 5.6.15]),
o each u∈ UT, as
Xξ,u
s=Ysξ+Zs
0
Ys
Y
u d , s ∈[0, T ],(2.4)
whe e
Ys:= eσBs−δ+σ2
2s, s ∈[0, T].(2.5)
In he mean- ield game s udied in his pape , he p ocess Xdesc ibes he p i a e s a e o a
ep esen a i e playe . In pa icula , i models he e olu ion o he /his p oduc ion capaci y, which
dep ecia es a a a e δand can be inc eased by choosing he in es men a e u∈ UT. No e ha
he s a is ical dis ibu ion o he s a e o he o he playe s in he economy does no in luence he
p oduc ion capaci y le el o he ep esen a i e playe .
Th oughou he pape , we wo k unde he ollowing assump ion.
Assump ion 2.1. The ini ial condi ion ξo SDE (2.1)is an F0-measu able, posi i e, and in eg able
andom a iable. Mo e p ecisely, ξ > 0,P-a.s., and 0<E[ξ]<+∞.
The assump ion abo e ensu es ha , o any u∈ UT,Xξ,u
s≥0,P-a.s., o all s∈[0, T]. Thus,
he p oduc ion capaci y le el o he ep esen a i e agen is ne e nega i e. Mo eo e , i g an s us
he ollowing esul , which will be used la e on.
Lemma 2.2. Unde Assump ion 2.1 and o any u∈ UT, he unique solu ion o SDE (2.1)has
ini e i s momen , gi en by
E[Xξ,u
s] = E[ξ]e−δs +EZs
0
e−δ(s− )u d , s ∈[0, T].(2.6)
P oo . Fix ξand uas abo e. Using he exp ession o Xξ,ugi en in (2.4) we ha e ha
E[Xξ,u
s] = E[Ysξ] + EZs
0
Ys
Y
u d , s ∈[0, T ].
Since he andom a iable Ysξis non-nega i e and Ysis independen o F0, o all s∈[0, T], we
can di ec ly compu e he i s summand
E[Ysξ] = E[ξ]E[Ys] = E[ξ]e−δs.
Fo he second summand, obse e ha he in eg and is non-nega i e and ha , o each ixed
s∈[0, T], he andom a iable Ys
Y is independen o F , o all ∈[0, s]. The e o e, applying he
Fubini-Tonelli heo em,
EZs
0
Ys
Y
u d =Zs
0
EYs
Y
u d =Zs
0
EYs
Y E[u ] d =EZs
0
e−δ(s− )u d ,
which is ini e hanks o he assump ions on u. The e o e, we ge (2.6). □
In ou mean- ield game we assume ha e e y playe aims a maximizing he discoun ed ne
p o i unc ional
JT,q(ξ, u):=EZT
0
e−ρs Xξ,u
sq−β
s−1
2u2
sds,(2.7)
A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 5
whe e ρ > 0is a discoun ac o , β > 0is a ixed pa ame e , and q= (qs)s∈[0,T]is a gi en
de e minis ic measu able unc ion. A equilib ium, he unc ion qwill iden i y wi h he a e age
p oduc ion capaci y o he whole popula ion o agen s. This is o malized in he nex de ini ion.
De ini ion 2.3. Fix a andom a iable ξunde Assump ion 2.1.
A pai (b
u,bq), whe e b
u∈ UTand bq: [0, T]→(0,+∞)is a measu able unc ion, is an equilib ium
o he mean- ield game i
(i) JT,bq(ξ, b
u)≥JT,bq(ξ, u), o all u∈ UT;
(ii) bqs=E[Xξ,b
u
s], o all s∈[0, T ].
In he nex sec ions we show ha he e exis s a unique equilib ium by adop ing a classic ixed
poin app oach. The i s s ep is o sol e an op imiza ion p oblem o a gi en measu able unc ion
q, ep esen ing he e olu ion o he a e age p oduc ion capaci y le el o he agen s in he economy.
We show ha he e exis s a unique explici and de e minis ic op imal con ol b
u∈ UT, depending
on q, which p o ides us wi h he bes esponse o he ep esen a i e agen o he dis ibu ion o
he s a es o he o he agen s in he economy, summa ized by he a e age q.
The second s ep is o de e mine he op imally con olled dynamics o he p oduc ion capaci y
le el o he ep esen a i e agen . Thanks o (2.6), we can compu e explici ly E[Xξ,b
u
s],s∈[0, T].
Since he op imal con ol b
udepends on q, he ixed poin a gumen ollows om condi ion (ii) in
De ini ion 2.3. We show ha p o ing ha he e exis s a unique equilib ium educes o inding he
unique solu ion o an in eg o-di e en ial equa ion.
Rema k 2.4. The p o i unc ion appea ing in (2.7)– i.e., he unc ion (x, p)7→ xp−β– is ela ed
o he isoelas ic demand ob ained om Spence-Dixi -S igli z p e e ences and can be mo i a ed as
ollows.1
Assume ha each playe in ou economy is a i m indexed by i s p oduc i i y (o size) x > 0.
We can conside he i ms’ p oduc ion capaci ies as a p oxy o his index and, hus, his se ing is
cohe en wi h he model in oduced abo e. Each i m p oduces a single good and aces he demand
unc ion π(x)
P−γ
,wi h γ > 1,
whe e π(x)is he p ice se by a i m wi h p oduc i i y le el x, and P=RRπ(x)1−γµ(dx)1/(1−γ)is
a p ice index, which is compu ed acco ding o he s a is ical dis ibu ion µo he i ms p oduc i i y
in he economy. He e we a e assuming ha he p ice se by wo i ms ha ing an equal p oduc i i y
le el is he same. In o he wo ds, he p ice policy o each i m is exclusi ely de e mined by i s
p oduc i i y.
Each i m (i.e., ix x > 0) aims a maximizing he unc ion
π7→ ππ
P−γ−1
xπ
P−γ.
I is easy o see ha he maximum p o i is
1
γ−1γ
γ−1−γ
xγ−1Pγ,(2.8)
which is a ained se ing he p ice π(x) = γ
(γ−1)x.
This en ails ha P=γ
(γ−1) RRxγ−1µ(dx)1/(1−γ), and hence, subs i u ing his exp ession in o (2.8),
we ge ha he p o i is
1
γ−1xγ−1p−γ,wi h p=ZR
xγ−1µ(dx)1
γ−1.(2.9)
The e o e, se ing β=γ= 2, we ge he p o i unc ion (x, p)7→ xp−βappea ing in ou model.
No e, howe e , ha in he discoun ed ne p o i unc ional in oduced in (2.7)we do no es ic he
choice o he pa ame e β o he one dic a ed by he economic applica ion ou lined abo e. Ins ead,
1See also he discussion in Achdou e al. [2, p. 7, Foo no e 5] conce ning he he model in Lu me [18].
6 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
we conside any possible alue β > 0, as he ma hema ical esul s ha we p o e can be s a ed in
his mo e gene al amewo k.
The discussion abo e highligh s ha i is possible o conside mo e gene al o ms o in e ac ion
be ween playe s in he p o i unc ion. In his pape we conside he a e age o he p oduc ion
capaci ies o he i ms in he economy, bu in (2.9) he geome ic a e age appea s. Mo e gene ally,
one can conside
p=FZR
(x)µ(dx),
o some F, :R→[0,+∞)s ic ly inc easing, see, e.g., [6]. Howe e , as will be clea la e on,
he analysis ca ied ou in his wo k hea ily elies on he ac ha he in e ac ion be ween playe s
is h ough he a e age o he p oduc ion capaci ies. Indeed, hanks o his ea u e, we will ind an
explici cha ac e iza ion o he solu ion o he mean- ield game in oduced abo e.
3. The op imiza ion p oblem
In his sec ion we conside he op imiza ion p oblem associa ed o he mean- ield game desc ibed
in Sec ion 2. We show ha , o each possible choice (in a sui able class o measu able unc ions)
o he unc ion qappea ing in he discoun ed ne p o i unc ional (2.7), he co esponding maxi-
miza ion p oblem has a unique solu ion and we compu e explici ly he associa ed op imal con ol.
We di ide ou analysis in o wo subsec ions, he i s de o ed o he ini e ime ho izon case, he
second one o he in ini e ime ho izon case.
3.1. The ini e ime ho izon case. Le us conside he ini e ime ho izon case, i.e., ix T < +∞.
Fo each ixed q: [0, T]→(0,+∞)such ha q−β∈L1((0, T]; (0,+∞)), we conside he p oblem o
maximizing he unc ional
JT,q(ξ, u):=EZT
0
e−ρs Xξ,u
sq−β
s−1
2u2
sds,(3.1)
whe e ξsa is ies Assump ion 2.1,uis chosen in he class o admissible con ols in oduced in (2.2),
and Xξ,uis he unique solu ion o SDE (2.1). We also in oduce he alue unc ion co esponding
o he op imiza ion p oblem, namely,
VT,q(ξ):= sup
u∈UT
JT,q(ξ, u), ξ ∈L1((Ω,F0,P); (0,+∞)).(3.2)
In he nex p oposi ion we a e going o show impo an p ope ies o he unc ional JT,q.
P oposi ion 3.1. Fix a andom a iable ξsa is ying Assump ion 2.1,u∈ UT, and q: [0, T]→
(0,+∞)such ha q−β∈L1((0, T]; (0,+∞)). Then, he unc ional JT,q de ined in (3.1)is ini e and
e i ies
JT,q(ξ, u) = E[ξ]z(T,q)
0+EZT
0
e−ρs z(T,q)
sus−1
2u2
sds,(3.3)
whe e z(T,q): [0, T]→[0,+∞)is he de e minis ic unc ion gi en by
z(T,q)
s:=ZT
s
e−(ρ+δ)( −s)q−β
d , s ∈[0, T ].(3.4)
Mo eo e , i ξ, ξ′bo h e i y Assump ion 2.1 and a e such ha E[ξ] = E[ξ′], hen JT,q(ξ;u) =
JT,q(ξ′;u), o any u∈ UT.
P oo . Fix ξ∈L1((Ω,F0,P); (0,+∞)), and u∈ UT. Since he assump ions o Lemma 2.2 a e
e i ied, we can use (2.6) and apply he Fubini-Tonelli heo em (no e ha he in eg and is non-
nega i e) o ge
EZT
0
e−ρsXξ,u
sq−β
sds=E[ξ]ZT
0
e−ρse−δsq−β
sds+EZT
0Zs
0
e−ρse−δ(s− )q−β
su d ds.
Clea ly, he i s summand o he las equali y is ini e, hanks o he assump ions on ξand q.
Mo eo e ,
E[ξ]ZT
0
e−ρse−δsq−β
sds=E[ξ]z(T,q)
0.
A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 7
Also he second summand is ini e, hanks o he assump ions on qand u. Exchanging he o de
o he wo ime in eg als we ge
EZT
0Zs
0
e−ρse−δ(s− )q−β
su d ds=EZT
0
eδ u ZT
e−(ρ+δ)sq−β
sdsd 
=EZT
0
e−ρ u ZT
e−(ρ+δ)(s− )q−β
sdsd =EZT
0
e−ρ z(T,q)
u d .
I easily ollows ha JT,q is ini e and ha Equa ion (3.3) holds, which also en ails he las s a emen
o he P oposi ion. □
Rema k 3.2. The las s a emen in P oposi ion 3.1 implies ha he alue unc ion VT,q depends on
he ini ial condi ion ξonly h ough i s a e age. Mo e p ecisely, i ξ, ξ′bo h e i y Assump ion 2.1
and a e such ha E[ξ] = E[ξ′], hen VT,q(ξ) = VT,q(ξ′).
Thanks o he P oposi ion abo e we can ind he op imal con ol o ou op imiza ion p oblem
and his, as a byp oduc , allows us o explici ly compu e he alue unc ion.
Theo em 3.3. Fix a andom a iable ξsa is ying Assump ion 2.1 and q: [0, T]→(0,+∞)such
ha q−β∈L1((0, T]; (0,+∞)). Then, b
u(T,q):=z(T,q)∈ UT, whe e z(T,q)is he unc ion de ined
in (3.4), is an op imal con ol o p oblem (3.2), which is de e minis ic and independen o ξ.
Mo eo e , b
u(T,q)is essen ially unique, i.e., i u(T,q)∈ UTis an op imal con ol o p oblem (3.2)
di e en om b
u(T,q), hen
u(T,q)
s=bu(T,q)
s, o P⊗Leb-a.e. (ω, s)∈Ω×[0, T].
Finally, he alue o he op imiza ion p oblem admi s he explici exp ession
VT,q(ξ) = E[ξ]z(T,q)
0+1
2ZT
0
e−ρs(z(T,q)
s)2ds, (3.5)
and he op imally con olled s a e p ocess Xξ,b
u(T,q)is gi en by
Xξ,b
u(T,q)
s=Ysξ+Zs
0
Ys
Y
z(T,q)
d , s ∈[0, T ],(3.6)
whe e Yis he p ocess de ined in (2.5).
P oo . Fix ξ∈L1((Ω,F0,P); (0,+∞)). F om (3.3) i immedia ely ollows ha
VT,q(ξ) = E[ξ]z(T,q)
0+ sup
u∈UT
EZT
0
e−ρs z(T,q)
sus−1
2u2
sds.(3.7)
Hence, i we can ind b
u(T,q)∈ UT ha maximizes he in eg and z(T,q)
sus−1
2u2
s, o P⊗Leb-a.e.
(ω, s)∈Ω×[0, T], hen b
u(T,q)i mus be op imal. Clea ly, he in eg and is maximized in he sense
abo e i we ake b
u(T,q)=z(T,q). I s admissibili y is a consequence o he ac ha z(T,q)is bounded.
The e o e, b
u(T,q)is op imal, and clea ly essen ially unique in he sense speci ied abo e. Subs i u ing
i s de ini ion in (3.7) we immedia ely ge (3.5) and (3.6). □
3.2. The in ini e ime ho izon case. We analyze now he in ini e ime ho izon case, i.e., we se
T= +∞. Fo each ixed q: [0,+∞)→(0,+∞)such ha q−β∈L1((0,+∞); (0,+∞)), we conside
he p oblem o maximizing he unc ional
J∞,q(ξ, u):=EZ+∞
0
e−ρs Xξ,u
sq−β
s−1
2u2
sds,(3.8)
whe e ξsa is ies Assump ion 2.1,uis chosen in he class o admissible con ols de ined in (2.3),
and Xξ,uis he unique solu ion o SDE (2.1). We also in oduce he alue unc ion co esponding
o he op imiza ion p oblem, namely,
V∞,q(ξ):= sup
u∈U∞
J∞,q(ξ, u), ξ ∈L1((Ω,F0,P); (0,+∞)).(3.9)
The nex esul is analogous o P oposi ion 3.1. The p oo p oceeds along he same lines and,
hus, we omi i .
14 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
We obse e ha , gi en he bounds in (4.22), he amily Zis equibounded. We wan o show
ha i is also equi-in eg able in L1
ρ+2δ. Fo any s≥0and h > 0we ha e ha
z(n)
s+h−z(n)
s=1[0,n−h](s)Zs+h
sz(n)
′d +1[n−h,n](s)Zn
sz(n)
′d .
Le K:=x−β
ρ+δ−δβ . Using he ac ha 0<z(n)
s′≤Keδ(1+β)s, o all s≥0, and all n≥1, we ge
ha , o any h > 0small enough,
∥z(n)
·+h−z(n)
·∥1,ρ+2δ=Z+∞
0
e−(ρ+2δ)s|z(n)
s+h−z(n)
s|ds
≤Zn−h
0
e−(ρ+2δ)sZs+h
sz(n)
′d ds+Zn
n−h
e−(ρ+2δ)sZn
sz(n)
′d
≤K(eδ(1+β)h−1)
δ(1 + β)Zn−h
0
e−(ρ+δ−δβ)sds
+Keδ(1+β)n
δ(1 + β)Zn
n−h
e−(ρ+2δ)sds−K
δ(1 + β)Zn
n−h
e−(ρ+δ−δβ)sds
| {z }
≤0
≤K(eδ(1+β)h−1)
δ(1 + β)(ρ+δ−δβ)[1 −e−(ρ+δ−δβ)(n−h)
| {z }
≤0
] + Ke−(ρ+δ−δβ)n
δ(1 + β)(ρ+ 2δ)[e(ρ+2δ)h−1]
≤K(eδ(1+β)h−1)
δ(1 + β)(ρ+δ−δβ)+K
δ(1 + β)(ρ+ 2δ)[e(ρ+2δ)h−1],∀n≥1.
The e o e, we ob ain ha
lim
h→0+sup
n≥1
∥z(n)
·+h−z(n)
·∥1,ρ+2δ= 0,
i.e., ha he amily Zis equi-in eg able in L1
ρ+2δ.
By he F eche -Kolmogo o heo em, i ollows ha Zis ela i ely compac , so we can ex ac
a subsequence {z(n)}n≥1⊂ Z, s ill labeled (wi h an abuse o no a ion) by {z(n)}n≥1, such ha
z(n)−→ z(∞)wi h espec o ∥·∥1,ρ+2δ, as n→ ∞. Then, we can ex ac a sub-subsequence, s ill
labeled (wi h an abuse o no a ion) by {z(n)}n≥1, such ha z(n)
s−→ z(∞)
s, o almos e e y s≥0,
as n→ ∞. Clea ly, he limi z(∞)belongs o L1
ρ+2δand, since z(n)∈ Cx,∞, o any n≥1, we
ha e ha αx,∞
s≤z(∞)
s≤αx,∞
s, o almos e e y s≥0, whe e αx,∞and αx,∞a e he unc ions
de ined in (4.20) and (4.21). Now we use (4.7) o ge ha also he de i a i e {z(n)′}o he las
subsequence {z(n)}con e ges a.e. o he unc ion
w(∞)
s:= e(ρ+2δ)sZ+∞
s
e−(ρ+δ−δβ)uz(∞)
u−βdu, s ≥0,
which belongs o L1
ρ+2δ. By s anda d compu a ions, i ollows ha {z(n)′} ⊂ L1
ρ+2δand ha
0<z(n)
s′≤x−β
ρ+δ−δβ eδ(1+β)s=:gs,∀s≥0,∀n≥1,
wi h g∈L1
ρ+2δ. The e o e, he sequence o de i a i es {z(n)′}con e ges in L1
ρ+2δ o w(∞). We de-
duce ha he subsequence {z(n)}con e ges in he weigh ed Sobole space W1,1
ρ+2δ(c . [4, Chap e 8,
Rema k 4]). By comple eness o his space, we ge ha w(∞)
s= (z(∞)
s)′, o a.e. s≥0. Mo eo e ,
using he same a gumen as in he p oo o [4, Theo em 8.2], we ge ha he e exis s a unc ion
z(∞)∈C([0,+∞); (0,+∞)) such ha z(∞)
s=z(∞)
s, o almos e e y s≥0, and
z(∞)
s=x+Zs
0
(z(∞)
u)′du=x+Zs
0
w(∞)
udu, ∀s≥0.

A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 15
Subs i u ing he exp ession o w(∞)in he p e ious equali y we ge ha
z(∞)
s=x+Zs
0
e(ρ+2δ) Z+∞
e−(ρ+δ−δβ)uz(∞)
u−βdud s ≥0,
ha is, z(∞)∈ Cx,∞and sol es (4.7).
Uniqueness. As we obse ed a he beginning o he p oo , any solu ion z o (4.7) is (a leas )
wice di e en iable. Di e en ia ing (4.7) wi h espec o he ime a iable and deno ing by ′and ′′
i s - and second-o de de i a i es wi h espec o his a iable, we ha e ha z e i ies
(z′′
s= (ρ+ 2δ)z′
s−eδ(1+β)sz−β
s, s ≥0,
z0=x . (4.25)
The equa ion abo e is a second-o de ODE wi h locally Lipschi z coe icien s o e (s, z)∈[0,+∞)×
(0,+∞), o which we can associa e he ollowing ini ial alue p oblem





z′′
s= (ρ+ 2δ)z′
s−eδ(1+β)sz−β
s, s ≥0,
z0=x ,
z′
0=ζ > 0.
(4.26)
No e ha we only need o conside ζ > 0, since solu ions o (4.7) a e such ha hei i s de i a i e
is s ic ly posi i e o all s≥0. Fo each ixed x > 0and ζ > 0, (4.26) has a unique solu ion zx,ζ
on he maximal in e al [0, τ∗(x, ζ)), wi h τ∗(x, ζ)≤+∞.
Since any solu ion z o (4.7) also e i ies (4.25), i also sa is ies (4.26) o some ζ > 0. The e o e,
i we show ha o each gi en x > 0, he e exis s a unique ζ > 0such ha (4.26) has a unique global
solu ion zx,ζ ∈ Cx,∞, hen necessa ily ζ=ζand zx,ζ mus also be he unique solu ion o (4.7). The
idea is, hus, o s udy, o each ixed x > 0, he dependence o he solu ion zx,ζ o (4.26) on ζ > 0.
Le us ix x > 0. S anda d esul s (see, e.g., [11, Theo em 7,5, Chap e 1]) ensu e ha he
solu ion zx,ζ o (4.26) and i s i s - and second-o de de i a i es wi h espec o he ime a iable
a e di e en iable wi h espec o ζand ha he o de o di e en ia ion can be exchanged.
The e o e, de ining bzs(ζ):=∂
∂ζ zx,ζ
s, o all s≥0, we ge om (4.26) ha bz(ζ)sol es he ini ial
alue p oblem 



bz′′
s(ζ)=(ρ+ 2δ)bz′
s(ζ) + βeδ(1+β)szs(ζ)−β−1bzs(ζ), s ≥0,
bz0(ζ) = 0 ,
bz′
0(ζ) = 1 ,
(4.27)
whe e zs(ζ):=zx,ζ
s. Le
s∗(ζ):= in {s≥0: bz′
s(ζ)≤0}>0.
By de ini ion z′
s(ζ)>0, o all s∈[0, s∗(ζ)), which implies
bzs(ζ)>0,∀s∈[0, s∗(ζ)).
The e o e, using (4.27) we ge ha bz′′
s(ζ)>0, o all s∈[0, s∗(ζ)). I ollows ha
bz′
s(ζ)≥1,∀s∈[0, s∗(ζ)),
which implies ha s∗(ζ) = +∞and, in u n, ha bzs(ζ)≥s, o all s≥0.
Using again (4.27), we hen ge
(bz′′
s(ζ)≥(ρ+ 2δ)bz′
s(ζ), s ≥0,
bz′
0(ζ) = 1 ,(4.28)
which, implies, by compa ison,
bz′
s(ζ)≥e(ρ+2δ)s,∀s≥0,
and, consequen ly,
bzs(ζ)≥1
ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0.
Thus, o e e y a<b, we ha e
16 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
zs(b)−zs(a)≥Zb
a
ˆzs(ζ) dζ≥(b−a)1
ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0.
We know ha , o some ζ > 0, he e exis s a global solu ion z(ζ)∈ Cx,∞ o (4.26), since any
solu ion z o (4.7) also e i ies his equa ion. Using he es ima e abo e and (4.22), we ge , o all
b > a :=ζ,
zs(b)≥zs(ζ)+(b−ζ)[e(ρ+2δ)s−1] ≥x+ (b−ζ)1
ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0.
This implies ha z(b)would g ow exponen ially a a a e a leas ρ+ 2δ > δ(1 + β), which
con adic s he ac ha solu ions in (4.22) need o g ow a a a e a mos δ(1 + β).
Simila ly, we ge , o all a < ζ =:b,
zs(a)≤zs(ζ)−(ζ−a)[e(ρ+2δ)s−1] ≤αx,∞
s−(ζ−a)1
ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0.
This implies ha z(a)becomes nega i e in ini e ime, which con adic s he ac ha solu ions
in (4.22) a e posi i e.
By a bi a iness o b > ζ and a < ζ, we conclude.
Con e gence. As ecalled a he beginning o he p oo , uniqueness o he solu ion o (4.7)
implies uniqueness o he solu ion o (4.6), which is also wice con inuously di e en iable. Di e -
en ia ing (4.6), we ge he second-o de ODE
y′′
s=ρy′
s+ (ρ+δ)δys−y−β
s, s ≥0.(4.29)
Hence, any solu ion o (4.6) is also a solu ion o (4.29). We obse e ha y∞is he unique solu ion
o he algeb aic equa ion
(ρ+δ)δy −y−β= 0.(4.30)
Hence, he unc ion ys=y∞, o all s≥0, is he unique s a iona y solu ion o (4.6).
Now, we show mono onici y and con e gence o he solu ion o (4.6) when x=y∞. We p o e
he case x>y∞, as he o he one can be es ablished by simila a gumen s.
Le ybe he unique solu ion o (4.6) and ix x>y∞. We p o e, i s , ha ys> y∞, o all s≥0.
Le us de ine
s0:= in {s≥0: ys=y∞},
and assume, by con adic ion, ha s0<+∞. We ha e wo cases:
(i) y′
s0<0;
(ii) y′
s0= 0.
Conside he i s case. By he low p ope y, he unc ion eys:=ys−s0,s≥s0, sol es (4.6) wi h
ini ial condi ion equal o y∞and i sa is ies he a p io i bounds gi en in (4.23) (wi h x=y∞).
Howe e , his solu ion is di e en om he cons an one, which is he unique solu ion o (4.6) wi h
ini ial condi ion y∞and sa is ying he a o emen ioned bounds. Hence we ha e a con adic ion.
The second case leads o a con adic ion as well. Indeed, we would ha e wo di e en solu ions
in he in e al [0, s0] o he Cauchy p oblem (4.29), wi h e minal condi ions ys0=y∞and y′
s0= 0,
which clea ly admi s unique solu ion, ha is, ys=y∞, o all s≥0.
Hence, we ha e p o ed ha ys> y∞, o all s≥0. This implies, by s ic mono onici y o he
map y7→ (ρ+δ)δy −y−βand (4.30), ha
(ρ+δ)δys−y−β
s>(ρ+δ)δy∞−y−β
∞= 0,∀s≥0,(4.31)
and hence, om (4.29), ha
y′′
s> ρy′
s,∀s≥0.(4.32)
Now we show ha y′
s<0, o all s≥0. Le
s1:= in {s≥0: y′
s≥0}.
A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 17
Assume, by con adic ion, ha s1<+∞. Then, om (4.32) we ge ha y′′
s1>0. I ollows ha
he e exis η, ε > 0such ha y′
s1+ε=η > 0. Using (4.32) again, we ge
(y′′
s> ρy′
s,∀s≥s1+ε,
y′
s1+ε=η > 0.(4.33)
The e o e, sol ing he co esponding Cauchy p oblem and using he compa ison, we ge
y′
s≥ηeρ[s−(s1+ε)] >0, s ≥s1+ε. (4.34)
Nex , di e en ia ing (4.29), we ge ha y∈C3([0,+∞); (0,+∞)) and ha i sa is ies
y′′′
s=ρy′′
s+ (ρ+δ)δy′
s+βy−β−1
sy′
s, s ≥0.(4.35)
Since, by (4.34), we know ha y′
s>0 o all s≥s1+ε, we deduce ha





y′′′
s> ρy′′
s+ (ρ+δ)δy′
s,∀s≥s1+ε,
y′
s1+ε=η,
y′′
s1+ε=κ,
wi h η > 0and κ > ηρ > 0, by (4.33). Sol ing he co esponding Cauchy p oblem and using he
compa ison, we ge
y′
s>ηδ +κ
ρ+ 2δe(ρ+δ)[s−(s1+ε)] +η(ρ+δ)−κ
ρ+ 2δe−δ[s−(s1+ε)],∀s≥s1+ε .
This implies ha yg ows, in he long un, a leas wi h a e ρ+δwhich is s ic ly bigge han
δβ, as p esc ibed by he admissibili y condi ion in (4.23). The con adic ion ollows, and hence we
p o ed ha y′
s<0, o all s≥0.
We a e now going o show ha y′′
s>0, o all s≥0. Since we p o ed ha y′
s<0, o all s≥0,
we deduce om (4.35) ha
y′′′
s< ρy′′
s,∀s≥0.(4.36)
Assume by con adic ion ha y′′
s2≤0 o some s2≥0. Then, om (4.36) we ge ha y′′′
s2<0. I
ollows ha he e exis ϑ < 0,τ > 0such ha y′′
s2+τ=ϑ < 0. Using again (4.36), we ge
(y′′′
s< ρy′′
s,∀s≥s2+τ,
y′′
s2+τ=ϑ.
The e o e, y′′
s≤ϑeρs <0, o all s≥s2+τ. Conside ing ha y′
s<0, o all s≥0, his implies
ha he g aph o s7→ yslies below a s aigh line wi h s ic ly nega i e slope, con adic ing he
ac ha ys> y∞, o all s≥0, as p e iously es ablished.
So, we ha e p o ed ha , o all s≥0,
ys> y∞, y′
s<0, y′′
s>0.
This implies ha he e exis s ¯y:= lim
s→∞ ys≥y∞and ha lim
s→∞ y′
s= 0.Howe e , i canno be ha
¯y > y∞, as i would imply, using (4.29), ha lim
s→∞ y′′
s>0, con adic ing he ac ha y′
s→0, as
s→+∞. The e o e, ¯y=y∞.□
We a e now eady o es ablish exis ence and uniqueness o an equilib ium o he mean- ield
game.
P oposi ion 4.13. Fo each ixed andom a iable ξsa is ying Assump ion 2.1, he e exis s a
unique equilib ium (b
u,bq) o he mean- ield game wi h in ini e ime ho izon, among all equilib ia
e i ying condi ion (4.3).
Mo e p ecisely, b
u=z(∞,bq), whe e z(∞,q)is he unc ion de ined in (3.10), and bqis he unique
solu ion o (4.6), wi h ini ial condi ion x=E[ξ].
18 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
P oo . Le bqbe he unique solu ion o (4.6), wi h x=E[ξ]. We need o check ha condi ion (4.3)
is sa is ied. We ha e he ollowing h ee cases.
Case x=y∞.F om Theo em 4.12, we know ha bqs=y∞, o all s≥0. The e o e,
z(∞,bq)
s=K:=y−β
∞
ρ+δ,∀s≥0,(4.37)
and hence bq e i ies condi ion (4.3).
Case x>y∞.F om Theo em 4.12, we know ha bqis mono one dec easing and con e ges o y∞.
This implies ha , 0< y∞<bqs≤x, o all s≥0, and ha z(∞,bq)is posi i e and bounded abo e
by he cons an Kde ined in (4.37). The e o e, bqsa is ies condi ion (4.3).
Case x<y∞.F om Theo em 4.12, we know ha bqis mono one inc easing and con e ges o y∞.
This implies ha , 0< x ≤bqs< y∞, o all s≥0, and ha z(∞,bq)is posi i e and bounded abo e
by x−β
ρ+δ. The e o e, bqsa is ies condi ion (4.3).
Applying Theo em 4.1-(i ) we ge he esul . □
Rema k 4.14. Also in his case he esul s o his sec ion can be easily ex ended o he case whe e
we conside an ini ial ime ime > 0.
Clea ly, we need o adap (in an ob ious way) he de ini ion o equilib ium gi en in De ini ion 2.3
o include he ini ial ime . The s a emen o Theo em 4.1 is also adap ed acco dingly.
The in eg o-di e en ial equa ion (4.6)becomes



d
dsys=−δys+ZT
s
e−(ρ+δ)(u−s)y−β
udu, s ∈[ , T],
y =x ,
(4.38)
while equa ion (4.7)becomes





d
dszs= e(ρ+2δ)sZT
s
e−(ρ+δ−δβ)uz−β
udu, s ∈[ , T],
z = eδ x .
(4.39)
The se s in which we look o solu ions o (4.39)(now dependen on ) a e
C ,x :={ ∈C([ , T]; (0,+∞)): eδ αx
s≤ s≤eδ αx
s,∀s∈[ , T]},(4.40)
in he ini e ime ho izon case, and
C ,x,∞:={ ∈C([ , +∞); (0,+∞)): eδ αx,∞
s≤ s≤eδ αx,∞
s,∀s≥ },(4.41)
in he in ini e ime ho izon case.
The s a emen s o Theo ems 4.9 and 4.12, and o P oposi ions 4.10 and 4.13, emain he same
(excep o mino modi ica ions). The bounds gi en in (4.14)become
e−δ(s− )x≤ys≤e−δ(s− )αx
s, s ∈[ , T],(4.42)
while hose gi en in (4.23)become
e−δ(s− )x≤ys≤e−δ(s− )x−x−β
δ(1 + β)(ρ+δ−δβ)+eδβ(s− )x−β
δ(1 + β)(ρ+δ−δβ), s ≥ . (4.43)
5. The de e minis ic model
In his sec ion we discuss he de e minis ic e sion o he mean- ield game p oblem in oduced
in Sec ion 2. The s uc u e o he p oblem will allow us o exploi he esul s gi en in Sec ions 3
and 4, o ge he unique equilib ium o he mean- ield game in an explici o m.
Le us conside he o dina y di e en ial equa ion (ODE)



d
dsXs=−δXs+us, s ∈(0, T],
X0=x ,
(5.1)
A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 19
whe e δ > 0is a gi en coe icien and he con ol u:= (us)s∈[0,T ]is chosen in ei he o he ollowing
wo classes o admissible con ols: i T < +∞,
UT:=u: [0, T]→[0,+∞)measu able and s. . ZT
0
u2
sds < +∞;(5.2)
i , ins ead, T= +∞,
U∞:=u: [0,∞)→[0,+∞)measu able and s. . Zs
0
u d < +∞,∀s≥0,
and Z+∞
0
e−ρsu2
sds < +∞.(5.3)
No e ha , o any x∈Rand any u∈ UT, Equa ion (5.1) has a unique solu ion Xx,u, gi en by
Xx,u
s= e−δsx+Zs
0
e−δ(s− )u d , s ∈[0, T ].(5.4)
As in he s ochas ic case, Xdesc ibes he e olu ion o he p oduc ion capaci y o a ep esen a i e
agen , which dep ecia es a a a e δand can be inc eased by choosing he in es men a e u∈ UT.
Nex , we conside he discoun ed ne p o i unc ional
JT,q(x, u):=ZT
0
e−ρs Xx,u
sq−β
s−1
2u2
sds, (5.5)
whe e ρ > 0is a discoun ac o , β > 0is a ixed pa ame e , and q= (qs)s∈[0,T]is a gi en
de e minis ic measu able unc ion.
We in oduce he ollowing assump ion.
Assump ion 5.1. The ini ial dis ibu ion o he agen s in he economy ν0has a densi y m0wi h
espec o he Lebesgue measu e on R, is suppo ed on (0,+∞), and has ini e i s momen . Mo e
p ecisely, he e exis s m0∈L1((0,+∞); (0,+∞)) such ha
ν0(A) = ZA
m0(x) dx, ∀A∈L(R),Z+∞
0
m0(x) dx= 1,Z+∞
0
x m0(x) dx < +∞.
Since he ep esen a i e agen is chosen andomly by picking he /his ini ial p oduc ion capaci y
le el xacco ding o he ini ial dis ibu ion ν0, he assump ion abo e ensu es ha , o any u∈ UT,
Xx,u
s≥0, o all s∈[0, T]. Thus, he p oduc ion capaci y le el o he ep esen a i e agen is ne e
nega i e.
To in oduce he de ini ion o equilib ium o he de e minis ic mean- ield game, we need o
conside he so-called con inui y equa ion (also known as Liou ille equa ion, c . [5]) associa ed o
ODE (5.1)



∂
∂sp(s, x) + ∂
∂x (p(s, x)[−δx +us]) = 0,(s, x)∈(0, T ]×(0,+∞),
p(0, x) = p0(x), x ∈(0,+∞),
(5.6)
whe e u= (us)s∈[0,T]∈ UTand p0is a gi en ini ial p obabili y densi y unc ion. This equa ion
desc ibes he e olu ion o p0unde he low de e mined by ODE (5.1).
Solu ions o (5.6) a e unde s ood in he weak sense, acco ding o he ollowing de ini ion.
De ini ion 5.2. A unc ion p∈L1([0, T]×(0,+∞); (0,+∞)) is a weak solu ion o (5.6)i , o
any unc ion φ∈C∞([0, T]×(0,+∞)) wi h compac suppo , and any > 0, we ha e
Z+∞
0
φ( , x)p( , x) dx(5.7)
=Z+∞
0
φ(0, x)p0(x) dx+Z
0Z+∞
0∂
∂sφ(s, x)+[−δx +us]∂
∂xφ(s, x)p(s, x) dxds .

20 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
Lemma 5.3. Fo any u∈ UTand any ini ial p obabili y densi y p0, suppo ed on (0,+∞)and
wi h ini e i s momen , he con inui y equa ion (5.6)has a unique solu ion, gi en by
pp0,u(s, x)=eδsp0eδsx−Zs
0
eδ u d ,(s, x)∈[0, T]×(0,+∞).(5.8)
Mo eo e , o each s∈[0, T], he p obabili y densi y pp0,u(s, ·)has ini e i s momen , gi en by
Z+∞
0
x pp0,u(s, x) dx= e−δs Z+∞
0
x p0(x) dx+Zs
0
e−δ(s− )u d s ∈[0, T].(5.9)
P oo . Fix u∈ UTand p0as abo e. The ac ha pp0,uis a solu ion o (5.6) can be e i ied ia
s anda d compu a ions (see, e.g., [5, Example 2.5.2] o [7, Lemma 4.15]). Uniqueness ollows om
an a gumen simila o he one used in he p oo o [7, Lemma 4.16].
Finally, using he change o a iables y= eδsx−Rs
0eδ u d , we ge ha , o each s∈[0, T],
Z+∞
0
x pp0,u(s, x) dx=Z+∞
0
xeδsp0eδsx−Zs
0
eδ u d dx
=Z+∞
0
e−δs y+Zs
0
eδ u d p0(y) dy= e−δs Z+∞
0
y p0(x) dy+Zs
0
e−δ(s− )u d ,
which is ini e hanks o he assump ions on p0and u. Thus, pp0,u(s, ·)has ini e i s momen , o
any s∈[0, T ], which sa is ies (5.9). □
We a e now eady o s a e he de ini ion o equilib ium o ou de e minis ic mean- ield game.
De ini ion 5.4. Fix an ini ial dis ibu ion ν0unde Assump ion 5.1 and le m0be i s densi y wi h
espec o he Lebesgue measu e.
A pai (b
u,bq), whe e b
u∈ UTand bq: [0, T]→(0,+∞)is a measu able unc ion, is an equilib ium
o he de e minis ic mean- ield game i
(i) JT,bq(x, b
u)≥JT,bq(x, u), o all u∈ UTand ν0-a.e. x > 0;
(ii) bqs=Z+∞
0
x mm0,b
u(s, x) dx, o all s∈[0, T ], whe e mm0,b
uis he unique solu ion o he
con inui y equa ion (5.6), gi en by (5.8), wi h ini ial condi ion m0and con ol b
u.
Also in his case, we can adop a ixed poin a gumen o show ha he e exis s a unique equi-
lib ium, by sol ing, i s , he p oblem o maximizing (5.5) o a gi en measu able unc ion q, hen
de e mining he op imally con olled dynamics o he p oduc ion capaci y le el o he ep esen a i e
agen and, inally, se ing he ixed poin a gumen om condi ion (ii) in De ini ion 5.4.
5.1. The op imiza ion p oblem. Le us conside he op imiza ion p oblem
VT,q(x):= sup
u∈UT
JT,q(x, u), x > 0,(5.10)
whe e JT,q is he discoun ed ne p o i unc ional de ined in (5.5), o any gi en and ixed qmea-
su able and de e minis ic unc ion.
In he ini e ime ho izon case, i.e. T < +∞, he ollowing esul holds, which is analogous o
Theo em 3.3.
Theo em 5.5. Fix x > 0and q: [0, T]→(0,+∞)such ha q−β∈L1((0, T]; (0,+∞)). Then,
b
u(T,q):=z(T,q)∈ UT, whe e z(T,q)is he unc ion de ined in (3.4), is an op imal con ol o p ob-
lem (5.10).
Mo eo e , b
u(T,q)is essen ially unique, i.e., i u(T,q)∈ UTis an op imal con ol o p oblem (5.10)
di e en om b
u(T,q), hen
u(T,q)
s=bu(T,q)
s, o Leb-a.e. s∈[0, T].
Finally, he alue unc ion o he op imiza ion p oblem admi s he explici exp ession
VT,q(x) = xz(T,q)
0+1
2ZT
0
e−ρs(z(T,q)
s)2ds, (5.11)
A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 21
and he op imally con olled s a e Xx,b
u(T,q)is gi en by
Xx,b
u(T,q)
s= e−δsx+Zs
0
e−δ(s− )z(T,q)
d , s ∈[0, T].(5.12)
P oo . Replica ing he a gumen o he p oo o P oposi ion 3.1, we ge ha
JT,q(x, u) = xz(T,q)
0+ZT
0
e−ρs z(T,q)
sus−1
2u2
sds .
Then, he esul ollows using he same easoning as in he p oo o Theo em 3.3.□
In he in ini e ime ho izon case, i.e. T= +∞, we ha e he ollowing s a emen , which is he
de e minis ic coun e pa o Theo em 3.6. We s a e i wi hou p oo .
Theo em 5.6. Fix x > 0and conside a unc ion q: [0,+∞)→(0,+∞)such ha q−β∈
L1
ρ+δ((0,+∞); (0,+∞)) and such ha he unc ion z(∞,q), de ined in (3.10)is bounded on [0,+∞).
Then, b
u(∞,q):=z(∞,q)∈ U∞is an op imal con ol o p oblem (5.10). Mo eo e , b
u(∞,q)is
essen ially unique, i.e., i u(∞,q)∈ U∞is an op imal con ol o p oblem (5.10)di e en om
b
u(∞,q), hen
u(∞,q)
s=bu(∞,q)
s, o Leb-a.e. s≥0.
Finally, he alue unc ion o he op imiza ion p oblem admi s he explici exp ession
V∞,q(x) = xz(∞,q)
0+1
2Z+∞
0
e−ρs(z(∞,q)
s)2ds, (5.13)
and he op imally con olled s a e Xx,b
u(∞,q)is gi en by
Xx,b
u(∞,q)
s= e−δsx+Zs
0
e−δ(s− )z(∞,q)
d , s ≥0.(5.14)
5.2. Exis ence and uniqueness o equilib ia. Also in he de e minis ic mean- ield game p e-
iously in oduced, he sea ch o an equilib ium boils down o inding a ixed poin o a sui able
map. Acco ding o De ini ion 5.4, his map is (qs)s∈[0,T ]7→ R+∞
0x mm0,b
u(T,q)(s, x) dxs∈[0,T ], whe e
b
u(T,q)is he op imal con ol o he op imiza ion p oblem (5.10) – whose exp ession is gi en ei he
in Theo em 5.5, in he case T < +∞, o in Theo em 5.6, in he case T= +∞– and mm0,b
u(T,q)is
he unique solu ion o (5.6), wi h ini ial p obabili y densi y m0and con ol b
u(T,q).
The nex esul , which is analogous o Theo em 4.1, shows ha also in he de e minis ic case
he ixed poin map is he solu ion map o he same in eg al equa ion s udied in Sec ion 4. We omi
i s p oo , which is a s aigh o wa d adap a ion o he p oo o Theo em 4.1.
Theo em 5.7. Le us ix an ini ial dis ibu ion ν0sa is ying Assump ion 5.1, and le m0be i s
densi y wi h espec o he Lebesgue measu e.
Conside he de e minis ic mean- ield game in he ini e ime ho izon case, i.e., T < +∞. Then,
(i) I he e exis s an equilib ium (b
u,bq)in he sense o De ini ion 5.4, such ha
bq−β∈L1((0, T]; (0,+∞)),(5.15)
hen bqis a solu ion o he in eg al equa ion
ys= e−δs Z+∞
0
y m0(y) dy+Zs
0
e−δ(s− )ZT
e−(ρ+δ)(u− )y−β
udud , s ∈[0, T ].(5.16)
(ii) Vice e sa, i he e exis a unique solu ion bq o (5.16)sa is ying (5.15), hen he e exis s
a unique equilib ium (b
u,bq) = (z(T,bq),bq)o he mean- ield game among all equilib ia (e
u,eq)
such ha eq e i ies (5.15), whe e z(T,q)is he unc ion de ined in (3.4).
Conside , ins ead, he de e minis ic mean- ield game in he in ini e ime ho izon case, i.e., T= +∞.
Then,
22 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI
(iii) I he e exis s an equilib ium (b
u,bq)in he sense o De ini ion 5.4, such ha
bq−β∈L1
ρ+δ((0,+∞); (0,+∞)) and z(∞,bq)is bounded on [0,+∞),(5.17)
whe e z(∞,q)is he unc ion de ined in (3.10), hen bqis a solu ion o he in eg al equa ion
ys= e−δs Z+∞
0
y m0(y) dy+Zs
0
e−δ(s− )Z+∞
e−(ρ+δ)(u− )y−β
udud , s ≥0.(5.18)
(i ) Vice e sa, i he e exis a unique solu ion bq o (5.18)sa is ying (5.17), hen he e exis s
a unique equilib ium (b
u,bq)=(z(∞,bq),bq)o he mean- ield game among all equilib ia (e
u,eq)
such ha eq e i ies (5.17).
Finally, we ge he ollowing esul , which es ablishes exis ence and uniqueness o an equilib ium
also o he de e minis ic mean- ield game. We omi i s p oo , since i ollows he same lines o he
p oo s o P oposi ions 4.10 and 4.13, excep o mino adap a ions.
P oposi ion 5.8. Fo each ixed ν0sa is ying Assump ion 5.1, he e exis s a unique equilib ium
(b
u,bq) o he de e minis ic mean- ield game wi h ini e ime ho izon ( esp., in ini e ime ho izon),
among all equilib ia e i ying he in eg abili y condi ion (5.15)( esp., (5.17)).
Mo e p ecisely, b
u=z(T,bq), whe e z(T,bq)is he unc ion de ined in (3.4)( esp. (3.10)), and bqis
he unique solu ion o (4.6), wi h ini ial condi ion x=R+∞
0y m0(y) dy, whe e m0is he densi y o
ν0wi h espec o he Lebesgue measu e.
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