Variational inequalities and smooth-fit principle for singular stochastic control problems in Hilbert spaces

Fede ico, Sal a o e; Fe a i, Gio gio; Riedel, F ank; Röckne , Michael
Wo king Pape
Va ia ional inequali ies and smoo h- i p inciple o
singula s ochas ic con ol p oblems in Hilbe spaces
Cen e o Ma hema ical Economics Wo king Pape s, No. 692
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Fede ico, Sal a o e; Fe a i, Gio gio; Riedel, F ank; Röckne , Michael (2024) :
Va ia ional inequali ies and smoo h- i p inciple o singula s ochas ic con ol p oblems in Hilbe
spaces, Cen e o Ma hema ical Economics Wo king Pape s, No. 692, Biele eld Uni e si y, Cen e
o Ma hema ical Economics (IMW), Biele eld,
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692
June 2024
Va ia ional Inequali ies and Smoo h-Fi
P inciple o Singula S ochas ic Con ol
P oblems in Hilbe Spaces
Sal a o e Fede ico, Gio gio Fe a i, F ank Riedel, and Michael R¨ockne
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
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VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE FOR
SINGULAR STOCHASTIC CONTROL PROBLEMS IN HILBERT SPACES
SALVATORE FEDERICO, GIORGIO FERRARI, FRANK RIEDEL, AND MICHAEL R ¨
OCKNER
Abs ac . We conside a class o in ini e-dimensional singula s ochas ic con ol p oblems.
These can be hough o as spa ial mono one ollowe p oblems and ind applica ions in spa ial
models o p oduc ion and clima e ansi ion. Le (D, M, µ) be a ini e measu e space and
conside he Hilbe space H:= L2(D, M, µ;R). Le hen Xbe an H- alued s ochas ic p ocess
on a sui able comple e p obabili y space, whose e olu ion is de e mined h ough an SPDE
d i en by a sel -adjoin linea ope a o Aand a ec ed by a cylind ical B ownian mo ion. The
e olu ion o Xis con olled linea ly ia an H- alued con ol consis ing o he di ec ion and he
in ensi y o ac ion, a eal- alued nondec easing igh -con inuous s ochas ic p ocess, adap ed o
he unde lying il a ion. The goal is o minimize a discoun ed con ex cos - unc ional o e an
in ini e ime-ho izon. By combining p ope ies o semiconca e unc ions and echniques om
iscosi y heo y, we i s show ha he alue unc ion o he p oblem Vis a C1,Lip(H)- iscosi y
solu ion o he co esponding dynamic p og amming equa ion, which he e akes he o m o
a a ia ional inequali y wi h g adien cons ain . Then, by allowing he decision make o
choose only he in ensi y o he con ol and equi ing ha he gi en con ol di ec ion ˆnis an
eigen ec o o he linea ope a o A, we es ablish ha he di ec ional de i a i e Vˆnis o class
C1(H), hence a second-o de smoo h- i p inciple in he con olled di ec ion holds o V. This
esul is ob ained by exploi ing a connec ion o op imal s opping and combining esul s and
echniques om con ex analysis and iscosi y heo y.
Keywo ds: in ini e-dimensional singula s ochas ic con ol; iscosi y solu ion; a ia ional
inequali y; in ini e-dimensional op imal s opping; smoo h- i p inciple.
MSC2020 subjec classi ica ion: 93E20, 37L55, 35D40, 49J40, 60G40, 91B72.
1. In oduc ion
Singula con ol and op imal s opping p oblems a ise equen ly in Economics, Finance, En-
ginee ing, and ela ed ields. Due o hei inhe en complexi y, much analysis ends o ocus on
one-dimensional p oblems, whe e ou unde s anding is ela i ely comp ehensi e. Howe e , con-
empo a y socie al challenges p esen complex s uc u es o singula con ol p oblems, ha ask
o a igo ous and sound ma hema ical basis. This pape in oduces a amewo k o add ess-
ing singula con ol p oblems in he con ex o s a e p ocesses go e ned by s ochas ic pa ial
di e en ial equa ions. By combining con ex-analy ic a gumen s and he heo y o iscosi y so-
lu ions, we show ha he p oblem’s alue unc ion Vis a C1,Lip(H)- iscosi y solu ion o he
co esponding dynamic p og amming equa ion. Fu he mo e, by exploi ing a connec ion o a
sui able amily o simple op imal s opping p oblems, we a e able o u he enhance egula i y
and p o e ha a second-o de smoo h- i p inciple holds o V. Finally, we discuss po en ial
applica ions in ields such as ene gy economics o clima e modeling.
Le us desc ibe he class o in ini e-dimensional singula s ochas ic con ol p oblems and ou
con ibu ions mo e p ecisely. Le (D, M, µ) be a ini e measu e space, and conside he Hilbe
space H:= L2(D, M, µ;R). The s a e a iable is desc ibed by a s ochas ic p ocess Xwi h
alues in H. I s e olu ion is de e mined by a s ochas ic pa ial di e en ial equa ion (SPDE)
d i en by a sel -adjoin linea ope a o Aand a cylind ical B ownian mo ion. Nex o echnical
equi emen s, we assume ha he ope a o Agene a es a C0-semig oup o posi i i y-p ese ing
Da e: June 10, 2024.
1
2 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
con ac ions. As a benchma k case, one can conside he sum o he Laplacian ope a o and a
mul iplica i e ope a o o he o m −δx, o δ > 0, ep esen ing a dep ecia ion o dissipa i e
e m. The e olu ion o Xis con olled linea ly h ough an H- alued con ol, encompassing bo h
di ec ion and in ensi y o ac ion, he la e being a eal- alued nondec easing igh -con inuous
s ochas ic p ocess adap ed o he unde lying il a ion. The objec i e is o minimize a discoun ed
con ex cos - unc ional o he o m
(1.1) J(x;I) := EZ∞
0−
e−ρ G(Xx,I
)d +⟨q, dI ⟩H,(x, I)∈H× I,
whe e Xx,I is he s a e p ocess s a ing a xand con olled ia I∈ I (c . (2.12) below), Gis
a unning cos unc ion (see Assump ion 2.8 below), q∈His a (s i ly posi i e) p opo ional
cos o ac ion, and ρ > 0 is an in e empo al discoun a e. The p oblem unde s udy can hus
be hough o as he in ini e-dimensional e sion o he mono one ollowe p oblems add essed
in [42,43,44], among o he s.
Ou analysis begins by es ablishing p elimina y egula i y p ope ies o he p oblem’s alue
unc ion, deno ed as V. Speci ically, assuming ha he unning cos unc ion Gis con ex
and semiconca e (wi h espec o he no m o H), we demons a e ha hese p ope ies a e
inhe i ed by V. Thus, by adap ing esul s om [14] o ou in ini e-dimensional se ing, we ind
ha V∈C1,Lip(H).
We p oceed by de i ing he dynamic p og amming equa ion associa ed wi h (1.1) and demon-
s a e ha Vis a C1,Lip(H)- iscosi y solu ion o i . The p oo o his esul elies on a Dynkin’s
o mula o es unc ions o semima ingales and u ilizes an equi alen ep esen a ion o V, de-
i ed om a ailo ed applica ion o he Radon-Nikodym heo em o ec o - alued measu es (see
Lemma 2.11 and (2.19) below). I is no ewo hy ha he p oo o he supe solu ion p ope y o
Vemploys a no el a gumen de i ed om an inequali y s emming om Dynkin’s o mula and
he dissipa i i y p ope y o he ope a o A. This app oach is applicable in ini e-dimensional
se ings as well, subs an ially educing he echnicali ies usually associa ed wi h demons a -
ing he supe solu ion (o subsolu ion) p ope y in minimiza ion (o maximiza ion) p oblems
in ol ing singula con ols (see, e.g., [16], [34, Ch. VIII], [38], and [49], among o he s).
To u he enhance he egula i y o V, we in oduce he assump ion ha he decision-make
can only con ol he in ensi y o ac ion. The di ec ion o ac ion, deno ed by ˆn∈H, is hen aken
o be an eigen ec o o he ope a o A. Unde his equi emen and u he echnical p ope ies
o G, we a e able o show ha he di ec ional de i a i e o Vin he di ec ion o con ol, Vˆn, is such
ha Vˆn∈C1(H). This esul can be ead as a second-o de smoo h i p ope y o V, a egula i y
esul o pa icula ele ance in singula s ochas ic con ol p oblems (see he discussion in Sec ion
5.1 below). The a o emen ioned smoo h- i p ope y is ob ained by iden i ying Vˆnas he alue
unc ion o an op imal s opping p oblem (in he spi i o he ini e- ini e dimensional con ibu ion
[4]) and subsequen ly examining he egula i y o i s (sub)g adien . In pa icula , unde a
sui able nondegene acy condi ion on he B ownian noise, assuming ha he di ec ional de i a i e
Gˆnis semiconca e, and combining a gumen s om iscosi y heo y and con ex analysis, we a e
able o show ha Vˆnis F ´eche di e en iable a any x∈Hand ha he g adien DVˆn∈
C(H;D(A)) (wi h he domain D(A) being endowed wi h he g aph no m). Fo u he de ails,
please e e o P oposi ion 4.17.
In Sec ion 6, we demons a e he ele ance o ou amewo k in economic applica ions. Fo
ins ance, we examine an i e e sible in es men p oblem in ene gy capaci y and an ene gy
balance clima e model inco po a ing human impac . In he o me , an ene gy p oduce seeks
o maximize he ne o al expec ed su plus esul ing om i e e sible in es men s in ene gy
p oduc ion. In he la e , empe a u e is inc eased by human ac i i ies h ough ca bon emissions
and a social planne aims o minimize an in e empo al expec ed cos c i e ion, penalizing
empe a u e de ia ions om an ideal le el, such as p e-indus ial empe a u es.
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 3
Le us now discuss ela ed li e a u e and ou con ibu ion o i . The o igin o singula s o-
chas ic con ol da es back o he ea ly con ibu ions by Ba he and Che no [6] , and la e by
Beneˇs, Shepp, and Wi senhausen [7] and Ka a zas [42,43]. Those seminal pape s deal wi h one-
dimensional p oblems o so-called mono one ollowe ype, in which a p ocess wi h mono one
pa hs (o , mo e gene ally, o bounded- a ia ion) has o be chosen in o de o ack he e olu ion
o a B ownian mo ion so ha an expec ed cos c i e ion is minimized. Since hen, he heo y
o singula s ochas ic con ol has a ac ed inc easing a en ion, also boos ed by i s connec ion
o op imal s opping (see [4], [9], and [44], jus o ci e a ew) and i s nume ous applica ions in
Economics and Finance. Among hose, p oblems o op imal capaci y expansion [4], op imal in-
es men wi h ansac ion cos s [58], op imal ha es ing [2], and op imal di idends’ dis ibu ion
[41].
Fo s a iona y one-dimensional p oblems, o o wo-dimensional degene a e p oblems wi h
a sui able s uc u e [31,51], explici solu ions can be expec ed. Typically, hese solu ions a e
ob ained h ough he guess-and- e i y app oach. This in ol es i s de e mining a smoo h so-
lu ion o he p oblem’s dynamic p og amming equa ion (in his case, a a ia ional inequali y
wi h g adien cons ain s), and hen e i ying i s op imali y using a e sion o I ˆo’s o mula.
Addi ionally, an op imal con ol is de e mined as a byp oduc o his analysis. This is gi en in
e ms o he solu ion o a Sko okhod e lec ion p oblem a he so-called ee bounda y, i.e. he
opological bounda y o he egion in which he g adien cons ain is no ac i e ( he so-called
no-ac ion o con inua ion egion).
Fo ime-dependen p oblems o o s a iona y p oblems in dimension la ge han one, he
guess-and- e i y app oach is no easible. This is because he dynamic p og amming equa ion
now becomes a pa ial di e en ial equa ion (PDE) wi h g adien cons ain s, o which explici
solu ions a e ypically no a ailable. As a consequence, di ec p obabilis ic and analy ical ap-
p oaches a e pu in place in o de o ob ain egula i y o he alue unc ion ( ypically unde
con exi y equi emen s; see, e.g., [39,40,50,57]) and, when possible, o cha ac e ize he op imal
con ol as he minimal amoun o e o needed o keep he unde lying s a e p ocess wi hin he
no-ac ion egion (see [23], [46], and e e ences he ein). As a ma e o ac , di e en ly o be o e,
in mul i-dimensional se ings, he ee bounda y is no explici and cons uc ing a solu ion o
he ela ed Sko okhod e lec ion p oblem is a om i ial. We e e o he in oduc ion o
[23] o a discussion on his aspec . The a o emen ioned challenges explain why he numbe o
con ibu ions on singula s ochas ic con ol p oblems in mul i-dimensional se ings is s ill e y
limi ed.
The heo y o egula s ochas ic con ol and o op imal s opping in in ini e-dimensional (no-
ably, Hilbe ) spaces ecei ed a la ge a en ion in he las decades (see, e.g., he monog aphy
[27] o con ol p oblems, and [5], [18], [19], [29], [35], [36], [59] o op imal s opping). As p e-
iously discussed, we con ibu e o ha bunch o li e a u e by p o iding he iscosi y p ope y
and C1- egula i y (smoo h- i ) o he alue unc ion o a class o op imal s opping p oblems in
Hilbe spaces. To he bes o ou knowledge, such a egula i y esul appea s he e o he i s
ime, and i is he e o e o independen in e es .
On he o he hand, he li e a u e on singula s ochas ic con ol in in ini e-dimensional spaces
is e y limi ed. The only h ee pape s b ough o ou a en ion a e [1] and [54], mo i a ed by
op imal ha es ing, and ou s [28]. In [54] he p oblem is posed o a qui e gene al con olled
SPDE, which also enjoys a space-mean dependence in [1]. The au ho s es ablish a necessa y
Maximum P inciple, which is also su icien assuming he conca i y o he Hamil onian unc ion
pe aining o he con ol p oblem unde conside a ion. Howe e , despi e hei signi ican con-
ibu ions, he e appea s o be a ounda ional conce n when dealing wi h (singula ly con olled)
SPDEs, pa icula ly ega ding he exis ence o a solu ion and he applica ion o I ˆo’s o mula
( e e o [47] o heo y and esul s on SPDEs). Speci ically, i is impo an o no ice ha in

4 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
in ini e-dimensional singula (s ochas ic) con ol p oblems, he p ecise in e p e a ion o he in e-
g al wi h espec o he ec o measu e ep esen ed by he con ol p ocess - and hus he exac
in e p e a ion o he con olled s a e equa ion - poses a nuanced issue ha wa an s ca e ul
conside a ion. Finally, ou p e ious wo k [28] de i es necessa y and su icien condi ions o a
class o singula s ochas ic con ol p oblems on an abs ac pa ially o de ed in ini e-dimensional
space. The main di e ences wi h espec o he p esen wo k a e in he amewo k, he me hod-
ology, and he na u e o he esul s. In [28], he con olled s a e p ocess is ully degene a e
and andomness comes in o he p oblem only in a pa ame ic o m, hus making he unde lying
op imiza ion p oblem no necessa ily Ma ko ian. Fu he mo e, he main esul is ob ained by
he exclusi e mean o con ex analy ic a gumen s, and no s a emen abou he egula i y o he
alue unc ion is made. In his wo k, we deal wi h a singula ly con olled SPDE and exploi he
dynamic p og amming app oach oge he wi h iscosi y heo y and con ex analysis in o de o
achie e egula i y esul s on he p oblem’s alue unc ion.
O ganiza ion o he Pape . The es o he pape is o ganized as ollows. In Sec ion 2we
p o ide he se ing and in oduce he p oblem. In Sec ion 3we hen conside he a ia ional
inequali y associa ed o he p oblem and p o e p elimina y egula i y and iscosi y p ope y o
i s alue unc ion V. Unde a sui able equi emen on he di ec ion o ac ion, in Sec ion 4a
connec ion o op imal s opping is de i ed and egula i y o he op imal s opping p oblem’s alue
unc ion is p o ed. As a byp oduc o ha , in Sec ion 5a second-o de smoo h- i p ope y o V
is hen ob ained. Finally, Sec ion 6p oposes wo applica ions in Economics, while Appendix A
collec s a esul on semiconca e and semicon ex unc ions and Appendix B echnical lemma a.
2. Se ing and P oblem Fo mula ion
2.1. Se ing. Le (D, M, µ) be a ini e s anda d Bo el measu e space and assume, wi hou loss
o gene ali y o wha ollows, ha µ(D) = 1. Conside he sepa able Hilbe space
H:= L2(D, M, µ;R).
The dual H∗is iden i ied wi h H ia he classical Riesz ep esen a ion o H∗. The nonnega i e
cone o His deno ed by
H+:= x∈H:x≥0.
We deno e by L(H) he space o linea bounded ope a o s on Hand by L+(H) he subspace o
posi i i y-p ese ing ope a o s o L(H); i.e., P∈ L+(H) i
x∈H+=⇒P x ∈H+.
Th oughou he pape , we conside a linea ope a o A:D(A)⊆H→Hsa is ying he
ollowing s anding equi emen s.
Assump ion 2.1. Ais sel -adjoin , closed, densely de ined, and such ha , o some δ > 0, we
ha e
⟨Ax, x⟩H≤ −δ|x|2
H,∀x∈H.
In pa icula (see, e.g., [26, Ch. II, Sec. 3] and [8, Ch. II-1, Sec. 2.10.1]), unde Assump ion 2.1,
he ope a o Agene a es a C0-semig oup o con ac ions (e A) ≥0⊆ L(H) and
|e A|L(H)≤e−δ ,∀ ≥0.
Mo eo e , 0 ∈ϱ(A) – wi h ϱdeno ing he esol en se – so ha Ais in e ible and
A−1∈ L(H).
We also assume he ollowing.
Assump ion 2.2. The C0-semig oup o con ac ions (e A) ≥0⊆ L(H)is posi i i y-p ese ing;
ha is, (e A) ≥0⊆ L+(H).
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 5
Rema k 2.3. Su icien condi ions gua an eeing posi i i y o semig oups can be ound, e.g., in
[3, Chap. C-II, Thm. 1.2, Thm. 1.8] and [20, Thm. 7.29 and P op. 7.46]
Fu he mo e, we impose he nex assump ion.
Assump ion 2.4. D(A),→L∞(D, M, µ;R).
Rema k 2.5. In he examples conside ed in Sec ion 6,D(A)will be he Sobole space W2,2(O)
wi h app op ia e bounda y condi ions, o Obeing an open, simply connec ed, and bounded se
o Rn(n < 4) wi h smoo h bounda y. Assump ion 2.4 is hen e i ied in his se ing by [12, Co .
9.15].
Le us now come o he p obabilis ic s uc u e o ou se up. We endow he ime-in e al
[0,∞) wi h he Bo el σ-algeb a B([0,∞)). Also, le (Ω,F,F,P) be a il e ed p obabili y space,
wi h il a ion F:= (F ) ∈[0,∞)sa is ying he usual condi ions, and le Wbe a cylind ical Wiene
p ocess on (Ω,F,F,P), aking alues in ano he Hilbe space K. Finally, o u u e use, we
deno e by T he se o all F-s opping imes.
In he ollowing, all he ela ionships in ol ing ω∈Ω as hidden andom pa ame e a e
in ended o hold P-almos su ely. Also, in o de o simpli y he exposi ion, o en we will no
s ess he explici dependence o he in ol ed andom a iables and p ocesses wi h espec o
ω∈Ω.
Le ∆ ⊆H+be a con ex cone o H+and se
M:= I: Ω ×[0,∞)→H+:I·is F−adap ed and such ha 7→ I
is c`adl`ag and wi h I −Is−∈∆∀s, ∈[0,∞) such ha ≥s.(2.1)
No ice ha , since any I∈ M akes alues in H+, igh -con inui y is in ended in he no m o H.
In he ollowing, we se I0−:= 0∈H+ o any I∈ M (see Rema k 2.6 below).
Any gi en I∈ M can be seen as a ( andom) coun ably addi i e ec o measu e
I:B([0,∞)) →H+
o local ini e a ia ion, de ined as
I([s, ]) := I −Is−∀s, ∈[0,∞), ≥s.
We deno e by |I| he a ia ion o I; i is a nonnega i e (op ional andom) measu e on ([0,∞),B([0,∞)))
ha , due o mono onici y o I, can be simply exp essed as
(2.2) |I|([s, ]) = |I −Is−|H,∀s, ∈[0,∞), s ≤ .
Rema k 2.6. By se ing I0−:= 0 o any I∈ M, we mean ha we ex end any I∈ M by
se ing I≡0on [−ε, 0), o a gi en and ixed ε > 0. In his way, he associa ed measu es ha e a
posi i e mass a ini ial ime o size I0. No ice ha his is equi alen wi h iden i ying any con ol
Iwi h a coun ably addi i e measu e I:B([0,∞)) →[0,∞)o local ini e a ia ion de ined as
I((s, ]) := I −Is, o e e y s, ∈[0,∞),s< , plus a Di ac-del a a ime 0o ampli ude I0.
Since His a e lexi e Banach space, by [24], Co olla y 13 a p. 76 (see also De ini ion 3 a p.
61), he e exis s a Bochne measu able unc ion ˆ
ϑ=ˆ
ϑ(ω) : [0,∞)→H+such ha
(2.3) Z[0,T]
|ˆ
ϑ |Hd|I| <∞ ∀T > 0 and dI =ˆ
ϑ d|I| ∀ ≥0.
No ice ha , seen as a s ochas ic p ocess, ˆ
ϑ= (ˆ
ϑ ) ≥0is F-adap ed, because so is I. Fu he mo e,
gi en ha he measu es Iand |I|a e equi alen by (2.2), one has
(2.4) ˆ
ϑ =0 o a.e. ≥0.
The p ocess ˆ
ϑis clea ly unique up o P× |I|−null measu e se s.
6 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
Then, o a gi en H+− alued F-adap ed p ocess := ( ) ∈[0,∞), ecalling (2.3), o any
∈[0,∞) we de ine
Z
0−
⟨ s,dIs⟩H:= Z[0, ]
⟨ s,ˆ
ϑs⟩Hd|I|s=Z[0, ]ZD
s(ξ)ˆ
ϑs(ξ)µ(dξ)d|I|s
=ZDZ[0, ]
s(ξ)ˆ
ϑs(ξ)d|I|sµ(dξ),(2.5)
whe e he las s ep is possible due o Fubini-Tonelli’s heo em. Wi h ega d o (2.3), we also se
(2.6) Z
0−
e( −s)AdIs:= Z
0−
e( −s)Aˆ
ϑsd|I|s, ≥0.
Thanks o (2.6), o any gi en I∈ M, we can hen in oduce he singula ly con inuous
con olled dynamics
(2.7) dXx,I
=AXx,I
d +σdW + dI , ≥0, Xx,I
0−=x∈H,
and de ine he unique mild solu ion o (2.7) as
(2.8) Xx,I
=e Ax+WA,σ
+Z
0−
e( −s)AdIs, ≥0.
He e,
(2.9) WA,σ
:= Z
0
e( −s)AσdWs, ≥0,
wi h σsa is ying he ollowing s anding condi ion.
Assump ion 2.7. σ∈ L2(K;H), whe e L2(K;H)deno es he space o Hilbe -Schmid ope a-
o s om K o H.
Deno ing by L1(H) he se o nonnega i e ace-class ope a o s on Hand endowing L1(H)
wi h he usual no m
|Q|L1(H):= T [Q] =
∞
X
k=0
⟨Qek, ek⟩H,
whe e (ek) is any o hono mal basis o H, we hen ha e unde Assump ion 2.7 ha
σσ∗∈ L1(H).
No ice ha Assump ions 2.1 and 2.7 imply ha he s ochas ic con olu ion (2.9) is well de ined
and con inuous (see [21, Ch. 5]). Fo u u e use, we also no e ha , because o Assump ions 2.1
and 2.7, o all m∈[1,∞) one has o some cm>0 (see [37])
(2.10) Esup
≥0
|WA,σ
|m
H≤cm,
which, deno ing he mild solu ion o (2.7) when Iis he null con ol by Xx,0
, implies
(2.11) Esup
≥0
|Xx,0
|m
H≤cp(1 + |x|m
H),∀x∈H,
o some o he cons an cm>0.
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 7
2.2. P oblem o mula ion. We now mo e on by in oducing he in ini e-dimensional singula
s ochas ic con ol p oblem which is he objec o ou s udy. Le
G:H→R
be a unning cos unc ion, sa is ying he ollowing equi emen s.
Assump ion 2.8.
(i) Gis con ex; The e exis s co, κ1, κ2>0and p≥2such ha
κ1|x|p
H−κ2≤G(x)≤co(1 + |x|p
H);
(ii) Gis semiconca e wi h semiconca i y cons an c1>0; ha is, he e exis s c1>0such ha
λG(x) + (1 −λ)G(y)−G(λx + (1 −λ)y)≤c1
2λ(1 −λ)|x−y|2
H,∀x, y ∈H, λ ∈[0,1].
No ice ha by Lemma A.1(ii) one has ha G∈C1,Lip(H).
Rema k 2.9. Benchma k examples sa is ying Assump ion 2.8 a e he quad a ic cos unc ion
G(x) = 1
2|x−x|2
H, x ∈H,
o some a ge le el x∈H, as well as
G(x) = 1
2⟨x, h⟩2
H,o G(x) = 1
2⟨Qx, x⟩H, x ∈H,
wi h h∈H, and wi h Qbeing posi i e semide ini e and symme ic.
Le now (c . (2.1))
I:= nI∈ M :EZT
0−
|ˆ
ϑs|Hd|I|sp<∞ o p≥2 as in
Assump ion 2.8 and ∀T > 0o
(2.12)
be he class o admissible con ols. Fo a discoun a e ρ > 0 and o q∈H+such ha q≥qo1
o some qo>0 (being 1∈H he cons an uni a y ec o o H), ecalling (2.5) we in oduce
he expec ed cos unc ional
(2.13) J(x;I) := EZ∞
0−
e−ρ G(Xx,I
)d +⟨q, dI ⟩H,(x, I)∈H× I,
which is well-de ined, al hough po en ially in ini e. The in ini e-dimensional singula s ochas ic
con ol p oblem unde s udy is hen
(2.14) V(x) := in
I∈I J(x;I), x ∈H.
Rema k 2.10. No ice ha he in eg abili y condi ion in (2.12)is no equi ed o he well
posedness o (2.13), bu i will be needed in he nex sec ion o he p oo o he iscosi y p ope y
o V.
Gi en he s uc u e o he cos unc ional (2.13), i is con enien o ew i e he decomposi ion
(2.3) in a ailo ed way based on he ins an aneous cos o con ol ⟨q, dI ⟩H. To ha end, ecall
ha ∆ ⊆H+is a con ex cone o H+(c . (2.12), conside he con ex se
(2.15) Θ := θ∈∆ : ⟨q, θ⟩H= 1,
and de ine
S:= {ν: Ω ×[0,∞)→[0,∞) : ν·is F−adap ed and such ha 7→ ν
is c`adl`ag and nondec easing}.(2.16)
14 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
one has
Dg(y) = 3|y−x|H(y−x), D2g(y) = 3|y−x|−1
H(y−x)⊗(y−x)+3|y−x|H·IdH.
In pa icula
T σσ∗D2g(y)≤ |D2g(y)|L(H)T [σσ∗]≤6|y−x|H|σσ∗|L1(H).
The claim hen ollows by a gumen s as in he p oo o P oposi ion 3.3, upon no icing ha o
he Yosida app oximan s ⟨AnXn;x,I
, Xn;x,I
⟩H≤0 (by dissipa i i y) and hen aking he lim sup
(in place o he lim) as n→ ∞ so o ge id o he unbounded e m ⟨Xx,I
,AXx,I
⟩H. See also
[27, P op. 1.166] o he egula con ol case. □
The nex esul is he dynamic p og amming p inciple o he singula s ochas ic con ol
p oblem (2.20). We a e no going o p o ide a p oo he e, as his would e y much ollow he
a gumen s used in he ini e-dimensional se ings (see [22], [38], and [49], among o he s). As a
ma e o ac , he key s eps in he p oo a e based on he low p ope y o he solu ion o he
con olled dynamics, which is, in he p esen se ing, gua an eed by he semig oup p ope y.
P oposi ion 3.5. [Dynamic P og amming P inciple o V] Recall (3.1). Fo each τ∈ T , we
ha e
(3.17) V(x) = in
I:=(ϑ,ν)∈ˆ
I0
EZτ
0−
e−ρ G(Xx,I
)d + dν +e−ρτ V(Xx,I
τ), x ∈H.
We a e hen eady o p o ide he de ini ion o iscosi y solu ion o (3.2) and o p o e he
iscosi y p ope y o Vas in (2.20).
De ini ion 3.6 (Viscosi y solu ion).
(i) We say ha ∈C(H)is a iscosi y supe solu ion o (3.2)a x∈Hi , o e e y φ∈ X
such ha 0 = (x)−φ(x) = min( −φ), one has
max n(ρ− G)φ(x)−G(x),sup
θ∈Θ− ⟨Dφ(x), θ⟩H−1o≥0.
(ii) We say ha ∈C(H)is a iscosi y subsolu ion o (3.2)a x∈Hi , o e e y φ∈ X such
ha 0 = (x)−φ(x) = max( −φ), one has
max n(ρ− G)φ(x)−G(x),sup
θ∈Θ− ⟨Dφ(x), θ⟩H−1o≤0.
(iii) We say ha ∈C(H)is a iscosi y solu ion o (3.2)a x∈Hi i is bo h a iscosi y
supe - and subsolu ion.
Theo em 3.7. Vis a iscosi y solu ion o (3.2)a all x∈H.
P oo . (Subsolu ion p ope y.) Le x∈H,φ∈ X be such ha 0 = V(x)−φ(x) = max(V−φ).
S ep 1. Fo θ∈Θ and ζ > 0, conside he con ol
I·= (ϑ·, ν·)≡(θ, ˆν)∈ˆ
I0,
wi h ˆν0−= 0 and ˆν =ζ o any ≥0. By P oposi ion 3.5, we ha e o h > 0
(3.18) V(x)≤EZh
0
e−ρ G(Xx,I
)d +ζ+e−ρhV(Xx,I
h).
We now aim a aking limi s as h→0+in (3.18). To ha end, no e ha he limi s in he igh -
hand side o (3.18) can be in e changed wi h he expec a ion: By he mono one con e gence
heo em o he in eg al e m; because o he domina ed con e gence heo em o he hi d
addend, since Vhas sub-polynomial g ow h (c . P oposi ion 3.1), and since Xx,I
h=Xx,0
h+ζθ

VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 15
(c . (2.7)) and E[sup ∈[0,T]|Xx,0
|p]<∞, o p≥2 and o any T > 0 by (2.11). Hence, due o
Xx,I
h→x+ζθ as h→0+, we ind
(3.19) V(x)≤ζ+V(x+ζθ).
Bu hen, (3.19) yields
φ(x)≤ζ+φ(x+ζθ),
which, di iding by ζand le ing ζ→0+, in u n gi es (c . P oposi ion 3.1)
−⟨Dφ(x), θ⟩H≤1.
Since he la e holds o e e y θ∈Θ, we ind
sup
θ∈Θ
{−⟨Dφ(x), θ⟩H−1} ≤ 0.
S ep 2. Le now I=0be he null con ol. Se ing
τR:= in { ≥0 : |Xx,0
|H≥R}
(wi h in ∅= +∞) and le ing h > 0, by P oposi ion 3.5 we ha e
φ(x)≤EZτR∧h
0
e−ρ G(Xx,0
)d +e−ρ(τR∧h)φ(Xx,0
τR∧h).
Using now P oposi ion 3.3, we ind
EZτR∧h
0
e−ρ (ρ− G)φ(Xx,0
)−G(Xx,0
)d ≤0.
Di iding by h, ecalling ha φ∈ X and using ha E[sup ∈[0,T ]|Xx,0
|]<∞, o any T > 0 by
(2.11), we can in oke he in eg al mean- alue and he domina ed con e gence heo ems when
le ing h→0+, and we ob ain
(ρ− G)φ(x)−G(x)≤0.
S ep 3. Combining he las wo s eps we ob ain he desi ed subsolu ion p ope y o V.
(Supe solu ion p ope y.) Le now x∈H,φ∈ X be such ha 0 = V(x)−φ(x) = min(V−φ).
Assume, by con adic ion ha he e exis s η > 0 such ha
(3.20) sup
θ∈Θ
{−⟨Dφ(x), θ⟩H−1}≤−2η
and
(3.21) (ρ− G)φ(x)−G(x)≤ −2η.
By con inui y, o a sui able ε > 0,
(3.22) sup
θ∈Θ
{−⟨Dφ(y), θ⟩H−1} ≤ −η, ∀y∈B|·|H(x, ε).
and
(3.23) (ρ− G)φ(y)−G(y)≤ −η, ∀y∈B|·|H(x, ε),
whe e
B|·|H(x, ε) := {y∈H:|y−x|H≤ε}.
Le now I= (ϑ, ν)∈ˆ
I0be a bi a y bu ixed, and se
τI
ε:= in { ≥0 : Xx,I
/∈B|·|H(x, ε)},
wi h he con en ion in ∅=∞, which s ill p o ides a sense o he o mulae below. In he
ollowing, we a e going simply o w i e τεins ead o τI
ε, unless i becomes impo an o s ess
16 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
he explici dependence on I. By Assump ion 2.4 and he ac ha φ∈ X, we ha e o a
cons an C > 0
sup
B|·|H(x,ε)
|Dφ|L∞≤C·sup
B|·|H(x,ε)
|Dφ|D(A)=: e
C < ∞.
Then, since q≥qo1, we ha e
EZτε
0−
e−ρ ⟨Dφ(Xx,I
), ϑ ⟩Hdν ≤e
CEZτε
0−
e−ρ ⟨1, ϑ ⟩Hdν 
=e
C
qo
EZ∞
0−
e−ρ ⟨qo1, ϑ ⟩Hdν ≤e
C
qo
EZ∞
0−
e−ρ ⟨q, ϑ ⟩Hdν 
(3.24)
=e
C
qo
EZ∞
0−
e−ρ dν <∞.
Simila ly, one ge s
(3.25) EZτε
0−
e−ρ ⟨Xx,I
−x, ϑ ⟩Hdν <∞.
Hence, employing (3.22), (3.23), and P oposi ions 3.3 and 3.4, whose equi emen s a e me
due o (3.24) and (3.25), we ind
V(x)−E[e−ρτεV(Xx,I
τε)] ≤φ(x)−E[e−ρτεφ(Xx,I
τε)]
=−Ehe−ρτεXx,I
τε−x3
Hi+Ehe−ρτεXx,I
τε−x3
Hi
+EZτε
0
e−ρ (ρ− G)φ(Xx,I
)d −Zτε
0−
e−ρ ⟨Dφ(Xx,I
), ϑ ⟩Hdν 
≤ −ε3Ee−ρτε1{τε<∞}+EZτε
0
e−ρ 3|σσ∗|L1(H)εd +Zτε
0−
3e−ρ ε2dν 
+EZτε
0−
e−ρ (G(Xx,I
)d + dν )−ηEZτε
0−
e−ρ (d + dν ).
The la e implies
V(x)−EZτε
0−
e−ρ (G(Xx,I
)d + dν ) + e−ρτεV(Xx,I
τε)
(3.26)
≤ −ε3Ee−ρτε+EZτε
0
e−ρ 3|σσ∗|L1(H)εd +Zτε
0−
3e−ρ ε2dν −ηEZτε
0−
e−ρ (d + dν ),
whe e we ha e used ha e−ρτε1{τε=+∞} = 0 by (3.6).
S essing now he dependency o τεwi h espec o he gi en and ixed I∈ˆ
I0, aking he
sup emum on bo h e ms o (3.26) wi h espec o I∈ˆ
I0, using (3.17), and conside ing he
iden i y
e−ρτI
ε+ρZτI
ε
0
e−ρ d = 1,
we ob ain
0≤sup
I∈ˆ
I0(−ε3E"ρ−ZτI
ε
0
e−ρ d #+E"ZτI
ε
0
e−ρ 3|σσ∗|L1(H)ε−ηd +ZτI
ε
0−
e−ρ (3ε2−η)dν #);
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 17
ha is,
0≤ −ρε3+ sup
I∈ˆ
I0
E"ZτI
ε
0
e−ρ 3ε|σσ∗|L1(H)+ε3−ηd +ZτI
ε
0−
e−ρ (3ε2−η)dν #,
Taking εsmall enough wi h espec o η, which can be done wi hou loss o gene ali y, he
in eg ands on he igh -hand side o he las display o mula a e nega i e, so ha he sup emum is
nonposi i e. Hence, we each a con adic ion and he supe solu ion p ope y o Vis p o ed. □
Rema k 3.8. I is wo h emphasizing ha he a gumen o he p oo o he supe solu ion p op-
e y can be clea ly applied also in he ini e-dimensional se ing. As a consequence, his no el
a gumen is able o educe subs an ially he echnicali ies ha a e ypically needed in Rn,n≥1,
in o de o show he supe solu ion ( espec i ely, subsolu ion) p ope y in minimiza ion p oblems
( espec i ely, maximiza ion p oblems) in ol ing singula con ols (see, e.g., [16],[34, Ch. VIII],
[38], and [49], among o he s).
Rema k 3.9. To he bes o ou knowledge, he e a e only wo con ibu ions in he li e a u e
in which he iscosi y app oach is de eloped o a ia ional inequali ies in in ini e dimensional
spaces: These a e [15]and [36]. In bo h o hem, he a ia ional inequali y is ela ed o an
obs acle p oblem and hus in ol es a cons ain on he solu ion i sel , and no on i s g adien
(as, ins ead, i is in ou case).
(i) In [15], a p oblem o op imal s opping o a s ochas ic p ocess wi h memo y is conside ed.
The in ini e-dimensional space in which he unde lying s a e p ocess akes alues is ha
o con inuous unc ions, and he e olu ion o he p ocess is subjec o he ac ion o an
unbounded ope a o ( he i s -o de de i a i e). Howe e , he B ownian noise is ini e-
dimensional in [15].
(ii) In [36], he aim is o p ice Ame ican op ions de ined on o wa d- a es models. The in e es
a e p ocess akes alues in an Hilbe space and i e ol es acco ding o nonlinea dynamics,
which howe e do no in ol e an unbounded ope a o . In his amewo k, he au ho s a e
able o p o e a compa ison esul and o apply i in o de o (uniquely) cha ac e ize he
alue unc ion o he unde lying in ini e-dimensional op imal s opping p oblem. Rema k
4.10 below will a icula e mo e on he ela ion be ween ou pape and [36].
4. Selec ing a speci ic di ec ion o ac ion: A ela ed op imal s opping p oblem
In o de o achie e u he egula i y o V, we now specialize o he case in which he con olle
can only choose he in ensi y ν∈ S appea ing in he decomposi ion (ϑ, ν)∈ I0o any admissible
con ol I∈ I.
In pa icula , we selec he con ex cone
∆ = Span{ˆn},
whe e ˆn∈H+ {0}and i is such ha , wi hou loss o gene ali y, ⟨q, ˆn⟩H= 1. Consequen ly
(c . (2.15)), Θ = {ˆn}and we make he ollowing assump ion, which will hold h oughou he
es o he pape wi hou u he men ion.
Assump ion 4.1. The ec o ˆn∈H+ {0}is an eigen ec o o Awi h eigen alue λ1.
Rema k 4.2. (i) I is wo h s essing ha Assump ion 4.1 does no imply ha he e exis s
a singula ly con olled componen o he s a e a iable X ha is decoupled om he es
o he in ini e-dimensional dynamics o X. To cla i y his, suppose ha H=R2and
conside
A=1 0
1 1 and ˆn=0
1.
1Clea ly, wi h λ≤ −δ < 0, whe e δis as in Assump ion 2.1.
18 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
No ice ˆnis an eigen ec o o A. In his case, he wo componen s o he s a e p ocess X
a e uly coupled h ough he ope a o Aso ha he ac ion along he di ec ion ˆna ec s
indi ec ly also he i s componen o X.
(ii) Su icien condi ions gua an eeing he alidi y o Assump ion 4.1 can be ound in [3,20].
We e e o Sec ion 4 in [13] o a de ailed discussion.
Fo u u e use, we ecall he e ha he uncon olled s a e-p ocess uniquely sol es in he mild
sense (c . (2.7))
dXx,0
=AXx,0
d +σdW , Xx,0
0=x∈H;
ha is (c . (2.9)),
(4.1) Xx,0
=e Ax+Z
0
e( −s)AσdWs=e Ax+WA,σ
, ≥0.
Fu he mo e, Xx,0has con inuous sample pa hs.
Fo ou subsequen analysis, we de ine
(4.2) Gˆn(x) := ⟨DG(x),ˆn⟩H, x ∈H,
and in oduce he op imal s opping p oblem
sup
τ∈T
EZτ
0
e−(ρ−λ) Gˆn(Xx,0
)d −e−(ρ−λ)τ, x ∈H,
whe e Tdeno es he se o he F-s opping imes. The nex esul ela es he p e ious op imal
s opping p oblem o he di ec ional de i a i e o Vas in (2.20), in he di ec ion ˆn. This can be
hough o as an in ini e-dimensional analogue o he esul in [4], whe e, in a ini e-dimensional
se ing, i is p o en ha a sui able op imal iming p oblem p o ides he ma ginal alue o an
i e e sible in es men p oblem o mono one ollowe ype.
Theo em 4.3. Fo any x∈H, one has
(4.3) Vˆn(x) := ⟨DV (x),ˆn⟩H= sup
τ∈T
EZτ
0
e−(ρ−λ) Gˆn(Xx,0
)d −e−(ρ−λ)τ.
P oo . The p oo is o ganized in wo s eps.
S ep 1. Le x∈H. He e we p o e ha he e exis s an op imal con ol I⋆:= (ˆn, ν⋆)∈ I0 o
V(x) as in (2.20). Fu he mo e, i Gis s ic ly con ex, I⋆is unique up o indis inguishabili y.
Le (Ik)k∈N:= (ˆn, νk)k∈N⊆ I0be a minimizing sequence o V(x). Le us deno e by Xx,k :=
Xx,Ik,k∈N. Gi en ha V(x)≤ˆco(1 + |x|p
H), x∈H, (c . P oposi ion 3.1) o p≥2 as in
Assump ion 2.8, we ha e o some ε > 0
(4.4) sup
k∈N
EZ∞
0−
e−ρ |Xx,k
|p
Hd ≤1
κ1ˆco(1 + |x|p
H) + ε+κ2
ρ.
Because
Xx,k
=e Ax+WA,σ
+bnZ
0−
eλ( −s)dνk
,
simple es ima es using λ < 0, (2.10), and |e A|L(H)≤e−δ , o all ≥0, gi e, o a cons an
C > 0 (changing om line o line),
(4.5) |bn|p
Heλ pE|νk
|p≤C|x|p
H+cp+E|Xx,k
|p.
This in u n yields by (4.4)
(4.6) sup
k∈N
EZ∞
0
e−(ρ−λp) |νk
|pd ≤C1 + |x|p
H+ sup
k∈N
EZ∞
0
e−ρ |Xx,k
|pd <∞.
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 19
Hence, by Banach-Saks heo em, he e exis a subsequence o (νk)k∈N- s ill deno ed by
(νk)k∈N- and some ν⋆∈Lp(Ω ×[0,∞); P⊗e−(ρ−λp) d ) such ha
(4.7) bνj:= 1
j+ 1
j
X
k=0
νk→ν⋆in Lp(Ω ×[0,∞); P⊗e−(ρ−λp) d ).
Ac ually, up o passing o a u he subsequence (again, elabeled in he ollowing), such a
con e gence can be ealized P⊗e−(ρ−λp) d -a.e. Then, by a guing as in [44, Lemma a 4.5-4.7],
he p ocess ν⋆admi s a modi ica ion - s ill deno ed by ν⋆- ha is igh -con inuous, nondec easing
and F-adap ed, and hus belongs o S. Fu he mo e, gi en ha ν⋆∈Lp(Ω ×[0,∞); P⊗
e−(ρ−λp) d ) and ha 7→ E[|ν⋆
|p] is nondec easing, i ollows ha he in eg abili y condi ion
equi ed in (2.17) is also me ( ecall ha , in his sec ion, ϑ ≡bn). Se hen b
Ij:= (ˆn, bνj)∈ I0,
j∈N, and I⋆:= (ˆn, ν⋆)∈ I0.
No ice now ha by making use o an in eg a ion by pa s in he in eg als wi h espec o dbνj,
one also has ha P⊗e−ρ d -a.e. (c . (2.9) and (2.8))
Xx,
b
Ij
=e Ax+WA,σ
+Z
0−
e( −s)Aˆndbνj
s=e Ax+WA,σ
+ ˆnZ
0−
eλ( −s)dbνj
s→Xx,I⋆
,
and also P-a.s. Z∞
0−
e−ρ dbνj
→Z∞
0−
e−ρ dν⋆
.
By con exi y o (x, ν)7→ J (x;I) (c . (2.19)) he sequence (b
Ij)j∈N:= (ˆn, bνj)j∈Nis minimizing
as well, and we conclude by Fa ou’s lemma and he p e ious con e gences ha
V(x) = lim in
j→∞ J(x;b
Ij)≥ J (x;I⋆), x ∈H,
om which he op imali y o I⋆= (ˆn, ν⋆)∈ I0 o V(x) ollows.
Finally, he uniqueness claim ollows om s ic con exi y o G, upon using a gumen s as
hose in he p oo o P oposi ion 3.4 in [30].
S ep 2. He e we p o e (4.3). Since he p oo e y much ollows he lines o hose o Lemma a
3 and 4 in [4], we only ske ch i .
Le x∈H,I⋆= (ˆn, ν⋆)∈ I0be op imal o V(x) (c . S ep 1 abo e), and o τ∈ T and ε > 0,
de ine
(4.8) ξ := (I⋆
,0≤ < τ,
I⋆
+εeλτ ˆn, ≥τ.
The p ocess ξ∈ I0and i has di ec ion o ac ion ˆnand in ensi y o ac ion νξsuch ha νξ
0−= 0
and dνξ
= dν⋆
+εeλ δ( −τ), o any ≥0. Fu he mo e, le ing Xx−εˆn,ξ be he solu ion o
(2.7) s a ed a ime 0− om le el x−εˆnand con olled by ξ, we ha e ha : Xx−εˆn,ξ ≡Xx−εˆn,I⋆
on [0, τ), while Xx−εˆn,ξ ≡Xx,I⋆on [τ, ∞).
Hence, by exploi ing he con exi y o G, Assump ion 2.2, and he p e ious obse a ions,
V(x)−V(x−εˆn)≥ J (x;I⋆)− J (x−εˆn;ξ)
=EZτ
0
e−ρ ⟨DG(Xx−εˆn,I⋆
), Xx,I⋆
−Xx−εˆn,I⋆
⟩Hd −εe−(ρ−λ)τ
=εEZτ
0
e−(ρ−λ) ⟨DG(Xx−εˆn,I⋆
),ˆn⟩Hd −e−(ρ−λ)τ
(4.9)
≥εEZτ
0
e−(ρ−λ) ⟨DG(Xx−εˆn,0
),ˆn⟩Hd −e−(ρ−λ)τ,

20 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
om which
(4.10) lim in
ε→0
1
εV(x)−V(x−εˆn)≥sup
τ∈T
EZτ
0
e−(ρ−λ) Gˆn(Xx,0
)d −e−(ρ−λ)τ.
Unde he usual con en ion in ∅= +∞, le now τ⋆:= in { ≥0 : ν⋆
>0}and τε:= in { ≥
0 : ν⋆
≥εeλ }, o ε > 0. Clea ly, τε↓τ⋆as ε↓0. We hen de ine
(4.11) η := (0,0≤ <τε,
I⋆
−εeλτεˆn, ≥τε,
and no ice ha deno ing by Xx+εˆn,η he solu ion o (2.7) s a ed a ime 0− om le el x+εˆn
and con olled by η, we ha e ha : Xx+εˆn,η ≡Xx+εˆn,0on [0, τε), while Xx+εˆn,η ≡Xx,I⋆on
[τε,∞).
Con exi y o G hen yields
V(x+εˆn)−V(x)≤ J (x+εˆn;η)− J (x;I⋆)
(4.12)
=EZτε
0
e−ρ ⟨DG(Xx+εˆn,0
), Xx+εˆn,0
−Xx,I⋆
⟩Hd −Zτε
0−
e−ρ dν⋆
−εe−λτε−ν⋆
τεe−ρτε.
No ice now ha
Xx+εˆn,0
−Xx,I⋆
= ˆneλ ε−Z
0−
e−λsdν⋆
s.
Plugging he la e ela ion in o (4.12), di iding by εand adding and subs ac ing e ms, one
a i es a
1
εV(x+εˆn)−V(x)≤EZτ⋆
0
e−(ρ−λ) ⟨DG(Xx,0
),ˆn⟩Hd −e−(ρ−λ)τ⋆#
+EZτ⋆
0
e−(ρ−λ) ⟨DG(Xx+εˆn,0
)−DG(Xx,0
),ˆn⟩Hd +Ehe−(ρ−λ)τ⋆−e−(ρ−λ)τε#
(4.13)
+EZτε
τ⋆
e−(ρ−λ) ⟨DG(Xx+εˆn,0
),ˆn⟩H1−1
εZ
0−
e−λsdν⋆
sd .
Taking limi s as ε↓0 i is no di icul o see ha all he addends on he igh -hand side o
(4.13) bu he i s con e ge o ze o. Hence,
lim sup
ε→0
1
εV(x+εˆn)−V(x)≤EZτ⋆
0
e−(ρ−λ) Gˆn(Xx,0
)d −e−(ρ−λ)τ⋆
≤sup
τ∈T
EZτ
0
e−(ρ−λ) Gˆn(Xx,0
)d −e−(ρ−λ)τ.(4.14)
Combining (4.10) and (4.14) and using he ac ha V∈C1,Lip(H) (c . P oposi ion 3.1( ))
we ob ain (4.3) and hus comple e he p oo . □
We assume he nex condi ion on he di ec ional de i a i e Gˆn.
Assump ion 4.4. Gˆn∈C1(H)and |DGˆn|H≤KGˆn.
Fo ou subsequen analysis, i is con enien o make an in eg a ion by pa s and exploi he
s ong Ma ko p ope y o w i e
Vˆn(x) = −1−Φ(x) + sup
τ∈T
Ehe−(ρ−λ)τΦ(Xx,0
τ)i, x ∈H,
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 21
wi h
(4.15) Φ(x) := −EZ∞
0
e−(ρ−λ) Gˆn(Xx,0
) + ρ−λd , x ∈H.
Unde Assump ion 4.4, using (4.1) and ha |e A|L(H)≤e−δ , one inds o any x1, x2∈H ha
( ecall λ≤ −δ < 0)
|Φ(x2)−Φ(x1)|H≤EZ∞
0
e−(ρ−λ) Gˆn(Xx2,0
)−Gˆn(Xx1,0
)Hd 
≤EZ∞
0
e−(ρ−λ) e A(x2−x1)Hd ≤KGˆn
ρ−λ+δ|x2−x1|H.(4.16)
Ac ually, gi en ha Gˆn∈C1(H), i can be easily shown by di ec calcula ions ha Φ ∈C1(H).
Wi h ega d o hose p ope ies o Φ, in o de o u he in es iga e he C1- egula i y o Vˆn
i hen su ices o conside
(4.17) U(x) := Vˆn(x) + 1 + Φ(x) = sup
τ∈T
Ehe−(ρ−λ)τΦ(Xx,0
τ)i, x ∈H.
P oposi ion 4.5. Le Assump ion 4.4 hold. Then, one has
|U(x2)−U(x1)|H≤KGˆn
ρ−λ+δ|x2−x1|H, x1, x2∈H.
P oo . Fo x1, x2∈H, ecalling (4.1) and using (4.16) as well as |e A|L(H)≤e−δ o any ≥0,
one has
|U(x2)−U(x1)| ≤ sup
τ∈T
Ehe−(ρ−λ)τΦ(Xx2,0
τ)−Φ(Xx1,0
τ)i
≤KGˆn
ρ−λ+δsup
τ∈T
Ehe−(ρ−λ)τeτA(x2−x1)Hi≤KGˆn
ρ−λ+δ|x2−x1|H.
□
By con inui y o U, he s opping egion {x∈H:U(x) = Φ(x)}is closed. Hence, by s anda d
heo y o op imal s opping (see, e.g., [55, Ch. I, Sec. 2.2], whose esul s hold o an unde lying
p ocess aking alues in a measu able space), one has P-a.s. ha
τ⋆:= in { ≥0 : U(Xx,0
) = Φ(Xx,0
)}, x ∈H,
is op imal.
P oposi ion 4.6 (Dynamic P og amming P inciple o U).Fo all s opping imes θ∈ T we
ha e
(4.18) U(x) = sup
τ∈T
Ehe−(ρ−λ)τ1τ<θΦ(Xx
τ) + e−(ρ−λ)θ1τ≥θU(Xx
θ)i,∀x∈H.
We e ain om p o iding a p oo o he p e ious esul . As a ma e o ac , his would
ollow he same lines and s eps as in he ini e-dimensional se ings (see, e.g., [56, Sec. 5.2] and
e e ences he ein) and exploi he low p ope y o X, which is ensu ed he e by he semig oup
p ope y.
Based on he dynamic p og amming p inciple, one has ha he di e en ial p oblem associa ed
o Uis he a ia ional inequali y
(4.19) min ((ρ−λ)− G)u, u −Φ= 0 on H,
wi h Gas in (3.3), and o which, ecalling he class (3.5), we now p o ide he de ini ion o
iscosi y solu ion.
De ini ion 4.7.
22 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
(i) We say ha u∈C(H)is a iscosi y supe solu ion o (4.19)a x∈Hi , o e e y φ∈ X
such ha 0 = u(x)−φ(x) = min(u−φ), one has
min ((ρ−λ)− G)φ, φ −Φ≥0.
(ii) We say ha u∈C(H)is a iscosi y subsolu ion o (4.19)a x∈Hi , o e e y φ∈ X
such ha 0 = u(x)−φ(x) = max(u−φ), one has
min ((ρ−λ)− G)φ, φ −Φ≤0.
(iii) We say ha u∈C(H)is a iscosi y solu ion o (4.19)a x∈Hi i is bo h a iscosi y
supe - and subsolu ion.
Lemma 4.8. Le x∈H,ε > 0, and le
θ:= in { ≥0 : Xx,0
/∈ B|·|H(x, ε)} ∧ 1,
wi h B|·|H(x, ε) := {y∈H:|y−x|H≤ε}.The e exis s mo>0such ha
(4.20) E"Zθ∧τ
0
e−(ρ−λ) d +e−(ρ−λ)τ
ρ−λ1τ<θ#≥mo,∀τ∈ T .
P oo . Fi s o all, no ice ha
E"Zθ∧τ
0
e−(ρ−λ) d +e−(ρ−λ)τ
ρ−λ1τ<θ#
=E"Zθ
0
e−(ρ−λ) d 1τ≥θ+Zτ
0
e−(ρ−λ) d 1τ<θ +e−(ρ−λ)τ
ρ−λ1τ<θ#
=E"1−e−(ρ−λ)θ
ρ−λ1τ≥θ+1
ρ−λ1τ<θ#.
Then assume, by aiming o a con adic ion, ha he e exis s a sequence (τn)⊆ T such ha
E"1−e−(ρ−λ)θ
ρ−λ1τn≥θ+1
ρ−λ1τn<θ#→0.
This means ha
(4.21) E"1−e−(ρ−λ)θ
ρ−λ1τn≥θ#→0 and E[1τn<θ]→0.
The second con e gence abo e yields, passing o a subsequence s ill labeled in he same way,
lim
n
1τn≥θ= 1 a.s..
Bu hen, he domina ed con e gence heo em gi es
E"1−e−(ρ−λ)θ
ρ−λ1τn≥θ#→E"1−e−(ρ−λ)θ
ρ−λ#>0,
whe e he s ic inequali y in he o mula abo e is clea ly due o con inui y o ajec o ies o he
p ocess Xx,0. Hence, we ha e ound a con adic ion wi h he i s con e gence in (4.21) and he
p oo is hus comple e. □
P oposi ion 4.9. Uis a iscosi y solu ion o (4.19)a all x∈H.
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 23
P oo . We ollow he ideas o he p oo in he ini e-dimensional se ing p oposed by [56, Thm.
5.2.1], bu we simpli y subs an ially he p oo o he subsolu ion p ope y hanks o Lemma 4.8.
Supe solu ion p ope y. Recall (3.5) and le φ∈ X be such ha
0 = U(x)−φ(x) = min(U−φ).
Clea ly,
φ(x) = U(x)≥Φ(x).
The e o e, i emains o show ha
((ρ−λ)− G)φ(x)≥0.
Se ing
τR:= in { ≥0 : |Xx,0
|H≥R}
(wi h in ∅= +∞) and le ing h > 0, by P oposi ion 3.5 we ha e
U(x)≥E[e−(ρ−λ)(τR∧h)U(Xx,0
τR∧h)],
so ha
0≥E[e−(ρ−λ)(τR∧h)U(Xx,0
τR∧h)] −U(x)≥E[e−(ρ−λ)(τR∧h)φ(Xx,0
τR∧h)] −φ(x).
On he o he hand, P oposi ion 3.3, applied in he case o I=0( he null con ol), yields
E[e−(ρ−λ)(τR∧h)φ(Xx,0
τR∧h)] = φ(x) + EZτR∧h
0
e−(ρ−λ) [(G − (ρ−λ))φ](Xx,0
)d .
Combining he las wo display equa ions and di iding by hwe ob ain
1
hEZτR∧h
0
e−(ρ−λ) h((ρ−λ)− G)φi(Xx,0
)d ≥0.
We conclude by aking h→0+and applying he in eg al mean- alue heo em and he domina ed
con e gence heo em, since φ∈ X.
Subsolu ion p ope y. Le φ∈ X be such ha 0 = U(x)−φ(x) = max(U−φ) and assume,
by con adic ion, ha he e exis s some η > 0 such ha
((ρ−λ)− G)φ(x)>2ηand U(x)−Φ(x)>2η
ρ−λ.
By con inui y,
((ρ−λ)− G)φ(y)> η and U(y)−Φ(y)>η
ρ−λ∀B|·|H(x, ε),
whe e we ecall ha B|·|H(x, ε) := {y∈H:|y−x|H≤ε}.
Le us de ine he s opping ime
θ:= in { ≥0 : Xx,0
/∈ B|·|H(x, ε)} ∧ 1.
Then, using Dynkin’s o mula o P oposi ion 3.3 (applied again in he case o Ibeing he null
con ol) we ob ain o each τ∈ T
Ehe−(ρ−λ)(θ∧τ)U(Xx,0
θ∧τ)i−U(x)≤Ehe−(ρ−λ)(θ∧τ)φ(Xx,0
θ∧τ)i−φ(x)
=EZθ∧τ
0
e−(ρ−λ) hG − (ρ−λ)φi(Xx,0
)d ,
30 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
6.2. An ene gy balance clima e model wi h human impac . Le us conside a one-
dimensional Ene gy Balance Clima e Model wi h Human Impac (see, e.g., [10]; he basic clima e
model da es back o Ge ald No h, see [53]).
We i s desc ibe quickly he basics o an ene gy balance clima e model. The ea h’s em-
pe a u e is aken o be he esul o incoming adia ion om he sun and ou going adia ion
h ough e lec ion. We conside empe a u e on he hemisphe e, modeled by he hal -ci cle ha
we iden i y wi h he in e al D:= [−1,1], whe e ξ∈[−1,1] is he sine o la i ude. Mis he
Bo el σ-algeb a on D, and µis aken o be he Lebesgue measu e on (D, M).
We now model he empe a u e e olu ion T (ξ) o e ime a loca ion ξ∈D. The incoming
adia ion a ξis deno ed by R(ξ) = QS(ξ)α(ξ). He e, S(ξ) is he sola ene gy a i ing a
la i ude ξ,Qis he sola cons an di ided by 4, and α(ξ) desc ibes he amoun o hea abso bed
a loca ion ξ(co-albedo); in gene al, i depends on empe a u e and loca ion, bu he e as in [53]
we assume ha i is jus a unc ion o ξ∈D. The ou going in a ed adia ion a loca ion ξis
linea in empe a u e, say γ+ηT (ξ), o wo cons an s γand η > 0. On he su ace, we ha e a
ypical hea anspo ha is modeled ia he second de i a i e d2
dξ2, o mo e gene ally, by he
ope a o
(B )(ξ) := d
dξD(ξ)d
dξ (ξ),
o some di usion coe icien Dd i ing he hea anspo . The o e all esul ing ene gy-balance
ope a o
(6.3) Q (ξ) := QS(ξ)α(ξ)−γ−η (ξ) + B (ξ)
desc ibes he ene gy balance o he ea h wi hou human impac . The equilib ium empe a u e
dis ibu ion T⋆(ξ) is gi en by he solu ion o he pa ial di e en ial equa ion QT⋆= 0, subjec
o app op ia e bounda y condi ions ( o ins ance, bo h ze o Neumann o pe iodic bounda y
condi ions can be chosen).
We now add human impac due o ca bon emissions. Le global cumula i e human ca bon
emissions be desc ibed by he ( eal- alued) p ocess ν∈ S (c . (2.16)).
The empe a u e e olu ion a ime is hen desc ibed by
(6.4) dT =QT d +σdW +1dν , T0−=x∈H,
wi h 1being he uni a y eco in H, and wi h σand Was speci ied in Sec ion 2. In pa icula ,
he B ownian mo ion W akes ca e o noise and unmodeled in luences.
The dynamics (6.4) does no i exac ly in o ou se ing because o he cons an (in ime)
d i e m
b(ξ) := QS(ξ)α(ξ)−γ,
bu his p oblem can be easily ixed. De ine
X := T −T⋆,A := −η +B ,
so ha
Q =A +b.
Recalling ha T⋆is an equilib ium dis ibu ion o he empe a u e, we may w i e
dX = d(T −T⋆) = QT d − QT⋆d +σdW +1dν
=AT d − AT⋆d +σdW +1dν
=AX d +σdW +1dν
and we a e back in ou se ing. In pa icula , he ope a o Awi h he a o emen ioned bounda y
condi ions sa is ies Assump ions 2.1,2.2, and 2.4 ( o he null Neumann bounda y condi ions
one, see, e.g., [48, Sec.3.1]; o he pe iodic ones, see Sec ion 5.2 in [28]).

VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 31
Assume now ha a decision make ( o ins ance, in his con ex , he Uni ed Na ions) has an
ideal p o ile o empe a u e b
Tin mind; his could be he p e-indus ial equilib ium empe a u e
T⋆o a empe a u e dis ibu ion in i s icini y. The planne hus aims a minimizing he
a e age squa e dis ance |T−b
T|2
H o ha ideal empe a u e and measu es he cos o in es men
in o capaci y by some p ice q > 0. Then, he esul ing minimiza ion p oblem in e ms o he
con olled p ocess Xνis (c . (2.12))
in
ν∈S
EZ∞
0
e−ρ Xν
+T⋆−b
T2
Hd +qdν ,
o some in e empo al discoun a e ρ > 0. This alls in o ou se ing o G(x) = |x+T⋆−b
T2
H.
Such a speci ica ion g ea ly simpli ies he ull economic model which would be beyond he scope
o he cu en pape . Compa e [11] o an a emp in ha di ec ion.
Acknowledgemen s. Funded by he Deu sche Fo schungsgemeinscha (DFG, Ge man Re-
sea ch Founda ion) - P ojec -ID 317210226 - SFB 1283. Sal a o e Fede ico is g a e ul o And zej
Swiech o he aluable discussions on iscosi y solu ions o a ia ional inequali ies in Hilbe
spaces ha led o he o mula ion o Rema ks 3.9 and 4.10.
Appendix A. A esul on semiconca e and semicon ex unc ions
We s a e he e some p ope ies o semiconca e and semicon ex unc ions on H.
Lemma A.1. (i) Le Hbe endowed wi h an a bi a y no m | · | and ake F:H→Rbe
semicon ex (wi h espec o he no m |·|); ha is, he e exis s cF>0such ha
λF(x) + (1 −λ)F(y)−F(λx + (1 −λ)y)≥ −cF
2λ(1 −λ)|x−y|2,∀x, y ∈S, λ ∈[0,1].
Then
x7→ F(x) + cF
2|x|2
is con ex.
(ii) Le F:H→Rbe bo h semiconca e and semicon ex (wi h espec o he no m | · |H);
ha is, he e exis s cF>0such ha
|λF(x) + (1 −λ)F(y)−F(λx + (1 −λ)y)| ≤ cF
2λ(1 −λ)|x−y|2
H,
o e e y x, y ∈Sand λ∈[0,1]. Then F∈C1,Lip(H).
P oo . P oo o (i). This ollows om [14, P op. 1.1.3] (equi alences (a)-(c)), whose p oo is no
a ec ed by he dimensionali y o he space unde conside a ion.
P oo o (ii). This ollows by adap ing he con ex-analy ic a gumen s o he ini e-dimensional
se ing in [14, Ch.3]. We epo hese he e o comple eness. Le x∈S. Since Fis semicon ex
and semiconca e, we ha e ha bo h he supe g adien D+F(x) and he subg adien D−F(x)
a e no emp y. We now show ha Fis di e en iable a x∈H. Suppose, by he aim o
con adic ion, ha p1∈D−F(x) and p2∈D+F(x) and ha p1=p2. Then, o h∈H, by
de ini ion o subg adien and by P oposi ion 3.3.1 in [14, Ch.3] (whose p oo does no su e he
dimensionali y o he conside ed space) in he case o linea modulus ω( ) = cF
2 , ≥0, we ind
(A-1) ⟨p1, h⟩H−cF
2|h|2
H≤F(x+h)−F(x)≤ ⟨p2, h⟩H+cF
2|h|2
H.
The e o e, we ob ain
0≥ ⟨p1−p2, h⟩H−cF|h|2
H.
32 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
Gi en he a bi a iness o h, we can now ake h=α(p1−p2), o α > cFso o achie e a
con adic ion. Hence, p1=p2, which, using (A-1), leads o
(A-2) |F(x+h)−F(x)− ⟨p1, h⟩H|
|h|H
≤cF
2|h|H,
and he e o e, in pa icula , o he di e en iabili y o Fa x∈Hwi h DF(x) = p1.
We now show ha DF :H→His Lipschi z con inuous. To ha end, we bo ow a gu-
men s om he p oo o Theo em 3.3.7 in [14, Ch.3], which, once mo e, does no su e he
dimensionali y o he conside ed space. By (A-2) wi h p1=DF (x), o any x, y ∈Hwe ha e
(A-3) |F(y)−F(x)− ⟨DF(x), y −x⟩H| ≤ cF
2|y−x|2
H.
Then, o each x, , w ∈H, we ha e
⟨DF(x+w), ⟩H≤F(x+w+ )−F(x+w) + cF
2| |2
H,
and
⟨DF(x), ⟩H≥F(x+ )−F(x)−cF
2| |2
H.
Hence, om he las wo display equa ions, we ob ain
⟨DF(x+w)−DF(x), ⟩H≤F(x+w+ )−F(x+w)−F(x+ ) + F(x) + cF| |2
H
=F(x+w+ )−1
2F(x+ 2 )−1
2F(x+ 2w)
−F(x+w) + 1
2F(x+ 2w) + 1
2F(x)
−F(x+ ) + 1
2F(x+ 2 ) + 1
2F(x) + cF| |2
H
≤2cF|w|2
H+| |2
H,
whe e in he las es ima e we used he semiconca i y and he semicon exi y o F. The la e
es ima e now implies he claimed Lipschi z p ope y, since
|DF(x+w)−DF(x)|H=1
|w|H
max
| |H=|w|H
⟨DF(x+w)−DF(x), ⟩H≤2cF|w|H.
□
Appendix B. Some echnical esul s
Lemma B.1. Le x∈H,I∈ I,(An)n∈Nbe he Yosida app oximan s o A, and deno e by
Xn;x,I he unique mild solu ion o
dXn;x,I
=AnXn;x,I
d +σdW + dI , X0−=x∈H;
ha is,
(B-1) Xn;x,I
=e Anx+WAn,σ
+Z
0−
e( −s)AndIs, ≥0.
Fo p≥2as in (2.12), o any T > 0and o some M > 0i holds:
(i) Esup ∈[0,T]Xn;x,I
p
H≤M1 + |x|p
H+EhRT
0ˆ
ϑsHd|I|spi;
(ii) limn↑∞ Esup ∈[0,T]Xn;x,I
−Xx,I
p
H] = 0.
VARIATIONAL INEQUALITIES AND SMOOTH-FIT PRINCIPLE IN HILBERT SPACES 33
P oo . We s a by p o ing (i). By (B-1), he Bu kholde -Da is-Gundy inequali y (see Theo em
1.111 in [27]), he ac ha |e An|L(H)≤e−nδ
n+δ (c . Equa ion (B-14) in [27]), and an es ima e
analogous o (2.10), we ha e, o some cons an M > 0 (possibly depending on pand changing
om line o line),
Esup
∈[0,T]Xn;x,I
p
H≤M|x|p
H+Ehsup
∈[0,T]Z
0
e( −s)AnσdWsp
Hi
+Ehsup
∈[0,T]Z
0
e( −s)Anˆ
ϑsd|I|sp
Hi
≤M|x|p
H+EhZT
0
|σσ∗|L1(H)dsip
2+EhZT
0ˆ
ϑsHd|I|spi
≤M1 + |x|p
H+EhZT
0ˆ
ϑsHd|I|spi,
whe e he las expec a ion is ini e due o he ac ha I∈ I (c . (2.12)). This p o es he i s
claim.
As o (ii) no ice ha , o a cons an M > 0 changing om line o line,
Ehsup
∈[0,T]Xn;x,I
−Xx,I
p
Hi≤Msup
∈[0,T]e An−e Axp
H
+Ehsup
∈[0,T]Z
0e( −s)An−e( −s)AσdWsp
Hi
+Ehsup
∈[0,T]Z
0e( −s)An−e( −s)Aˆ
ϑsd|I|sp
Hi.
The i s wo addends on he igh -hand side o he la e display equa ion con e ge o ze o as
n↑ ∞ as in he p oo o [27, Thm. 1.131]. In o de o deal wi h he hi d addend, de ine
ψn(s) := sup
∈[s,T]e( −s)An−e( −s)Aˆ
ϑsH, s ∈[ , T ],
which is such ha ψn(s)→0 as n↑ ∞ P-a.s. by [27, P op. B.34]. Since now |ψn(s)| ≤ 2|ˆ
ϑs|H,
and, o any T > 0, RT
0|ˆ
ϑs|Hd|I|s<∞P-a.s. by (2.3) and E[|RT
0|ˆ
ϑs|Hd|I|s|p]<∞because
I∈ I, he domina ed con e gence heo em gi es
lim
n↑∞
Ehsup
∈[0,T]Z
0e( −s)An−e( −s)Aˆ
ϑsd|I|sp
Hi≤lim
n↑∞
Ehsup
∈[0,T]Z
0
ψn(s)d|I|sp
Hi= 0
□
By a guing as in he p oo o Lemma B.1(i) one can also p o e he ollowing.
Lemma B.2. Le x∈H,I∈ I and le Xx,I deno e he unique mild solu ion o (2.7); ha is,
Xx,I
=e Ax+WA,σ
+Z
0−
e( −s)AdIs, ≥0.
Le p≥2as in (2.12). Then, o any T > 0and o some M > 0i holds:
Esup
∈[0,T]Xx,I
p
H≤M1 + |x|p
H+EhZT
0ˆ
ϑsHd|I|spi.
34 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
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36 FEDERICO, FERRARI, RIEDEL, AND R¨
OCKNER
S. Fede ico: Dipa imen o di Ma ema ica, Uni e si `
a di Bologna, Piazza di Po a S. Dona o 5,
40126, Bologna, I aly
Email add ess:[email p o ec ed]
G. Fe a i: Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y, Uni e si ¨
a ss asse
25, 33615, Biele eld, Ge many
Email add ess:[email p o ec ed]
F. Riedel: Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y, Uni e si ¨
a ss asse
25, 33615, Biele eld, Ge many
Email add ess:[email p o ec ed]
M. R¨
ockne : Facul y o Ma hema ics, Biele eld Uni e si y, Uni e si ¨
a ss asse 25, 33615,
Biele eld, Ge many
Email add ess:[email p o ec ed]