SINGULAR FORWARD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND EMISSIONS DERIVATIVES

Transcription

1 he Annal of Applied Probabiliy 213, Vol. 23, No. 3, DOI: /12-AAP865 Iniue of Mahemaical Saiic, 213 SINGULAR FORWARD BACKWARD SOCHASIC DIFFERENIAL EQUAIONS AND EMISSIONS DERIVAIVES BY RENÉ CARMONA 1,FRANÇOIS DELARUE, GILLES-EDOUARD ESPINOSA 2 AND NIZAR OUZI 2 Princeon Univeriy, Univerié de Nice Sophia-Anipoli, Ecole Polyechnique and Ecole Polyechnique We inroduce wo imple model of forward backward ochaic differenial equaion wih a ingular erminal condiion and we explain how and why hey appear naurally a model for he valuaion of CO 2 emiion allowance. Single phae cap-and-rade cheme lead readily o erminal condiion given by indicaor funcion of he forward componen, and uing fine parial differenial equaion eimae, we how ha he exience heory of hee equaion, a well a he properie of he candidae for oluion, depend rongly upon he characeriic of he forward dynamic. Finally, we give a fir order aylor expanion and how how o numerically calibrae ome of hee model for he purpoe of CO 2 opion pricing. 1. Inroducion. hi paper i moivaed by he mahemaical analyi of he emiion marke, a implemened, for example, in he European Union (EU) emiion rading cheme (ES). hee marke mechanim have been hailed by ome a he mo co efficien way o conrol green houe ga (GHG) emiion. hey have been criicized by oher for being a ax in diguie and adding o he burden of indurie covered by he regulaion. Implemenaion of cap-and-rade cheme i no limied o he implemenaion of he Kyoo proocol. he ucceful US acid rain program i a cae in poin. However, a widepread lack of underanding of heir properie and miinformaion campaign by advocacy group more inereed in puhing heir poliical agenda han uing he reul of objecive cienific udie have muddied he waer and add o he confuion. More mahemaical udie are needed o increae he underanding of hee marke mechanim and raie he level of awarene of heir advanage a well a heir horcoming. hi paper wa prepared in hi piri. In a fir par, we inroduce imple ingle-firm model inpired by he working of he elecriciy marke (elecric power generaion i reponible for mo of Received December 21; revied March Suppored in par by NSF Gran DMS Suppored by he Chair Financial Rik of he Rik Foundaion ponored by Sociéé Générale, he Chair Derivaive of he Fuure ponored by he Fédéraion Bancaire Françaie, he Chair Finance and Suainable Developmen ponored by EDF and CA-CIB. MSC21 ubjec claificaion. 6H3, 91G8. Key word and phrae. Sochaic analyi, forward backward ochaic differenial equaion, emiion derivaive. 186

2 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 187 he CO 2 emiion worldwide). Depie he pecificiy of ome aumpion, our reamen i quie general in he ene ha individual rik avere power producer chooe heir own uiliy funcion. Moreover, he financial marke in which hey rade emiion allowance are no aumed o be complee. While marke incompleene preven u from idenifying he opimal rading raegy of each producer, we how ha, independen of he choice of he uiliy funcion, he opimal producion or abaemen raegy i wha we expec by proving mahemaically, and in full generaliy (i.e., wihou auming compleene of he marke), a folk heorem in environmenal economic: he equilibrium allowance price equal he marginal abaemen co, and marke paricipan implemen all he abaemen meaure whoe co are no greaer han he co of compliance (i.e., he equilibrium price of an allowance). he nex ecion pu ogeher he economic aciviie of a large number of producer and earche for he exience of an equilibrium price for he emiion allowance. Such a problem lead naurally o a forward ochaic differenial equaion (SDE) for he aggregae emiion in he economy, and a backward ochaic differenial equaion (BSDE) for he allowance price. However, hee equaion are coupled ince a nonlinear funcion of he price of carbon (i.e., he price of an emiion allowance) appear in he forward equaion giving he dynamic of he aggregae emiion. hi feedback of he emiion price in he dynamic of he emiion i quie naural. For he purpoe of opion pricing, hi approach wa decribed in 5] where i wa called deailed rik neural approach. Forward backward ochaic differenial equaion (FBSDE) of he ype conidered in hi ecion have been udied for a long ime (ee, e.g., 13] or17]). However, he FBSDE we need o conider for he purpoe of emiion price have an unuual peculiariy: he erminal condiion of he backward equaion i given by a diconinuou funcion of he erminal value of he ae driven by he forward equaion. We ue our fir model o prove ha hi lack of coninuiy i no an iue when he forward dynamic are rongly ellipic in he neighborhood of he ingulariie of he erminal condiion, in oher word, when he volailiy of he forward SDE i bounded from below in he neighborhood of he diconinuiie of he erminal value. However, uing our econd equilibrium model, we alo how ha when he forward dynamic are degenerae (even if hey are hypoellipic), diconinuiie in he erminal condiion and lack of rong ellipiciy in he forward dynamic can conpire o produce poin mae in he erminal diribuion of he forward componen a he locaion of he diconinuiie. hi implie ha he erminal value of he backward componen i no given by a deerminiic funcion of he forward componen, for he forward cenario ending a he locaion of jump in he erminal condiion, and juifie relaxing he definiion of a oluion of he FBSDE. Even hough we only preen a deailed proof for a very pecific model for he ake of definiene, we believe ha our reul i repreenaive of a large cla of model. Since from he poin of view of he definiion of aggregae emiion

3 188 CARMONA, DELARUE, ESPINOSA AND OUZI (he degeneracy of he forward dynamic i expeced) hi eemingly pahological reul hould no be overlooked. Indeed, i hed new ligh on an abolue coninuiy aumpion made repeaedly in equilibrium analye, even in dicree ime model (ee, e.g., 3]and4]). hi aumpion wa regarded a an annoying echnicaliy, bu in ligh of he reul of hi paper, i look more inrinic o hee ype of model. In any cae, i fully juifie he need o relax he definiion of a oluion of a FBSDE when he erminal condiion of he backward par jump. A vibran marke for opion wrien on allowance fuure/forward conrac ha recenly developed and increaed in liquidiy (ee, e.g., 5] for deail on hee marke). Reduced form model have been propoed o price hee opion (ee 5] or 6]). Several aemp have been made a maching he mile (or lack hereof) conained in he quoe publihed daily by he exchange. Secion 5 develop he echnology needed o price hee opion in he conex of he equilibrium framework developed in he preen paper. We idenify he opion price in erm of oluion of nonlinear parial differenial equaion and we prove when he dynamic of he aggregae emiion are given by a geomeric Brownian moion, a aylor expanion formula when he nonlinear abaemen feedback i mall. We derive an explici inegral form for he fir order aylor expanion coefficien which can eaily be compued by Mone Carlo mehod. We believe ha he preen paper i he fir rigorou aemp o include he nonlinear feedback erm in he dynamic of aggregae emiion for he purpoe of emiion opion pricing. he final Secion 5 wa moivaed by he deire o provide pracical ool for he efficien compuaion of opion price wihin he equilibrium framework of he paper. Indeed, becaue of he nonlinear feedback creaed by he coupling in he FBSDE, opion price compued from our equilibrium model differ from he linear price compued in 6, 19] and5] in he framework of reduced form model. We derive rigorouly an approximaion baed on he fir order aympoic in he nonlinear feedback. hi approximaion can be ued o numerically compue opion price and ha he poenial o efficienly fi he implied volailiy mile preen in recen opion price quoe. he final Secion 5.3 numerically illurae he properie of our approximaion. 2. wo imple model of green houe ga emiion conrol. We fir decribe he opimizaion problem of a ingle power producer facing a carbon capand-rade regulaion. We aume ha hi producer i a mall player in he marke in he ene ha hi acion have no impac on price and ha a liquid marke for polluion permi exi. In paricular, we aume ha he price of an allowance i given exogenouly, and we ue he noaion Y = (Y ) for he (ochaic) ime evoluion of he price of uch an emiion allowance. For he ake of impliciy we aume ha,] i a ingle phae of he regulaion and ha no banking or borrowing of he cerificae i poible a he end of he phae. For illuraion purpoe, we analyze wo imple model. Srangely enough, he fir ep of hee analye, namely, he idenificaion of he opimal abaemen and producion

4 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 189 raegie, do no require he full force of he ophiicaed echnique of opimal ochaic conrol Modeling fir he emiion dynamic. We aume ha he ource of randomne in he model i given by W = (W ), a finie family of independen one-dimenional Wiener procee W j = (W j ),1 j d. Inoherword, W = (W 1,...,Wd ) for each fixed,]. All hee Wiener procee are aumed o be defined on a complee probabiliy pace (, F, P), and we denoe by F ={F, } he Brownian filraion hey generae. Here, >i a fixed ime horizon repreening he end of he regulaion period. We will evenually exend he model o include N firm, bu for he ime being, we conider only he problem of one ingle firm whoe producion of elecriciy generae emiion of carbon dioxide, and we denoe by E he cumulaive emiion up o ime of he firm. We alo denoe by Ẽ he percepion a ime (e.g., he condiional expecaion) of wha he oal cumulaive emiion E will be a he end of he ime horizon. Clearly, E and Ẽ can be differen ochaic procee, bu hey have he ame erminal value a ime,hai,e = Ẽ. We will aume ha he dynamic of he proxy Ẽ for he cumulaive emiion of he firm are given by an Iô proce of he form (1) Ẽ = Ẽ + (b ξ )d+ σ dw, where b repreen he (condiional) expecaion of wha he rae of emiion would be in a world wihou carbon regulaion, in oher word, in wha i uually called buine a uual (BAU), while ξ i he inananeou rae of abaemen choen by he firm. In mahemaical erm, ξ repreen he conrol on emiion reducion implemened by he firm. Clearly in uch a model, he firm only ac on he drif of i perceived emiion. For he ake of impliciy we aume ha he procee b and σ are adaped and bounded. Becaue of he vecor naure of he Brownian moion W, he volailiy proce σ i in fac a vecor of calar volailiy procee (σ j ) 1 j d. For he purpoe of hi ecion, we could ue one ingle calar Wiener proce and one ingle calar volailiy proce a long a we allow he filraion F o be larger han he filraion generaed by hi ingle Wiener proce. hi fac will be needed when we udy a model wih more han one firm. Noice ha he formulaion (1) doe no guaranee he poiivene of he perceived emiion proce, a one would expec i o be. hi iue will be dicued in Propoiion 3 below, where we provide ufficien condiion on he coefficien of (1) in order o guaranee he poiivene of he proce Ẽ. Coninuing on wih he decripion of he model, we aume ha he abaemen deciion i baed on a co funcion c : R R which i aumed o be coninuouly differeniable (C 1 in noaion), ricly convex and aify Inada-like condiion: (2) c ( ) = and c (+ ) =+.

5 19 CARMONA, DELARUE, ESPINOSA AND OUZI Noe ha (c ) 1 exi becaue of he aumpion of ric convexiy. Since c(x) can be inerpreed a he co o he firm for an abaemen rae of level x, wihou any lo of generaliy we will alo aume c() = min c =. Noice ha (2) implie ha lim x ± c(x) =+. EXAMPLE 1. A ypical example of abaemen co funcion i given by he quadraic co funcion c(x) = αx 2 for ome α>uedin19], or more generally, he power co funcion c(x) = α x 1+β for ome α>andβ>. he firm conrol i deiny by chooing i own abaemen chedule ξ a well a he quaniy θ of polluion permi i hold hrough rading in he allowance marke. For hee conrol o be admiible, ξ and θ need only be progreively meaurable procee aifying he inegrabiliy condiion (3) E θ 2 + ξ 2 ] d <. We denoe by A he e of admiible conrol (ξ, θ). Given i iniial wealh x,he erminal wealh X of he firm i given by (4) X = X ξ,θ = x + θ dy c(ξ )d E Y. he fir inegral in he righ-hand ide of he above equaion give he proceed from rading in he allowance marke. Recall ha we ue he noaion Y for he price of an emiion allowance a ime. he nex erm repreen he abaemen co and he la erm give he co of he emiion regulaion. Recall alo ha a hi age we are no inereed in he exience or he formaion of hi price. We merely aume he exience of a liquid and fricionle marke for emiion allowance, and ha Y i he price a which each firm can buy or ell one allowance a ime. he rik preference of he firm are given by a uiliy funcion U : R R, whichiaumedobec 1, increaing, ricly concave and aifying he Inada condiion (5) (U) ( ) =+ and (U) (+ ) =. he opimizaion problem of he firm can be wrien a he compuaion of V(x)= up EU ( X ξ,θ ) (6), (ξ,θ) A where E denoe he expecaion under he hiorical meaure P, and A i he e of abaemen and rading raegie (ξ, θ) admiible o he firm. he following imple reul hold. PROPOSIION 1. he opimal abaemen raegy of he firm i given by ξ = c ] 1 (Y ).

6 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 191 REMARK 1. Noice ha he opimal abaemen chedule i independen of he uiliy funcion. he beauy of hi imple reul i i powerful inuiive meaning: given a price Y for an emiion allowance, he firm implemen all he abaemen meaure which make ene economically, namely, all hoe coing le han he curren marke price of one allowance (i.e., one uni of emiion). PROOF OF PROPOSIION 1. By an immediae inegraion by par in he expreion (4) of he erminal wealh, we ee ha ) Ẽ Y = Y (Ẽ + b d + σ dw Y ξ d = Y (Ẽ + b d + o ha X = A θ + B ξ wih A θ = θ dy Y (Ẽ + ) σ dw Y ξ d ( b d + σ dw ), where he modified conrol θ i defined by θ = θ + ξ d,and B ξ = x ] c(ξ ) Y ξ d. ) ξ d dy Noice ha B ξ depend only upon ξ wihou depending upon θ while A θ depend only upon θ wihou depending upon ξ.heea of admiible conrol i equivalenly decribed by varying he couple (θ, ξ) or ( θ,ξ), o when compuing he maximum up (θ,ξ) A EU(X ) = up ( θ,ξ) A EU ( A θ + B ξ ), one can perform he opimizaion over θ and ξ eparaely, for example, by fixing θ and opimizing wih repec o ξ before maximizing he reul wih repec o θ. he proof i complee once we noice ha U i increaing and ha for each,] and each ω, he quaniy B ξ i maximized by he choice ξ = (c ) 1 (Y ). REMARK 2. he above reul argue neiher exience nor uniquene of an opimal admiible e (ξ,θ ) of conrol. In he conex of a complee marke, once he opimal rae of abaemen ξ i implemened, he opimal invemen raegy θ hould hedge he financial rik creaed by he implemenaion of he abaemen raegy. hi fac can be proved uing he claical ool of porfolio opimizaion in he cae of complee marke model. Indeed, if we inroduce he convex dual Ũ of U defined by { } Ũ(y):= up U(x) xy x

7 192 CARMONA, DELARUE, ESPINOSA AND OUZI and he funcion I by I = (U ) 1 o ha Ũ(y) = U I(y) yi(y) and if we denoe by E and E Q, repecively, he expecaion wih repec o P and he unique equivalen meaure Q under which Y i a maringale (we wrie Z for i volailiy given by he maringale repreenaion heorem), hen from he a.. inequaliy U ( X ξ,θ ) dq y dp Xξ,θ U I ( y dq dp valid for any admiible (ξ, θ),andy R,wege EU ( X ξ,θ ) EU I (y dq dp ) y dq dp I ( ) + ye Q X ξ,θ I ( y dq dp y dq dp afer aking expecaion under P. Compuing E Q X ξ,θ by inegraion by par we ge EU ( X ξ,θ ) EU I (y dq ) dp + y x E Q ( c c ) 1 ( (Y ) + Y b ( c ) 1 (Y ) ) ] + σ Z d ), )] ( E Q I y dq dp if we ue he opimal rae of abaemen. So if we chooe y =ŷ R a he unique oluion of ( E Q I ŷ dq ) = x E Q c ( c ) 1 ( (Y ) + Y b ( c ) 1 (Y ) ) + σ Z d, dp i follow ha ( E Q X ˆξ,θ = E Q I ŷ dq ), dp and finally, if he marke i complee, he claim I(ŷ dq dp ) i aainable by a cerain θ. he proof i complee Modeling he elecriciy price fir. We conider a econd model for which again, par of he global ochaic opimizaion problem reduce o a mere pah-by-pah opimizaion. A before, he model i impliic, epecially in he cae of a ingle firm in a regulaory environmen wih a liquid fricionle marke for emiion allowance. However, hi model will become very informaive laer on when we conider N firm ineracing on he ame marke, and we ry o conruc he allowance price Y by olving a FBSDE. he model concern an economy wih one ingle good (ay, elecriciy) whoe producion i he ource of a negaive exernaliy (ay, GHG emiion). I price (P ) evolve according o he following Iô ochaic differenial equaion: (7) dp = μ(p )d + σ(p )dw, )],

8 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 193 where he deerminiic funcion μ and σ are aumed o be C 1 wih bounded derivaive. A each ime,], he firm chooe i inananeou rae of producion q and i producion co are c(q ) where c i a funcion c : R + R whichiaumedobec 1 and ricly convex. Wih hi noaion, he profi and loe from he producion a he end of he period,] are given by he inegral P q c(q ) ] d. he emiion regulaion mandae ha a he end of he period,], hecumulaive emiion of each firm be meaured, and ha one emiion permi be redeemed per uni of emiion. A before, we denoe by (Y ) he proce giving he price of one emiion allowance. For he ake of impliciy, we aume ha he cumulaive emiion E up o ime are proporional o he producion in he ene ha E = ɛq where he poiive number ɛ repreen he rae of emiion of he producion echnology ued by he firm, and Q denoe he cumulaive producion up o and including ime, Q = q d. A he end of he ime horizon, he co incurred by he firm becaue of he regulaion i given by E Y = ɛq Y. he firm may purchae allowance; we denoe by θ he amoun of allowance held by he firm a ime. Under hee condiion, he erminal wealh of he firm i given by X = X q,θ (8) = x + θ dy + P q c(q ) ] d ɛq Y, where, a before, we ued he noaion x for he iniial wealh of he firm. he fir inegral in he righ-hand ide of he above equaion give he proceed from rading in he allowance marke, he nex erm give he profi from he producion and he ale of elecriciy and he la erm give he co of he emiion regulaion. We aume ha he rik preference of he firm are given by a uiliy funcion U : R R,whichiaumedobeC 1, increaing, ricly concave and aifying he Inada condiion (5) aed earlier. A before, he opimizaion problem of he firm can be wrien a (9) V(x)= up (q,θ) A EU ( X q,θ ), where E denoe he expecaion under he hiorical meaure P, anda i he e of admiible producion and rading raegie (q, θ). hi problem i imilar o hoe udied in 2] where he equilibrium iue i no addreed. A before, for hee conrol o be admiible, q and θ need only be adaped procee aifying he inegrabiliy condiion E θ 2 + q 2 ] (1) d <.

9 194 CARMONA, DELARUE, ESPINOSA AND OUZI PROPOSIION 2. he opimal producion raegy of he firm i given by q = ( c ) 1 (P ɛy ). REMARK 3. A before, he opimal producion raegy q i independen of he rik averion (i.e., he uiliy funcion) of he firm. he inuiive inerpreaion of hi reul i clear: once a firm oberve boh price P and Y, i compue he price for which i can ell he good minu he price i will have o pay becaue of he emiion regulaion, and he firm ue hi correced price o chooe i opimal rae of producion in he uual way. PROOF OF PROPOSIION 2. bounded variaion) give A imple inegraion by par (noice ha E i of (11) Q Y = Y dq + Q dy = Y q d + Q dy, o ha X = A θ + B q wih A θ = which depend only upon θ and θ dy wih θ = θ ɛ q d, B q = x + (P ɛy )q c(q ) ] d, which depend only upon q wihou depending upon θ. Since he e A of admiible conrol i equivalenly decribed by varying he couple (q, θ) or (q, θ),when compuing he maximum up (q,θ) A E { U(X ) } = up (q, θ) A E { U ( A θ + B q )}, one can perform he opimizaion over q and θ eparaely, for example, by fixing θ and opimizing wih repec o q before maximizing he reul wih repec o θ. he proof i complee once we noice ha U i increaing and ha for each,] and each ω, he quaniy B q i maximized by he choice q = (c ) 1 (P ɛy ). 3. Allowance equilibrium price and a fir ingular FBSDE. he goal of hi ecion i o exend he fir model inroduced in Secion 2 o an economy wih N firm, and olve for he allowance price.

10 SINGULAR FBSDES AND EMISSIONS DERIVAIVES Swiching o a rik neural framework. A before, we aume ha Y = (Y ), ] i he price of one allowance in a one-compliance period cap-and-rade model, and ha he marke for allowance i fricionle and liquid. In he abence of arbirage, Y i a maringale for a meaure Q equivalen o he hiorical meaure P. Becaue we are in a Brownian filraion, dq dp = exp α dw 1 ] α 2 d 2 for ome equence α = (α ), ] of adaped procee. By Giranov heorem, he proce W = ( W ), ] defined by W = W α d i a Wiener proce for Q o ha equaion (1), giving he dynamic of he perceived emiion of a firm, now read dẽ = ( b ξ )d + σ d W under Q, where he new drif b i defined by b = b + σ α for all,] Marke model wih N firm. We now conider an economy compriing N firm labeled by {1,...,N}, and we work in he rik neural framework for allowance rading dicued above. When a pecific quaniy uch a co funcion, uiliy, cumulaive emiion, rading raegy,... depend upon a firm, we ue a upercrip i o emphaize he dependence upon he ih firm. So in equilibrium (i.e., whenever each firm implemen i opimal abaemen raegy), for each firm i {1,...,N} we have dẽ i = { b i ( c i) ] 1 (Y ) } d + σ i d W wih given iniial perceived emiion Ẽ i. Conequenly, he aggregae perceived emiion Ẽ defined by aifie N Ẽ = Ẽ i i=1 dẽ = ( b f(y ) ) d + σ d W, where N b = b i, σ = N σ i i=1 i=1 and f(x)= N ( c i ) ] 1 (x). i=1

11 196 CARMONA, DELARUE, ESPINOSA AND OUZI Again, ince we are in a Brownian filraion, i follow from he maringale repreenaion heorem ha here exi a progreively meaurable proce Z = (Z ), ] uch ha dy = Z d W and E Q Z 2 d <. Furhermore, in order o enerain a concree exience and uniquene reul, we aume ha W i one dimenional and ha here exi deerminiic coninuou funcion,] R (, e) b(,e) R and,] R σ(,e) R uch ha b = b(,ẽ ) and σ = σ(,ẽ ),forall,], Q-a.. Conequenly, he procee Ẽ, Y and Z aify a yem of FBSDE under Q, which we reae for he ake of laer reference: { dẽ = ( b(,ẽ ) f(y ) ) d + σ(,ẽ )d W, wih given Ẽ R, (12) dy = Z d W, Y = λ1,+ ) (Ẽ ). he fac ha he erminal condiion for Y i given by an indicaor funcion reul from he equilibrium analyi of hee marke (ee 3] and4]). i he global emiion arge e by he regulaor for he enire economy. I repreen he cap par of he cap-and-rade cheme. λ i he penaly ha firm have o pay for each emiion uni no covered by he redempion of an allowance. Currenly, hi penaly i 1 euro in he EU ES. Noice ha ince all he co funcion c i are ricly convex, f i ricly increaing. We hall make he following addiional aumpion: (13) (14) (15) b(,e) and σ(,e)are Lipchiz in e uniformly in, here exi an open ball U R 2, U (, ), uch ha inf σ 2 (, e) >, (,e) U, ] R f i Lipchiz coninuou (and ricly increaing). We denoe by H he collecion of all R-valued progreively meaurable procee on,] R, and we inroduce he ube } H 2 := {Z H ; E Q Z 2 d < and S 2 { := Y H ; E Q up Y 2] } < Solving he ingular equilibrium FBSDE. he purpoe of hi ubecion i o prove exience and uniquene of a oluion o FBSDE (12).

12 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 197 HEOREM 1. If aumpion (13) o (15) hold for a given R, hen, for any λ>, FBSDE (12) admi a unique oluion (Ẽ,Y,Z) S 2 S 2 H 2. Moreover, for any,], Ẽ i nonincreaing wih repec o λ and nondecreaing wih repec o. PROOF. For any funcion ϕ : R R, we wrie FBSDE(ϕ) forhefbsde (12) when he funcion g = λ1,+ ) appearing in he erminal condiion in he backward componen of (12) i replaced by ϕ. (i) We fir prove uniquene. Le (Ẽ,Y,Z) and (Ẽ,Y,Z ) be wo oluion of FBSDE (12). Clearly i i ufficien o prove ha Y = Y.Leue δe := Ẽ Ẽ, δy := Y Y, δz := Z Z, β := b(,ẽ ) b(,ẽ ) 1 {δe }, := σ(,ẽ ) σ(,ẽ ) 1 {δe }. δe δe Noice ha (β ) and ( ) are bounded procee. By direc calculaion, we ee ha where d(b δe δy ) = B δy ( f(y ) f ( Y )) d + B δe δz d W, ( ( 2 ) B := exp 2 β d d W ). Since δe = andδe δy = (Ẽ Ẽ )(g(ẽ ) g(ẽ )), becaue g i nondecreaing, hi implie ha E Q ( B δy f(y ) f ( ] Y )) d. Since B > andf i (ricly) increaing, hi implie ha δy = d dq-a.e. and herefore Y = Y by coninuiy. (ii) We nex prove exience. Le (g n ) n 1 be an increaing equence of mooh nondecreaing funcion wih g n,λ] and uch ha g n g = λ1 (, ). (ii-1) We fir prove he exience of a oluion when he boundary condiion i given by g n.foreveryn 1, he FBSDE(g n ) aifie he aumpion of heorem 5.6 and 7.1 in 12] wih b 3 =, f 1 = f 2 = f 3 =, σ 2 = σ 3 =, b 2 by (15)] and h = (inceg n i nondecreaing) o ha Condiion (5.11) in 12] hold wih λ = andf (, ) = foranyε>. By heorem 7.1 in 12], he FBSDE(g n ) ha a unique oluion (Ẽ n,y n,z n ) S 2 S 2 H 2. Moreover, i hold Y n = u n (, Ẽ n),, for ome deerminiic funcion un. In conra wih 12], he funcion u n i no a random field bu a deerminiic funcion ince he coefficien of he FBSDE are deerminiic. We refer o 15] forhe general conrucion of u n when he coefficien are deerminiic. Since he equence (g n ) n 1 i increaing, we deduce from he comparion principle 12], heorem 8.6, which applie under he ame aumpion a 12], heorem 7.1, ha, for

13 198 CARMONA, DELARUE, ESPINOSA AND OUZI any,], he equence of funcion (u n (, )) n 1 i nondecreaing. By 12], heorem 8.6, again, u n i nondecreaing in λ and nonincreaing in. Sinceg n i,λ]-valued and u n (, e) = E Q g n (Ẽ n ) Ẽn = e], we deduce ha u n i,λ]- valued a well. Since he equence of funcion (u n ) n 1 i nondecreaing, we may hen define u(, e) := lim n un (, e),,],e R. Clearly, u i,λ]-valued and u(, ) i a nondecreaing funcion for any,]. Moreover, u i nondecreaing in λ and nonincreaing in. By 12], heorem 6.1(iii) and 7.1(i), we know ha, for every n 1, he funcion u n i Lipchiz coninuou wih repec o e, uniformly in,]. Acually, we claim ha, for any δ (,), he funcion u n (, ) i Lipchiz coninuou in e, uniformly in, δ] and in n 1. he proof follow again from 12], heorem 6.1(iii) and 7.1(i). o be more pecific, we need o eablih a uniform upper bound for he bounded oluion ȳ o he fir ODE in 12], (3.12), aociaed wih an arbirary poiive erminal condiion ȳ = h>, namely, for given bounded (meaurable) funcion b 1 :,] b 1 () R + and b 2 :,] b 2 () R +, wih inf, ] b 2 () >, we are eeking an upper bound for any bounded (ȳ ) aifying ȳ =ȳ + ( b1 () ȳ b 2 ()ȳ 2 ) d; ȳ = h>. Here b 1 () i underood a an upper bound for he derivaive of b wih repec o x, andb 2 a a lower bound for he derivaive of f wih repec o y. A long a ȳ doe no vanih, we deduce from a imple compuaion ha ( )( 1 ( ) 1 ȳ = exp b 1 () d + b 2 () exp b 1 (r) dr d). ȳ Since he righ-hand ide above i alway (ricly) poiive, we conclude ha i i indeed a oluion for any,]. herefore, here exi a conan C, independen of ȳ, uch ha ȳ C/( ) for any,).by12], heorem 6.1(iii) and 7.1(i), we deduce ha, for any δ (,], he funcion u n (, ) i Lipchiz coninuou wih repec o e, uniformly in, δ] and n 1. Leing n end o +, we deduce ha he ame hold for u. Noice ha he proce Ẽ n olve he (forward) ochaic differenial equaion dẽ n = ( b (,Ẽ n ) f u n (,Ẽ n )) ( d + σ,ẽ n ) d W,,), where here and in he following, we ue he noaion f u for he compoiion of he funcion f and u. Sincef i increaing and he equence (u n ) n 1 i nondecreaing, i follow from he comparion heorem for (forward) ochaic differenial equaion ha he equence of procee (Ẽ n ) n 1 i nonincreaing. We may hen define Ê := lim n Ẽn for,].

14 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 199 (ii-2) o idenify he dynamic of he limiing proce Ê, we inroduce he proce Ẽ defined on,)a he unique rong oluion of he ochaic differenial equaion dẽ = (b f u)(, Ẽ )d + σ(,ẽ )d W,,); Ẽ =. he fac ha he funcion u i bounded and Lipchiz coninuou in pace (locally in ime), ogeher wih our aumpion on b, f and σ guaranee he exience and uniquene of uch a rong oluion. Since b i a mo of linear growh and u i bounded, he oluion canno explode a end o, o ha he proce (Ẽ ) < can be exended by coninuiy o he cloed inerval,]. Sinceu i Lipchiz coninuou wih repec o e, uniformly in, δ] for any δ (,),we deduce from he claical comparion reul for ochaic differenial equaion ha Ẽ n Ẽ for any,). Leing end o, i alo hold Ẽ n Ẽ. Since, for any n 1, u n (, e) = E Q g n (Ẽ n ) Ẽn = e], for(, e),) R, andg n i a nondecreaing funcion, we deduce ha u n (, ) i a nondecreaing funcion a well. Obviouly, he ame hold for u(, ). We hen ue he fac ha Ẽ n Ẽ ogeher wih he increae of u n (, ) o compue, uing Iô formula, ha, for any,]: (Ẽn ) 2 (Ẽn )(( Ẽ = 2 Ẽ b f u n )(,Ẽ n ) (b f u)(, Ẽ ) ) d (16) + C + 2 σ (,Ẽ n ) σ(,ẽ ) 2 d + 2 (Ẽn )( ( Ẽ σ,ẽ n ) σ(,ẽ ) ) d W Ẽ n Ẽ 2 (Ẽn )( d + 2 Ẽ f u f u n ) (, Ẽ )d (C + 1) + 2 (Ẽn )( ( Ẽ σ,ẽ n ) σ(,ẽ ) ) d W Ẽ n Ẽ 2 d + ( f u f u n) (, Ẽ ) 2 d (Ẽn )( ( Ẽ σ,ẽ n ) σ(,ẽ ) ) d W by he Lipchiz propery of he coefficien b and σ. aking expecaion, we deduce E Q( Ẽ n ) 2 ] Ẽ (C + 1)E Q Ẽ n Ẽ 2 d + E Q ( f u f u n) (, Ẽ ) 2 d.

15 11 CARMONA, DELARUE, ESPINOSA AND OUZI hen E Q( Ẽ n ) 2 ] Ẽ (C + 1) E Q( Ẽ n Ẽ ) 2 ] d + ε n, where ε n := E Q (f u f u n )(, Ẽ ) 2 d], by he dominaed convergence heorem. herefore i follow from Gronwall inequaliy ha up E Q (Ẽ n Ẽ ) 2 ] an end o +. Repeaing he argumen, bu uing in addiion he Burkhölder Davi Gundy inequaliy in (16), we deduce ha Ẽ n Ẽ in S 2, and a a conequence, Ê = Ẽ. (ii-3) he key poin o pa o he limi in he backward equaion i o prove ha QẼ = ]=. Given a mall real δ>, we wrie QẼ = ]=Q Ẽ =, (, Ẽ ) δ U ] (17) + Q Ẽ =, δ, ] : (, Ẽ )/ U ], where U i a in (14). (Here, he noaion (, Ẽ ) δ U mean ha (, Ẽ ) U for any δ, ].) On he even {(, Ẽ ) δ U}, he proce (Ẽ ) δ coincide wih (X ) δ, oluion o X = Ẽ δ + δ ( b(,x ) f u(, X ) ) + σ(,x )d W, δ, δ where σ :,] R R i a given bounded and coninuou funcion which i Lipchiz coninuou wih repec o e, which aifie inf, ] R σ >, and which coincide wih σ on U. Since σ 1 i bounded and f i bounded on,λ], we may inroduce an equivalen meaure Q Q under which he proce B := W σ 1 (, X )(f u)(, X ), δ, ], i a Brownian moion. hen X olve he ochaic differenial equaion (18) dx = b(,x )d + σ(,x )d B, δ, ]; X δ = Ẽ δ. By 14], heorem 2.3.1, he condiional law, under Q,ofX given he iniial condiion X δ ha a deniy wih repec o he Lebegue meaure. Conequenly, QX = ]=, and he ame hold rue under he equivalen meaure Q. herefore, Q Ẽ =, (, Ẽ ) δ U ] =. By (17), we deduce QẼ = ]=Q Ẽ =, δ, ] : (, Ẽ )/ U ] Q (, Ẽ ) (, Ẽ ) di ( (, ), U )]. up δ

16 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 111 A δ end o, he righ-hand ide above end o, o ha (19) QẼ = ]=, which implie ha we can ue g = λ1 (, ) inead of g = λ1, ) in (12). Moreover, we alo have lim n Q Ẽ n > F ] (2) = QẼ > F ] for each <. he fac ha g n g implie Y n = E Q g n ( Ẽ n )] E Q (Ẽn )] g E Q g(ẽ ) ] a n by (2). On he oher hand, ince Ẽ n Ẽ, i follow from he nondecreae of g n, he dominaed convergence heorem and (2) ha Y n = E Q g n ( Ẽ n )] E Q g n (Ẽ ) ] E Q g(ẽ ) ]. Hence, Y n Y := E Q g(ẽ )].Now,leZ H 2 be uch ha Y = g(ẽ ) Z d W,,]. Noice ha Y ake value in,λ], and herefore Y S 2. Similarly, uing he increae and he decreae of he equence (u n ) n 1 and (E n ) n 1, repecively, ogeher wih he increae of he funcion u n (, ) and u(, ) and he coninuiy of he funcion u(, ) for,), we ee ha for,) u(, Ẽ ) = lim n un (, Ẽ ) lim inf n un(,ẽ n lim n u(,ẽ n ) = u(, Ẽ ). ) lim up n un(,ẽ n Since Y n = u n (, Ẽ n), hi how ha Y = u(, Ẽ ) on,), and he proof of exience of a oluion i complee. Impac on he model for emiion conrol. A expeced, he previou reul implie ha he ougher he regulaion (i.e., he larger λ and/or he maller ), he higher he emiion reducion (he lower Ẽ ). In paricular, in he abence of regulaion which correpond o λ =, he aggregae level of emiion i a i highe. We alo noice ha he aumpion in heorem 1 can be pecified in uch a way ha he aggregae perceived emiion proce Ẽ ake nonnegaive value, a expeced from he raionale of he model. PROPOSIION 3. Le he condiion of heorem 1 hold rue. Aume furher ha f() = and here exi r>uch ha σ(,) =, b(, ) on,r], and b(, ) on r, ]. hen: (i) for any Ẽ, he proce Ẽ in (12) i nonnegaive; )

17 112 CARMONA, DELARUE, ESPINOSA AND OUZI (ii) if in addiion Ẽ >, hen Ẽ > for all,). PROOF. By (15), we know ha f(y) fory,λ]. Since he proce (Y ) i,λ]-valued, we deduce from he comparion principle for forward SDE ha he forward proce (Ẽ ) i dominaed by he oluion (X ) o he SDE X = Ẽ + b(,x )d+ σ(,x )d W,. Oberve ha our condiion on b and σ imply ha, whenever Ẽ, we have X and herefore Ẽ. hen Y = λ1,+ ) (Ẽ ) =, o ha u(, Ẽ ) = E(Y ) =. Similarly, u(, e) =, for any,] and e. A a conequence, for any iniial condiion Ẽ, we can wrie (f (Y )) < in he forward equaion in (12) a f(y ) = f ( u(, Ẽ ) ) = f ( u(, Ẽ ) ) f ( u(, ) ) = f(u(,ẽ )) f(u(,)) Ẽ Ẽ 1 {Ẽ }, where he raio (f (u(, e)) f(u(,)))/e, fore, i uniformly bounded in e R \{} and in in compac ube of,)ince u i Lipchiz coninuou in pace, uniformly in ime in compac ube of,)ee poin (ii-1) in he proof of heorem 1]. Similarly, he procee β := b(,ẽ ) Ẽ 1 {Ẽ } and := σ(,ẽ ) Ẽ 1 {Ẽ } are adaped and bounded by he Lipchiz propery of he coefficien b,σ in e uniformly in and he fac ha b(,) = σ(,) =. We hen deduce ha (Ẽ ) < may be expreed a ( ( Ẽ = Ẽ exp β ϕ 1 ) 2 2 d + d W ), <, wih ϕ =f(y )/Ẽ ]1 {Ẽ }, <. REMARK 4. Uing for u addiional eimae from he heory of parial differenial equaion, we may alo prove ha ϕ appearing in he above proof of Propoiion 3 grow up a mo a ( ) 1/2 when. hi implie ha ϕ i inegrable on he whole,] and hu, ha Ẽ > awellwhenẽ >. Since hi reul i no needed in hi paper, we do no provide a deailed argumen. REMARK 5. he nondegeneracy of σ in he neighborhood of (, ) ee (14)] i compaible wih he condiion σ(,) = of Propoiion 3, ince, whichi he regulaory emiion cap in pracice, i expeced o be (ricly) poiive.

18 SINGULAR FBSDES AND EMISSIONS DERIVAIVES Enlighening example of a ingular FBSDE. We aw in he previou ecion ha he erminal condiion of he backward equaion can be a diconinuou funcion of he erminal value of he forward componen wihou hreaening exience or uniquene of a oluion o he FBSDE when he forward dynamic are nondegenerae in he neighborhood of he ingulariy of he erminal condiion. In hi ecion, we how ha hi i no he cae when he forward dynamic are degenerae, even if hey are hypoellipic and he oluion of he forward equaion ha a deniy before mauriy. We explained in he Inroducion why hi eemingly pahological mahemaical propery hould no come a a urprie in he conex of equilibrium model for cap-and-rade cheme. Moivaed by he econd model given in Secion 2.2, we conider he FBSDE dp = dw, (21) de = (P Y )d, dy = Z dw,, wih he erminal condiion (22) Y = 1, ) (E ) for ome real number. Here, (W ), ] i a one-dimenional Wiener proce. hi unrealiic model correpond o quadraic co of producion, and chooing appropriae uni for he penaly λ and he emiion rae ɛ o be 1. For noaional convenience, he maringale meaure i denoed by P inead of Q a in Secion 3, and he aociaed Brownian moion by (W ) inead of ( W ).] Below, we will no dicu he ign of he emiion proce E a we did in Propoiion 3 above for he fir model. Our inere in he example (21) and(22) i he oucome of i mahemaical analyi, no i realim! We prove he following unexpeced reul. HEOREM 2. Given (p, e) R 2, here exi a unique progreively meaurable riple (P,E,Y ) aifying (21) ogeher wih he iniial condiion P = p and E = e and (23) 1 (, ) (E ) Y 1, ) (E ). Moreover, he marginal diribuion of E i aboluely coninuou wih repec o he Lebegue meaure for any <, bu ha a Dirac ma a when =. In oher word, P{E = } >. In paricular, (P,E,Y ) may no aify he erminal condiion P{Y = 1, ) (E )}=1. However, he weaker form (23) of erminal condiion i ufficien o guaranee uniquene.

19 114 CARMONA, DELARUE, ESPINOSA AND OUZI Before we engage in he echnicaliie of he proof we noice ha he ranformaion (24) (P,E ) ( Ē = E + ( )P ) map he original FBSDE (21) ino he impler one { (25) dē = Y d + ( )dw, dy = Z dw, wih he ame erminal condiion Y = 1, ) (Ē ). Moreover, he dynamic of (E ) can be recovered from hoe of (Ē ) ince (P ) in (21) i purely auonomou. In paricular, excep for he proof of he abolue coninuiy of E for <, we reric our analyi o he proof of heorem 2, forē oluion of (25) incee and Ē have he ame erminal value a ime. We emphaize ha yem (25) i doubly ingular a mauriy ime ; he diffuion coefficien of he forward equaion vanihe a end o and he boundary condiion of he backward equaion i diconinuou a. ogeher, boh ingulariie make he emiion proce accumulae a nonzero ma a a ime.hi phenomenon mu be een a a ochaic reidual of he hock wave oberved in he invicid Burger equaion (26) v(,e) v(,e) e v(,e) =,,),e R, wih v(,e) = 1,+ ) (e) a boundary condiion. A explained below, equaion (26) i he fir-order verion of he econd-order equaion aociaed wih (25). Indeed, i i well known ha he characeriic of (26) may mee a ime and a poin. By analogy, he rajecorie of he forward proce in (25)mayhi a ime wih a nonzero probabiliy, hen producing a Dirac ma. In oher word, he hock phenomenon behave like a rap ino which he proce (E ) or equivalenly, he proce (Ē ) ] may fall wih a nonzero probabiliy. I i hen well underood ha he noie plugged ino he forward proce (Ē ) may help ecape he rap. For example, we aw in Secion 3 ha he emiion proce did no ee he rap when i wa rongly ellipic in he neighborhood of he ingulariy. In he curren framework, he diffuion coefficien vanihe in a linear way a ime end o mauriy; i decay oo fa o preven almo every realizaion of he proce from falling ino he rap. A before, we prove exience of a oluion o (25) by fir moohing he ingulariy in he erminal condiion, olving he problem for a mooh erminal condiion and obaining a oluion o he original problem by a limiing argumen. However, in order o prove he exience of a limi, we will ue PDE a priori eimae and compacne argumen inead of comparion and monooniciy argumen. We call mollified equaion he yem (25) wih a erminal condiion (27) Y = φ(ē ), given by a Lipchiz nondecreaing funcion φ from R o, 1] which we view a an approximaion of he indicaor funcion appearing in he erminal condiion (22).

20 SINGULAR FBSDES AND EMISSIONS DERIVAIVES Lipchiz regulariy in pace. PROPOSIION 4. Aume ha he erminal condiion in (25) i given by (27) wih a Lipchiz nondecreaing funcion φ wih value in, 1]. hen, for each (,e),] R,(25) admi a unique oluion (Ē,e,Y,e,Z,e ) aifying Ē,e = e and Y,e = φ(ē,e ). Moreover, he mapping (, e) v(,e) = Y,e i, 1]-valued, i of cla C 1,2 on,) R and ha Hölder coninuou firorder derivaive in ime and fir and econd-order derivaive in pace. Finally, he Hölder norm of v, e v, 2 e,e v and v on a given compac ube of,) R do no depend upon he moohne of φ provided φ i, 1]-valued and nondecreaing. Specifically, he fir-order derivaive in pace aifie (28) e v(,e) 1,,). In paricular, e v(,e) i nondecreaing for any,). Finally, for a given iniial condiion (,e), he procee (Y,e ) and (Z,e ) <, oluion o he backward equaion in (25)(wih φ a boundary condiion), are given by Y,e = v (,Ē,e), ; (29) Z,e = ( ) e v (,Ē,e), <. (3) PROOF. he problem i o olve he yem { dē = Y d + ( )dw, dy = Z dw, wih ξ = φ(ē ) a erminal condiion and (,e) a iniial condiion. he drif in he fir equaion, ha i, (, y),] R y, i decreaing in y and Lipchiz coninuou, uniformly in. By Peng and Wu16], heorem 2.2 (wih G = 1, β 1 = andβ 2 = 1 herein), we know ha equaion (3) admi a mo one oluion. Unforunaely, Peng and Wu 16], heorem 2.6 (ee alo Remark 2.8 herein) doe no apply o prove exience direcly. o prove exience, we ue a variaion of he inducion mehod in Delarue 7]. In he whole argumen, and for he generic iniial ime a which he proce Ē ar. he proof coni of exending he local olvabiliy propery of Lipchiz forward backward SDE a he diance increae, o ha he value of will vary in he proof. Recall indeed from 7], heorem 1.1, ha exience and uniquene hold in mall ime. Specifically, we can find ome mall poiive real number δ, poibly depending on he Lipchiz conan of φ, uch ha (3) admi a unique oluion when belong o he inerval δ, ]. Remember

21 116 CARMONA, DELARUE, ESPINOSA AND OUZI ha he iniial condiion i Ē = e. A a conequence, we can define he value funcion v : δ, ] R (,e) Y,e.By7], Corollary 1.5, i i known o be Lipchiz in pace uniformly in ime a long a he iniial ime parameer remain in δ, ]. he diffuion coefficien in (3) being uniformly bounded away from on he inerval, δ], by7], heorem 2.6, (3) admi a unique oluion on, δ] when i aumed o be in, δ). herefore, we can conruc a oluion o (3) inwoepwhen < δ: wefirolve (3) on, δ] wih Ē = e a iniial condiion and v( δ, ) a giving he erminal condiion, he oluion being denoed by (Ē,Y,Z ) δ; hen, we olve (3)on δ, ] wih he previou Ē δ a iniial condiion and wih φ a giving he erminal condiion, he oluion being denoed by (Ē,Y,Z ) δ. We already know ha Ē δ mache Ē δ. o pach (Ē,Y,Z ) δ and (Ē,Y,Z ) δ ino a ingle oluion over he whole ime inerval,], i i ufficien o check he coninuiy propery Y δ = Y δ a done in Delarue 7]. hi coninuiy propery i a raighforward conequence of 7], Corollary 1.5: on δ, ], (Y ) δ ha he form Y = v(,ē ). In paricular, Y δ = v( δ,ē δ ) = v( δ,ē δ ) = Y δ. hi prove he exience of a oluion o (3) wih Ē = e a iniial condiion. We conclude ha, for any (,e),(3) admi a unique oluion (Ē,e Z,e ) aifying Ē,e = e and Y,e = φ(ē,e ). In paricular, he value func- i.e., he value a ime of he oluion (Y ) under he iniial condiion Ē = e] can be defined on he whole,] R. From 7], Corollary 1.5, and he dicuion above, we know ha he mapping e v(,e) i Lipchiz coninuou when i le han δ and ha, for any ion v : (,e) Y,e,Y,e,,], Y,e ha he form Y,e = v(,ē,e ) when i le han δ. In paricular, on any, δ ], δ being le han δ,(3) may be een a a uniformly ellipic FBSDE wih a Lipchiz boundary condiion. By Delarue and Guaeri 8], heorem 2.1 (ogeher wih he dicuion in Secion 8 herein), we deduce ha v belong o C (,] R) C 1,2 (,) R),ha e v(, ) i bounded on he whole,] and ha ee 2 v(, ) i bounded on every compac ube of,). 3 Moreover, (29) hold. By he maringale propery of (Y,e ), i i well een ha v i, 1]- valued. o prove ha i i nondecreaing (wih repec o e), we follow he proof of heorem 1. We noice ha (Ē,e ) aifie he SDE dē,e = v (,Ē,e) d + ( )dw,, 3 Specifically, 8], heorem 2.1, ay ha v belong o C (,) R) and ha e v(, ) i bounded on every compac ube of,). In fac, by Delarue 7], Corollary 1.5, we know ha v belong o C ( δ, ] R) and ha e v(, ) i bounded on δ, ] for δ mall enough.

22 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 117 which ha a Lipchiz drif wih repec o he pace variable. In paricular, for e e, Ē,e Ē,e,ohav(,e)= Eφ(Ē,e ) Eφ(Ē,e ) = v(,e ). We now eablih (28). For, he forward equaion in (3)haheform (31) Ē,e = e v (,Ē,e ) d + ( )dw. Since v i C 1 in pace on,) R wih bounded Lipchiz fir-order derivaive, we can apply andard reul on he differeniabiliy of ochaic flow (ee, e.g., Kunia monograph 1]). We deduce ha, for almo every realizaion of he randomne and for any,), he mapping e Ē,e (32) e Ē,e = 1 e v (,Ē,e ) e Ē,e d. i differeniable and In paricular, ( e Ē,e = exp e v ( ),Ē ) (33),e d. Since v i nondecreaing, we know ha e v on,) R o ha e Ē,e belong o, 1]. Since e v i alo bounded on he whole,) R, we deduce by differeniaing he righ-hand ide in (31) wih = ha e Ē,e exi a well and ha e Ē,e = lim e Ē,e, 1]. o complee he proof of (28), we hen noice ha for any,], d ( )Y,e Ē,e] = ( )dy,e ( )dw = ( ) Z,e 1 ] dw, o ha aking expecaion we ge ( )v(,e) e = E Ē,e]. Now, differeniaing wih repec o e,wehave ( ) e v(,e)= 1 E e Ē,e] 1, which conclude he proof of (28). I now remain o inveigae he Hölder norm (boh in ime and pace) of v, e v, ee 2 v and v. We fir deal wih v ielf. For <<<, v(,e) v(,e) = v(,e) v (,Ē,e = v(,e) v (,Ē,e = v(,e) v (,Ē,e ) ( + v,ē,e ) v(,e) ) + Y,e ) + Y,e Z,e r db r.

23 118 CARMONA, DELARUE, ESPINOSA AND OUZI From (28), we deduce v(,e) v(,e) 1 E Ē,e 1 1 ( + + e + E Zr,e db r ) 1/2 ] ] ( r) 2 dr + E Z,e 1/2 2 r dr ( ) 1/2 ] ( r) 2 dr + ( ) 1/2, ince Zr,e = ( r) e v(r,ēr,e ), 1].Soforɛ >, v i 1/2-Hölder coninuou in ime, ɛ], uniformly in pace and in he moohne of φ. Now, by Delarue and Guaeri 8], heorem 2.1, we know ha v aifie he PDE ( )2 (34) v(,e) + ee 2 2 v(,e) v(,e) ev(,e) =,,),e R, wih φ a boundary condiion. On, ɛ] R, ɛ>, equaion (34) i a nondegenerae econd-order PDE of dimenion 1 wih v a drif, hi drif being C 1/2,1 -coninuou independenly of he moohne of φ. By well-known reul in PDE (o-called Schauder eimae; ee, e.g., Krylov 9], heorem ) for any mall η>, he C (3 η)/2,3 η -norm of v on, ɛ] R i independen of he moohne of φ. REMARK 6. A announced, equaion (34) i of Burger ype. In paricular, i ha he ame fir-order par a equaion (26) Boundary behavior. Sill in he framework of a erminal condiion given by a mooh (i.e., nondecreaing Lipchiz) funcion wih value in, 1], we inveigae he hape of he oluion a approache. PROPOSIION 5. Aume ha here exi ome real + uch ha φ(e) = 1 on +, + ). hen, here exi a univeral conan c> uch ha for any δ>, v (, δ ) ( 1 exp δ 2 ), <. (35) c ( ) 3 In paricular, v(,e) 1 a uniformly in e in compac ube of ( +, + ). Similarly, aume ha here exi an inerval (, ] uch ha φ(e)= on (, ]. hen, for any δ>, v (, δ ) ( δ 2 ) (36) exp c ( ) 3. In paricular, v(,e) a uniformly in e in compac ube of (, ).

24 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 119 PROOF. We only prove (35) a he proof of (36) i imilar. o do o, we fix (,e),) R and conider he following yem: { de = d + ( )dw, dy = Z dw,, wih E = e a iniial condiion for he forward equaion and Y erminal condiion for he backward par. he oluion (Ē,e,Y,e = φ(e ) a,z,e ) given by Propoiion 4 wih Ē,e = e and Y,e = φ(ē,e ) aifie Y,e, 1] for any,] o ha E Ē,e almo urely for,]. Now,inceφ i nondecreaing, φ(e ) φ(ē,e ) almo urely, namely, Y Y,e. Seing v (,e)= Y (recall ha Y i deerminiic) we ee ha (37) Now, ince v (,e)= Eφ ( E v (,e) v(,e) 1. ( ) ) = Eφ e ( ) + ( )dw ( )dw + wih φ 1 +,+ ), by chooing e = + + ( ) + δ a in he aemen of Propoiion 5 we ge Eφ ( ( E ) ) = Eφ + + δ + ( )dw P + + δ + ] ] = P ( )dw δ = 1 P ] ( )dw δ and we complee he proof by applying andard eimae for he decay of he cumulaive diribuion funcion of a Gauian random variable. Noe indeed ha var( ( )dw ) = ( ) 3 /3 if we ue he noaion var(ξ) for he variance of a random variable ξ. he following corollary elucidae he boundary behavior beween and + + ( ) wih and + a above. COROLLARY 1. Chooe φ a in Propoiion 5. If here exi an inerval +, + ) on which φ(e)= 1, hen for α>and e< + + ( )+ ( ) 1+α we have v(,e) e + ( ) c (38) exp ( ) 1 2α ( ) α

25 111 CARMONA, DELARUE, ESPINOSA AND OUZI for he ame c a in he aemen of Propoiion 5. Similarly, if here exi an inerval (, ] on which φ(e) =, hen for α> and e> ( ) 1+α we have (39) PROOF. we have v(,e) e ( c + exp ( ) 1 2α ) + ( ) α. We fir prove (38). Since v(, ) i 1/( )-Lipchiz coninuou, v (, + + ( )+ ( ) 1+α) v(,e) + e + ( )+ ( ) 1+α herefore, = + e ( ) α. v(,e) v (, + + ( )+ ( ) 1+α) 1 ( ) α + e and applying (35), v(,e) e + exp ( c( ) 2α 1) ( ) α. For he upper bound, we ue he ame raegy. We ar from o ha v(,e) v (, ( ) 1+α) e + ( ) α, v(,e) e + exp ( c( ) 2α 1) + ( ) α Exience of a oluion. We now eablih he exience of a oluion o (25) wih he original erminal condiion. We ue a compacne argumen giving he exience of a value funcion for he problem. PROPOSIION 6. here exi a coninuou funcion v :,) R, 1] aifying: (1) v belong o C 1,2 (,) R) and olve (34), (2) v(, ) i nondecreaing and 1/( )-Lipchiz coninuou for any,), (3) v aifie (35) and (36) wih = + =, (4) v aifie (38) and (39) wih = + =,

26 SINGULAR FBSDES AND EMISSIONS DERIVAIVES 1111 and for any iniial condiion (,e),) R, he rong oluion (Ē,e ) < of (4) Ē = e v(,ē )d+ ( )dw, <, i uch ha (v(, Ē,e )) < i a maringale wih repec o he filraion generaed by W. PROOF. Chooe a equence of, 1]-valued mooh nondecreaing funcion (φ n ) n 1 uch ha φ n (e) = fore 1/n and φ n (e) = 1fore + 1/n, n 1, and denoe by (v n ) n 1 he correponding equence of funcion given by Propoiion 4. By Propoiion 4, we can exrac a ubequence, which we will ill index by n, converging uniformly on compac ube of,) R. We denoe by v uch a limi. Clearly, v aifie (1) in he aemen of Propoiion 6. Moreover, i alo aifie (2) becaue of Propoiion 4, (3) by Propoiion 5 and (4) by Corollary 1. Having Lipchiz coefficien, he ochaic differenial equaion (4) ha a unique rong oluion on,) for any iniial condiion Ē = e. If we denoe he oluion by (Ē,e ) <, Iô formula and (34) imply ha he proce (v(, Ē,e )) < i a local maringale. Since i i bounded, i i a bona fide maringale. We finally obain he deired oluion o he FBSDE in he ene of heorem 2. PROPOSIION 7. v and (Ē,e ) < being a above and eing Y,e = v (,Ē,e), Z,e = ( ) e v (,Ē,e), <, he proce (Ē,e ) < ha an a.. limi Ē,e a end o. Similarly, he proce (Y,e ) < ha an a.. limi Y,e a end o, and he exended proce (Y,e ) i a maringale wih repec o he filraion generaed by W. Moreover, P-a.., we have (41) and (42) 1 (, ) (Ē,e ) Y,e (Ē 1,e), ) Y,e = Y,e + Z,e dw. Noice ha Z,e i no defined for =. PROOF OF PROPOSIION 7. he proof i raighforward now ha we have colleced all he neceary ingredien. We ar wih he exenion of (Ē,e ) <

27 1112 CARMONA, DELARUE, ESPINOSA AND OUZI up o ime. he only problem i o exend he drif par in (4), bu ince v i nonnegaive and bounded, i i clear ha he proce ( v ( ),Ē ),e d < i almo urely increaing in, o ha he limi exi. he exenion of (Y,e ) < up o ime follow from he almo ure convergence heorem for poiive maringale. o prove (41), we apply (3) in he aemen of Propoiion 6. IfĒ,e = lim Ē,e >, hen we can find ome δ> uch ha Ē,e > + ( )+ δ for cloe o,ohay,e = v(,ē,e ) 1 exp cδ 2 /( ) 3 ] for cloe o,hai,y,e 1. Since Y,e 1, we deduce ha Ē,e > Y,e = 1. Inheameway, Ē,e < Y,e =. hi prove (41). Finally, (42) follow from Iô formula. Indeed, by Iô formula and (34), Y,e = Y,e + Z,e dw, <. By definiion, Z,e = ( ) e v(,ē,e ), <. By par (2) in he aemen of Propoiion 6, iiin, 1]. herefore, he Iô inegral dw Z,e make ene a an elemen of L 2 (, P). hi prove (42) Improved gradien eimae. Again uing andard reul on he differeniabiliy of ochaic flow (ee again Kunia monograph 1]), we ee ha formulae (32) and(33) ill hold in he preen iuaion of a diconinuou erminal condiion. We alo prove a repreenaion for he gradien of v of Malliavin Bimu ype. PROPOSIION 8. For,), e v(,e)admi he repreenaion e v(,e)= 2( ) 2 E lim v( δ,ē,e ) (43) δ δ e Ē,e dw ]. In paricular, here exi ome conan A> uch ha (44) up up e v(,e) < +. e >A

28 E PROOF. v ( δ,ē,e δ SINGULAR FBSDES AND EMISSIONS DERIVAIVES 1113 For δ>, Propoiion 7 yield ) ] e Ē,e dw = E = E δ Z,e δ ] dw e Ē,e dw ( ) e v (,Ē,e ] ) e Ē,e d. he bound we have on e v and ( e Ē,e ) < juify he exchange of he expecaion and inegral ign. We obain E v ( δ,ē,e ) ] δ δ e Ē,e dw = ( )E ( e v,ē,e)]] d. Similarly, we can exchange he expecaion and he parial derivaive o ha E v ( δ,ē,e ) ] δ δ e Ē,e ( dw = ( ) e Ev,Ē,e)] d. Since (v(, Ē,e )) δ i a maringale, we deduce E v ( δ,ē,e ) ] δ e Ē,e dw = e v(,e) δ ( )d = 1 2 ( δ )( + δ ) e v(,e). Leing δ end o zero and applying dominaed convergence, we complee he proof of he repreenaion formula of he gradien. o derive he bound (44), we emphaize ha, for e away from (ay, e.g., ) hi i very mall and decay exponenially fa a end o. On he complemen, ha i, for up Ē,e <, we know ha v(,ē,e ) end o a end o. Specifically, following he proof of Propoiion 5, here exi a univeral conan c > uch ha for any e 1 and,), ( ) 2 e v(,e) 2( ) 1/2 P 1/2 up Ē ],e e ), he probabiliy ha (Ē,e ] 2( ) 1/2 P 1/2 1 + up ( )dw ] 2( ) 1/2 P 1/2 up ( )dw 1 2( ) 1/2 exp ( c ( ) 3 he la line following from Roger and William 18], maximal inequaliy (IV.37.12). ),