CHALMERS GÖEBORG UNIVERSIY MASER S HESIS On Weak Diffeeniabiliy of Backwad SDE and Co Hedging of Inuance Deivaive ADAM ANDERSSON Depamen of Mahemaical Saiic CHALMERS UNIVERSIY OF ECHNOLOGY GÖEBORG UNIVERSIY Göebog, Sweden 28
hei fo he Degee of Mae of Science (3 cedi) On Weak Diffeeniabiliy of Backwad SDE and Co Hedging of Inuance Deivaive Adam Andeon CHALMERS GÖEBORG UNIVERSIY Depamen of Mahemaical Saiic Chalme Univeiy of echnology and Göebog Univeiy SE 412 96 Göebog, Sweden Göebog, Sepembe 28
Abac hi hei deal wih diibuional diffeeniabiliy of he oluion (X,Y,Z) o a quadaic non-degeneae fowad-backwad SDE. he diffeeniabiliy i conideed wih epec o he iniial value of he oluion X o he coupled fowad SDE. I i poved ha he oluion poce Y i weakly diffeeniable, and ha he oluion poce Z can be epeened uing he diibuional gadien of Y. hi eul i new in he way ha i elaxe echnical condiion impoed by peviou auho in a ignifican way and in a way ha i impoan e.g., in he applicaion decibed below. he poof make ue of Diichle pace echnique o conclude ha Y i a membe of a local Sobolev pace. Ou eul ae applied o deive new eul in mahemaical finance and inuance heoy. When deivaive ae wien on non-adeable undelying ae, uch a weahe, a ongly coelaed adeable ae pice poce i ued inead of he non-adeable one o paially hedge he ik of he deivaive. hi concep i known a co hedging. Applicaion fo non-diffeeniable Euopean ype pay off funcion ae given and explici hedging aegie ae deived uing a diibuional gadien. Keywod: Backwad ochaic diffeenial equaion; Diibuional diffeeniabiliy; Diichle pace; Co hedging; Inuance deivaive; Weahe deivaive; Explici hedging aegy. i
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Acknowledgemen I would vey much like o hank my upevio pofeo Paik Albin a Chalme fo good advice and encouagemen duing he wok a well a fo hi excellen coue in ochaic calculu and ochaic pocee. I would alo like o hank pofeo Boualem Djehiche a KH fo fi giving me he uggeion o wok wih co hedging. he ubjec wa pefec fo me. Mo of all I would like o hank pofeo Pee Imkelle, doco Sefan Ankichne and PhD uden Gonçalo Do Rei a Humbold Univeiy in Belin fo inviing me fo a wo day dicuion on co hedging and BSDE in Belin and epecially o Sefan Ankichne and Gonçalo Do Rei fo giving me fuiful feedback on my wok via mail. I vey much appeciae he way mahemaic i done in Belin. hank you alo PhD uden Maia Sundén and pofeo Sig Laon fo anweing my queion in a clea and pecie way. Finally, I would like o hank my wife Feeheh and my daughe Alice fo hei uppo and paience. iii
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Conen 1 Inoducion 1 2 Backwad ochaic diffeenial equaion 5 2.1 Fowad-backwad ochaic diffeenial equaion............... 5 2.2 BSDE wih andom Lipchiz geneao.................... 7 2.3 Hioy....................................... 8 3 Weak deivaive and Sobolev pace 9 3.1 he Sobolev pace H 1 loc.............................. 9 3.2 wo Diichle pace d and d........................... 1 3.3 Deniie and non-degeneacy of SDE...................... 11 4 Weak diffeeniabiliy of quadaic non-degeneae FBSDE 13 4.1 Aumpion.................................... 13 4.2 Some ueful eul................................. 14 4.3 Main eul..................................... 16 5 Applicaion o inuance and finance: Opimal co hedging 27 5.1 Aumpion and make model.......................... 27 5.2 Soluion o he opimal co hedging poblem via a FBSDE.......... 3 5.3 Explici hedging aegy uing he weak pice gadien............ 32 6 Concluion and dicuion 33 v
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Chape 1 Inoducion Imagine ha you ae he owne of a Swedih ice ceam facoy. hen you ae expoed o weahe ik. A wam and unny umme, people will pend hei ime on he beach, wih an ice ceam in hei hand. A cold o ainy umme, hey will complain ove he Swedih weahe o pehap avel o a wame place. hey will anyway no ea ice ceam o a gea exen. I i clea ha you economic gain, fom elling ice ceam, depend on he weahe. So, how can you poec youelf again uch ik? Inuance ae ofen fo maeial o economical loe, no fo defaul income, due o bad weahe. One way could be o wie a financial conac, like an opion, on ome weahe index. Accumulaed aveage empeaue o un hou duing a umme could be uiable indice. hen he ik will be pead. In cae of unfavoable umme weahe, you ge money accoding o he conac. Unde favoable weahe you ge nohing. In eihe cae you pay he pemium fo he conac. You expoe youelf of a lowe ik. Say, ha you chooe o buy, o ge ho in, a Euopean call opion wih ike pice K, wien on ome (aificial) unhine index. he index will have value X a ime of mauiy. he andom income a ime i given by (X K) + = max(, X K) := F (X ). How hall he conac be piced? Now, we mu un he pepecive o he elle of he deivaive, having he long poiion. He obligaion o pay F (X ) a ime, implie a ik. If he undelying X wee a adable ae he would hedge he ik of he deivaive. hi would be done by inveing in he undelying ae accoding o an opimal hedging aegy. he hedged ik would be conideed when he e he pemium, by he machiney of Black-Schole picing heoy. I would be nice if unhine wa adable, bu i in. She can no buy heelf a pofolio of unhine. Picing wihou hedging would imply a geae ik, and he would no la long unle he ge a high pemium. A way o ge aound hi, fo he, could be o inve in a adable ae, coelaed o unhine. In uch a way he ik could paially be hedged by inveing in he coelaed ae. Ae poibly coelaed o unhine ae fo example heaing oil fuue o eleciciy fuue. he concep i known a co hedging, and he kind of deivaive, wien on non-adable undelying, ae called inuance deivaive. Aco moe likely o buy inuance deivaive ae enegy companie, eniive o cold umme and wam wine. Chicago Mecanile Exchange wa he fi in 1997 o offe deivaive wien on accumulaed heaing degee day (chdd) and cooling degee day (ccdd). A heaing degee day (HDD) and (CDD) i given by 1
HDD = max(, 18 ) and CDD = max(, 18), epecively, whee i he aveage empeaue duing a day. Saiic ha hown ha when he aveage empeaue i 18 degee Celiu he enegy conumpion i he lowe. When i i highe, enegy i ued fo cooling and when i i lowe, enegy i ued fo heaing. he chdd index i given by 31 chdd = HDD i i=1 whee HDD i i he daily HDD 31 day back in ime. I can be een a a moving aveage poce. he ccdd index i defined analogouly. Hioical weahe daa i oday ued o pice chdd. he diibuion of he oucome of he index i eimaed and heeafe he diibuion of he payoff. he deivaive i piced by he expeced payoff, dicouned a he ik fee ae [8. hi doe no involve co hedging o any hedging a all. Hedging can be aic. hen an invemen i done a ime zeo, and no invemen i done heeafe. Hedging can alo be dynamic. hen he hedge inve accoding o a hedging aegy. he numbe of hae inveed change in ime, a infomaion i evealed, and new compuaion can be done. An appoach o dynamic co hedging and picing i by eing up a ochaic conol poblem. An opimal aegy i ough ha maximize he expeced uiliy of ha invemen. hi can be done analyically by olving he Hamilon-Jacobi- Bellman paial diffeenial equaion [1 o by a ochaic appoach uing fowad-backwad ochaic diffeenial equaion (FBSDE) [2. he fome appoach i limied o he cae of one-dimenional ae. he lae appoach can be applied wih muliple-dimenion boh fo he adable and he non-adable ae. One dawback of he FBSDE appoach i ha he payoff funcion mu be mooh. Hence, Euopean pu and call opion can no be piced and hedged wih ha appoach. In hi hei mahemaical eul ae poved, ha make i poible o elax he moohne popey of he payoff funcion. Wih hee eul Euopean pu and call opion can be piced and hedged, a lea heoeically, when he non-adable poce aifie a non-degeneacy condiion. Suiable numeic mu be ued o implemen hi. he mahemaical eul ae abou diffeeniabiliy in he weak ene of FBSDE. hey e on ochaic calculu, ome diibuion heoy, meaue heoy and he heoy of wo pecific Diichle pace. Knowledge in ochaic calculu i aumed, of he eade, a well a ome knowledge abou meaue heoy and Lebegue inegaion. Familiaiy wih Sobolev pace and diibuion heoy make he eading conideably eaie. he wok i oganized a follow: Chape 2 deal wih BSDE and FBSDE. he fi ecion inoduce he ubjec. he econd ecion conain moe pecific eul needed in hi hei. he chape end wih ome hioy of he heoy. Chape 3 fi give an inoducion o weak deivaive and Sobolev pace. Secion wo inoduce Diichle pace, eenial fo he poof of Chape 4. he chape end wih a ecion abou deniie fo ochaic diffeenial equaion and explain he concep of non-degeneacy of SDE. Chape 4 conain he mahemaical eul in hi hei. I a wih aumpion and a ecion wih ueful eul. In Secion 3 he main eul ae aed and poved. 2
Chape 5 i abou opimal co hedging. he chape peen he make model and oluion appoach of finding pice and co hedging aegie of inuance deivaive. In he la ecion he eul fom Chape 4 ae applied o deive an explici expeion fo he hedging aegie. Chape 6 conain a dicuion and concluion. 3
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Chape 2 Backwad ochaic diffeenial equaion Backwad ochaic diffeeniable equaion have poved o be ueful in opimal ochaic conol heoy, mahemaical finance and paial diffeenial equaion (PDE). In finance he conol pocee uually ae omehow elaed o invemen aegie. he fi ecion of hi chape inoduce BSDE and FBSDE peen he inimae connecion beween BSDE and he maingale epeenaion heoem. hi connecion i cucial fo he undeanding of BSDE. Secion 2 inoduce BSDE wih andom Lipchiz coninuou geneao. An exience and uniquene eul i alo peened. All eul peened in hi chape ae well known. he la ecion peen ome hioy of BSDE. 2.1 Fowad-backwad ochaic diffeenial equaion Le W be a d-dimenional Wiene poce on a fileed pobabiliy pace (Ω, F, {F } [,, P). he filaion {F } [, i he naual filaion of W compleed by he P-null e of Ω. A fowad backwad ochaic diffeenial equaion (FBSDE) i a yem of equaion, dx = b(, X )d + σ(, X )dw, dy = f(, X, Z )d + Z dw, [,, X = x, Y = g(x ). (2.1) he coefficien b : [, R m R m and σ : [, R m R m d ae uppoed o be meauable and aify Lipchiz condiion and linea gowh condiion in he pace vaiable, i.e. C : { b(, x) b(, x) + σ(, x) σ(, x) C x x, (2.2) b(, x) + σ(, x) C(1 + x ) (, x, x) [, R m R m. he equaion fo X i a fowad Iô SDE. Noice ha σ(, X )dw i a maix muliplicaion. he nom σ i he Fobeniu nom σ = ace(σ σ), (2.3) 5
whee σ i he anpoe of σ. he equaion fo Y i known a a backwad ochaic diffeenial equaion. he funcion f : Ω [, R m R d R i called he geneao of he FBSDE and g : R m R deemine he eminal value. he equaion fo Y i alo a fowad SDE, bu he conol poce Z conol i o aify he eminal value. I will be hown below ha Y ha a deeminiic iniial value. he poce X i m-dimenional, he poce Y i one-dimenional and he poce Z i d-dimenional. he iple (X, Y, Z) i called he oluion of FBSDE (2.1). o be able o undeand BSDE ecall he maingale epeenaion heoem. heoem 2.1. [2 he maingale epeenaion heoem. Suppoe ha M i a quae inegable maingale w... F. hen hee exi a unique, pedicable, quae inegable and d-dimenional poce Z uch ha: almo uely, fo all [, ). M = E[M + Z dw he pocee Y and Z ae defined a follow. Define he maingale [ M = E g(x ) + f(, X, Z )d F fo [,. (2.4) Hee, M i unde uiable aumpion, peened lae, quae inegable. he maingale epeenaion heoem heefoe implie he exience of a unique pedicable d-dimenional poce Z uch ha: Le now, M = M + Z dw. (2.5) I follow ha: Y = M Y = M f(, X, Z )d. f(, X, Z )d + [ M = E g(x ) + f(, X, Z )d Z dw (2.6) a in (2.1). Equaion (2.4) and (2.5) conideed ogehe i he hid equaion ha deemine (X, Y, Z). Noice ha Y ha a deeminiic iniial value M. Since f(, X, Z )d i F - meauable (2.6) and (2.4) give ha [ Y = E g(x ) + f(, X, Z )d F. (2.7) Fuhe, Y = M f(, X, Z )d + Z dw = g(x ) fom (2.6) and (2.7). Adding = g(x ) M + f(, X, Z )d Z dw o (2.6) he mo common fom of a BSDE i obained: 6
Y = g(x ) + f(, X, Z )d Z dw, [,. (2.8) he poce {X, Y, Z } [, i a Makov poce. Condiioned on X = x, fo (, x) [, R m, he FBSDE i given by: X,x = x + Y,x = g(x,x ) + b(, X,x )d + f(, X,x, Z,x σ(, X,x )dw )d Z,x dw, [,. (2.9) Equaion (2.9) will be denoed FBSDE(b, σ, g, f). he iniial value i uppeed in he noaion ince i houghou hi hei will be an abiay veco xr m. Le ξ be a quae inegable and F -meauable andom vaiable and f : Ω [, R d R be a geneao funcion. he BSDE Y = ξ + f(, Z )d Z dw, [,. (2.1) will be denoed BSDE(ξ, f). he poce Z i deemined analogouly a fo a FBSDE. When nohing i ele i aid he oluion o FBSDE(b, σ, g, f) and he oluion o BSDE(ξ, f) will be denoed (X, Y, Z) and (Y, Z), epecively. A fowad SDE wih coefficien b and σ will be denoed SDE(b, σ). 2.2 BSDE wih andom Lipchiz geneao Le {H } [, be an inegable, pedicable and poiive poce. he poce H dw i called a bounded mean ocillaion (BMO) maingale if [ E τ H 2 d F τ D (2.11) almo uely, fo all opping ime τ [, and ome conan D >. he malle D > ha aifie (2.11) i called he BMO nom of H dw and will be denoed H dw BMO. Le inoduce he following poce pace: S p (R k ) i he pace of all k-dimenional pedicable pocee {Y } [, uch ha E[up [, Y p <, H p (R d ) i he pace of all d-dimenional pedicable pocee {Z } [, uch ha E[( Z 2 d) p/2 <, S (R) = p>2 S p (R), hoe pocee ae bounded fo all, P-almo uely. H (R d ) = p>2 H p (R d ). 7
heoem 2.2. [6 (Exience and uniquene fo BSDE wih andom Lipchiz condiion) Aume ha BSDE(ξ, f) aifie he andom Lipchiz condiion f(, z) f(, ẑ) H z ẑ, (, z, ẑ) [, R d R d, (2.12) whee H i inegable, pedicable, non-negaive and H dw i a BMO-maingale. Fuhe aume ha fo ome p > 1 i hold ha [ ( ) p E ξ p + f(, ) d < (2.13) and f(, z) g() + H z, (, z) [, R d, whee g : [, Ω R + i a funcion aifying [ ( ) p E g()d <. hen hee exi a unique oluion (Y, Z) S p (R) H p (R d ) fo 1 < p < p. 2.3 Hioy In 1978 Bimu [3 inoduced a linea BSDE, a he adjoin equaion o he maximum pinciple, in opimal ochaic conol heoy. Fi, in 199, Padoux and Peng publihed a pape [21 wee hey poved he exience of an adaped oluion, o a BSDE, in he cae of Lipchiz coninuou geneao. Afe ha, he ubjec gew apidly. he awaene, of he poibiliy o ue BSDE in finance, inceaed. In 1997 Kaoui, Peng and Quenez publihed an impoan and long efeence pape on BSDE [12, conaining heoy a well a applicaion in finance. I i ill fequenly efeed o in pape publihed oday. In 2 Kobylanki publihed an impoan pape [13 on quadaic BSDE and he connecion o vicoiy and Sobolev oluion o non linea paabolic PDE. ha pape oo i ill fequenly efeed o in pape dealing wih quadaic BSDE. A lage amoun of ohe pape ha been publihed on he ubjec. hee ae a lo of vaiaion, BSDE diven by Levy pocee, BSDE wih jump o delay, numeic of BSDE, ec.. Much of he moivaion ha come fom mahemaical finance. BSDE ae ueful in uiliy maximizaion poblem in incomplee make, i.e., make wee hee ae ik no poible o hedge compleely. So fa hee han been wien any book abou he ubjec. Howeve hee ae wo confeence ex namely [11 fom 1997 and [16 fom 1999. he pape in hi line ha ha i fundamenal fo hi hei i [2 Picing and hedging of deivaive baed on non-adable undelying by Ankichne, Imkelle and Do Rei. 8
Chape 3 Weak deivaive and Sobolev pace In he fi ecion of hi chape o called weak o diibuional deivaive will be inoduced. Weak deivaive need no be funcion bu ahe diibuion. he funcion pace ha funcion wih weak paial deivaive live in, namely Sobolev pace i alo inoduced. he main ool fo poving ha he oluion Y of a FBSDE i a membe of a Sobolev pace i he concep of Diichle pace. hoe will be inoduced in he econd ecion ogehe wih a ueful eul. In he hid ecion, he eul of ecion wo will be applied and exended o ochaic diffeenial equaion. Reul fo degeneae SDE wih andom iniial value will be peened ogehe wih an inoducion o non-degeneae SDE. All he eul in hi chape ae well-known. 3.1 he Sobolev pace H 1 loc A funcion ψ : R m R i aid o have compac uppo if hee i a compac e K R m uch ha ψ = ouide K. A coninuouly diffeeniable funcion wih compac uppo i called a e funcion. If g : R R i a quae inegable coninuouly diffeeniable funcion, hen inegaion by pa applie K g dψ dx = dx i K dg dx i ψdx. (3.1) he bounday em diappea ince ψ vanihe on he bounday of K. hi opeaion i obviouly poible when g i coninuouly diffeeniable. If g i no coninuouly diffeeniable bu hee exi a funcion dg/dx i in { } L 2 (K) = f : K R : f(x) 2 dx < K uch ha (3.1) hold, hen dg/dx i i called a weak paial deivaive. If hi deivaive exi, hen i i unique in L 2 (K) by he Riez epeenaion heoem. he pace L 2 loc (Rm ) i he pace of funcion fom R m o R ha ae Lebegue quae inegable on evey compac ube K R m. Define he local Sobolev pace Hloc 1 (Rm ) by H 1 loc (Rm ) = { f L 2 loc (Rm ) : x i f L 2 loc (Rm ), 1 i m }. 9
(ee [15). hi pace i of inee in hi hei a i will be poved ha Y H 1 loc (Rm ) whee (X, Y, Z) i he oluion o a ceain quadaic FBSDE. he non-diffeeniable funcion conideed in hi ex will be coninuouly diffeeniable almo eveywhee. A ube of hoe funcion ae he locally Lipchiz coninuou funcion, and a ube of hoe funcion in un ae he globally Lipchiz coninuou funcion. Example 3.1. he weak deivaive of he payoff funcion of a Euopean call opion F (x) = max(, x K), K R, i he equivalence cla of funcion df dx (x) =, x < K C, x = K 1, x > K fo abiay C R. hi make bee ene in L 2 (R) whee df/dx i unique ahe han abiay ince objec in L 2 ae equivalence clae up o Lebegue almo eveywhee equaliy. On he ohe hand, if C i fixed hen df/dx i called a veion. 3.2 wo Diichle pace d and d Le h : R m R be a fixed, coninuou and poiive funcion aifying R h(x)dx = 1 and R m x 2 h(x)dx <. he pace d i defined by m whee L 2 (R m, h) = d = { { f L 2 (R m, h) : f : R m R : x i f L 2 (R m, h), 1 i m }, ( ) 1/2 f(x) R 2 h(x)dx := f L2(R m,h) < }. m he deivaive i conideed in he weak ene. he pace d equipped wih he nom f d = [ f 2 L 2 (R m,h) + m i=1 1/2 f 2 L x 2 i (R m,h) i a o called claical Diichle pace. hi pace i a Hilbe pace and a ubpace of he local Sobolev pace Hloc 1 (Rm ). Nex, define an enlaged pobabiliy pace ( Ω, F, P), whee Ω = Ω R m, F i he poduc σ-algeba of F and he Boel σ-algeba of R m and P i he poduc meaue P hdx. he expeced value on ( Ω, F, P) will be denoed Ẽ. Le L 2 ( Ω) = { f : Ω R m R : f L2 (e Ω) = Ẽ[ f 2 1/2 = ( ) 1 } E[ f(x) R 2 2 h(x)dx <, m and D i be he pace of funcion u : Ω R m R ha ha a veion ũ uch ha ε ũ(ω, x+εe i ) i locally aboluely coninuou (ω, x) Ω, ε R, 1 i m. Hee e i i he i-h uni veco in R m. Locally aboluely coninuou funcion ae coninuouly diffeeniable 1
almo eveywhee. Lipchiz coninuou funcion aifie hi popey. Fuhe define he opeao i on D i by i u(x, ω) = lim ε ũ(x + εe i, ω) ũ(x, ω) ε fo ũ being a locally aboluely coninuou veion of u. Now we ae eady o define a econd Diichle pace d on ( Ω, F, P) by { d = f L 2 ( Ω) ( m ) } D i : i f L 2 ( Ω), 1 i m, he pace d equipped wih he nom e d = [ i=1 2 L 2 (e Ω) + m i=1 i 2 L 2 (e Ω) 1 2 i a o called geneal Diichle pace. In Chape 4 d will be ued in he poof ha he poce Y of he oluion (X, Y, Z) of a FBSDE belong o d. he following popoiion connec he wo pace: Popoiion 3.2. [5 If u d, hen u(, ω) d and x i u(x, ω) = i u(x, ω) P a.., 1 i m. 3.3 Deniie and non-degeneacy of SDE hi ecion exend he eul fom he peviou ecion, o ochaic diffeenial equaion. I alo explain impoan popeie of ceain SDE. Conide fo [, he SDE(b, σ). he coefficien b : [, R m R m and σ : [, R m R m d aifie global Lipchiz and linea gowh condiion (2.2). he SDE i aid o be non-degeneae if, fo ome conan C >, ξ σ(, x)σ (, x)ξ C ξ 2, (, x, ξ) [, R m R m (3.2) hold. Hee, σ and ξ denoe he anpoe of σ and ξ. heoem 3.3. [4 Given aumpion (3.2), X x ha a deniy fo all (, x) (, R m. Example 3.4. Conide he SDE: ( 1 dx = 1 ) ( dw, d X 1 = 1 ) dw, fo a 2-dimenional Wiene poce W and X = X =. he fi ha oluion X = (W 1, W 2 ) and he econd X = (W 1, W 1 ). I i clea ha X will evolve feely in he enie R 2 -plane and ha X will evolve along he line L = {(x, x) : x R} R 2. X i nondegeneae and X i degeneae. hi i a ivial example, bu given he non-degeneacy condiion (3.2), he pah of he oluion o SDE(b, σ), will no be limied o any Lebegue null-e of R m. 11
Example 3.5. Lae in he hei, non-degeneacy will be ued o conclude ha E [ ξ 1 (X ) ξ 2 (X ) =, (3.3) whee ξ 1 : R m R and ξ 2 : R m R aifie ξ 1 (x) = ξ 2 (x) excep a Lebegue nulle of R m. Suppoe ha X and X ae he pocee in he peceding example and ha ξ 1 (x) = ξ 2 (x) excep on he line L. hen (3.3) hold fo X bu no fo X. he concluion would be impoible fo any degeneae SDE egadle of wha null-e ξ 1 (x) and ξ 2 (x) diffe on. 12
Chape 4 Weak diffeeniabiliy of quadaic non-degeneae FBSDE In hi chape he main mahemaical conibuion of hi hei will be peened. Reul fo claical diffeeniabiliy of he oluion poce Y of a quadaic FBSDE, poved in [2, will be genealized. In he fi ecion ou echnical aumpion will be peened and in he econd ecion a collecion of eul ha will be needed ae lied. In he hid ecion, weak diffeeniabiliy of Y will be aed and poved when he coupled fowad SDE i nondegeneae. A ueful epeenaion eul i alo poved. In he la ecion he ame will be poved when he fowad SDE i degeneae. In ha cae a lighly diffeen FBSDE will be conideed and finally poved o epeen he weak gadien of Y. 4.1 Aumpion Conide FBSDE(b, σ, g, f). he Coefficien b and σ ae aumed o aify Lipchiz and linea gowh condiion (2.2) ogehe wih non-degeneacy condiion (3.2). he eminal funcion g : R m R i aumed o be deeminiic, bounded, meauable and Lipchiz coninuou. he geneao f : [, R m R d R i aumed o be meauable, coninuouly diffeeniable in x and z and aify f(, x, z) C(1 + z 2 ) a.., f(, x, z) f(, x, z) C(1 + z ) x x a.., z f(, x, z) C(1 + z ) a.., x f(, x, z) x f(, x, z) C(1 + z + z )( x x + z z ) a.., fo ome conan C >, (, x, x, z, z) [, R m R m R d R d. When hee aumpion hold he FBSDE i aid o aify aumpion (A). We call he FBSDE(b, σ, g, f) unde aumpion (A) quadaic, o diinguih i fom geneao globally Lipchiz coninuou in z. 13
4.2 Some ueful eul he following momen eimae will be he main ool when poving ou main eul: Lemma 4.1. [2(Momen eimae fo BSDE wih andom Lipchiz geneao) Conide he BSDE(ξ, f). Suppoe ha condiion (2.12) hold and ha fo all β 1 we have f(, ) d Lβ (P). Le p > 1. hen hee exi conan q > 1 and C >, depending only on p, and he BMO-nom of H d whee {H } [, i he andom Lipchiz bound, uch ha we have E [ [ ( [ ) 1 up Y 2p + E ( Z 2 d) p C E ξ 2pq + ( f(, ) d) 2pq q. [, he following hee eul will alo be of gea impoance. Lemma 4.2. [17 Conide he FBSDE(b, σ, g, f). Given aumpion (A), he poce Z,x dw i a BMO-maingale. he BMO-nom only depend on he eminal value, he funcion f(, X,x, ), and he duaion. Lemma 4.2 how ha if Z,x i he andom Lipchiz bound fo a geneao of a BSDE, hen he momen eimae Lemma 4.1 can be applied. hi will be ued fequenly in he poof of he main eul of hi hei. Popoiion 4.3. Unde aumpion (A) he FBSDE(b, σ, g, f) aifie a andom Lipchiz condiion wih BMO bound. Moeove, he oluion (X, Y, Z) i unique wih (X, Y, Z) S (R m ) S (R) H (R d ). Poof. Fi, X aifie he uual Iô condiion and i hence well defined and unique. Popoiion 4.7 guaanie X S p (R m ) fo all p 2, i.e., X S (R m ). Nex, he geneao i diffeeniable and hence by he mean value heoem and aumpion z f(, x, z) C(1+ z ), λ [, 1 : f(, x, z) f(, x, ẑ) z f(, x, λz + (1 λ)ẑ) z ẑ C(1 + λz + (1 λ)ẑ ) z ẑ C(1 + z + ẑ ) z ẑ. hi implie ha he geneao aifie a andom Lipchiz condiion wih Lipchiz bound C(1 + Z,x + Ẑ,x ). Lemma 4.2 implie ha Z,x dw i a BMO maingale and hence he bound aifie he aumpion of heoem 2.2. Moeove he boundedne aumpion on g and f(,, ) implie (Y, Z) S (R) H (R d ) by heoem 2.2. Lemma 4.4. Conide he FBSDE(b, σ, g, f). Given aumpion (A), he mapping x Y,x i Lipchiz coninuou fo all [, and [,. Poof. he Lemma a Lemma 6.3 in [2 wa aed unde he onge aumpion of heoem 4.5. Howeve, hi wa done fo noaional impliciy. he poof caie ove o ou eing wihou change. Nex follow wo impoan eul on claical diffeeniabiliy fo quadaic FBSDE. 14
heoem 4.5. [2 Conide he FBSDE(b, σ, g, f). Aume (A), wih he addiional equiemen ha he eminal funcion g i wice coninuouly diffeeniable and b and σ ae coninuouly diffeeniable in x wih Lipchiz coninuou fi deivaive. hen fo evey fixed [, X,x and Y,x ae coninuou in and coninuouly diffeeniable in x. Moeove, hee exi a poce x Z,x H 2 (R d ) uch ha ( x Y,x, x Z,x ), fo [,, i he oluion o he BSDE: x Y,x = x g(x,x ) xx,x + [ x f(, X,x x Z,x dw, Z,x ) x X,x + z f(, X,x, Z,x ) x Z,x d heoem 4.6. Le he aumpion of he peviou heoem hold. hen fo [, and u(, x) := Y,x Z,x fo almo all [,, P-almo uely. = x u(, X,x )σ(, X,x ) Poof. he heoem wa aed [2 wih he exa aumpion of he exience of a equence {f n } n 1, of geneao, Lipchiz coninuou in z, conveging locally unifomly o f. he aumpion i no needed ince i i alway poible o find uch a equence, when f i quadaic. he following eimae fo claical SDE will be needed. heoem 4.7. [14 Conide he SDE(b, σ) wih iniial value x : Ω R m. Aume ha b : Ω [, R m R m and σ : Ω [, R m R m d ae Lipchiz coninuou in he pace vaiable. hen fo any p 2, hee exi a conan C, only depending on p, and he Lipchiz bound of b and σ, uch ha: E [ up X p C [, ( [ ) E [ x p + E ( b(, ) p + σ(, ) p )d. Finally, he following inequaliy will be ued fequenly. Lemma 4.8. Fo x i,..., x k V, whee (V, ) i a nomed veco pace i hold ha fo p, x 1 +... + x k p k p ( x 1 p +... + x k p ). Poof. x 1 +... + x k p (k max( x 1,..., x k )) p k p ( x 1 p +... + x k p ). 15
4.3 Main eul In hi Secion weak diffeeniabiliy of FBSDE(b, σ, g, f) will be aed and poved unde aumpion (A). he following FBSDE i known a he vaiaional equaion of FBSDE(b, σ, g, f). Φ,x i, = e i + Ψ,x i, = xg(x,x )Φ,x + z f(, X,x d x b(, X,x )Φ,x i, d + j=1 i, + ( x f(, X,x, Z,x x σ j (, X,x )Φ,x i, dw j )Φ,x i,, Z,x )Γ,x i, )d Γ,x i, dw, i = 1,..., m. (4.1) fo [,. I will be denoed i FBSDE(b, σ, g, f), i = 1,..., m, componenwie o FBSDE(b, σ, g, f) ohewie. Hee, x b, x σ and x g ae he gadien of b, σ and g in he weak ene. he index i denoe he i:h column of Φ, Ψ and Γ. Fuhe, x σ j denoe he gadien of he j h ow of σ. If g, σ, g, X, Y and Z wee diffeeniable w... x hen Φ, Ψ and Γ would be he gadien of X, Y and Z. he following heoem ae ha Y,x i weakly diffeeniable wih epec o x and ha Ψ,x i i weak gadien. heoem 4.9. Le aumpion (A) hold. hen, (i) he funcion, x Y,x belong o H 1 loc (Rm ) P-a.., [,, [,. (ii) he weak gadien x Y,x = Ψ,x, fo almo all x P-a.., [,, [,, whee (Φ,x, Ψ,x, Γ,x ) S (R m m ) S (R 1 m ) H (R d m ) i he unique oluion o FBSDE(b, σ, g, f). he poof will be divided ino fou ep. he main idea i o pove ha Y, L 2 ( Ω) and x Y, L 2 ( Ω), i.e. ha Y,x belong o he Diichle pace d, [,, [,, and ha Equaion 4.1 ha a well defined oluion. Afe ha, he eul follow eaily in he la ep of he poof. he poof echnique ae mainly hoe of [19 and [2. he fi of hee pape [19 give a imila poof fo weak diffeeniabiliy fo BSDE wih Lipchiz coninuou geneao in x and z and m = d. he econd of hee pape [2 conain eul and echnique fo woking wih quadaic BSDE and BSDE aifying a andom Lipchiz condiion. Poof. Sep 1: Le φ : R m R be a infiniely coninuouly diffeeniable and nonnegaive funcion wih uppo in he uni ball and R φ(x)dx = 1. hen he funcion, known a m mollifie, defined by φ n (x) = n m φ(nx), n 1 ha he ame popeie fo all n, only ha he uppo i vanihing a n. By convoluion, define he funcion b n (, x) = (b φ n )(, x) = b(, x ξ)φ R n (ξ)dξ, m σ n (, x) = (σ φ n )(, x) = σ(, x ξ)φ R n (ξ)dξ, (4.2) m g n (x) = (g φ n )(x) = 16 R m g(x ξ)φ n (ξ)dξ.
I i known ha b n, σ n and g n ae in C, n Z + and convege unifomly o b, σ and g and ha x b n, x σ n and x g n convege dx-a.e. o x b, x σ and x g a n [9. Conide fo (, x, n) [, R m Z + and [, he equence of FBSDE(b n, σ n, g n, f) wih coeponding oluion (X,x,n, Y,x,n, Z,x,n ). Fi, conide he convegence fo X,x,n. Le he diffeence aifie X,x,n := X,x,n X,x. X,x,n = (b n (, X,x,n ) b(, X,x ))d + (σ n (, X,x,n By uing heoem 4.7 i follow ha fo all p 2, C > uch ha E [ up X,x,n p [, ) σ(, X,x ))dw. [ CE [ b n (, ) b(, ) p + σ n (, ) σ(, ) p d (4.3) a n. Hee he unifom conan C exi ince i only depend on p, and he Lipchiz bound fo b, b n, σ and σ n. he Lipchiz bound of b n and σ n in highe han ha of b and σ and p and ae fixed. Fuhe he convegence in (4.3) i bounded. By aumpion, b(, ) and σ(, ) ae bounded fo all [,. he ame hold fo b n (, ) and σ n (, ), (n, ) Z + [,. he coefficien b n and σ n convege unifomly o b and σ a n. Le he diffeence poce ( Y,x,n Y,x,n Z,x,n g,x,n, Z,x,n := Y,x,n := Z,x,n Y,x, Z,x, := g n (X,x,n ) g(x,x ). ) aifie he BSDE( g(x,x ), f n ) wih geneao f n := f(, X,x,n, Z,x,n ) f(, X,x, Z,x ) = f(, X,x,n, Z,x,n ) f(, X,x, Z,x,n ) + f(, X,x, Z,x,n ) f(, X,x, Z,x ). Line inegal anfomaion ae ued, o how ha he geneao aifie a andom Lipchiz condiion. he chain ule give, f(, X,x,n, Z,x,n ) f(, X,x, Z,x,n ) = = 1 1 d f(, X,x + θ(x,x,n dθ x f(, X,x + θ(x,x,n X,x X,x ), Z,x,n )dθ ), Z,x,n )dθ X,x,n := J n X,x,n 17
and imilaly f(, X,x, Z,x,n ) f(, X,x, Z,x ) = 1 z f(, X,x, Z,x + θ(z,x,n Z,x ))dθ Z,x,n Hence := H n Z,x,n. f n (, v) = H n v + J n X,x,n. he geneao clealy aifie he andom Lipchiz condiion, f n (, v) f n (, v ) H n v v fo all (, v, v ) [, R d R d. Now, by aumpion and he mean value heoem, H n = 1 C z f(, X,x, Z,x + θ(z,x,n 1 (1 + Z,x + θ(z,x,n Z,x ))dθ Z,x ) )dθ (4.4) = C(1 + Z,x + ξ(z,x,n C(1 + Z,x + Z,x,n ) Z,x ) ) fo ome ξ 1 and fo each n Z +. Lemma 4.2 ae ha Z,x dw and Z,x,n dw ae BMO-maingale n Z + and i follow fom (4.4) ha o i alo Hn dw, n Z +. Nex, a unifom bound fo he BMO-nom of Z,x,n dw, ae ough fo all n Z +. By Lemma 4.2 he BMO-nom only depend on g n (X,x,n ), f(, X,x,n, ), [,, and. he eminal funcion g i bounded, and he mollifie φ n inegae o one, hence he convoluion (4.2) doe no inceae he bound. I follow ha he g n ae unifomly bounded. Nex, by he aumpion f(, x, z) C(1 + z 2 ) i follow ha f(, X,x,n, ) i unifomly bounded. Finally, i he ame fo all n. I follow ha he BMO-nom of Z,x,n dw BMO < β, fo ome β, n Z +. heefoe he ame hold fo Hn dw fom he eimae (4.4). Now, he momen eimae Lemma 4.1 will be applied on Y,x,n and Z,x,n. he conan C n > and q n > 1, appeaing in evey eimaion of Y,x,n and Z,x,n, n Z +, only depend on p, and he BMO-nom. Since a unifom bound of he BMO-nom ha been poved, i follow ha q n and C n, ae unifomly bounded by ome conan C > and q > 1. Lemma 4.1 heefoe give ha, fo any p > 1 and evey n Z + hee exi conan q > 1 and C > uch ha [ [ ( ) p E up Y,x,n 2p + E Z,x,n 2 d [, C ( E [ g,x,n 2pq [ ) 1/q + E ( J n X,x,n d) 2pq 18 (4.5)
Fi, by Lemma 4.8 E [ g,x,n 2pq [ CE g n (X,x,n ) g n (X,x ) 2pq + g n (X,x ) g(x,x ) 2pq a n. he convegence i bounded and follow ince g n fom (4.3) and he boundedne of g and g n. he diffeeniabiliy of f in x and aumpion f(, x, z) f(, ˆx, z) C(1 + z ) x ˆx implie ha g, X,x,n X,x x f(, x, z) C(1 + z ). (4.6) Second, by (4.6), Cauchy-Schwaz inequaliy and uing he fac ha f() p d ( ) up f() p (4.7) [, i follow ha [ E ( J n X,x,n d) 2pq [ CE ( (1 + Z,x,n ) 2 d) 2pq 1/2 E [ 1/2 up X,x,n 4pq [, (4.8) a n. he fi faco of (4.8) i unifomly bounded, ince i can be eimaed by Lemma 4.1, wih unifom conan C and q by he ame agumen a above. he convegence o zeo of he econd faco follow fom (4.3). I ha now been poved ha Y,x,n Y,x in S (R) and ha Z,x,n Z,x in H (R d ). Recall fom Secion 3.2 ha h : R m R i a, coninuou poiive funcion aifying R m h(x)dx = 1 and R m x 2 h(x)dx <. I follow ha uch a funcion mu be bounded. Boh Y,x,n and Y,x ae in S (R) by Popoiion 4.3 and hence bounded, fo almo all [,, almo uely. Now, by bounded convegence [ lim E Y n R,x,n Y,x 2p h(x)dx =. m Hence, Y,,n Y, in L 2p ( Ω) L 2 ( Ω), [,. Sep 2: he funcion b, σ and g ae all Lipchiz coninuou. Hence, hey ae coninuouly diffeeniable almo eveywhee, i.e., he weak paial deivaive equal he claical paial deivaive excep a a e of Lebegue meaue zeo. he queion of hi ep in he poof i o pove ha he oluion (Φ,x, Ψ,x, Γ,x ), o he vaiaional equaion FBSDE(b, σ, g, f), i well defined, i.e., doe no depend on Boel (dx a.e.) veion of he weak gadien x b, x σ and x g. Fo all [,, le x b 1 = x b 2, x σ 1 = x σ 2 and x g 1 = x g 2 excep a a e N [, R m wih Lebegue meaue zeo. 19
Le (Φ 1 i,, Ψ1 i,, Γ1 i, ) and (Φ2 i,, Ψ2 i,, Γ2 i, ), i = 1,..., m, be he oluion o ifbsde(b, σ, g, f), i = 1,..., m, wih veion ( x b 1, x σ 1, x g 1 ) and ( x b 2, x σ 2, x g 2 ) of he weak gadien epecively. he equaion mu be conideed componenwie o be able o ue momen eimae Lemma 4.1. he upecip,x i omied fo noaional impliciy. I will be poved ha he oluion ae idenical. he uniquene of Φ 1 and he ideniy Φ 1 = Φ 2 in S 2 (R m m ), ha been poved unde he non-degeneacy aumpion (3.2) [19. Denoe he coefficien of Φ,x i,, α(, φ) = x b(, X,x )φ, β(, φ) = d j=1 xσ j (, X,x )φ, fo (ω,, φ) Ω [, R m m. I i known ha he Fobeniu nom (2.3) i ubmuliplicaive, i.e., i aifie AB = A B. he coefficien heefoe aifie α(, φ) α(, φ) x b(, X,x ) φ φ and m 2 C 2 φ φ. β(, φ) β(, φ) d j=i xσ j (, X,x ) φ φ dm 2 C 2 φ φ. he bound follow ince each elemen in he maice x σ j and x b ae bounded by he common Lipchiz conan C of b and σ. he coefficien ae hence Lipchiz coninuou. heoem 4.7 can be applied o conclude ha Φ,x i, := Φ1 i, = Φ2 i,, i = 1,..., m, ae unique in S (R m ). I emain o pove ha Ψ 1 i, = Ψ2 i, and Γ1 i, = Γ2 i, and ha hey ae unique in S (R 1 m ) and H (R d m ) epecively, and hence well defined. Le Ψ i, := Ψ 1 i, Ψ2 i,, Γ i, := Γ 1 i, Γ2 i,, ξ i := ( x g 1 (X,x ) xg 2 (X,x ))Φ,x i,. he poce ( Ψ i,, Γ i, ) aifie BSDE( ξ i, f) fo f(, v) = z f(, X,x I aifie he andom Lipchiz condiion:, Z,x )v. f(, v) f(, v) z f(, X,x, Z,x ) v v C(1 + Z,x ) v v (, v, v) [, R m m R m m, by aumpion. Fom Lemma 4.2 Z,x dw i a BMOmaingale. I hen follow fom Lemma 4.1 and Cauchy Schwaz inequaliy ha fo any p > 1 hee exi a q > 1 and C > uch ha: 2
E [ up Ψ i, 2p + E [, [ ( ) p Γ i, 2 d [ CE x g 1 (X,x ) xg 2 (X,x ) 4pq 1/(2q) [ E =. Φ,x i, 4pq 1/(2q) he fi faco vanihe ince x g 1 (x) x g 2 (x) = fo almo all x R m and ince X,x ha a deniy fom heoem 3.3. he econd faco i finie ince Φ,x i, ha finie momen. Hence fo all [, and i = 1,..., m, Ψ,x i, := Ψ1 i, = Ψ2 i, and Γ,x i, := Γ1 i, = Γ2 i,. o be able o conclude ha he oluion (Φ, Ψ, Γ) i unique in S (R m m ) S (R 1 m ) H (R d m ) and hence well defined, he condiion (2.13) of heoem 2.2 mu be checked fo p > 1 and evey componen. Fi, [ E x g(x,x )Φ,x i, p <, i = 1,..., m, ince g i Lipchiz coninuou and hence ha bounded paial deivaive and Φ,x i, ha finie momen. Denoe he geneao of i FBSDE(b, σ, g, f), by f i : Ω [, R d m R. I can be idenified fom (4.1). I aifie f i (, ) = x f(, X,x, Z,x )Φ,x i,. Uing Cauchy Schwaz inequaliy, (4.6) and (4.7) he econd em of (2.13) can be eimaed by E [ ( ) p f i (, ) d = E [ ( x f(, X,x, Z,x )Φ,x ) p i, d E E [ ( [ ( p 1/2 x f(, X,x, Z,x ) d) 2 E C(1 + Z,x ) 2 d) p 1/2 E <, i = 1,..., m. [ [ ( up Φ,x i, 2p [, ) p 1/2 Φ,x i, 2 d he finiene of he fi faco follow ince Z,x H (R d ) fom Popoiion 4.3. he finiene of he econd faco follow fom heoem 4.7. I hold fo any p > 1 and hence (Ψ,x, Γ,x ) i unique in S (R 1 m ) H (R d m ) fom heoem 2.2. I can be concluded ha, Γ,x ) ae well defined pocee. (Ψ,x Sep 3: Again conide he appoximaing funcion b n, σ n and g n. Define, fo (, x, n) [, R m Z + and [,, an appoximaing equence FBSDE(b n, σ n, g n, f) of vaiaional equaion wih oluion (Φ,x,n, Ψ,x,n, Γ,x,n ). I will be poved in hi ep ha 1/2 21
Φ,x,n Φ,x in S (R d m ), Ψ,,n n. Ψ, I ha been poved [19 ha, (, x) [, R m, [ lim E n in L 2 ( Ω) and ha Γ,x,n Γ,x in H (R d m ) a up Φ,x,n Φ,x 2p = (4.9) [, fo p = 1. In hi poof p 1 i needed. he genealizaion of he poof [19 i a mae of noaion and will no be peened. Le (Φ,x,n i, Ψ,x,n i, Γ,x,n i ) be he oluion o i FBSDE(b n, σ n, g n, f), i = 1,..., m. Denoe he diffeence by Ψ,x,n i, Γ,x,n i, := Ψ,x,n i, := Γ,x,n i, Ψ,x i,, Γ,x i,, ξ,x,n i := x g n (X,x,n )Φ,x,n i, x g(x,x )Φ,x i, he poce ( Ψ,x,n i,, Γ,x,n ) aifie BSDE( ξ,x,n, f i n) fo i, f n i (, v) := xf(, X,x,n +( z f(, X,x,n + z f(, X,x,n, Z,x,n, Z,x,n i )Φ,x,n i,, Z,x,n )v. x f(, X,x ) z f(, X,x, Z,x, Z,x )Φ,x i, ))Γ,x i, he geneao f n i aifie a andom Lipchiz condiion f n i (, v) f n i (, v) = zf(, X,x,n, Z,x,n ) v v C(1 + Z,x,n ) v v by aumpion. Recall fom Lemma 4.2 ha Z,x,n db i a BMO-maingale and ha i BMO-nom ae unifomly bounded fom ep 1. Hence momen eimae Lemma 4.1 can be applied wih unifom bound fo he conan C and q ogehe wih Lemma 4.8. Fo any p > 1 hee exi conan q > 1 and C > uch ha, n Z + : [ [ ( ) p E Ψ,x,n i, 2p + E Γ,x,n i, 2 d whee up [, C ( [ E ξ,x,n i 2pq + I,x,n i (4.1) ) 1/q + J,x,n i, 22
I,x,n i := E J,x,n i := E [ ( [ ( ( z f(, X,x,n x f(, X,x,n, Z,x,n ) z f(, X,x, Z,x, Z,x,n )Φ,x,n i, Fi, by Lemma 4.8 and Cauchy Schwaz inequaliy x f(, X,x ))Γ,x, Z,x ) 2pq i, d ) 2pq )Φ,x i, d. [ E ξ,x,n i 2pq [ = E x g n (X,x,n C )Φ,x,n i, x g(x,x )Φ,x i, 2pq ( [ 1/2 [ 1/2 E x g n (X,x,n ) 4pq E Φ,x,n i, Φ,x i, 4pq [ 1/2 [ +E x g n (X,x,n ) x g n (X,x ) 4pq E [ 1/2 [ + E x g n (X,x ) xg(x,x ) 4pq E a n. 1/2 Φ,x i, 4pq Φ,x i, 4pq 1/2 ) he dominaed convegence heoem applie ince x g and x g n ae bounded by he Lipchiz conan of g and g n, n Z +, epecively and Φ,x,n i, and Φ,x i, have finie momen, i = 1,..., m. he convegence o zeo follow ince X,x,n X,x in S (R m ), Φ,x,n i Φ,x i in S (R m m ) fom ep 1 and x g n x g fo almo all x R m. he final value X,x ha a deniy by heoem 3.3 and will hence no aain i value a undefined poin of x g. Nex, le eimae I,x,n i by Lemma 4.8 and Cauchy Schwaz inequaliy: I,x,n i CE ( [ ( E [ ( + E ) 2pq 1/2 Γ,x i, 2 d [ ( a n. x f(, X,x,n x f(, X,x,n, Z,x,n ) x f(, X,x,n, Z,x, Z,x ) x f(, X,x, Z,x ) 2pq ) 2 d ) 2pq ) 1/2 ) 2 d he fi faco i finie ince Γ,x i H (R d ) fom ep 2. he econd faco i by uing (4.6) bounded by CE [ ( (1 + Z,x ) 2pq + Z,x,n ) 2 d <. 23
he finiene follow ince Z,x, Z,x,n H (R d ) fom Popoiion 4.3. Dominaed convegence, he coninuiy of x f in x and z and he convegence X,x,n X,x in S (R m ) and Z,x,n Z,x in H (R d ) implie ha lim n I,x,n i =. Finally, lim n J,x,n i =, i = 1,..., m, by imila ue of Lemma 4.8, Cauchy Schwaz inequaliy, he aumpion, convegence eul and dominaed convegence. Hence, he column of Ψ,x,n and Γ,x,n convege o he column of Ψ,x and Γ,x in S (R) and H (R d ) epecively. I follow ha Ψ,x,n Ψ,x in S (R m ) and Γ,x,n Γ,x in H (R d m ) a n. he funcion h, defined in ecion 3.2, i coninuou wih inegal one and hence bounded. Moeove Ψ,x,n heoem ha and Ψ,x and in paicula Ψ,,n ae eenially bounded. I follow fom he bounded convegence lim E n [ up Ψ R,x,n m [, Ψ,x 2p h(x)dx Ψ, in L 2p ( Ω) L 2 ( Ω), [,. Sep 4: Finally, le pu he eul fom ep 1 and 3 ogehe wih heoem 4.5: Y,,n Y, L 2 ( Ω) [, Ψ,,n Ψ, L 2 ( Ω) [, = x i Y,x,n = Ψ,x,n i, 1 i m [,, x R m. Lemma 4.4 ae ha x Y,x i Lipchiz coninuou, which in un implie he weake condiion of abolue coninuiy of ε Y,x+εe i, 1 i m. Hence Y, ( m D i=1 i ) and i Y,x i well defined. he convegence hen hold wih epec o he Diichle- d nom e d = [ 2 L 2 (e Ω) + Hence Y, d. Popoiion 3.2 ell ha Y, poved. (ii) follow immediaely. m i=1 i ( ) 2 L 2 (e Ω) 1 2. d = Y,x d H 1 loc (Rm ). Hence (i) i Coollay 4.1. Aume (A). hen fo u(, x) := Y,x i hold ha u(, ) H 1 loc (Rm ) and Z,x = x u(, X,x )σ(, X,x ) (4.11) fo Lebegue a.e. [,, P-almo uely, whee x u i he weak gadien of u. Poof. Fi u(, ) := Y, d Hloc 1 (Rm ), [, fom heoem 4.9. he coollay hold fom heoem 4.6 unde aumpion (A) and he addiional equiemen ha he eminal funcion g i wice coninuouly diffeeniable and b and σ ae coninuouly diffeeniable in x. he appoximaing coefficien in he poof of he peviou heoem aifie hee condiion. Hence, fo n Z +, Lebegue a.a. [, and P-a.. Z,x,n = x u n (, X,x,n )σ n (, X,x,n ). 24
Since u(, x) := Y,x and u n (, x) := Y,x,n i hold, fom eul in he poof of he peviou heoem, ha u n (, x) u(, x) and x u n (, x) x u(, x) in L 2 ( Ω). Alo, X,x,n X,x and σ n σ a n. Hence, x u n (, X,x,n )σ n (, X,x,n ) x u(, X,x )σ(, X,x ) a n. Moeove Z,x,n Z,x in H (R d ) a n. Hence he eul hold in he limi. I i eay o check, by uing he Lipchiz condiion on u in x and he linea gowh condiion on σ in x, ha he igh hand ide of (4.11) i in H (R d ). 25
26
Chape 5 Applicaion o inuance and finance: Opimal co hedging he make fo financial deivaive ha exploded he la 25 yea. Mo ofen he conac ae wien on adable undelying uch a ock, gain o oil ec.. In ha cae he deivaive ae piced by ceaing a eplicaing pofolio, conaining hae of he undelying uch ha he value of he pofolio equal ha of he deivaive. he fai pice of he deivaive i hen he ame a he co o ceae he eplicaing pofolio. he pupoe of buying he deivaive can be eihe fo hedging, peculaion o abiage pupoe [8. hee i an inceaing make fo deivaive wien on indice uch a empeaue, ain, now fall o economic lo o ohe non-adable indice. he main pupoe o wie uch conac i fo inuance o fo inuance companie a an alenaive o claical einuance. Rik can in uch a way be moved o he financial make. Since he indice ae non-adable i impoible o ceae eplicaing pofolio o pice deivaive wien on hem. I i alo by he ame eaon impoible o hedge he ik of he deivaive diecly. he way o ackle hi i by co hedging, i.e. o find a ongly coelaed and adable ae and ue i fo hedging. I i of coue impoible o hedge all ik ince he undelying and he coelaed ae ae no compleely coelaed. he make i aid o be incomplee. he appoach ofen aken when picing and hedging in incomplee make i ha of maximizing he uiliy of an invemen by chooing an opimal hedging aegy. hi i alo he appoach aken hee. In he fi ecion he aumpion and make model will be peened and alo ome example. In he econd he oluion appoach by olving a FBSDE i peened. So fa nohing ha been new bu ahe aken fom [2 and [7. In he fouh ecion he main eul of hi hei i applied. An explici expeion fo he opimal co hedging aegy i deived, in em of he weak gadien of a FBSDE. he gain of hi i ha, Euopean pu and call opion o ohe deivaive wih non-diffeeniable payoff funcion can be wien. 5.1 Aumpion and make model Le (Ω, F, {F } [,, P), >, be a fileed pobabiliy pace wih a d-dimenional Wiene poce W. F i he naual filaion of W compleed by he P-null e of Ω. A financial deivaive wih mauiy and payoff funcion F : R m R i wien on a m-dimenional non-adable ik poce X. Fo x R m he dynamic of X i given by: 27
X = x + b(, X )d + σ(, X )dw [,. he coefficien b : [, R m R m and σ : [, R m R m d ae aumed o aify a global Lipchiz and linea gowh condiion, i.e. hee exi a C > uch ha { b(, x) b(, x) + σ(, x) σ(, x) C x x, b(, x) + σ(, x) C(1 + x ). (, x, x) [, R m R m. Moeove b(, ) and σ(, ) ae aumed bounded [, and σ i aumed o aify he non-degeneacy condiion ξ σ(, x)σ (, x)ξ C ξ 2, (, x, ξ) [, R m R m (5.1) fo ome C >. he andom income a ime of mauiy of he deivaive i F (X ). F i aumed o be bounded and Lipchiz coninuou. he boundedne of F eem unnaual fo many deivaive bu he bound can be choen abiaily high and doen imply any poblem in pacice. Condiioned on he infomaion X = x, fo x R m, he non-adable poce will be denoed X,x, and aify: X,x = x + b(, X,x )d + σ(, X,x )dw, [,. Since X i non-adable i impoible o hedge he ik aociaed wih he deivaive diecly. heefoe a coelaed and adable ae pice poce i ued o paially hedge he ik. he k-dimenional ae pice poce i given by: S i = i + S i (α i (, X )d + β i (, X )dw ), i = 1,..., k. Hee α i and β i denoe he i:h ow of he funcion α : [, R m R k and β : [, R m R k d, i = 1,..., k. he coefficien α i aumed o be bounded and β i aumed o aify he condiion εi k k β(, x)β (, x) KI k k (5.2) fo ome < ε < K, (, x) [, R m, whee β i he anpoe of β and I k k i he ideniy maix in R k. I implie ha β(, x)β (, x) i inveible and bounded. Boh α and β aify a global Lipchiz condiion: α(, x) α(, x) + β(, x) β(, x) C x x (, x, x) [, R m R m, C >. Moeove α and β ae aumed o be coninuouly diffeeniable in x. Noice ha X and S ae diven by he ame Wiene poce. hei coelaion i deemined by σ and β. o ule ou abiage oppouniie he aumpion d k mu hold, i.e. hee mu be moe ouce of unceainy han numbe of adable ae. When all he aumpion above hold aumpion (B) will be aid o be fulfilled. he following example i aken fom [2. 28
Example 5.1. A company poducing keoene (ke) fom cude oil (co) i eniive again udden inceae in he pice of cude oil. I heefoe inve in o called cack pead. hey ae Euopean opion on he diffeence of he cude oil pice and he keoene pice, i.e. deivaive wih payoff funcion F (X co, Xke ) = [Xco Xke K+ whee K i he ike pice. i he ime o mauiy. he make fo ading keoene i no liquid enough o waan fuue conac on i. heefoe ome liquid and ongly coelaed ae mu be ued o paially hedge he ik aociaed wih he cack pead. Heaing oil (ho) ha hi popey and i i liquid. Hee co hedging applie. In [2 he following model fo he indice i peened: dx ke = X ke (b 1 d + γ 2 dw 1 + γ 3 dw 2 + γ 4 dw 3 ) dx co = X co (b 2 d + γ 1 dw 1 ) ds ho = S ho (b 3 d + β 1 dw 1 + β 2 dw 2 ) fo b 1, b 2, b 3 R, γ 1, γ 2, γ 3, γ 4, β 1, β 2 R \ {} and [,. Wih he eul of chape 4 an explici expeion of he opimal co hedging aegy can be obained. hi wa no poible befoe ince F i no coninuouly diffeeniable. See he la ecion below. An invemen aegy i a k-dimenional pedicable poce λ ha aifie λi ds i < S i, i = 1,..., k. λ i i he value of he pofolio inveed in he i:h ae a ime [,. he oal gain fom inveing accoding o λ in he ime ineval [, i G λ, he gain condiioned on X = x, fo x R m, i denoed G λ,,x G λ,,x = k i=1 = k i=1 and i given by: λ i [α i (, X,x )d + β i (, X,x )dw. Le A,x denoe he pace of all aegie λ aifying E[ λi ds i. S i λ β(, X,x ) 2 d < and uch ha he family {e ηgλ,,x τ : τ [, i a opping ime} i unifomly inegable, η >. Saegie in A,x ae called admiible. I eem naual o eek fo an opimal aegy ha in ome ene maximize he gain of he invemen. he appoach hee i o maximize he expeced exponenial uiliy. he exponenial uiliy funcion i given by: U(y) = e ηy. he ik aveion coefficien η >, y R. he maximal expeced uiliy, condiioned on X = x, fo ik level x R m a ime [, and wealh v R, of an invemen wihou he deivaive i [ V (, x, v) := up E U(v + G λ,,x ). λ A,x In em of ochaic conol heoy V i called he value funcion of a ochaic conol poblem. I ha been hown in [7 ha hee exi an almo uely unique opimal aegy π A,x uch ha [ V (, x, v) = E U(v + G π,,x ). 29
Remak 5.2. he exponenial uiliy funcion punihe loe ongly and ewad gain modeaely. Hence he aegy π i he aegy ha minimize ik fo loe. π doe of coue depend on he ik aveion coefficien η. he malle η > i he moe ewaded ae high gain bu moe likely ae loe. U i a concave uiliy funcion and all uch ae ik avee. On he ohe ide convex uiliy funcion ae ik-eeking, i.e. vey high gain ae pefeed even hough hey ae unlikely o occu. A linea uiliy funcion maximize he expeced value and he concep of uiliy i gone. If he pofolio conain he deivaive F (X ) he condiional maximal expeced uiliy and value funcion become [ V F (, x, v) := up E U(v + G λ,,x + F (X,x )). λ A,x Alo in hi cae hee exi an almo uely unique invemen aegy π ha aifie [ V F (, x, v) = E U(v + G bπ,,x + F (X,x )). he diffeence of he wo aegie = π π i called he deivaive hedge and i ued o hedge he deivaive. In a lae ubecion explici expeion fo he deivaive hedge will be deived via he diibuional gadien of a quadaic FBSDE. I i a genealizaion of he claical -hedge in he Black-Schole model. In he cae of a complee make, i.e. when d = k and S = R, deivaive hedge coincide wih Black-Schole -hedge. Nex, how hall he deivaive be piced? hi i olved by calculaing he o called indiffeence pice p(, x) a ime condiioned on X = x and wealh v, given by: V F (, x, v p(, x)) = V (, x, v). I i he pice ha make he buye of he deivaive indiffeen, in a uiliy poin of view, o wehe he hould buy he deivaive o no. I will lae be een ha he deivaive hedge can be expeed in em of he diibuional gadien of he indiffeence pice p(, x). he mahemaical poblem o find he opimal invemen aegy i an opimal ochaic conol poblem. I i ofen ackled by olving he o called Hamilon-Jacobi-Bellman PDE. he appoach hee i aken fom [2 and [7 and ue FBSDE. 5.2 Soluion o he opimal co hedging poblem via a FB- SDE In hi ecion a FBSDE will be ued o find explici expeion fo he indiffeence pice and opimal co hedging aegie peened above. he eul have been poved in [2 and [7. Fix (, x) [, R m. Aumpion (5.2) implie ha β(, x)β (, x) i inveible hence he mapping β(, x) : R k R d i one-o-one. Recall ha a ading aegy λ i a k-dimenional poce, coeponding o he value of a pofolio inveed in he i:h ae a ime. In he oluion appoach hee he d-dimenional image aegie given by λ β(, x) will be conideed inead. Le 3
{ C(, x) = β(, x) : R k}. be he conain e fo he image aegie. he maix β(, x) i no neceaily ono. Hence C(, x) i in fac a conain e. I i cloed and convex. Le ϑ(, x) = β (, x)(β(, x)β (, x)) 1 α(, x). he poce ϑ i bounded ince α and β ae bounded and ββ KI k k fom aumpion 5.2. Le di(z, C) = min{ z u : u C} be he diance of a veco z R d o he cloed and convex e C. Define he geneao of a FBSDE by: f : [, R m R d R, (, x, z) zϑ(, x) + 1 2η ϑ(, x) 2 η 2 di2 (z + 1 ϑ(, x)), C(, x)). η hee exi a unique oluion o he FBSDE Ŷ,x = F (X,x ) f(, X,x, Ẑ,x )d Ẑ,x dw, [,, ince f ha quadaic gowh in z. he value funcion, defined in he peviou ecion, fo an invemen wih he deivaive i given by: V F (, x, v) = e η(v b Y,x ). Le C(,x) (z) denoe he ohogonal pojecion of z Rd ono he ubpace C(, x). Condiioned on X = x he opimal co hedging aegy i given by: π β(, R,x ) = [Ẑ,x + 1 ϑ(, X,x ), η C(,x) [,. Analogouly, an invemen wihou he deivaive give ie o he FBSDE: wih value funcion Y,x = f(, X,x, Z,x )d Z,x dw, V,x η(v Y (, x, v) = e ) and opimal aegy: π β(, R,x ) = [Z,x + 1 ϑ(, X,x ), η C(,x) [,. he pojecion opeao i linea hence he deivaive hedge i given by β(, X,x ) = [Ẑ,x Z,x. C (,x) Recall he definiion of he indiffeence pice p(, x) and noice ha, 31
V F (, x, v p(, x)) = e η(v p(,x) Y b,x ),x η(v Y = e ) = V (, x, v) which in un implie ha p(, x) = Y,x Ŷ,x. 5.3 Explici hedging aegy uing he weak pice gadien In hi ecion he eul on weak diffeeniabiliy of FBSDE fom Chape 4 will be applied o he opimal co hedging poblem. he majo advanage of he new eul i ha he payoff funcion no longe mu be coninuouly diffeeniable. hi implie ha pu and call opion can be wien on he undelying and explici hedging aegie deived via he weak gadien of he indiffeence pice. Anohe i ha he coefficien of he non-adable ae poce no longe need o be diffeeniable bu only Lipchiz coninuou wih linea gowh. heoem 5.3. Unde aumpion (B) he funcion û(, x) := Ŷ,x weakly diffeeniable wih epec o x. and u(, x) := Y,x Poof. Aumpion (B) implie aumpion (A) of chape (4). he ickie pa of he poof concen he geneao and wa made in [2. heoem 4.9(i) applie ince (A) hold. Since p(, x) = Y,x Ŷ,x he following coollay hold. Coollay 5.4. Unde aumpion (B) he indiffeence pice p(, x) i weakly diffeeniable wih epec o x. heoem 5.5. Aume (B). hen he deivaive hedge, fo ik level x R m a ime [,, i given by (, x) = C(,x) [ x p(, x)σ(, x)β (, x)(β(, x)β (, x)) 1 fo (, x) [, R m, whee x p i he pice gadien conideed in he weak ene and C(,x) (z) i he ohogonal pojecion of z Rd ono C(, x) := {β(, x) : R k }. Poof. Recall ha which implie (, x) = β(, X,x ) = [Ẑ,x Z,x, C(,x) [Ẑ,x C(,x) Z,x β (, x)(β(, x)β (, x)) 1. ae Coollay 4.1 implie ha Ẑ,x diibuional ene. Hence, = x û(, x)σ(, x) and Z,x = x u(, x)σ(, x) in he Ẑ,x Z,x and he eul follow. = ( x û(, x) x u(, x))σ(, x) = x (Ŷ,x Y,x }{{} = p(,x) )σ(, x) 32