An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

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1 See icuion, a, an auho pofile fo hi publicaion a: hp:// An FBSDE appoach o he Skookho embeing poblem fo Gauian pocee wih non-linea if Aicle in ELECTRONIC JOURNAL OF PROBABILITY Augu 214 Impac Faco:.77 DOI: /EJP.v Souce: axiv CITATIONS 2 READS 28 3 auho: Alexane Fomm Humbol-Univeiä zu Belin 4 PUBLICATIONS 7 CITATIONS Pee Imkelle Humbol-Univeiä zu Belin 32 PUBLICATIONS 43 CITATIONS SEE PROFILE SEE PROFILE Davi Johanne Pömel ETH Zuich 9 PUBLICATIONS 17 CITATIONS SEE PROFILE Available fom: Davi Johanne Pömel Reieve on: 8 May 216

2 An FBSDE appoach o he Skookho embeing poblem fo Gauian pocee wih non-linea if Alexane Fomm, Pee Imkelle, Davi J. Pömel axiv: v1 [mah.pr 27 Aug 214 Iniu fü Mahemaik Humbol-Univeiä zu Belin Une en Linen Belin Gemany Augu 28, 214 Abac We olve he Skookho embeing poblem fo a cla of Gauian pocee incluing Bownian moion wih non-linea if. Ou appoach elie on olving an aociae ongly couple yem of Fowa Backwa Sochaic Diffeenial Equaion FBSDE, an inveigaing he egulaiy of he obaine oluion. Fo hi pupoe we exen he exience, uniquene an egulaiy heoy of o calle ecoupling fiel fo Makovian FBSDE o a eing in which he coefficien ae only locally Lipchiz coninuou. MSC 21: Pimay: 6G4, 6H3; econay: 93E2. Key wo an phae: BMO poce, BSDE, ecoupling fiel, fowa backwa ochaic iffeenial equaion, FBSDE, Skookho embeing, vaiaional iffeeniaion. 1 Inoucion The Skookho embeing poblem SEP imulae eeach in pobabiliy heoy now fo ove 5yea. Theclaical goal ofhesepconi infining,foagivenbownianmoion W an a pobabiliy meaue ν, a opping ime τ uch ha W τ poee he law ν. I wa fi fomulae an olve by Skookho [Sko61, Sko65 in Since hen hee appeae many iffeen conucion fo he opping ime τ an genealizaion of he oiginal poblem in he lieaue. Ju o name ome of he mo famou oluion o he SEP we efe o Roo [Roo69, Ro [Ro71 an Azéma-Yo [AY79. A compehenive uvey can be foun in [Ob l4. Recenly, he Skookho embeing aie aiional inee becaue of ome applicaion in financial mahemaic, a fo inance o obain moel-inepenen boun on lookback opion [Hob98 o on opion on vaiance [CL1, CW13, OR13. An inoucion o hi cloe connecion of he Skookho embeing poblem an obu financial mahemaic can A.F. an D.J.P. wee financially uppoe by Ph.D. cholahip of he DFG Reeach Taining Goup 1845 Sochaic Analyi wih Applicaion in Biology, Finance an Phyic. 1

3 be foun in [Hob11. In hi pape we conuc a oluion o he Skookho embeing poblem fo Gauian poce G of he fom G := G + α + β W, whee G R i a conan an α,β: [, R ae uiable funcion. Epecially, hi cla of pocee inclue Bownian moion wih non-linea if. The SEP fo Bownian moion wih linea if wa fi olve in he echnical epo [Hal68 an 3 yea lae again in [GF an [Pe. Technique evelope in hee wok can be exene o imehomogeneou iffuion, a one in[pp1, an can be een a genealizaion of he Azéma-Yo oluion. Howeve, o he be of ou knowlege hee exi no oluion o fa fo he cae of a Bownian moion wih non-linea if. The pii of ou appoach i elae o he one by Ba [Ba83, who employe maingale epeenaion o fin an alenaive oluion of he SEP fo he Bownian moion. Thi appoach wa fuhe evelope fo he Bownian moion wih linea if in [AHI8 an fo ime-homogeneou iffuion in [AHS13. I e upon he obevaion ha he SEP may be viewe a he weak veion of a ochaic conol poblem: he goal i o ee G in uch a way ha i ake he iibuion of a pecibe law, which in cae of zeo if i cloely elae o he maingale epeenaion of a anom vaiable wih hi law. We heefoe popoein hi pape o fomulae anolve hesep fo Gin em of afully couple Fowa Backwa Sochaic Diffeenial Equaion FBSDE. In geneal em, he ynamic of a yem of FBSDE i expee by he equaion X = X + Y = ξx T µ,x,y,z + f,x,y,z σ,x,y,z W, Z W, [,T, wih coefficien funcion µ,σ of he fowa pa, eminal coniion ξ an ive f of he backwa componen. In ecen ecae he heoy of FBSDE wih i cloe connecion o quai-linea paial iffeenial equaion an hei vicoiy oluion ha been popagae exenively, in paicula in i numeou aea of applicaion a ochaic conol an mahemaical finance ee [EPQ97 o [PW99. Thee ae mainly hee meho o how he exience of a oluion fo a yem of FBSDE: he conacion meho [An93, PT99, he fou ep cheme [MPY94 an he meho of coninuaion [HP95, Yon97, PW99. A a unifie appoach, [MWZZ11 ee alo [Del2 eigne he heoy of ecoupling fiel fo FBSDE, which wa ignificanly efine in [FI13. I can pimaily be een a an exenion of he conacion meho. In ou appoach of he SEP via FBSDE, we hall focu on he ubcla of Makovian one fo which all involve coefficien funcion ξ,µ,σ,f ae eeminiic. We, howeve, have o allow fo no globally, bu only locally Lipchiz coninuou coefficien µ, σ, f in he conol vaiable z, an heefoe o evelop an exience, uniquene an egulaiy heoy fo FBSDE in hi famewok. Equippe wih hee ool we olve he FBSDE yem euling fom he SEP. We fi conuc a weak oluion, i.e. we obain a Gauian poce of he above fom an an inegable anom ime uch ha, oppe a hi ime, he poce poee he given iibuion ν. Une uiable egulaiy on he given meaue ν an he poce, hi conucion will be caie ove o he oiginally given Gauian poce G. Thi olve he SEP fo G. 2

4 The pape i oganize a follow: in Secion 2 we elae he SEP o a fully couple yem of FBSDE, an in Secion 3 we eablih geneal eul fo ecoupling fiel of FBSDE. The Skookho embeing poblem i olve in Secion 4, in i weak an in i ong veion. Secion A ecall ome auxiliay eul fo BMO pocee. 2 An FBSDE appoach o he Skookho embeing poblem We conie a filee pobabiliy pace Ω,F,F [,,P lage enough o cay a oneimenional Bownian moion W. The filaion F [, i aume o be geneae by he Bownian moion an i aume o be augmene by P-null e. We alo aume ha F = σ = F. We a by fomulaing he Skookho embeing poblem in he moifie veion SEP: Fo given pobabiliy meaue ν on R an a Gauian poce X on [, of he fom X := X + α + β W, 1 whee X R i ome peeemine conan an α,β: [, R ae eeminiic meauable pocee uch ha α + β2 < fo all, fin a F -opping ime τ.. E[τ < ogehe wih a aing poin c R uch ha c+x τ ha he law ν. In oe o have a uly ochaic poblem β houl no vanih an ν houl no be a Diac meaue. In fac we will aume ha β i boune away fom zeo lae on. Ou meho of olving hi poblem i bae on he obevaion ha i may be viewe a he weak veion of a ochaic conol poblem: We wan o ee X in uch a way ha i ake he iibuion of a pecibe law. The pii of ou appoach i elae o an appoach o he oiginal Skookho embeing poblem by Ba [Ba83 ha wa lae exene o he Bownian moion wih linea if in [AHI8. The poceue of boh pape can be biefly ummaize an ivie ino he following fou ep. 1. Conuc a funcion g: R R uch ha gw 1 ha he given law ν. 2. Ue he maingale epeenaion popey of he Bownian moion fo α an β 1 o BSDE echnique fo α κ an β 1 o olve Y = gw 1 κ 1 1 Z 2 Z W, [, Apply he anom ime-change of Dambi, Dubin an Schwaz in he quaaic vaiaion cale. Z2 o anfom he maingale. Z W ino a Bownian moion B. Thi alo povie a anom ime τ := 1 Z2 fulfilling B τ +κ τ +Y = gw 1, which i why B τ +κ τ +Y ha he law ν. 3

5 4. Show ha τ i a opping ime wih epec o he filaion geneae by B hough an explici chaaceizaion uing he unique oluion of an oinay iffeenial equaion. Wih hi ecipion anfom he embeing wih epec o B ino one wih epec o he oiginal Bownian moion W o obain he opping ime τ a he analogue o τ. The fi ep of he algoihm ju keche i faily eay. Le F: R [,1 uch ha Fx := ν, x i he cumulaive iibuion funcion aociae wih ν an efine F 1 :,1 R via F 1 y := inf{x R : Fx y}. DenoingbyΦheiibuionfuncionofheananomaliibuion, weefineg: R R a gx := F 1 Φx. I i aighfowa o pove ha g ha he following popeie. Lemma 1. The funcion g i meauable an non-eceaing. Moeove, if ν i no a Diac meaue, hen g i no ienically conan an gw 1 ha he law ν. Poof. Since Φ an F 1 ae meauable an non-eceaing, hei compoiion g i alo meauable an non-eceaing. Clealy, g can only be conan if F 1 i conan, which can only happen if F aume value in {,1}. Thi only happen in cae ν i a Diac meaue. In oe o ee ha gw 1 ha he law ν, noe ha PgW 1 x = PF 1 ΦW 1 x = PW 1 Φ 1 Fx = ΦΦ 1 Fx = Fx fo all x R. Now efine a meauable funcion ˆδ: [, R via ˆδ := X + α uch ha X = ˆδ+ β W. Obviouly, ˆδ i weakly iffeeniable. Conveely, fo evey weakly iffeeniable funcion ˆδ: [, R we can e X := ˆδ an α := ˆδ. Fuhemoe, efine H: [, [, via H := β 2. Noe ha H i weakly iffeeniable, monoonically inceaing an a a. If we aume ha β i boune away fom, H become icly inceaing an inveible uch ha he invee funcion H 1 i monoonically inceaing an Lipchiz coninuou. In hi cae we can efine δ := ˆδ H 1. If β 1, hen H = I an hu δ = ˆδ. Fo he econ ep we aume ha β i boune away fom an obeve ha he anom ime change, which un he maingale Z W ino a Gauian poce of he fom β B imulaneouly un he cale poce. Z2 ino β2 = H. Thi mean we have o moify he claical maingale epeenaion of gw 1 o 1 [ 1 1 gw 1 + H ˆδ 1 Z 2 E gw 1 + H ˆδ 1 Z 2 = Z W, 4

6 which amoun o fining a oluion Y,Z o he equaion 1 Y = gw 1 δ Z 2 1 Z W, [,1. 3 Fo δ hi woul be ju he uual maingale epeenaion wih epec o he Bownian moion. Alo fo a linea if δ = κ an β 1 equaion 3 can be ewien a Ỹ := Y +κ Z 2 = gw 1 κ 1 Z 2 1 Z W, [,1, which i exacly he BSDE 2 elae o he SEP a ae in [AHI8. In he cae of a Bownian moion wih geneal if equaion 3 woul be a BSDE wih ime-elaye eminal coniion. Unfounaely, he heoy of BSDE wih ime-elay a inouce by Delong an Imkelle in [DI1 an exene by Delong [Del12 fo ime-elaye eminal coniion eache i limi in ou iuaion. Alenaively, we will unean equaion 3 a an FBSDE an evelop new echnique o olve i. Thi will be one in Secion 3 an 4. Befoe we ackle he olvabiliy of equaion 3, we how ha i eally lea o he eie eul in he hi ep of ou algoihm. To be mahemaically igoou we inouce S 2 R a he pace of all pogeively meauable pocee Y : Ω [,1 R aifying up [,1 E[ Y 2 <, H 2 R a he pace of all pogeively meauable pocee Z: Ω [,1 R aifying E[ 1 Z 2 <, whee enoe he Eucliean nom on R. Fo he e of he pape we aume ha β i boune away fom, i.e. inf [, β >. Lemma 2. Suppoe Y,Z S 2 R H 2 R i a oluion of 3. Then hee exi a Bownian moion B an a anom ime τ wih E[ τ < uch ha τ τ Y +X + α + β B = gw 1. Poof. Noe ha Y i a maingale wih quaaic vaiaion poce Z2 fo [,1 ince Z H 2 R. Now chooe anohe Bownian moion B which i inepenen of Y. If neceay we exen ou pobabiliy pace uch ha i accommoae he Bownian moion B. Se τ := H 1 1 Z2, an efine he ime-change of he ype of Dambi, Dubin an Schwaz by { { inf : σ := Z2 > } β2 if < τ 1 if τ. Obeve ha he coniion < τ i equivalen o β2 < 1 Z2. Since Y σ i a coninuou maingale wih quaaic vaiaion H = β2, we can efine a Bownian moion B by τ 1 B := B B τ + Y σ, <. β 5

7 We fin an fuhe τ 1 β B + ˆδ τ+y = Y 1 Y +δ Z 2 +Y = gw 1, 1 E[ τ = E [H 1 Z 2 <, whee we ue ha Z H 2 R an H 1 i Lipchiz coninuou. A an immeiae conequence of he peviou lemma we obeve he following fac. If we have a oluion Y,Z S 2 R H 2 R of equaion 3, we obain a weak oluion o he Skookho embeing poblem, i.e. a Gauian poce of he fom 1, a aing poin c, an an inegable anom ime uch ha ou poce oppe a hi ime poee a given iibuion. A a fi glance equaion 3 migh look eay. We, howeve, have o eal wih a fully couple FBSDE which in aiion poee a no globally Lipchiz coninuou coefficien in he fowa componen. 3 Decoupling fiel fo fully couple FBSDE The heoy of FBSDE, cloely connece o he heoy of quai-linea paial iffeenial equaion an hei vicoiy oluion, eceive i geneal inee fom numeou aea of applicaion among which ochaic conol an mahemaical finance ae he mo vivi one in ecen ecae ee [EPQ97 o [PW99. Owing o hei geneal ignificance, we ea he heoy of FBSDE an hei ecoupling fiel in a moe geneal famewok han migh be neee o obain a oluion o ou equaion 3. Alhough in Secion 3.2 we will focu on he Makovian cae, which mean ha all involve coefficien ae puely eeminiic, le u well in a moe geneal eing fi. 3.1 Geneal ecoupling fiel Fo a fixe ime hoizon T >, we conie a complee filee pobabiliy pace Ω,F,F [,T,P, whee F conain all null e, W [,T i a -imenional Bownian moion inepenen of F, an F := σf,w [, wih F := F T. The ynamic of an FBSDE i claically given by X = X + Y = ξx T µ,x,y,z + f,x,y,z σ,x,y,z W, Z W, fo, [,T an X R n, whee ξ,µ,σ,f ae meauable funcion. Moe peciely, ξ: Ω R n R m, µ: [,T Ω R n R m R m R n, σ: [,T Ω R n R m R m R n, f: [,T Ω R n R m R m R m, 6

8 fo n,m, N. Thoughou he whole ecion µ, σ an f ae aume o be pogeively meauable wih epec o F [,T, i.e. µ1 [,,σ1 [,,f1 [, ae B[,T F BR n BR m BR m -meauable fo all [,T. A ecoupling fiel come wih an even iche ucue han ju a claical oluion. Definiion 3. Le [,T. A funcion u: [,T Ω R n R m wih ut, = ξ a.e. i calle ecoupling fiel fo ξ,µ,σ,f on [,T if fo all 1, 2 [,T wih 1 2 an any F 1 -meauable X 1 : Ω R n hee exi pogeive pocee X,Y,Z on [ 1, 2 uch ha X = X 1 + Y = Y 2 1 µ,x,y,z + 2 f,x,y,z 1 σ,x,y,z W, 2 Z W, Y = u,x, 4 fo all [ 1, 2. In paicula, we wan all inegal o be well-efine an X,Y,Z o have value in R n, R m an R m, epecively. Some emak abou hi efiniion ae in place. The fi equaion in 4 i calle he fowa equaion, he econ he backwa equaion an he hi will be efee o a he ecoupling coniion. The equiemen ha X houl a a X 1 i efee o a he iniial coniion. By a ligh abue of noaion we will omeime efe o X 1 ielf a he iniial coniion. Noe ha, if 2 = T, we ge Y T = ξx T a.. a a conequence of he ecoupling coniion ogehe wih ut, = ξ. A he ame ime Y T = ξx T ogehe wih ecoupling coniion implie ut, = ξ a.e. If 2 = T we can ay ha a iple X,Y,Z olve he FBSDE, meaning ha i aifie he fowa an he backwa equaion, ogehe wih Y T = ξx T. Thi elaionhip Y T = ξx T i efee o a he eminal coniion. By an abueof noaion he funcion ξ ielf i alo omeime efeeo a heeminal coniion. Someime we will ecibe he elaionhip ut, = ξ a.e. wih hi em. In cona o claical oluion of FBSDE, ecoupling fiel on iffeen ineval can be pae ogehe. Lemma 4 Lemma 1 in [FI13. Le u be a ecoupling fiel fo ξ,µ,σ,f on [,T an ũ be a ecoupling fiel fo u,,µ,σ,f on [,, fo < < T. Then, he map û given by û := ũ1 [, +u1,t i a ecoupling fiel fo ξ,µ,σ,f on [,T. We wan o emak ha, if u i a ecoupling fiel an ũ i a moificaion of u, i.e. fo each [,T he funcion u,ω, an ũ,ω, coincie fo almo all ω Ω, hen ũ i alo a ecouplingfielo heamepoblem. Soucoul alo beefeeo a acla of moificaion. Some of he epeenaive of he cla migh be pogeively meauable, ohe no. A we ee below a pogeively meauable epeenaive oe exi if he ecoupling fiel i Lipchiz coninuou in x: 7

9 Lemma 5 Lemma 2 in [FI13. Le u: [,T Ω R n R m be a ecoupling fiel o ξ,µ,σ,f which i Lipchiz coninuou in x R n in he ene ha hee exi a conan L >.. fo evey [,T: u,ω,x u,ω,x L x x x,x R n, fo a.a. ω Ω. Then u ha a moificaion ũ which i pogeively meauable an Lipchiz coninuou in x in he ong ene ũ,ω,x ũ,ω,x L x x [,T, ω Ω, x,x R n. Le I [,T bean ineval an u : I Ω R n R m a map uch ha u, i meauable fo evey I. We efine L u,x := upinf{l fo a.a. ω Ω : u,ω,x u,ω,x L x x fo all x,x R n }, I whee inf :=. We alo e L u,x := if u, i no meauable fo evey I. One can how ha L u,x < i equivalen o u having a moificaion which i uly Lipchiz coninuou in x R n. WeenoebyL σ,z helipchizconan ofσ w... heepenenceonhela componen z an w... he Fobeniu nom on R m an R n. We e L σ,z = if σ i no Lipchiz coninuou in z. By L 1 σ,z = 1 L σ,z we mean 1 L σ,z if L σ,z > an ohewie. Definiion 6. Le u: [,T Ω R n R m be a ecoupling fiel o ξ,µ,σ,f. We ay u o be weakly egula if L u,x < L 1 σ,z an up [,T u,, <. Thi i a naual efiniion ue o Lemma 5. In pacice, howeve, i i impoan o have explici knowlege abou he egulaiy of X, Y, Z. Fo inance, i i impoan o know in which pace he pocee live, an how hey eac o change in he iniial value. Specifically, i can be vey ueful o have iffeeniabiliy of X, Y, Z w... he iniial value. In he following we nee fuhe noaion. Fo an inegable eal value anom vaiable F he expeion E [F efe o E[F F, while Eˆ, [F efe o eupe[f F, which migh be, bu i alway well efine a he infimum of all conan c [, uch ha E[F F c a.. Aiionally, we wie F fo he eenial upemum of F. Definiion 7. Leu: [,T Ω R n R m beaweaklyegulaecouplingfieloξ,µ,σ,f. We call u ongly egula if fo all fixe 1, 2 [,T, 1 2, he pocee X,Y,Z aiing in 4 ae a.e unique an aify [ 2 up E 1, [ X 2 + up E 1, [ Y 2 +E 1, Z 2 <, 5 [ 1, 2 [ 1, 2 1 foeach conan iniialvaluex 1 = x R n. Inaiionheymubemeauableafuncion of x,,ω an even weakly iffeeniable w... x R n uch ha fo evey [ 1, 2 he mapping X an Y ae meauable funcion of x,ω an even weakly iffeeniable w... x uch ha eup x R n up up E 1, [ x X 2 <, v S n 1 [ 1, 2 8 v

10 eup x R n eup x R n up v S n 1 [ 1, 2 up v S n 1 E 1, up E 1, [ x Y 2 <, v [ 2 x Z 2 <. 6 We ay ha a ecoupling fiel on [,T i ongly egula on a ubineval [ 1, 2 [,T if u eice o [ 1, 2 i a ongly egula ecoupling fiel fo u 2,,µ,σ,f. Une ceain coniion a ich exience, uniquene an egulaiy heoy fo ecoupling fiel can be evelope. We will ummaize he main eul, which ae poven in [FI13: Aumpion SLC: ξ,µ, σ, f aifie ana Lipchiz coniion SLC if 1. µ,σ,f ae Lipchiz coninuou in x,y,z wih Lipchiz conan L, 2. µ + f + σ,,,, <, 3. ξ: Ω R n R m i meauable uch ha ξ, < an L ξ,x < L 1 σ,z. Theoem 8 Theoem 1 in [FI13. Suppoe ξ,µ,σ,f aifie SLC. Then hee exi a ime [,T uch ha ξ,µ,σ,f ha a unique up o moificaion ecoupling fiel u on [,T wih L u,x < L 1 σ,z an up [,T u,, <. A bief icuion of exience an uniquene of claical oluion can be foun in Remak 3 in [FI13. Fo lae efeence we give he following emak cf. Remak 1 an 2 in [FI13. Remak 9. I can be obeve fom he poof ha he upemum of all h = T, wih aifying he popeie equie in Theoem 8 can be boune away fom by a boun, which only epen on he Lipchiz conan of µ,σ,f w... he la 3 componen, T, L ξ an L ξ L σ,z < 1, an which i monoonically eceaing in hee value. Remak 1. I can be obeve fom he poof ha ou ecoupling fiel u on [,T aifie L u,,x L ξ,x +CT 1 4, whee C i ome conan which oe no epen on [,T. Moe peciely, C epen only on T, L, L ξ,x, L ξ,x L σ,z an i monoonically inceaing in hee value. We can yemaically exen hi local heoy o obain global eul. Thi i bae on a imple agumen which we will efe o a mall ineval inucion. Lemma 11 Lemma 11 an 12 in [FI13. Le T 1 < T 2 be eal numbe an le S [T 1,T Fowa: If T 1 S an hee exi an h >.. [,+h [T 1,T 2 S fo all S, hen S = [T 1,T 2 an in paicula T 2 S. 2. Backwa: If T 2 S an hee exi an h >.. [ h, [T 1,T 2 S fo all S, hen S = [T 1,T 2 an in paicula T 1 S. Uing hee imple eul we obain global uniquene an global egulaiy of a ecoupling fiel. 1 v 9

11 Theoem 12 Coollay 1 an 2 in [FI13. Suppoe ha ξ,µ,σ,f aifie SLC. 1. Global uniquene: If hee ae wo weakly egula ecoupling fiel u 1,u 2 o he coeponing poblem on ome ineval [,T, hen we have u 1 = u 2 up o moificaion. 2. Global egulaiy: If hee exi a weakly egula ecoupling fiel u o hi poblem on ome ineval [,T, hen u i ongly egula. Noice ha Theoem 12 only povie uniquene of weakly egula ecoupling fiel, no uniquene of pocee X, Y, Z olving he FBSDE in he claical ene. Howeve, uing global egulaiy in Theoem 12 one can how: Coollay 13 Coollay 3 in [FI13. Le ξ,µ,σ,f fulfill SLC. If hee exi a weakly egula ecoupling fiel u of he coeponing FBSDE on ome ineval [,T, hen fo any iniial coniion X = x R n hee i a unique oluion X,Y,Z of he FBSDE on [,T aifying [ up E[ X 2 + up E[ Y 2 +E Z 2 <. [,T [,T 3.2 Makovian ecoupling fiel A yem of FBSDE given by ξ,µ,σ,f i ai o be Makovian if hee fou coefficien funcion ae eeminiic, ha i, if hey epen only on, x, y, z. In he Makovian iuaion we can omewha elax he Lipchiz coninuiy aumpion an ill obain local exience ogehe wih uniquene. Wha make he Makovian cae o pecial i he popey Z = u x,x σ,x,y,z, which come fom he fac ha u will alo be eeminiic. Thi popey allow u o boun Z by a conan if we aume ha σ i boune. Lemma 14 Lemma 14 in [FI13. Le µ,σ,f,ξ aify SLC an aume in aiion ha hey ae eeminiic. Aume ha we have a weakly egula ecoupling fiel u on an ineval [,T. Then u i eeminiic in he ene ha i ha a moificaion which i a funcion of,x [,T R n only. An applicaion of Lemma 14 i he following vey funamenal eul. Lemma 15 Lemma 15 in [FI13. Le ξ,µ, σ, f aify SLC an uppoe ha hee coefficien funcion ae eeminiic. Le u be a weakly egula ecoupling fiel on an ineval [,T. Chooe 1 < 2 fom [,T an an iniial coniion X 1. Then he coeponing Z aifie Z L u,x σ. If Z <, we alo have Z L u,x σ,,, 1 L u,x L σ,z 1. Nex we inveigae he coninuiy of u a a funcion of ime an pace. Lemma 16 Lemma 16 in [FI13. Aume ha µ,σ,f have linea gowh in x,y in he ene µ + σ + f,ω,x,y,z C1+ x + y,x,y,z [,T R n R m R m, 1

12 fo a.a. ω Ω, whee C [, i ome conan. If u i a ongly egula an eeminiic ecoupling fiel o ξ,µ,σ,f on an ineval [,T, hen u i coninuou in he ene ha i ha a moificaion which i a coninuou funcion on [,T R n. Thi bounene of Z in he Makovian cae moivae he following efiniion. I will allow u o evelop a heoy fo non-lipchiz poblem via uncaion. Definiion 17. Le [,T. We call a funcion u: [,T Ω R n R m wih ut,ω, = ξω, fo a.a. ω Ω a Makovian ecoupling fiel fo ξ,µ,σ,f on [,T if fo all 1, 2 [,T wih 1 2 an any F 1 -meauable X 1 : Ω R n hee exi pogeive pocee X,Y,Z on [ 1, 2 uch ha he equaion in 4 hol a.. fo all [ 1, 2, an aiionally Z <. We emak ha a Makovian ecoupling fiel i alway a ecoupling fiel in he ana ene a well. The only iffeence i ha we ae only ineee in iple X,Y,Z, whee Z i a.e. boune. Regulaiy fo Makovian ecoupling fiel i efine vey imilaly o ana egulaiy. Definiion 18. Le u: [,T Ω R n R m be a Makovian ecoupling fiel o ξ,µ,σ,f. We call u weakly egula if L u,x < L 1 σ,z an up [,T u,, <. We call a weakly egula u ongly egula if fo all fixe 1, 2 [,T, 1 2, he pocee X, Y, Z aiing in he efining popey of a Makovian ecoupling fiel ae a.e. unique fo each conan iniial value X 1 = x R n an aify 5. In aiion hey mu be meauable a funcion of x,, ω an even weakly iffeeniable w... x R n uch ha fo evey [ 1, 2 he mapping X an Y ae meauable funcion of x,ω, an even weakly iffeeniable w... x uch ha 6 hol. We ay ha a Makovian ecoupling fiel on [,T i ongly egula on a ubineval [ 1, 2 [,T if u eice o [ 1, 2 i a ongly egula Makovian ecoupling fiel fo u 2,,µ,σ,f. Now we efine a cla of poblem fo which an exience an uniquene heoy will be evelope. Aumpion MLLC: ξ,µ, σ, f fulfill a moifie local Lipchiz coniion MLLC if 1. he funcion µ,σ,f ae a eeminiic, b Lipchiz coninuou in x,y,z on e of he fom [,T R n R m B, whee B R m i an abiay boune e, c an fulfill µ,,,, f,,,, σ,,,,l σ,z <, 2. ξ: R n R m aifie L ξ,x < L 1 σ,z. We a a poviing a local exience eul. 11

13 Theoem 19. Le ξ,µ,σ,f aify MLLC. Then hee exi a ime [,T uch ha ξ,µ,σ,f ha a unique weakly egula Makovian ecoupling fiel u on [,T. Thi u i alo ongly egula, eeminiic, coninuou an aifie up 1, 2,X 1 Z <, whee 1 < 2 ae fom [,T an X 1 i an iniial value ee he efiniion of a Makovian ecoupling fiel fo he meaning of hee vaiable. Poof. Fo any conan H > le χ H : R m R m be efine a χ H z := 1 { z <H} z + H z 1 { z H}z. I i eay o check ha χ H i Lipchiz coninuou wih Lipchiz conan L χh = 1 an boune by H. Fuhemoe, we have χ H z = z if z H. We implemen an inne cuoff by efining µ H,σ H,f H via µ H,x,y,z := µ,x,y,χ H z, ec. The bounene of χ H ogehe wih i Lipchiz coninuiy make µ H,σ H,f H Lipchiz coninuou wih ome Lipchiz conan L H. Fuhemoe, L σh,z L σ,z. Alo µ H,σ H,f H have linea gowh in y,z a equie by Lemma 16. Accoing o Theoem 8 we know ha he poblem given by ξ,µ H,σ H,f H ha a unique weakly egula ecoupling fiel u on ome mall ineval [,T whee [,T. We alo know ha hi u i ongly egula, u i eeminiic by Lemma 14, an coninuou by Lemma 16. We will how ha fo ufficienly lage H an [,T i will alo beamakovian ecoupling fiel o he poblem ξ,µ,σ,f. Uing Remak 1 L u,,x L ξ,x +C H T 1 4 [,T, whee C H < i a conan which oe no epen on [,T. Fo any 1 [,T an F 1 - meauable iniial value X 1 conie he coeponing unique X,Y,Z on [ 1,T aifying he fowa equaion, he backwa equaion an he ecoupling coniion fo µ H,σ H,f H an u. Uing Lemma 15 we have Z L u,x σ H L u,x σ,,, +L σ,z H < an, heefoe, Z up [ 1,TL u,,x σ,,, 1 up [1,TL u,,x L σ,z L ξ,x +C H T σ,,, 1 L ξ,x L σ,z L σ,z C H T = L ξ,x σ,,, 1 L ξ,x L σ,z L σ,z C H T C HT σ,,, 1 L ξ,x L σ,z L σ,z C H T fo T 1 mall enough. Now we only nee o chooe H lage enough uch ha L ξ,x σ,,, 1 L ξ,x L σ,z become malle han H 4, an hen in he econ ep chooe cloe enough o T, uch ha L σ,z C H T 1 4 become malle han L ξ,xl σ,z, C H σ,,, T L ξ,x L σ,z become malle han H 4. 12

14 Conieing 7 hi implie ha if 1 [,T he poce Z a.e. oe no leave he egion in which he cuoff i paive, i.e. he ball of aiu H. Theefoe, u eice o he ineval [,T i a ecoupling fiel o ξ,µ,σ,f, no ju o ξ,µ H,σ H,f H. I i even a Makovian ecoupling fiel ue o he bounene of Z. A a Makovian ecoupling fiel i i weakly egula, becaue i i weakly egula a a ecoupling fiel o ξ,µ H,σ H,f H. Uniquene: Aume han hee i anohe weakly egula Makovian ecoupling fiel ũ o ξ,µ,σ,fon [,T. Chooe a 1 [,T an an x R n a iniial coniion X 1 = x, an conie he coeponing pocee X,Ỹ, Z ha aify he coeponing FBSDE on [ 1,T, ogehe wih he ecoupling coniion via ũ. A he ame ime conie X,Y,Z olving he ame FBSDE on [ 1,T, bu aociae wih he Makovian ecoupling fiel u. Since Z,Z ae boune, he wo iple X,Ỹ, Z an X,Y,Z alo olve he Lipchiz FBSDE given by ξ,µ H,σ H,f H on [ 1,T fo H lage enough. The wo coniion Ỹ = ũ, X an Y = u,x imply by Remak 3 in [FI13 ha boh iple ae pogeively meauable pocee on [ 1,T Ω.. [ up E, X 2 [ + up E, Y 2 [ +E, Z 2 [ 1,T [ 1,T 1 < an coincie. In paicula, ũ 1,x = Ỹ 1 = Y 1 = u 1,x. Song egulaiy of u a a Makovian ecoupling fiel o ξ,µ,σ,f follow iecly fom he above agumen abou uniquene of X, Y, Z fo eeminiic iniial value an boune Z, an he ong egulaiy of u a ecoupling fiel o ξ,µ H,σ H,f H. Remak 2. We obeve fom he poof ha he upemum of all h = T wih aifying he hypohee of Theoem 19 can be boune away fom by a boun, which only epen on L ξ,x, L ξ,x L σ,z, σ,,,, T, L σ,z, he value L H H [, whee L H i he Lipchiz conan of µ,σ,f on [,T R n R m B H w... o he la 3 componen, whee B H R m enoe he ball of aiu H wih cene, an which i monoonically eceaing in hee value. The following naual concep inouce a ype of Makovian ecoupling fiel fo non- Lipchiz poblem non-lipchiz in z, o which nevehele ana Lipchiz eul can be applie. Definiion 21. Le u be a Makovian ecoupling fiel fo ξ,µ,σ,f. We call u conolle in z if hee exi a conan C > uch ha fo all 1, 2 [,T, 1 2, an all iniial value X 1, he coeponing pocee X,Y,Z fom he efiniion of a Makovian ecoupling fiel aify Z ω C, fo almo all,ω [,T Ω. If fo a fixe iple 1, 2,X 1 hee ae iffeen choice fo X,Y,Z, hen all of hem ae uppoe o aify he above conol. We ay ha a Makovian ecoupling fiel on [,T i conolle in z on a ubineval [ 1, 2 [,T if u eice o [ 1, 2 i a Makovian ecoupling fiel fo u 2,,µ,σ,f ha i conolle in z. 13

15 A Makovian ecoupling fiel u on an ineval,t i ai o be conolle in z if i i conolle in z on evey compac ubineval [,T,T wih C poibly epening on. Remak 22. Ou Makovian ecoupling fiel fom Theoem 19 i obviouly conolle in z: conie 7 ogehe wih he choice of 1 mae in he poof. Remak 23. Le ξ,µ,σ,f aify MLLC, an aume ha we have a Makovian ecoupling fiel u on ome ineval [,T, which i weakly egula an conolle in z. Then u i alo a oluion o a Lipchiz poblem obaine hough a cuoff a in Theoem 19. A uch i i ongly egula Theoem 12 an eeminiic Lemma 14. Bu now Lemma 16 i alo applicable, ince ue o he ue of a cuoff we can aume he ype of linea gowh equie hee. So u i alo coninuou. Lemma 24. Le ξ,µ,σ,f aify MLLC. Fo < < T le u be a weakly egula Makovian ecoupling fiel fo ξ,µ,σ,f on [,T. If u i conolle in z on [, an T i mall enough a equie in Theoem 19 ep. Remak 2, hen u i conolle in z on [,T. Poof. Clealy, u i no ju conolle in z on [,, bu alo on [,T wih a poibly iffeen conan, accoing o Remak 22. Define C a he maximum of hee wo conan. We only nee o conol Z by C fo he cae 1 2 T, he ohe wo cae being ivial. Fo hi pupoe conie he pocee X,Y,Z on he ineval [ 1, 2 coeponing o ome iniial value X 1 an fulfilling he fowa equaion, he backwa equaion an he ecoupling coniion. Since he eicion of hee pocee o [ 1, ill fulfill hee hee popeie we obain Z ω C fo almo all [ 1,, ω Ω. A he ame ime, if we eic X,Y,Z o [, 2, we obeve ha hee eicion aify he fowa equaion, he backwa equaion an he ecoupling coniion fo he ineval [, 2 wih X a iniial value. Theefoe Z ω C hol fo a.a. [, 2, ω Ω a well. The following impoan eul allow u o connec he MLLC-cae o SLC. Theoem 25. Le ξ,µ,σ,f be uch ha MLLC i aifie an aume ha hee exi a weakly egula Makovian ecoupling fiel u o hi poblem on ome ineval [,T. Then u i conolle in z. Poof. Le S [,T be he e of all ime [,T,.. u i conolle in z on [,. Clealy S: Fo he ineval [, = {} one can only chooe 1 = 2 = an o Z: [, Ω R m i P-a.e., inepenenly of he iniial value X 1. So we can ake fo C any poiive value. Le S be abiay. Accoing o Lemma 24 hee exi an h >.. u i conolle in z on [,+h T ince u+h T, < an L u+h T, < L 1 σ,z. Conieing Remak 2 an he equiemen u <, L u,x < L 1 σ,z, we can chooe h inepenenly of. Thi how S = [,T by mall ineval inucion. 14

16 Noe ha Theoem 25 implie ogehe wih Remak 23 ha a weakly egula Makovian ecoupling fiel o an MLLC poblem i eeminiic an coninuou. Such a u will be a ana ecoupling fiel o an SLC poblem if we uncae µ,σ,f appopiaely. We can heeby exen he whole heoy o MLLC poblem: Theoem 26. Le ξ,µ,σ,f aify MLLC. 1. Global uniquene: If hee ae wo weakly egula Makovian ecoupling fiel u 1,u 2 o hi poblem on ome ineval [,T, hen u 1 = u Global egulaiy: If ha hee exi a weakly egula Makovian ecoupling fiel u o hi poblem on ome ineval [,T, hen u i ongly egula. Poof. 1. We know ha u 1 an u 2 ae conolle in z. Chooe a paive cuoff ee poof of Theoem 19 an apply 1. of Theoem u i conolle in z. Chooe a paive cuoff ee poof of Theoem 19 an apply 2. of Theoem 12. Lemma 27. Le ξ,µ,σ,f aify MLLC an aume ha hee exi a weakly egula Makovian ecoupling fiel u of he coeponing FBSDE on ome ineval [, T. Then fo any iniial coniion X = x R n hee i a unique oluion X,Y,Z of he FBSDE on [,T uch ha up E[ X 2 + up E[ Y 2 + Z <. [,T [,T Poof. Exience follow fom he fac ha u i alo ongly egula accoing o 2. of Theoem 26 an conolle in z accoing o Theoem 25. Uniquene follow fomcoollay 13: Aumeheeae wo oluion X,Y,Z an X,Ỹ, Z o he FBSDE on [,T boh aifying he afoemenione boun. Bu hen hey boh olve anslc-confom FBSDE obaine hough a paive cuoff. So hey mu coincie accoing o Coollay 13. Definiion 28. Le I M max [,T fo ξ,µ,σ,f be he union of all ineval [,T [,T uch ha hee exi a weakly egula Makovian ecoupling fiel u on [,T. Unfounaely, he maximal ineval migh vey well be open o he lef. Theefoe, we nee o make ou noion moe pecie in he following efiniion. Definiion 29. Le < T. We call a funcion u:,t R n R m a Makovian ecoupling fiel fo ξ,µ,σ,f on,t if u eice o [,T i a Makovian ecoupling fiel fo all,t. A Makovian ecoupling fiel u on,t i ai o be weakly egula if u eice o [,T i a weakly egula Makovian ecoupling fiel fo all,t. A Makovian ecoupling fiel u on,t i ai o be ongly egula if u eice o [,T i ongly egula fo all,t. Theoem 3 Global exience in weak fom. Le ξ,µ,σ,f aify MLLC. Then hee exi a unique weakly egula Makovian ecoupling fiel u on I M max. Thi u i alo eeminiic, conolle in z an ongly egula. Moeove, eihe I M max = [,T o IM max = M min,t, whee M min < T. 15

17 Poof. Le Imax. M Obviouly, hee exi a Makovian ecoupling fiel ǔ on [,T aifying Lǔ,x < L 1 σ,z an up [,T ǔ,, <. ǔ i conolle in z an ongly egula ue o Theoem 25 an 26. We can fuhe aume w.l.o.g. ha ǔ i a coninuou funcion on [,T R n accoing o Remak 23. Thee i only one uch ǔ accoing o Theoem 26. Fuhemoe, fo, Imax M he funcion ǔ an ǔ coincie on [,T becaue of Theoem 26. Define u, := ǔ, fo all Imax. M Thi funcion u i a Makovian ecoupling fiel on [,T, ince i coincie wih ǔ on [,T. Theefoe, u i a Makovian ecoupling fiel on he whole ineval Imax M an aifie L u [,T,x < L 1 σ,z, up [,T u [,T,, < fo all Imax. M Uniquene of u follow iecly fom Theoem 26 applie o evey ineval [,T Imax. M Aeing he fom of Imax M, we ee ha IM max = [,T wih,t i no poible: Aume ohewie. Accoing o he above hee exi a Makovian ecoupling fiel u on [,T.. L u,x < L 1 σ,z an up [,T u,, <. Bu hen u can be exene a lile bi o he lef uing Theoem 19 an Lemma 4, heeby conaicing he efiniion of Imax M. Thefollowing eulbaically ae ha foaingulaiy M min o occuu x ha o exploe a M min. Lemma 31. Le ξ,µ,σ,f aify MLLC. If Imax M = M min,t, hen lim L u,,x = L 1 σ,z, M min whee u i he Makovian ecoupling fiel accoing o Theoem 3. Poof. We ague iniecly. Aume ohewie. Then we can elec ime n M min, n uch ha upl un,,x < L 1 σ,z. n N Bu hen we may chooe an h > accoing o Remak 2 which oe no epen on n an hen chooe n lage enough o have n M min < h. So u can be exene o he lef o a lage ineval [ n h,t conaicing he efiniion of Imax. M 4 Soluion o he Skookho embeing poblem In hi ecion we peen a oluion o he Skookho embeing poblem a ae in SEP a he beginning of Secion 2 bae on oluion of he aociae yem of FBSDE. 4.1 Weak oluion Le u heefoe eun o ou FBSDE 3 ha can be ewien lighly moe geneally a X 1 = x 1 + 1W, X 2 = x 2 + Z 2, Y = gx 1 T δx2 T Z W, u,x 1,X 2 = Y, 8 16

18 fo [,T an x = x 1,x 2 R 2. So uing he noaion of Secion 3 we have µ,ω,x,y,z =,z 2, σ,ω,x,y,z = 1,, f,ω,x,y,z =, ξω,x = gx 1 δx 2, fo all,ω,x,y,z [,T Ω R 2 R R an = 1, n = 2 an m = 1. In paicula, he poblem aifie MLLC. Noice ha by chooing x := x 1,x 2 :=, an T = 1 we will have X 1 1 = W 1 an X 2 1 = 1 Z2, which make he FBSDE equivalen o 3. WihhegenealeulofSecion3.2ahanweaecapableoolvehiyemofequaion. In ohe wo, we ae able o pefom he econ ep of ou algoihm o olve he SEP. Lemma 32. Aume δ an g ae Lipchiz coninuou. Then fo he FBSDE 8 hee exi a unique weakly egula Makovian ecoupling fiel u on [,T. Thi u i ongly egula, conolle in z, eeminiic an coninuou. In paicula, equaion 3 ha a unique oluion Y,Z uch ha Z <. Poof. Uing Theoem 3 we know ha hee exi a unique weakly egula Makovian ecoupling fiel u on Imax. M Thi u i fuhemoe ongly egula, conolle in z, eeminiic an coninuou. I emain o pove Imax M = [,T. Due o Lemma 31 i i ufficien o how he exience of a conan C [, uch ha L u,,x C < L 1 σ,z fo all Imax. M In ou cae L 1 σ,z =, o we have o pove ha he weak paial eivaive of u wih epec o x1 an x 2 ae boh unifomly boune. Fix Imax M an conie he coeponing FBSDE on [,T: Fi noice ha he aociae iple X,Y,Z epen on he iniial value x = x 1,x 2 R 2, even in a weakly iffeeniable way wih epec o he iniial value x, accoing o he ong egulaiy of u. Fo moe on ule egaing woking wih weak eivaive conul Secion 2.2 of [FI13. Le u fi look a he maix xx. We have x 1X1 = 1, x 2X1 =, x 1X2 = 2Z x 2X2 = 1+ x 1Z, 2Z x 2Z, a.. fo [,T, fo almo all x = x 1,x 2 R 2. In paicula, he 2 2-maix x X i inveible if an only if X 2 x 2 i no. We will ee lae ha i emain poiive on he whole ineval allowing u o apply he chain ule of Lemma 5 in oe o wie x u,x x X. Bu le u fi pocee by iffeeniaing he backwa equaion in 8 wih epec o x 2 : x 2Y = δ X 2 T T x 2X2 T x 2Z W. To be pecie he above hol a.. fo evey [,T, fo almo all x = x 1,x 2 R 2. Now efine a opping ime τ via { } τ := inf [,T : x 2X2 T. 17

19 Fo [,τ we have x u,x x X accoing o he chain ule of Lemma 5 an in paicula u,x 1 x 2,X 2 X 2 x 2 = Y x 2. Le u e V := x 2u,X1,X 2, [,T an Z := Z x 2 X 2 x 2 1 { [,τ}. Then he ynamic of X 2 1 x 2 can be expee by 1 τ 1 x τ 2X2 = 1 2Z Z x 2X2, 9 fo an abiay opping ime τ < τ wih value in [,T. We alo have Y x 2 = V an heefoe V = x 2 Y X 2 x 2, [,τ. X 2 x 2 Applying Iô fomula an uing he ynamic of Y an X 2 we eaily obain an x 2 x 2 equaion ecibing he ynamic of V τ : τ 1 τ 1 V τ = V + 2Z Z x 2X2 x 2Y + x 2Z x 2X2 W = V + τ 2Z V Z + τ Z W 1 fo any opping ime τ < τ wih value in [,T. Noe ha, ince V an 2ZV ae boune pocee, Z1[ τ i in BMOP accoing o Theoem 49 wih a BMOP-nom which oe no epen on τ < τ, an o in paicula E[ τ 2Z Z 2 <. Fom 9 we can acually euce ha τ = T mu hol almo uely. Inee, 9 implie ha 1 τ x τ 2X2 = exp 2Z Z o equivalenly τ x 2X2 τ = exp 2Z Z fo all opping ime τ < τ wih value in [,T. Uing coninuiy of whichgive uτ = T a.. becaue{τ < T} τ x 2X2 τ = exp 2Z Z >, { } X 2 x 2 τ = x 2 X 2 we obain, ueoconinuiy of x 2 X 2. So we have X 2 i poiive on he whole [,T an heefoe x 2 xx i inveible on [,T. Seing W := W 2Z V, [,T we can efomulae 1 o V = V + 18 Z W.

20 Thi mean ha V can be viewe a he coniional expecaion of V T = x 2uT,X1 T,X2 T = δ X 2 T wih epec o F an ome pobabiliy meaue, which un W ino a Bownian moion on [,T. Noeheeha 2Z V ibouneon[,t becaue Z <. Hence, weconclueha V an heefoe u,x 1,x 2 i boune by δ x 2 fo almo all x = x 1,x 2 R 2. Thi value i inepenen of. Seconly, we have o boun u,x 1,x 2. To hi en we iffeeniae he equaion in x 1 8 wih epec o x 1 : an efine Noe ha x 1X1 = 1, x 1X2 = 2Z x 1Z, x 1Y = g X 1 T δ X 2 T T x 1X2 T x 1Z W, x 1u,X1,X 2 + x 2u,X1,X 2 x 1X2 = x 1Y, U := x 1u,X1,X 2, Ž := x 1Z Z x 1X2. x 1X2 = U = x 1Y V x 1X2, 2Z Ž + Z x 1X2, which allow u o euce he ynamic of U fom he ynamic of Y, X 2 an V x 1 x 1 uing Iô fomula: U =U + 1 x 1Y V x 1X2 x 1X2 V =U + x 1Z W 2 V Z Ž + Z 2Z V Z + Z W x 1X2 x 1X2 whee he make em eihe mege ino one o cancel ou an he equaion implifie o U =U + =U + 2Z V Ž + Ž W. Ž W. 11 By heame agumen a fo he poce V weeuce ha U an heefoe x 1 u,x 1,x 2 i boune by g = L g fo almo all x 1,x 2, whee L g i he Lipchiz conan of g, 19

21 i.e. he infimum of all Lipchiz conan. Thi how ha Imax M = [,T. Finally, Lemma 27 how ha hee i a unique oluion X,Y,Z o he FBSDE on [,T fo any iniial value X 1,X2 = x 1,x 2 R 2 uch ha up E[ X 2 + up E[ Y 2 + Z <, [,T [,T which i equivalen o he imple coniion Z < a we claim: If Z <, hen accoing o he fowa equaion X 2 x 2 +T Z 2 <, up E[ X 2 = x up E[ W 2 = x 1 2 +T <, [,T [,T an accoing o he backwa equaion ogehe wih he Minkowki inequaliy 1 [ 1 up E[ Y 2 2 = up E[ E gx T δx2 T F [,T [,T E [ gx 1 T δx2 T E g +L g E [ X 1 T whee L g,l δ ae Lipchiz conan of g,δ δ +Lδ E [ gx 1 T 2 1 [ 2 δx 2 + E T [ X 2 T 21 2 <, Fo he following eul we ue he noaion of Secion 2. A befoe we aume ha β i boune away fom. Une hi coniion H 1 i well efine an Lipchiz coninuou. Theefoe, δ = ˆδ H 1 i Lipchiz coninuou if ˆδ i Lipchiz coninuou, which i equivalen o α being boune. Lemma 33. Suppoe g an δ ae boh Lipchiz coninuou wih Lipchiz conan L g an L δ. Then hee exi a Bownian moion B, a anom ime τ H 1 L 2 g an a conan c R uch ha c+ τ α + τ β B ha law ν. Poof. Fi we olve FBSDE 3 uing Lemma 32 uch ha he coeponing Z i boune. Accoing o Lemma 36, which we pove a bi lae, we can even aume ha Z i boune by L g. Now we e c := Y an conuc B an τ a in he poof of Lemma 2. Moeove, τ = H 1 1 Z2 i boune by H 1 L 2 g ince Z i boune by L g an H 1 i inceaing. Remak 34. I i a pioi no clea ha he anom ime τ i alo a opping ime wih epec o F B [, := σb, [, [, a alo menione in Remak 1.2 in [AHI8. Howeve, we will pove a ufficien cieion fo hi in em of egulaiy popeie of he Makovian ecoupling fiel u. Remak 35. The bounene of he opping ime olving he Skookho embeing poblem ha no been inveigae o fequenly. Howeve, vey ecenly i gaine aenion in [AS11 an [AHS13. Epecially, i economic inee come fom i applicaion in he conex of game heoy ee [SS9. 2

22 4.2 Song oluion Thi ubecion i evoe o he fouh ep of ou algoihm, i.e. o anlae he eul of he peceing ecion ino a oluion of he Skookho embeing poblem in he ong ene. Ou main goal i o how ha if g,δ ae ufficienly mooh, hen τ an B conuce o fa will have he popey ha τ i inee a opping ime w... filaion F B [, geneae byhebownian moion B, anhu afuncional of heajecoie of B. Theame funcional applie o he ajecoie of he oiginal Bownian moion W will hen povie he equie ong oluion. Fo hi pupoe, we will aume ha g an δ ae hee ime weakly iffeeniable wih boune eivaive. We will alo equie ha g i non-eceaing an no conan. Ou agumen hall be bae on a eep analyi of egulaiy popeie of he aociae ecoupling fiel u. Fi le u fi pove he following vey ueful eul abou he oluion Y,Z o FBSDE 3 conuce in Lemma 32. Lemma 36. Aume δ an g ae Lipchiz coninuou. Le u be he unique weakly egula Makovian ecoupling fiel aociae o he poblem 8 on [, T conuce in Lemma 32. Then fo any [,T an iniial coniion X 1,X 2 = x 1,x 2 R 2 he aociae poce Z on [,T aifie Z L g = g. Fuhemoe, if he weak eivaive u ha a veion which i coninuou in he fi wo x 1 componen,x 1 on [,T R 2 hen Z ω = x 1u,X 1 ω,x 2 ω fo almo all,ω [,T Ω. Poof. We aleay know ha Z i boune accoing o Lemma 32, bu no in he fom of he moe explici boun Z L g. 1 +h Noice ha lim h h Z ω = Z ω fo almo all ω, Ω [,T ue o he funamenal heoem of Lebegue inegal calculu. 1 +h Now ake ome [,T.. lim h h Z = Z almo uely. Almo all [,T have hi popey. Chooe any h >. +h < T an conie he expeion 1 h E[Y +hw +h W F fo mall h >. On he one han we can wie uing Iô fomula which lea o Y +h W +h W = +h Y W + 1 h E[Y +hw +h W F = 1 h E +h [ +h W W Z W + On he ohe han we can ue he ecoupling coniion o wie Y +h W +h W =u +h,x 1 +h,x2 +h W +h W =u +h,x 1 +h,x2 W +h W + u +h,x 1 +h,x2 +h u +h Z, Z F Z fo h. +h,x 1 +h,x2 W +h W. 21

23 Afe applying coniional expecaion o boh ie of he above equaion we inveigae he wo umman on he igh han ie epaaely. Fi umman: Recall: X 1 an X 2 ae F -meauable, X 1 +h = X1 +W +h W, W +h W i inepenen of F, u i eeminiic, i.e. can be aume o be a funcion of,x 1,x 2 [,T R R only. Thee popeie imply [ E u +h,x 1 +h,x2 W +h W F = u +h,x 1 +z h,x 2 z h 1 e 1 2 z2 z R 2π = x 1u +h,x 1 +z h,x 2 h 1 e 1 2 z2 z, 2π which mean 1 [ lim h h E u +h,x 1 +h,x2 W +h W F R = x 1u,X 1,X 2, if u i coninuou in he fi wo componen on [,T R 2. Hee we ue ha u i x 1 x 1 boune by g accoing o he poof of Lemma 32. Bu even if u i no coninuou x 1 in he fi wo componen, we can ill a lea conol he value by g. 1 [ h E Secon umman: Recall: u +h,x 1 +h,x2 W +h W F u i alo Lipchiz coninuou in he la componen an δ eve a a Lipchiz conan, X 2 +h = X2 + +h Z 2. Thee popeie allow u o eimae 1 [ E u +h,x 1 h +h,x2 +h 1 [ u h E +h,x 1 +h,x2 +h 1 [ +h h E δ Z 2 which clealy en o a h. u u +h,x 1 +h,x2 +h,x 1 +h,x2 W +h W F W+h W F W +h W F 1 h δ h Z 2 E[ W +h W, 22

24 Concluion: We have hown 1 Z = lim h h E[Y 1 [ +hw +h W F = lim h h E u +h,x 1 +h,x2 W +h W F, which i ienical wih x 1 u,x 1,X 2 componen on [,T R 2 an boune by g ohewie. a.. if x 1 u i coninuou in he fi wo FoheequelleubeheuniqueweaklyegulaMakovianecouplingfielohepoblem 8 conuce in Lemma 32. A lea fo he following eul we aume fo convenience T = 1. We alo ue efiniion an noaion fom he poof of Lemma 2. Theoem 37. Aume ha x 1 u i Lipchiz coninuou in he fi wo componen on compac ube of [,1 R 2, R\{} - value on [,1 R 2. Then τ i a opping ime wih epec o he filaion F B = F B [,. Poof. We conie he yem 8 fo = an x 1 = x 2 =. Accoing o Lemma 36 we can aume Z = x 1u,X 1,X 2 an, heeby, have X 2 = Z 2 = fo all [,T. So, we can aume ha X 1 2 x 1u,X 1,X 2 i Lipchiz coninuou an icly inceaing in ue o poiiviy of [,1 R 2, 2 u x on 1 a in. Theefoe, fo evey ω Ω he pah H 1 X 2 ω : [,1 [, i alo Lipchiz coninuou an icly inceaing in ime an, heefoe, ha a coninuou an icly inceaing invee funcion on he ineval [,H 1 X 2 1 ω = [, τω. I i aighfowa o ee ha hi invee i given by he poce σ fom he poof of Lemma 2. We can now calculae he weak eivaive of σ: Fily, noe H 1 x = 1 H H 1 x an 23

25 alo H 1 X 2 σ ω = o equivalenly X 2 σ ω = H. So, we can calculae σ = 1 H 1 X 2 =σ = = 1 H 1 X 2 σ Z 2 σ H u σ x 1,X σ 1,X σ 2 2 = on {σ < 1}. Obeve a hi poin ha { {σ < 1} = < H 1 } X 2 1 = { < τ}. If we efine σ := 1 fo > τ, hen σ i ill coninuou an we have τ = inf{ [, σ 1}. x 1 u β 2 2σ,W σ,h I i alo aighfowa o ee Z σ = uσ x 1,W σ,h fo [, τ. Now, emembe B = 1 β Y σ fo [, τ an alo Y Y = Z W fo [,1, o β Z σ B = β Z σ 1 β Y σ = So, if we efine Σ := W σ, we have ynamic Σ = β uσ x 1,Σ,H B, fo [, τ. So, o um up σ,σ fulfill on [, τ he ynamic σ = + β 2 u x 1 Σ = + + 2σ +,Σ,H β 1 Z σ Z σ W σ = W σ. uσ x 1,Σ,H B, B, whee [, τ. Noe ha hi ynamical yem i locally Lipchiz coninuou in σ,σ. Now, fo any K 1,K 2 > an K 3,1 efine a boune anom vaiable τ K1,K 2,K 3 via τ K1,K 2,K 3 := K 1 inf{ [, Σ K 2 } inf{ [, σ K 3 }. Noe ha σ an Σ boh emain boune on [,τ K1,K 2,K 3. Theefoe, on [,τ K1,K 2,K 3 he pai σ,σ coincie wih he unique oluion σ K 1,K 2,K 3,Σ K 1,K 2,K 3 o a Lipchiz poblem, which i auomaically pogeively meauable w... he filaion F B. Noe ha τ K1,K 2,K 3 = K 1 inf { [, Σ K 1,K 2,K 3 K 2 } inf { [, σ K 1,K 2,K 3 K 3 }, which i clealy a opping ime w... F B. Fuhemoe, ue o coninuiy of Σ an σ τ = up τ K1,K 2,K 3, K 3,1,K 1,K 2 > which make i a opping ime w... F B a well

26 In oe o euce ufficien coniion fo Theoem 37 o hol we nee o inveigae highe oe eivaive of u. Aume ha g, δ, g an δ ae Lipchiz coninuou, an conie he following MLLC yem wih = 1, n = 2 an m = 3: X 1 = x 1 + 1W, X 2 = x 2 + Z 2, Y = gx 1 T δx2 T Z W, u,x 1,X 2 = Y, Y 1 = g X 1 T Y 2 = δ X 2 T Z 1 W Z 2 W 2Z Y 2 Z 1, u 1,X 1,X 2 = Y 1, 2Z Y 2 Z 2, u 2,X 1,X 2 = Y 2. Theoem 38. Fo he above poblem 13 we have Imax M = [,T. Fuhemoe, u = u, u 1 = x 1u an u2 = x 2u, a.e., whee u i he unique weakly egula Makovian ecoupling fiel o he poblem 8. In paicula, u i wice weakly iffeeniable w... x wih unifomly boune eivaive. Poof. The poof i in pa akin o he poof of Lemma 32 an we will eek o keep hee pa ho. Le u i, i =,1,2 be he unique weakly egula Makovian ecoupling fiel on Imax M. We can aume u i o be coninuou funcion on Imax M R2 Theoem 3. Le Imax M. Fo an abiay iniial coniion x R2 conie he coeponing pocee X 1,X 2,Y,Y 1,Y 2,Z,Z 1,Z 2 on [,T. Noe ha X 1,X 2,Y,Z olve FBSDE 8, which implie ha hey coincie wih he pocee X 1,X 2,Y,Z fom 8 if we aume 2 i=1 up [,T E, [ X i 2 + up E, [ Y 2 + Z + [,T 2 i= up [,T E, [ Y i Z i <, accoing o Lemma 27. Thi coniion i fulfille ue o ong egulaiy an he fac ha we wok wih Makovian ecoupling fiel. Now, Y = Y implie u,x = u,x fo all Imax M,x R2, whee Imax M i he maximal ineval fo he poblem given by 13. We now claim ha Y 1,Y 2 ae boune pocee: Uing he backwa equaion we have an, heefoe, [ [ Y 2 = E δ X 2 T E Y 2 δ + 2Z Y 2 Z 2 2 Z Z 2 E [ Y 2 25, i=

27 fo [,T, which uing Gonwall lemma implie [ Y 2 Y 2 = E δ exp 2T Z Z 2. Thi in un auomaically implie bounene of Y 1 accoing o i ynamic. Fuhemoe, Y 1,Z 1 an Y 2,Z 2 aify he BSDE which i alo fulfille by he pocee U,Ž an V, Z fom he poof of Lemma 32 ee 1 an 11 an o in paicula Y 2 V = = Z 2 Z W Z 2 Z W 2Z 2Z Y 2 Z 2 V Z Y 2 V Z 2 +V Z 2 Z. Uing he bounene of Z, Z 2 an V hi implie uing Lemma 48 ha Y 2 V i almo eveywhee. Theefoe, afe eing W := W 2Z V, [,T we ge fom he above equaion T Z 2 Z W = a.. fo [,T. Since W i a Bownian moion une ome pobabiliy meaue equivalen o P we alo have Z 2 Z = a.e. Similaly, one how ha Y 1 an U a well a Z 1 an Ž coincie o Y 1 = U, Y 2 = V, Z 1 = Ž an Z2 = Z a.e. Now, emembe U = u,x 1 x 1,X 2. Togehe wih u 1,X 1,X 2 = Y 1 an Y 1 = U hi yiel u 1, = u, an, heefoe, u 1 = u a.e. on I x 1 x max. M 1 Similaly, we ge u 2 = u. Now, noe ha u 1 = u i coninuou. Thi make x 2 x 1 Lemma 36 applicable, o Z = Z = U = Y 1 a.e. 14 Theeby Y 1, Y 2 aify he following ynamic: Y 1 = g X 1 T Y 2 = δ X 2 T Z 1 W Z 2 W which implie uing he chain ule of Lemma 5: 1 x iy =g X 1 an T 2 2 x iy = δ X 2 x ix1 x iz1 W T 1 x iy Y 2 +Y 1 T 2 2Y 1 Y 2 Z 1, 15 2Y 1 Y 2 Z 2, [,T, 16 2 x iy x iz2 W x ix2 T 1 x iy Y 2 +Y 1 2 x iy Z 1 +Y 1 Y 2 Z 2 +Y 1 Y 2 x iz1, x iz2, 26

28 fo i = 1,2. Le u ecall ome aemen abou he fowa poce obaine in he poof of Lemma 32: an x 2X2 >, x 2X2 x 1X2 = x 1X1 = 1, 1 = 1 2Y 1 2Y 1 Z 2 Z 1 +Z 2 x 2X1 =, a.e., x 2X2 x 1X2 Uing he chain ule of Lemma 5 an he ecoupling coniion, we have Now, efine i x 1Y = x 1ui,X 1,X 2 i x 2Y = x 2ui,X 1,X 2 Y 12 Y 22 Y 11 Y 21 := x 2u1,X 1,X 2 = x := x 2u2,X 1,X 2 = := x 1u1,X 1,X 2 = x := x 1u2,X 1,X 2 = x + x 2ui,X 1 x 2X2, i = 1,2. 1 2Y 2 x 2Y 1, ,X 2 x 2X2 x 2X2 x 1X2, 1, 19 1, 1 1Y Y 12 x 1X2 2 1Y Y 22 x 1X2., 2 We can apply he Iô fomula o euce ynamic of Y 12 an Y 11 fom ynamic of x 2 Y 1, Le u efine Z 12 := Y 12 = 1, X 2 Y 1 an X 2 : x 2 x 1 x 1 Z 1 1, x 2 X 2 x 2 o we can wie uing 19 +Y 1 Y 2 Z 12 W { 2 x 2Z1 x 2X2 1 x 2Y Y 2 +Y Uing he efiniion of Y 12, Y 22, Z 12 we can implify hi o Y 12 = Z 12 W 2 x 2Y Z 1 1 x 2Y Y 1 Z 2 x 2X2 1 } 2 Y 12 Y 2 +Y 1 Y 22 Z 1 +Y 1 Y 2 Z 12 +Y 12 Y 1 Z

29 Le u now efine Z 11 := Z 1 x 1 Z 12 X 2 x 1, o we can wie uing 2 Y 11 =g X 1 T Z 11 { T 2 2 Y 12 2 Y 1 W 1 x 1Y Y 2 +Y 1 Y 12 Y 2 +Y 1 Y 22 Z 1 +Z 2 2 x 1Y Z 1 +Y 1 Y 2 Z 1 +Y 1 Y 2 Z 12 +Y 12 Y 1 Z 2 }. x 1X2 The wo make em above can be effecively mege ino one uing 2: Y 11 =g X 1 T 2 { 2 Z 11 W Y 11 Y 2 +Y 1 2 x 1Y Z 1 +Y 1 Y 2 Y 1 Y 22 Z 1 +Y 1 Y 2 Z 12 +Y 12 Y 1 Z 2 }. Y 12 2 Y 1 Z 1 +Z 2 x 1X2 x 1Z1 x 1X2 x 1Z1 x 1X2 Similaly, he fou make em can be mege ino only wo uing he ucue of Y 21 an Z 11.. Y 11 =g X 1 T 2 { 2 Z 11 W Y 11 Y 2 +Y 1 Y 21 Z 1 +Y 1 Y 2 Z 11 Y 12 Y 1 Z 2 x 1X2 Y 12 2 Y 1 Z 1 + Z 2 whee he wo make em effecively cancel each ohe ou: Y 11 =g X 1 T 2 Z 11 W } x 1X2, Y 11 Y 2 +Y 1 Y 21 Z 1 +Y 1 Y 2 Z 11 +Y 12 Y 1 Z 1. Analogouly o Y 12 we can euce ynamic of Y 22 : Y 22 = δ X 2 T Z 22 W 2 Y 12 Y 2 +Y 1 Y 22 Z 2 +Y 1 Y 2 Z 22 +Y 22 Y 1 Z 2. 28

30 Fom hee we can, analogouly o Y 11, euce ynamic of Y 21 : Y 21 = Z 21 W 2 Y 11 Y 2 +Y 1 Y 21 Z 2 +Y 1 Y 2 Z 21 +Y 22 Y 1 Z 1. An o we have finally obaine he complee ynamic of he 4-imenional poce Y ij, i,j = 1,2, which ae clealy linea in i. Fuhemoe, emembe: Y 1,Y 2 aeunifomlybouneinepenenlyof,xueoheecouplingconiion, u i = x i u, i = 1,2 an Lemma 32, Z 1, Z 2 ae BMOP pocee wih unifomly boune BMOP-nom inepenenly of,x ue o 15, 16 an Theoem 49, Y ij, i,j = 1,2 ae boune accoing o hei efiniion wih a boun which may epen on,x a hi poin, Z ij, i,j = 1,2 ae in BMOP accoing o Theoem 49, Y ij T i,j=1,2 i unifomly boune by g + δ <. Theefoe, Lemma 48 i applicable an Y ij i,j=1,2 i unifomly boune, inepenenly of,x. In paicula, Y ij = u i,x, i,j = 1,2 can be conolle inepenenly of x j Imax, M x R 2, while u,x, j = 1,2 ha he ame popey a we aleay know. x j Thi how Imax M = [,T uing Lemma 31. Lemma 39. Aume ha g, δ, g, δ ae Lipchiz coninuou. Le u i be he unique i=,1,2 weakly egula Makovian ecoupling fiel o he poblem 13 conuce in Theoem 38. Aume ha u i, i =,1,2, ha a veion which i coninuou in he fi wo componen x 1,x 1 on [,T R 2 fo ome [,T. Then fo any iniial coniion X 1,X 2 = x 1,x 2 = x R 2 he aociae pocee Z i, i =,1,2, on [,T aify Z i ω = x 1ui,X 1 ω,x 2 ω, i =,1,2, fo almo all,ω [,T Ω. Fuhemoe, in hi cae he pocee x 1X2, x 2X2 an 1 x 2X2 on [,T, can be boune unifomly, i.e. inepenenly of, x. Poof. The fi pa of he poof wok analogouly o he poof of Lemma 36. So we keep ou agumen ho. Fo i =,1,2 we conie 1 i E[Y h +h W +h W F 29

31 fo mall h >. A in he poof of Lemma 36, we ue Iô fomula applie o 13 o obain an alo Y i +h W +h W = Y +h W +h W = which lea o an 1 i E[Y h +h +h + +h Y i W + W W Y W + +h W W Z i W 2Z Y 2 Z i + +h 1 E[Y h +h W +h W F = 1 [ +h h E +h W +h W F = 1 h E [ +h W W Z W + F Z +h +h Z i, Z, Z fo h, Z i 1+W W 2Z Y 2 F Z i a h fo i = 1,2. The agumen ae vali fo almo all [,T. On he ohe han we can ue he ecoupling coniion o ewie Y i +h W +h W =u i +h,x 1 +h,x2 W +h W + u i +h,x 1 +h,x2 +h u i +h,x 1 +h,x2 W +h W. Le u eal epaaely wih he wo umman. Fo he fi one ecall ha X 1 an X 2 ae F -meauable, X 1 +h = X1 +W +h W, W +h W i inepenen of F, u i eeminiic, i.e. i aume o be a funcion of,x 1,x 2 [,T R 2. A combinaion of hee popeie lea o 1 [ lim h h E u i +h,x 1 +h,x2 W +h W F = x 1ui,X 1,X 2, if u i i coninuou in he fi wo componen on [,T R 2, whee we ue ha x 1 i boune. Fo he econ umman ecall ha x 1 u i u i i alo Lipchiz coninuou in he la componen wih ome Lipchiz conan L, X 2 +h = X2 + +h 2. Z 3