On g Evaluations with L p Domains under Jump Filtration

Transcription

1 O g valuaos wh L Domas uder Jum Flrao Sog Yao Absrac Gve 1,, he uque L soluos of backward sochasc dffereal equaos wh jums BSDJs allow us o exed he oo of g evaluaos, arcular g execaos, o he jum case wh L domas. We exlore may mora roeres of he exeded g evaluaos cludg ooal samlg, ucrossg equaly, Doob-Meyer decomoso, geeraor rereseao ad Jese s equaly. Mos of hese resuls are mora for he furher develome of jum-flrao cosse olear execaos wh L domas 95. Keywords: Backward sochasc dffereal equao wh jums, L soluos, g evaluao, g execao, ooal samlg, ucrossg equaly, Doob-Meyer decomoso, geeraor rereseao, reverse comarso heorem of BSDJs, Jese s equaly. 1 Iroduco Le 1, ad T,. Gve a Lschz geeraor g, 94 showed ha for each egrable ermal daa ξ, he real-valued backward sochasc dffereal equao wh jums BSDJ T T Y = ξ + gs, Y s, Z s, U s ds Z s db s U s xñds, dx,, T 1.1,T ha s drve by a Browa moo B ad a deede valued Posso o rocess adms a uque L soluo Y ξ, Z ξ, U ξ. I arcular, he rocess Y ξ ca be regarded as he so-called codoal g execao of ξ: g ξ F := Y ξ,, T. The g execao { g ξ F },T ca be furher geeralzed as g evaluaos { g τ,γ ξ }, by cosderg BSDJ wh radom horzo. Such a g evaluaos are closely relaed o a large class τ<γ of cohere or covex rsk measures for egrable facal osos whch may o be square-egrable a marke wh jums. I hs aer, we show ha as olear execaos wh L domas uder jum flrao he flrao geeraed by B ad Posso radom measure N, he g evaluaos her may mora margale roeres from he classc lear execaos such as ooal samlg, ucrossg equaly, Doob-Meyer decomoso, Jese s equaly ad ec. Mos of hese resuls wll asss us o sudy face markes wh jums usg olear evaluao crera or rsk measureme. The well-kow Allas aradox suggess eole o develo a olear-execao verso of he vo Neuma- Morgeser s axomac sysem of execed ules, a fudameal oo he moder ecoomcs. Movaed by such a geeralzao, Peg 77, 8 roduced he coces of g execaos ad g evaluaos va backward sochasc dffereal equaos BSDs. These wo semal works ad some followg research 3, 15,, 81, 86 amog ohers show ha he g evaluaos are closely relaed o axom-based cohere ad covex rsk measures see 4, 39 mahemacal face: Whe he geeraor g s osvely homogeeous or covex y, z, he ρ g ξ:= g ξ F defes a cohere or covex rsk measure. Reversely, uder cera domao codo see 4.1 of 3, a cohere or covex rsk measure {ρ },T wh L doma uder Browa flrao ca be rereseed by some g execao or he soluo of a BSD wh geeraor g ad square-egrable ermal daa ξ. L 67 ad Royer 87 exeded he g execaos o he jum case ad obaed a Doob-Meyer decomoso for g execaos wh L domas uder jum flrao. Uder a smlar domao codo o 4.1 of 3, 87 Dearme of Mahemacs, Uversy of Psburgh, Psburgh, PA 156; emal: sogyao@.edu.

2 g valuaos wh L Domas uder Jum Flrao also showed ha a rsk measure wh L doma s sll a g execao a facal marke wh jums. O he oher had, Ma ad Yao 68 geeralzed he g evaluaos o he quadrac case.e. he geeraor g has a quadrac growh z whle Hu e al. 45 derved a rereseao of covex rsk measures by quadrac g execaos uder a dffere domao codo. Recely, 54 eve exeded he quadrac g execaos o he jum case ad demosraed ha he corresodg margale roeres sll hold, such as Doob-Meyer decomoso ad dowcrossg equaly. Based o hese feaures, hey rovded a dual rereseao for dyamc rsk measures wh jums. The rese aer sars wh a src comarso heorem for L soluos of BSDJs Theorem. uder a addoal codo A3 u. Theorem. ogeher wh he uqueess resul of BSDJs L sese mles ha he corresodg g evaluaos wh L domas uder he jum flrao hers src mooocy, cosa reservg, me-cossecy, zero-oe law, raslao varace, covexy, osve homogeey g1 g7 from lear execaos ad hus reserves some classc margale roeres such as ooal samlg, ucrossg equaly ad Doob-Meyer decomoso Prooso 4.1, Prooso 4., Theorem 4.1. I arcular, he roof of he Doob-Meyer decomoso for g suermagales also deeds o a moooc lm heorem of egrable jum dffuso rocesses wh jums Theorem A.1 as well as a a ror L esmae of a geeralzed BSDJ Prooso 4.3. Moreover, we exlore oher ce roeres of g evaluaos: Usg a resul of 94, we ca rerese a geeraor g as he lm of he dfferece quoes of he corresodg g evaluaos see Prooso 5.1, whch gves rse o a reverse comarso heorem of BSDJs Theorem 5.1. Prooso 5.1 also esablshes a equvalece bewee he covexy res. osve homogeey of g y, z, u ad he covexy res. osve homogeey of g evaluaos, as well as a equvalece bewee he deedece of g o y varable ad he raslao varace of g evaluaos Prooso 5.. Whe he geeraor s covex z, u, we ca use he comarso heorem of BSDJs aga o derve Jese s equaly of g evaluaos Theorem 5.. Ma Corbuos. Gve U U loc, ulke he case of Browa sochasc egrals, he Burkholder-Davs-Gudy equaly s o alcable for he / h ower of he Posso sochasc egral, Y su s xñds, dx,, T see e.g. Theorem VII.9 of 34:.e. su, Y su s xñds, dx cao be domaed by,t Y s U x N d,,t dx 4. So o derve a a ror L esmae for BSDJs, we could o follow he classcal argume he { roof of 16, Prooso 3., eher could we emloy he sace U, T := U : U x νdxd } < { or he sace Ũ, := U :,T U x N d, dx } < Acually oe may o be able o comare,t U x N d, dx T wh U x νdxd. I 94, we sared wh a geeralzao of he Posso sochasc egral for a radom feld U U by cosrucg a càdlàg uformly egrable margale M U :=, U sxñds, dx,, T, whose quadrac varao M U, M U s sll, U sx N ds, dx,, T. I dervg he key L ye equaly see Lemma 3.1 of 94 abou he dfferece Y = Y 1 Y of wo egrable soluos o BSDJs wh dffere arameers, our delcae aalyss showed ha he varaoal jum ar Y s Y s Y s 1, Y s he dyamcs of Y wll s eveually bol dow o he erm T U 1 x U x νdxd, whch jusfes our choce of U over U, or Ũ, as he sace for jum dffuso. The esmao course of he varaoal jum s full of aalycal sublees, bu we maaged o overcome hem by ulzg some ew echques ad secal reames see of 94 for deals. I he rese aer, we develoed hese echques o hadle smlar bur more comlcaed echcal hurdles whe we are dervg he a ror L esmae for a secal BSDJ Prooso 4.3 see , or whe we are measurg he L dsace of a creasg sequece of jum dffuso rocesses Y from s lm Y Theorem A.1 see A.4 A.47 or A.57 A.6. As aforemeoed, boh Prooso 4.3 ad Theorem A.1 are crucal rovg he Doob-Meyer decomoso for g suermagales our ma Theorem 4.1.

3 1. Iroduco 3 Our aalyss he aer also heavly reles o he follow equaly T M U, M U = U x νdxd U x N d, dx=,t T U x νdxd. Alhough may of our resuls look smlar o hose wh L domas he o-jum case 3 or jum case 87, we have o do more delcae aalyss o overcome varous echcal sublees rased he L jum case. For sace, o demosrae he moooc lm heorem Theorem A.1, we orvally exed Lemma.3 of 78 o he L jum case, see Lemma A.4. All margale roeres of g evaluaos L jum case, esecally he Doob-Meyer decomoso ad he moooc lm heorem, wll lay mora roles our sudy of a geeral class of jum-flrao cosse olear execaos wh L domas, whch ecomasses may cohere or covex me-cosse rsk measures ρ = {ρ },T. Uder cera domao codo, we show 95 ha he olear execao reserves may mora margale roeres of lear execaos cludg ooal samlg ad Doob-Meyer decomoso, ad hus ca be rereseed by some g execao. Cosequely, oe ca ulze he BSDJ heory o sysemacally aalyze he rsk measure ρ wh L domas ad emloy umercal schemes of BSDJs o ru smulao for facal roblems volvg ρ a marke wh jums. I aoher of our accomay aer 93, we aalyze a BSDJ wh a egrable reflecg barrer L whose geeraor g s Lschz couous y, z, u. We show ha such a refleced BSDJ wh egrable arameers adms a uque L soluo, ad hus solves he corresodg omal sog roblem uder he g execao or some domaed rsk measure wh L doma: he Y comoe of he uque soluo s exacly he Sell eveloe of rocess L uder he g execao ad he frs me mees L s a omal sog me for maxmzg he g execao of reward L or mmzg he rsk measure of facal oso L. Releva Leraure. The backward sochasc equao BSD was roduced by Bsmu 1 as he adjo equao for he Poryag maxmum rcle sochasc corol heory. Laer, Pardoux ad Peg 76 commeced a sysemacal research of BSDs. Sce he, he BSD heory has grow radly ad has bee aled o varous areas such as mahemacal face, heorecal ecoomcs, sochasc corol ad omzao, aral dffereal equaos, dffereal geomery ad ec, see he refereces 38, L ad Tag 9 roduced o he BSD a jum erm ha s drve by a Posso radom measure deede of he Browa moo. These auhors obaed he exsece of a uque soluo o a BSDJ wh a Lschz geeraor ad square-egrable ermal daa. The Barles, Buckdah ad Pardoux 19, 7 showed ha he wellosedess of BSDJs gves rse o a vscosy soluo of a semlear arabolc aral egro-dffereal equao PID ad hus rovdes a robablsc erreao of such a PID. Laer, Pardoux 75 relaxed he Lschz codo of he geeraor o varable y by assumg a mooocy codo o varable y sead. Su 89 ad Mao ad Y 96 eve degeeraed he mooocy codo of he geeraor o a weaker verso so as o remove he Lschz codo o varable z. Durg he develome of he BSD heory, some effors were made relaxg he square egrably o he ermal daa so as o be comable wh he fac ha lear BSDs are well-osed for egrable ermal daa or ha lear execaos have L 1 domas: l Karou e al. 38 showed ha for ay egrable ermal daa, he BSD wh a Lschz geeraor adms a uque L soluo. The Brad ad Carmoa 14 reduced he Lschz codo of he geeraor o varable y by a srog mooocy codo as well as a olyomal growh codo o varable y. Laer, Brad e al. 16 foud ha he olyomal growh codo s o ecessary f oe uses he mooocy codo smlar o ha of 75. We aalyzed L soluos of mul-dmesoal BSDJs uder a moooe codo 94, whle Kruse ad Poer 61, 63 suded a smlar L soluo roblem of BSD uder a rgh-couous flrao whch may be larger ha he jum flrao: T T T Y =ξ+ fs, Y s, Z s, U s ds Z s db s U s xñds, dx dm s,, T, 1.,T where M s a local margale orhogoal o he jum flrao. Also, Klmsak suded L soluos of refleced BSDs uder a geeral rgh-couous flrao 57.

4 g valuaos wh L Domas uder Jum Flrao 4 3 The researches o BSDs over geeral flered robably saces have recely araced more ad more aeo. A seres of works 18, 36, 38, 17,, 66, 1 are dedcaed o he heory of BSDs 1. bu drve by a càdlàg margale uder a rgh-couous flrao ha s also quas-lef couous. Laely, 13, 74 removed he quaslef couy assumo from he flrao so ha he quadrac varao of he drvg margale does o eed o be absoluely couous. O he oher had, based o a geeral margale rereseao resul due o Davs ad Varaya 3, Cohe ad llo 5, 6 dscussed he case where he drvg margales are o a ror chose bu mosed by he flrao; see Hassa ad Ouke 44 for a smlar aroach o a BSD form of a geerc ma from a sace of semmargales o he saces of margales ad hose of fe-varao rocesses. Also, Maa ad Tevzadze 69 ad Jeablac e al. 48 suded BSDs for semmargales ad her alcaos o mea-varace hedgg. As o BSDs drve by oher dscouous radom sources, a 9 ad Bad 6 suded BSDs drve by a radom measure; Coforola e al. 8, 9 cosdered BSDs drve by a marked o rocess; 73, 5, 84, 4 aalyzed BSDs drve by Lévy rocesses;, 88, 55 dscussed BSDs drve by a rocess wh a fe umber of marked jums. 4 There are also ley of researches o quadrac BSDJs: To sudy he exoeal uly maxmzao roblem wh a addoal lably, Becherer 1 exeded Kobylask 6 s moooe sably aroach o a jum-dffuso model ad obaed a uque bouded soluo o a relaed BSD drve by a radom measure whose geeraor may o be Lschz couous u. Becherer e al. 11 recely geeralzed hs resul for radom measures of fe acvy wh a o-deermsc comesaor. Meawhle, Morlas 7 ulzed a smlar moooe sably aroach ad dyamc rogrammg o show ha a secal quadrac BSDJ wh bouded ermal daa has a uque soluo, whose Y comoe s he value rocess of a exoeal uly maxmzao roblem wh jums. Morlas 71 eve obaed a exsece resul for such quadrac BSDJs wh exoeally egrable ermal daa. For geeral quadrac BSDJs wh ubouded ermal daa, Ngoueyou 7 ad l Karou e al. 37 exeded Barreu ad l Karou 8 s quadrac semmargales aroach o he jum case. They maaged o oba a exsece resul for quadrac-exoeal BSDJs.e. quadrac BSDJs whose geeraors have a exoeal growh u wh ubouded ermal daa. Also, Jeablac e al. 49 descrbed he value rocess of a uly omzao roblem uder Kgha-uceray a jum seg as a class of quadrac-exoeal BSDJs. Whe geeraors of quadrac-exoeal BSDJs are allowed o be locally-lschz, Fuj ad Takahash 4 rovded a suffce codo for he Mallav s dffereably of such BSDJs wh bouded ermal daa whle 3 could sll emloy 6 s moooe sably aroach o show he wellosedess of such BSDJs. As o dffere mehods o quadrac BSDJs, Kaz-Ta e al. 51, 54 exloed he fxed-o aroach as Tevzadze 91 ad a exquse slg echque o demosrae he wellosedess of quadrac-exoeal BSDJs wh bouded ermal daa ad aled hs resul o sudy he relaed olear execaos; Laeve ad Sadje 64 ook a dualy aroach o characerze he value of a omal orfolo valuao roblem as he uque soluo o a BSDJ wh a covex geeraor whch has a mos quadrac growh z. 5 I s worh meog ha 53, 5 recely made a very eresg develome of secod-order BSDs wh jums, ad rovded a robablsc erreao for he relaed fully-olear PIDs. For ocs of BSDJs oher drecos, see Cohe ad llo 3, 4, 7 for BSDs drve by Markov chas; see Kharroub e al. 56 for mmal soluos o BSDs wh cosraed jums ad relaed quas-varaoal equales; see Aazz ad Ouke 1 for a class of cosraed BSDJs ad s alcao rcg ad hedgg Amerca oos; see Klmsak ad Rozkosz 58, 59 for a geeral o-markova BSD ad a relaed semlear ellc equao wh measure daa whose oeraor s assocaed wh a regular sem-drchle form; see 6, 43 for BSDJs wh sgular ermal daa ad her alcaos o omal oso argeg ad a o-markova lqudao roblem resecvely; see also 65, 41, 35 for umercal smulao of BSDJs amog oher. The res of he aer s orgazed as follows: We roduce some oaos Seco 1.1. I Seco, afer makg basc assumos o geeraor g, we revew some roeres of L soluos o BSDJs wh geeraor g cludg he wellosedess resul, he margale rereseao heorem as well as a a ror esmae, ad rove a src comarso heorem for hese L soluos. I Seco 3, we defe he g evaluao wh doma L uder jum flrao accordg o he wellosedess of BSDJs wh geeraor g L sese. The we show ha he g evaluao reserves may basc roeres of lear execaos. I Seco 4, we obaed some margale

5 1.1 Noao ad Prelmares 5 roeres of he g evaluao such as ooal samlg, ucrossg equaly ad Doob-Meyer decomoso. Seco 5 dscuss some oher fe roeres of g evaluaos cludg a geeraor rereseao va g evaluaos, some of s cosequeces ad Jese s equaly of he g evaluaos. The roofs of our resuls are relegaed o Seco 6. We geeralze 79 s moooc lm heorem for egrable jum dffuso rocesses wh jums he aedx as s eresg s ow rgh. 1.1 Noao ad Prelmares Throughou hs aer, we fx a me horzo T, ad le Ω, F, P be a comlee robably sace o whch a d dmesoal Browa moo B s defed. For a geerc càdlàg rocess, we deoe s corresodg jum rocess by :=,, T wh :=. Gve a measurable sace, F, le be a valued Posso o rocess o Ω, F, P ha s deede of B. For ay scearo ω Ω, he se D ω of all jum mes of ah ω s a couable subse of, T see e.g. Seco 1.9 of 46. We assume ha for some fe measure ν o, F, he coug measure N d, dx of o, T has comesaor N d, dx = νdxd. The corresodg comesaed Posso radom measure Ñ wll be deoed by Ñd, dx:=n d, dx νdxd. For ay, T, we defe sgma-felds F B := σ { B s ; s }, F N := σ { } N, s, A; s, A F, F :=σ F B F N ad augme hem by all P ull ses of F. Clearly, he jum flrao F = {F },T s comlee ad rghcouous.e. sasfes he usual hyoheses, see e.g., 8. Deoe by P res. P he F rogressvely measurable res. F redcable sgma-feld o, T Ω, ad le T be he colleco of all F sog mes. For ay τ T, we se T τ :={γ T : γ τ, P a.s.}. Recall ha a uformly egrable càdlàg margale M s sad o be a BMO Bouded Mea Oscllao margale f here exss C > such ha for ay τ T M, M T M, M τ F τ C ad Mτ C, P a.s. The followg saces of fucos wll be used he sequel: 1 For ay 1,, le L +, T be he sace of all measurable fucos ψ :, T, wh T d<. ψ For 1,, le L ν := L, F, ν; R be he sace of all real-valued, F measurable fucos u wh u L ν := ux νdx 1 <. For ay u 1, u L ν, we say u 1 =u f u 1 x=u x for ν a.s. x. 3 For ay sub-sgma-feld { G of F, le L G be he sace of all real-valued, G measurable radom varables ad se { L G := ξ L G : ξ L G := } ξ } 1 < for ay 1, ; { } L G := ξ L G : ξ L G := esssu ξω <. ω Ω 4 Le D be he sace of all real-valued, F adaed càdlàg rocesses, ad le K be a subsace of D ha cludes all F redcable càdlàg creasg rocesses wh =. 5 Se Z loc :=L loc, T Ω, P, d dp ; R d, he sace of all R d valued, F redcable rocesses Z wh T Z d <, P a.s. 6 For ay { 1,, we le D := D : D := { } } 1 <, where := su <.,T K :=K D = { K K : K T < }. { Z, := {Z Z loc : Z Z, := T Z d } 1 } <. For ay Z Z,, he Burkholder-Davs-Gudy equaly mles ha T su Z s db s c Z s ds < 1.3,T for some cosa c > deedg o. So { Z sdb s s a uformly egrable margale. },T

6 g valuaos wh L Domas uder Jum Flrao 6 U loc := loc L, T Ω, P F, d dp νdx; R be he sace of all P F measurable radom felds U :, T Ω R such ha T U x νdxd= T U d<, P a.s. { L ν U := U U loc : U U :={ T U x νdxd } } 1 < =L, T Ω, P F, d dp νdx; R. For ay U U loc res. U, holds for d dp a.s., ω, T Ω ha U, ω L ν. Accordg o Seco 1. of 94, we ca defe a Posso sochasc egral of U: M U := U s xñds, dx,, T, 1.4, whch s a càdlàg local margale res. uformly egrable margale wh quadrac varao M U, M U =, U sx N ds, dx,, T. The jum rocess of M U s M U ω = 1 { Dω }U, ω, ω,, T. For ay U U, a aalogy o 5.1 of 94 shows ha U x N d, dx s U x νdxd, <s T. 1.5,s We smly deoe D Z, U by S. As usual, we se x := x, x + := x for ay x R, ad use he coveo f :=. Gve,, he followg wo equales wll be frequely aled hs aer: For ay a, b R, a ± b ± a b. 1.6 For ay fe subse {a 1,, a } of,, 1 1 a a 1 1 a. 1.7 Also, we le c deoe a geerc cosa deedg oly o arcular, c sads for a geerc cosa deedg o ohg, whose form may vary from le o le. =1 =1 =1 L Soluos of BSDs wh Jums From ow o, we fx 1, ad se q := 1. A mag g :, T Ω R R d L ν R s called a geeraor f s P BR BR d B L ν /BR measurable. For ay τ T, g τ, ω, y, z, u:=1 {<τω} g, ω, y, z, u,, ω, y, z, u, T Ω R R d L ν s also P BR BR d B L ν /BR measurable. We say a geeraor g s covex y, z, u f holds P a.s. ha for ay, α, T, 1 ad y, z, u R R d L ν, =1, g, αy 1 +1 αy, αz 1 +1 αz, αu 1 +1 αu αg, y1, z 1, u 1 +1 αg, y, z, u..1 Also, we say a geeraor g s osvely homogeeous y, z, u f holds P a.s. ha g, αy, αz, αu= αg, y, z, u,, α, T,, y, z, u R R d L ν.. Defo.1. Gve 1,, le ξ L F T ad g be a geeraor. A rle Y, Z, U D Z loc U loc s called a soluo of a backward sochasc dffereal equao wh jums ha has ermal daa ξ ad geeraor g BSDJ ξ, g for shor f T gs, Y s, Z s, U s ds<, P a.s. ad f 1.1 holds P a.s. We shall make he followg sadard assumos o geeraors g: A1 T g,,, d L F T. A There exs wo, valued, B, T F T measurable rocesses β, Λ wh T β q Λ d L F T such ha for d dp a.s., ω, T Ω g, ω, y 1, z 1, u g, ω, y, z, u β, ω y 1 y +Λ, ω z 1 z, y 1, z 1, y, z R R d, u L ν.

7 . L Soluos of BSDs wh Jums 7 A3 There exss a fuco h:, T Ω R R d L ν L ν L q ν such ha h s P BR BR d BL ν BL ν/bl q ν measurable; There exs κ 1 1, ad κ κ 1 such ha for ay, ω, y, z, u 1, u, x, T Ω R R d L ν L ν κ 1 h, ω, y, z, u 1, u x κ ; I holds for d dp a.s., ω, T Ω ha g, ω, y, z, u 1 g, ω, y, z, u u1 x u x h, ω, y, z, u 1, u x νdx, y, z, u 1, u R R d L ν L ν..3 We refer o Ξ:=β, Λ, κ 1, κ as a coeffce se. Remark.1. Le 1, ad le g be a geeraor. 1 By A3, ad Hölder s equaly, A ad A3 mly A There exs wo, valued, B, T F T measurable rocesses β, Λ wh T ha for d dp a.s., ω, T Ω β q Λ d L F T such g, ω, y 1, z 1, u 1 g, ω, y, z, u β, ω y 1 y + u 1 u L ν +Λ, ω z1 z, y, z, u R R d L ν, =1,. If g sasfes A ad T g,,, d<, P a.s., he a aalogy o Remark.1 of 94 shows ha for ay Y, Z, U D 1 Z loc U loc, oe has T gs, Y s, Z s, U s ds<, P a.s. 3 If g sasfes A1, A res. A, he g, ω, y, z, u := g, ω, y, z, u,, ω, y, z, u, T Ω R R d L ν ad g τ, τ T are also geeraors sasfyg A1, A res. A. If g furher sasfes A3, so do g ad g τ. 4 We eed he assumo κ 1 > 1 A3 for a src comarso heorem of BSDJs Theorem. ad he ucrossg equaly of g suermargales Prooso 4.. Acually, s ecessary for he Doléas-Dade exoeals M 6.6 ad M D 6.4 o be srcly osve margales see e.g. 5, whch he allows us o aly Grsaov Theorem o chage robables he roofs of Theorem. ad Prooso 4.. For smlcy, we se Ĉ := T 1 β q Λ d L, ad le C be a geerc cosa deedg o T, ν,, F T Ĉ ad κ f ecessary, whose form may vary from le o le. For L soluos of BSDs wh jums, we frs quoe a wellosedess resul, he corresodg margale rereseao heorem as well as a a ror esmae from Remark 4.1, Prooso 3.1, Corollary 4.1, Corollary.1 ad Lemma 3.1 of 94. Theorem.1. Gve 1,, Le g be a geeraor sasfyg A1 ad A. BSDJ ξ, g adms a uque soluo Y ξ,g, Z ξ,g, U ξ,g S, whch sasfes Y ξ,g + D Z ξ,g + Z U ξ,g C ξ +, U T For ay ξ L F T, he g,,, d..4 I arcular, for ay τ T ad ξ L F τ, he uque soluo Y ξ,gτ, Z ξ,gτ, U ξ,gτ of he BSDJ ξ, gτ S sasfes ha P { Y ξ,gτ =Y ξ,gτ τ,, T } =1 ad ha Z ξ,gτ, U ξ,gτ =1{ τ} Z ξ,g τ, U ξ,gτ, d dp a.s. Remark.. Gve 1,, le g be a geeraor sasfyg A1 ad A. I holds for ay ξ L F T ha P { Y ξ,g = Y ξ,g,, T } =1. Corollary.1. Le 1,. For ay ξ L F T, here exss a uque ar Z, U Z, U such ha P a.s. ξ F = ξ + Z s db s + U s xñds, dx,, T., Prooso.1. Le 1,. For = 1,, le ξ L F T, g be a geeraor, ad Y, Z, U be a soluo of BSDJ ξ, g such ha Y 1 Y D. If g sasfes A, he Y 1 Y + Z 1 Z + U 1 U T C ξ D Z, U 1 ξ + g 1, Y, Z, U g, Y, Z, U d..5

8 g valuaos wh L Domas uder Jum Flrao 8 Moreover, we have he followg src comarso heorem for BSDJs, whch wll lay a key role he aer. Theorem.. Le 1,, τ T ad γ T τ. For = 1,, le ξ L F T, le g be a geeraor, ad le Y, Z, U be a soluo of BSDJ ξ, g such ha Y 1 Y D ad ha Yγ 1 Yγ, P a.s. For eher =1 or =, f g sasfes A, A3, ad f g 1, Y 3, Z 3, U 3 g, Y 3, Z 3, U 3, d dp a.s. o τ, γ, he holds P a.s. ha Y 1 Y for ay τ, γ. If oe furher has Yτ 1 = Yτ, P a.s., he holds P a.s. ha Y 1 =Y for ay τ, γ; holds d dp a.s. o τ, γ ha Z 1, U 1 =Z, U ad g 1, Y, Z, U =g, Y, Z, U, = 1,. 3 g valuaos wh L Domas The wellosedess resul of BSDs wh jums L sese Theorem.1 gves rse o a olear execao, called g evaluaos wh L domas, whch geeralzes he oe roduced 77 ad 8: Defo 3.1. Gve 1,, le g be a geeraor sasfyg A1, A, ad le τ T, γ T τ. g evaluao τ,γ g : L F γ L F τ by Defe g τ,γξ:=y ξ,gγ τ, ξ L F γ. If γ = T, we call g ξ F τ := g τ,t ξ he codoal g execao of ξ L F T a me τ. By Theorem.1, holds for ay ξ L F γ ha 1 {τ=γ} g τ,γξ=1 {τ=γ} Y ξ,gγ τ =1 {τ=γ} Yγ ξ,gγ =1 {τ=γ} Y ξ,gτ T = 1 {τ=γ} ξ, P a.s. 3.1 Lemma 3.1. Gve 1,, le g be a geeraor sasfyg A ad ha d dp a.s. g, y,, =, y R. 3. For ay τ T ad γ T τ, holds for ay ξ L F γ ha g τ,γξ = g ξ F τ, P a.s. I arcular, whe g, he g evaluao degeeraes o he classc lear execao: for ay τ T ad γ T τ, holds for ay ξ L F γ ha g τ,γξ = ξ F τ, P a.s. Le 1, ad le g be a geeraor sasfyg A1 ad A. Oe ca deduce from he uqueess resul ad he comarso heorem of L soluos o BSDJs Theorem.1 ad. as well as Lemma 3.1 ha he g evaluaos wh L domas ossess he followg basc roeres cf. 81: Le τ T, γ T τ ad ξ L F γ. g1 Src Mooocy : If g furher sasfes A3, he for ay η L F γ wh ξ η, P a.s. oe has τ,γξ g τ,γη, g P a.s.; Moreover, f furher holds ha τ,γξ= g τ,γη, g P a.s., he ξ =η, P a.s. g Cosa Preservg : Uder 3., f ξ s F τ measurable, he τ,γξ=ξ, g P a.s. g ζ,γ ξ = g τ,γξ. g3 Tme Cossecy : For ay ζ T wh τ ζ γ, P a.s., holds P a.s. ha g τ,ζ g4 Zero Oe Law : For ay A F τ, we have 1 A τ,γ1 g A ξ = 1 A τ,γξ, g P a.s.; I addo, f g,,, =, d dp a.s., he τ,γ1 g A ξ=1 A τ,γξ, g P a.s. g5 Traslao Ivarace : If g s deede of y, he τ,γξ+η= g τ,γξ+η, g P a.s. for ay η L F τ. g6 Covexy : If g s covex y, z, u, he τ,γαξ+1 αη α g τ,γξ+1 α g τ,γη, g P a.s. for ay η L F γ ad α, 1. g7 Posve Homogeey : If g s osvely homogeeous y, z, u, he τ,γαξ g = ατ,γξ, g P a.s. for ay α,. Now, le us cosder wo secfc geeraors sasfyg A1 A3 ad her corresodg g evaluaos: xamle 3.1. Gve 1,, le Ξ be a coeffce se. The fucos g Ξ, ω, y, z, u := β, ω y +Λ, ω z κ 1 u xνdx+κ g Ξ, ω, y, z, u := g Ξ, ω, y, z, u, u + xνdx,, ω, y, z, u, T Ω R R d L ν are wo geeraors sasfyg A1 A3 wh resec o he same coeffce se Ξ, where u ± x:=ux ±. The Ξ τ,γ := gξ τ,γ ad Ξ τ,γ := gξ τ,γ, τ T, γ T τ are wo g evaluaos wh L domas.

9 4. g Margales 9 I lgh of he comarso heorem for BSDJs Theorem., we ca boud he varao of a g evaluao by g Ξ evaluao ad g Ξ evaluao as follows. Prooso 3.1. Gve 1,, le g be a geeraor sasfyg A1 A3 wh resec o some coeffce se Ξ. For ay τ T, γ T τ ad ξ, η L F T, holds P a.s. ha Ξ τ,γξ η g τ,γξ g τ,γη Ξ τ,γξ η. 4 g Margales Le g be a geeraor sasfyg A1 ad A. We ca defe margales wh resec o he g evaluaos ha have L domas uder jum flrao. Defo 4.1. Gve 1,, le g be a geeraor sasfyg A1 ad A. A real-valued, F adaed rocess s called a g submargale res. g suermargale or g margale f for ay s T, s < ad g,s s res. or =, P a.s. The g margales rea may classc roeres such as ooal samlg, ucrossg equaly ad Doob-Meyer decomoso. Le us sar wh he ooal samlg heorem of g margales, whch s mora for he Doob-Meyer decomoso of g margales Theorem 4.1. Prooso 4.1. Ooal Samlg Gve 1,, le g be a geeraor sasfyg A1 A3. Le be a g submargale res. g suermargale wh < ad le τ T, γ T τ. If s rgh-couous or f τ, γ are fely valued, he g τ,γ γ res. τ, P a.s. The roof of Prooso 4.1 deeds o he followg lemma. Lemma 4.1. Gve 1,, le g be a geeraor sasfyg A1 ad A. Le τ T akg values a fe se {= 1 < < =T } wh. If <s +1 for some {1, 1}, he for ay ξ L F τ s g τ,τ sξ = 1 {τ }ξ + 1 {τ +1} g,sξ, P a.s. 4.1 To rese he ucrossg equaly of g margales, we recall he oo of umber of ucrossgs: Gve a real-valued rocess ad wo real umbers a < b, for ay fe subse D = { 1 < < m } of, T, we defe he umber of ucrossgs U D a, b; ω of erval a, b by he samle ah { ω} D as follows: Se m := m ad τ := 1. For =1,, m, we recursvely defe 1 ω := m{ D : > ω, ω < a} m T ω := m{ D : > 1 ω, ω > b} m T, ad 4. wh he coveo m =. The U D a, b; ω s se o be he larges eger such ha ω< m. To w, m U D a, b; ω = 1 {τω< m}. =1 Prooso 4.. Ucrossg Iequaly Gve 1,, le g be a geeraor sasfyg A1 A3 wh resec o some coeffce se Ξ, ad le be a g suermargale wh <. For ay real umbers a<b ad ay fe subse D ={ 1 < < m } of, T, he ucrossg umber U D a, b; of erval a, b sasfes l 1+U D a, b; { e 3Ĉ l b a, Ξ m m a + m } gs,,, ds + a e3ĉ b a Ĉ+ κ l1+κ 1 ν T. The Doob-Meyer decomoso of g margales wll lay a crucal role for rereseg jum-flrao cosse olear execaos wh doma L F T by g execaos our accomayg aer 95. Theorem 4.1. Doob-Meyer Decomoso Gve 1,, le g be a geeraor sasfyg A. ha g also sasfes A3 wh T Λ Assume d L F T f 1,, or wh Λ κ Λ, f =. If D s

10 g valuaos wh L Domas uder Jum Flrao 1 a g suermargale res. g submargale ad f T g,,, d <, he here exs uque rocesses Z, U, K Z, U K such ha P a.s. = T + T gs, s, Z s, U s ds T Z s db s U s xñds, dx+k T K res. K T +K,, T. 4.3,T The roof of Theorem 4.1 reles o a moooc lm heorem of jum dffuso rocesses over D see Theorem A.1 as well as he followg a ror L esmae o a secal BSDJ: Prooso 4.3. Gve 1, ad ξ L F T, le g be a geeraor ad le be a real-valued, F adaed càdlàg rocess wh + D. Le Y, Z, U, K D Z loc U loc K sasfes ha P a.s. Y =ξ+ T T gs, Y s, Z s, U s ds+k T K 1 {Y > }dk =. T Z s db s U s xñds, dx,, T,T If here exs hree, valued, B, T F T measurable rocesses f, β, Λ wh T f d L F T, T β q Λ d L F T such ha he Z, U Z, U ad 4.4 g, Y, Z, U f +β Y + U L ν +Λ Z, d dp a.s., 4.5 Y D + Z Z, + U U +K T C ξ + 5 Oher Fe Proeres of g valuaos T + + f d. 4.6 I hs seco we wll exed some fe roeres of g evaluaos o he jum case wh L domas. These roeres have bee exlored for dffere reasos uder Browa flrao, ad hus form a mora grede of he olear-execao heory. I lgh of Prooso 4.1 of Arv verso of 94, we ca frs rerese geeraors g as he lm of he dfferece quoes of he corresodg g evaluaos: Prooso 5.1. Gve 1, ad κ g >, le g be a geeraor sasfyg A1 ad A here exss some, valued, B, T F T measurable rocess β wh T βq d L + F T such ha for d dp a.s., ω, T Ω g, ω, y 1, z 1, u 1 g, ω, y, z, u β, ω y 1 y +κ g z1 z + u 1 u L ν, y, z, u R R d L ν, =1,. Le, y, z, u, T R R d L ν such ha lm gs, y, z, u=g, y, z, u, P a.s. ad s + 1 ε + ε The holds P a.s. ha g, y, z, u = lm r,s uxñdr, dx, s, T. su s,+δ gs, y,, < for some δ =δ, y, T. 5.1 g,+ε y + V, + ε, z, u y, where V, s, z, u := zbs B + A smle alcao of Prooso 5.1 gves rse o he followg reverse o Theorem.. Theorem 5.1. A Reverse Comarso Theorem of BSDJs Gve 1,, κ g > ad = 1,, le g be a geeraor sasfyg A1 ad A. Le, y, z, u, T R R d L ν such ha boh g 1 ad g sasfy 5.1. If here exss δ, y, T such ha,sξ g1,sξ, g P a.s. for ay s, +δ, y ad ξ L F s, he holds P a.s. ha g 1, y, z, u g, y, z, u.

11 6. Proofs 11 As oher cosequeces of Prooso 5.1, we have he followg reverse g5 g7 roeres of g evaluaos, whch show ha he covexy res. osve homogeey of g y, z, u s equvale o he covexy res. osve homogeey of g evaluaos ad ha he deedece of g o y varable s equvale o he raslao varace of g evaluaos. Prooso 5.. Gve 1,, assume ha L ν s a searable sace. Le g be a geeraor such ha for some κ g >, A holds wh Λ =κ g,, T ad ha for P a.s. ω Ω g, ω, y, z, u s rgh couous, T for ay y, z, u R R d L ν If g also sasfes A1, A ad f for ay, y, T R, su s,+δ δ =δ, y, T, he g s covex res. osvely homogeeous y, z, u f ad oly f gs, y,, < for cera g,s s a covex res. osvely homogeeous oeraor o L F s for ay s T. 5.3 If g also sasfes 3. ad A3, he g s deede of y f ad oly f g, ξ + c = g, ξ + c,, T, ξ L F, c R. 5.4 Wha ex s a Jese s equaly of g evaluaos wh L domas. Before dscussg, we recall some basc feaures of covex fucos see 85 for he relaed oos: Le f : R R be a covex fuco ad x R. Oe has { fλx λfx + 1 λf, f λ, 1, fλx λfx + 1 λf, f λ, 1 c =, 1,. 5.5 Also, he subdffereal fx of f a x s he erval f x, f +x, where f x ad f +x are lef dervaves ad rgh dervaves of f a x resecvely. Theorem 5.. Jese s Iequaly Le f : R R be a covex fuco. Gve 1,, le g be a geeraor deede of y such ha A, A3 hold ad ha g,, =, d dp a.s. 5.6 Gve τ T ad γ T τ, le ξ L F γ such ha fξ < ad ha f g τ,γξ, 1 c, P a.s. If holds for d dp a.s., ω τ, γ ha g, ω, z, u s covex z, u R d L ν, 5.7 he f g τ,γξ g τ,γ fξ, P a.s. 6 Proofs 6.1 Proofs of Seco Proof of Remark.1 1: We ca deduce ha for d dp a.s., ω, T Ω g, ω, y, z, u 1 g, ω, y, z, u κ u 1 x u x νdx So oe ca ake β :=β 1 κ ν q,, T. Le Y, Z, U D 1 Z loc U loc. Fx N. Defe { τ :=f, T : gs,,, ds+ κ ν 1 q u 1 u L ν, y, z, u 1, u R R d L ν L ν. 6.1 Zs ds+ } U s x νdxds> T T.

12 g valuaos wh L Domas uder Jum Flrao 1 Hölder s equaly ad A mly ha τ g, Y, Z, U d T + Y 1 β q d τ g,,, +β Y +Λ Z +β U L ν d + τ +Ĉ Y D 1 + Ĉ Ĉ 1 q <, 1 τ Λ d Z d 1 + τ β q 1 τ q d U 1 L d ν whch shows ha τ g, Y, Z, U d< exce o a P ull se N. Sce T g,,, d<, P a.s. ad sce Z, U Z loc U loc, here exss a P ull se N such ha for ay ω N c, τ ω=t for some =ω N. Now, for ay ω N, c oe ca deduce ha T N {} g, ω, Y ω, Z ω, U ω τω d= g, ω, Y ω, Z ω, U ω d<. 3 Le τ T. If g sasfes A1, A res. A, he g ad g τ are clearly P BR BR d B Lν /BR measurable fucos sasfyg A1, A res. A. Assume furher ha g sasfes A3. The g τ sasfes A3 wh h τ, ω, y, z, u 1, u x:=1 {<τω} h, ω, y, z, u1, u x κ 1, κ,, ω, y, z, u 1, u, x, T Ω R R d L ν L ν. O he oher had, for ay, ω, y, z, u 1, u, T Ω R R d L ν L ν, oe ca deduce ha g, ω, y, z, u 1 g, ω, y, z, u = g, ω, y, z, u g, ω, y, z, u 1 u x+u 1 x h, ω, y, z, u, u 1 xνdx. 6. So g sasfes.3 wh h, ω, y, z, u 1, u x = h, ω, y, z, u, u 1 x κ 1, κ, x. The P BR BR d BL ν BL ν/bl q ν measurably of mag h easly mles ha of mags h τ ad h. Proof of Theorem.1: Le ξ L F T. Uder A1 ad A, he wellosedess of BSDJ ξ, g drecly follows from Remark 4.1 of 94. By A, holds d dp a.s. ha g, Y ξ,g, Z ξ,g, U ξ,g g,,, + β Y ξ,g + U ξ,g L ν + Λ Z ξ,g. So we see ha he codo 3.1 of 94 holds for ξ 1, f 1, Y 1, Z 1, U 1 = ξ, g, Y ξ,g, Z ξ,g, U ξ,g, ξ, f, Y, Z, U =,,,, ad g, Φ, Λ, Γ, Υ = g,,,, β, Λ, β,,, T. The Lemma 3.1 ad Corollary 4.1 of 94 yelds.4 ad he remag cocluso. Proof of Remark.: Gve ξ L F T, le Y, Z, U S be he uque soluo of BSDJ ξ, g. Mullyg 1 o BSDJ ξ, g shows ha Y, Z, U S solves BSDJ ξ, g. The we see from Remark.1 3 ad Theorem.1 ha P { Y ξ,g = Y ξ,g,, T } =1. Proof of Prooso.1: By A, holds d dp a.s. ha g 1, Y 1, Z 1, U 1 g, Y, Z, U g 1, Y, Z, U g, Y, Z, U +β Y 1 Y + U 1 U L ν +ΛZ 1 Z whch shows ha he codo 3.1 of 94 holds for ξ, f, Y, Z, U =ξ, g, Y, Z, U, =1, ad g, Φ, Λ, Γ, Υ = g 1, Y, Z, U g, Y, Z, U, β, Λ, β,,, T. The Lemma 3.1 of 94 gves rse o.5. Proof of Theorem.: Whou loss of geeraly, we suose ha g 1 sasfes A, A3 ad ha g 1, Y, Z, U g, Y, Z, U, d dp a.s. o τ, γ Se Y, Z, U:=Y 1 Y, Z 1 Z, U 1 U ad cosder he followg F rogressvely measurable rocesses:, a b g 1, Y 1, Z 1, U 1 g 1, Y, Z 1, U 1 := 1 {Y }, Y Θ := e asds >, ad g 1, Y, Z 1, U 1 g 1, Y, Z, U 1 := 1 {Z } Z Z,, T.

13 6.1 Proofs of Seco 13 By A, holds d dp a.s. ha Defe H :=h, Y, Z, U 1, U ad M := b s db s + a β ad b Λ. 6.4, H s xñds, dx,, T. 6.5 T Sce Λ d + T H x νdxd Ĉ +κ ν T <, we see from 1.3 ad 1.4 ha M s a uformly egrable margale. For ay ζ T, 6.4 ad A3 aga yeld ha Mζω, ω=1 {ζω Dω }H ζω, ω, ζω, ω κ 1, κ, ω Ω, ad ha M, M T M, M ζ F ζ = T ζ b s ds+ T = b s ds+ ζ ζ,t T ζ Hs x N ds, dx F ζ H s x νdxdsf ζ Ĉ +κ ν T <. Thus, M s a BMO margale. I vrue of 5, he Doléas-Dade exoeal of M M:=e M 1 M c 1+ M s e Ms >,, T 6.6 <s s a uformly egrable margale, where M c deoe he couous ar of M. Defe a robably measure Q by dq dp := T M, whch sasfes dq dp F := M,, T. The Grsaov s Theorem e.g. 47, 8 he shows ha B Q := B b sds,, T s a Q Browa moo ad Ñ Q, A := Ñ, A, H sxνdxds,, T, A F s a Q comesaed Posso radom measure. By 6.4, Θ e T βd eĉ, P a.s. ad hus Q a.s. 6.7 Now, we fx s T ad N. Defe γ := f { r τ, T : r τ Z r dr + r τ U r x νdxdr > } γ T τ ad se ζ :=τ γ, ς :=τ s γ. Alyg Iô s formula o Θ r Y r o ζ, ς = τ, γ, s yelds ha ς ς Θ ζ Y ζ = Θ ς Y ς + Θ r gr a r Y r dr Θ r Z r db r Θ r U r xñdr, dx, P a.s. 6.8 ζ ζ ζ,ς where g r :=g 1 r, Yr 1, Zr 1, Ur 1 g r, Yr, Zr, Ur. By A3 ad 6.3, holds dr dp a.s. o τ, γ ha g r =a r Y r +b r Z r +g 1 r, Yr, Zr, Ur 1 g r, Yr, Zr, Ur a r Y r +b r Z r + H r xu r xνdx. Pluggg hs equaly back o 6.8 leads o ha Θ ζ Y ζ Θ ς Y ς M s M +Ms M, P a.s. ad hus Q a.s., 6.9 where M r := r 1 {r τ,γ }Θ r Z r db Q r ad M r :=,r 1 {r τ,γ }Θ r U r xñ Q dr, dx, r, T. We ca deduce from he Burkholder-Davs-Gudy equaly, 1.5 ad 6.7 ha Q su r,t M r + su r,t M r c Q γ τ Θ r Z r dr + c e Ĉ γ Q Z r dr τ + γ τ τ,γ Θ r U r x N dr, dx U r x νdxdr c e Ĉ + <, hus M ad M are wo uformly egrable Q margales. Takg codoal execao Q F ζ 6.9 yelds ha Q a.s. Θ ζ Y ζ Q Θς Y ς F ζ =1{γ<τ γ} Q Θς Y ς F γ +1{γ τ γ} Q Θς Y ς F τ γ :=η 1 +η. 6.1

14 g valuaos wh L Domas uder Jum Flrao 14 As Z, U Z loc U loc, oe has T Zr + U r L ν dr <, P a.s. ad hus Q a.s. So for Q a.s. ω Ω here exss a N ω N such ha for ay N ω, γ ω = γω ad hus η 1 ω = I follows ha lm η 1 =, Q a.s. O he oher had, he frs equaly 6.11 also shows ha lm Θ ζ Y ζ = Θ τ γ Y τ γ ad lm Θ ς Y ς =Θ τ s γ Y τ s γ, Q a.s. eve hough he rocess Y may o be lef-couous. For ay N, 6.7 shows ha Θ ς Y ς eĉy, P a.s. Sce a slgh exeso of 83, Prooso A.1 a shows ha q T M <, we ca deduce from Hölder s equaly ha Q Y = T MY T M Lq F T Y D <. 6.1 As 6.1, a codoal-execao verso of domaed covergece heorem ad 6.11 yeld ha Θ τ γ Y τ γ lm Q Θς Y ς F τ γ = Q Θτ s γ Y τ s γ F τ γ, Q a.s Takg s = T shows ha Θ τ γ Y τ γ Q Θγ Y γ F τ γ, Q a.s. ad hus P a.s. I follows ha Y τ γ, P a.s. By he rgh couy of rocesses Y 1 ad Y, holds P a.s. ha Y 1 Y for ay τ, γ. Suose furher ha Yτ 1 = Yτ, P a.s. For ay, T, as Θ τ γ Y τ γ, Q a.s., alyg 6.13 wh, s =, shows ha = Θ τ Y 1 τ Y τ Q Θτ γ Y τ γ F τ, Q a.s. So Q Θτ γ Y τ γ = Q Q Θτ γ Y τ γ F τ =, whch haes oly f Y τ γ =, P a.s. sce Θ τ γ >. Usg he rgh τ γ τ couy of Y 1, Y aga shows ha P { Y 1 =Y, τ, γ } =1. I he follows from 1.1 ha P a.s. g 1 r, Y 1 r, Z 1 r, U 1 r g r, Y r, Z r, U r dr = τ γ τ Zr 1 Zr db r + τ,τ γ U 1 r x U r xñdr, dx,, T. As couous fe-varaoal rocesses, couous margales ad dscouous jum margales are of dffere aures, ay wo of hem oly ersec a. So holds d dp a.s. o τ, γ ha Z 1 = Z ad U 1 x = U x, d dp νdx a.s. o o τ, γ Sce he laer s equvale o U 1 = U, d dp a.s. o τ, γ, we furher see ha d dp a.s. o τ, γ g 1, Y 1, Z 1, U 1 =g, Y, Z, U ad hus g 1, Y j, Z j, U j =g, Y j, Z j, U j, j =1,. 6. Proofs of Seco 3 Proof of Lemma 3.1: Le γ T. I suffces o show ha he uque soluo Y, Z, U = Y ξ,gγ, Z ξ,gγ, U ξ,gγ of BSDJ ξ, g γ s also he uque soluo of BSDJ ξ, g S : Se M := Z sdb s +, U sxñds, dx,, T. Sce Z, U =1{ γ} Z, U, d dp a.s. by Theorem.1, we ca deduce from 3. ha P a.s. T Y = ξ+ = ξ+ T T 1 {s<γ} gs, Y s, Z s, U s ds M T +M =ξ+ 1 {s γ} gs, Y s, Z s, U s ds M T +M g s, Y s, 1 {s γ} Z s, 1 {s γ} U s ds MT +M =ξ+ whch shows ha Y, Z, U s he uque soluo of BSDJ ξ, g. T g s, Y s, Z s, U s ds MT +M,, T, Proof of g1 g7: 1 Le g sasfes A3 ad le η L F γ wh ξ η, P a.s. Alyg Theorem. wh g 1 = g = g γ yelds ha P { Y ξ,gγ Y η,gγ, τ, γ } =1. I arcular, τ,γξ=y g τ ξ,gγ Yτ η,gγ =τ,γη, g P a.s. Moreover, f furher holds ha Yτ ξ,gγ = τ,γξ g = τ,γη g = Yτ η,gγ, P a.s., Theorem. aga shows ha P { Y ξ,gγ = Y η,gγ, τ, γ } = 1. The Theorem.1 mles ha ξ = Y ξ,gγ T = Yγ ξ,gγ = Yγ η,gγ = Y η,gγ T = η, P a.s., rovg g1. Le g sasfes 3.. For ay ξ L F τ L F γ, Lemma 3.1 ad 3.1 mly ha τ,γξ= g g ξ F τ =τ,τ g ξ=ξ, P a.s., rovg g.

15 6. Proofs of Seco Se Y, Z, U := Y ξ,gγ, Z ξ,gγ, U ξ,gγ ad Y, Z, U := Y η,g ζ, Z η,g ζ, U η,g ζ wh η := g ζ,γ ξ L F ζ. We defe Y := 1 {<ζ} Y +1 { ζ} Y ad Z, U := 1{ ζ} Z, U +1 {>ζ} Z, U,, T. Oe ca deduce ha Y, Z, U belog o S ad ha P a.s. T Y ζ = ξ + = ξ + ζ T ζ T g γ s, Y s, Z s, U s ds g γ s, Y s, Z s, U s ds ζ T ζ Z s db s Z s db s ζ,t ζ,t U s xñds, dx U s xñds, dx,, T Le, T. Sce Theorem.1 shows ha Z s, U s =1 {s ζ} Z s, U s =1 {s ζ} Z s, U s, ds dp a.s., akg =ζ 6.14 yelds ha T Y ζ = η+ = Y ζ + = ξ+ ζ ζ ζ T ζ T 1 {s<ζ} gs, Y s, Z s, U s ds 1 {s<γ} gs, Y s, Z s, U s ds g γ s, Y s, Z s, U s ds T ζ ζ ζ Z s db s ζ Z s db s ζ,t Z s db s ζ,t ζ,ζ U s xñds, dx U s xñds, dx U s xñds, dx, P a.s Mullyg 1 { ζ} o 6.14 ad mullyg 1 {<ζ} o 6.15 leads o ha T Y = 1 {<ζ} Y +1 { ζ} Y = ξ+ g γ s, Y s, Z s, U s ds T Z s db s U s xñds, dx,,t P a.s. The we see from he rgh couy of rocess Y ha P a.s. T Y =ξ+ g γ s, Y s, Z s, U s ds T Z s db s U s xñds, dx,, T.,T So Y, Z, U solves BSDJ ξ, g γ. By uqueess, oe has P { Y = Y,, T } = 1. I arcular, alyg 3.1 wh τ, γ, ξ = τ, ζ, g ζ,γ ξ yelds ha τ,γξ g = Y τ = Y τ = 1 {τ<ζ} Y τ +1 {τ=ζ} Y ζ = 1 {τ<ζ} g τ,ζ η+1 {τ=ζ} g ζ,γ ξ = g τ,ζ g ζ,γ ξ, P a.s. Hece, g3 holds. 4a Fx A F τ. Se Y 1, Z 1, U 1 := Y ξ,gγ, Z ξ,gγ, U ξ,gγ ad Y, Z, U := Y 1 Aξ,g γ, Z 1 Aξ,g γ, U 1 Aξ,g γ. Gve = 1,, alyg Corollary.1 wh ξ = 1 A Y L F τ shows ha here exss a uque ar Z, U Z, U such ha P a.s., Y := 1 A Y F = 1A Y + Z sdb s +, U sxñds, dx,, T. We defe Y := 1 {<τ} Y +1 { τ} 1 A Y, Z, U := 1{ τ} Z, U +1 {>τ} 1 A Z, U,, T, ad ca deduce ha Y, Z, U belog o S. For ay, T, sce {τ } F τ, we see ha A {τ } F ad hus A {τ < γ} F. The { } 1A 1 {τ <γ} s a F adaed càdlàg rocess. I follows ha,t g A, ω, y, z, u:=1 {ω A} 1 {τω <γω} g, ω, y, z, u,, ω, y, z, u, T Ω R R d L ν s a P BR BR d B L ν /BR measurable mag ha sasfes A1 ad A. Gve, T, mullyg 1 A o he BSDJ ξ, g γ over erod τ, T yelds ha T 1 A Yτ 1 = 1 A ξ + = 1 A ξ + = 1 A ξ + τ T τ T τ 1 A 1 {s<γ} gs, Ys 1, Zs 1, Us 1 ds T τ 1 A Zs 1 db s 1 A 1 {τ s<γ} gs, 1 A Ys 1, 1 A Zs 1, 1 A Us 1 ds 1 g A s, Y s, Z 1 s, U 1 T s ds τ Z 1 sdb s T τ,t τ τ,t Z 1 sdb s 1 A U 1 s xñds, dx τ,t U 1 sxñds, dx U 1 sxñds, dx, P a.s. 6.16

16 g valuaos wh L Domas uder Jum Flrao 16 Smlarly, mullyg 1 A o he BSDJ 1 A ξ, g γ over erod τ, T yelds ha T 1 A Yτ =1 A ξ + g A s, Y s, Z s, U T s ds Z sdb s U sxñds, dx, τ τ τ,t P a.s Fx =1,. The rgh couy of rocess Y, 6.16 ad 6.17 shows ha P a.s. T 1 A Yτ =1 A ξ + g A s, Y s, Z s, U T s ds Z sdb s U sxñds, dx,, T τ τ τ,t Le, T. Sce Y = 1 A Y F τ =1A Y, P a.s. akg = 6.18 yelds ha τ τ Yτ = Y ZsdB s UsxÑds, dx=1 A Y Z sdb s = 1 A ξ+ τ T τ τ,τ g A s, Y s, Z s, U sds T τ Z sdb s τ,t τ τ,τ U sxñds, dx U sxñds, dx, P a.s Mullyg 1 { τ} o 6.18 ad mullyg 1 {<τ} o 6.19 leads o ha T Y = 1 {<τ} Y +1 { τ} 1 A Y = 1 A ξ+ g A s, Y s, Z s, U sds T Z sdb s U sxñds, dx,,t P a.s. The rgh couy of rocess Y he mles ha P a.s. T T Y =1 A ξ+ g A s, Y s, Z s, U sds Z sdb s,t U sxñds, dx,, T. Thus boh Y 1, Z 1, U 1 ad Y, Z, U solve BSDJ 1A ξ, g A. By uqueess, oe has P { Y 1 =Y,, T } =1. I follows ha 1 A g τ,γξ=1 A Y 1 τ =Y 1 τ =Y τ =1 A Y τ =1 A g τ,γ1 A ξ, P a.s. 4b Nex, suose ha g,,, =, d dp a.s. Se Y, Z, U := Y ξ,gγ, Z ξ,gγ, U ξ,gγ. Sce η := 1A Y τ L F τ, Theorem.1 shows ha he BSDJ η, g τ adms a uque soluo Y, Z, U S. We defe Y := 1 {<τ} Y + 1 { τ} 1 A Y, Z, U :=1{ τ} Z, U +1 {>τ} 1 A Z, U,, T. Lke Y, Z, U, he rocesses Y, Z, U also belog o S. Gve, T, smlar o 6.16, mullyg 1 A o he BSDJ ξ, g γ over erod τ, T aga yelds ha T 1 A Y τ = 1 A ξ + = 1 A ξ + = 1 A ξ + τ T τ T τ T 1 A g γ s, Y s, Z s, U s ds τ g γ s, 1 A Y s, 1 A Z s, 1 A U s ds 1 A Z s db s T τ T g γ s, Y s, Z s, U s ds Z s db s By he rgh couy of rocess Y, holds P a.s. ha T T 1 A Y τ =1 A ξ + g γ s, Y s, Z s, U s ds Z s db s τ τ τ τ,t Z s db s τ,t τ,t τ,t 1 A U s xñds, dx U s xñds, dx U s xñds, dx, Le, T. Takg = 6. ad usg a aalogy o 6.15 yeld ha T T Y τ = η+ 1 {s<τ} gs, Y s, Z s, U s ds Z s db s U s xñds, dx = 1 A Y τ + τ τ τ τ 1 {s<γ} gs, Y s, Z s, U s ds T = 1 A ξ+ g γ s, Y s, Z s, U s ds τ T τ τ τ Z s db s τ,t Z s db s τ,t P a.s. U s xñds, dx,, T. 6. τ,τ U s xñds, dx U s xñds, dx, P a.s. 6.1

17 6. Proofs of Seco 3 17 Mullyg 1 { τ} o 6. ad mullyg 1 {<τ} o 6.1 leads o ha Y = 1 {<τ} Y +1 { τ} 1 A Y = 1 A ξ+ T g γ s, Y s, Z s, U s ds T Z s db s U s xñds, dx,,t P a.s. The rgh couy of rocess Y he shows ha P a.s. Y =1 A ξ+ T g γ s, Y s, Z s, U s ds T Z s db s U s xñds, dx,, T.,T So Y, Z, U solves BSDJ 1 A ξ, g γ. By uqueess, oe has P { Y = Y 1 Aξ,g γ,, T } = 1. I follows ha τ,γ1 g A ξ=y 1 Aξ,g γ τ =Y τ =1 A Y τ =1 A τ,γξ, g P a.s., rovg g4. 5 Assume ha g s deede of y. Se Y, Z, U := Y ξ,gγ, Z ξ,gγ, U ξ,gγ ad le η L F τ. I lgh of Theorem.1, he BSDJ Y τ +η, g τ adms a uque soluo Y, Z, U S. We defe Y := 1 {<τ} Y +1 { τ} Y +η ad Z, U :=1{ τ} Z, U +1 {>τ} Z, U,, T. Oe ca deduce ha Y, Z, U belog o S. Gve, T, addg η o he BSDJ ξ, g γ over erod τ, T aga yelds ha T Y τ +η = ξ+η+ = ξ+η+ τ T τ T g γ s, Z s, U s ds τ Z s db s T g γ s, Zs, U s ds Z s db s τ τ,t τ,t U s xñds, dx U s xñds, dx, P a.s. By he rgh couy of rocess Y, holds P a.s. ha T T Y τ +η =ξ+η+ g γ s, Zs, U s ds Z s db s U s xñds, dx,, T. 6. τ τ τ,t Le, T. Sce Theorem.1 shows ha Z s, U s =1 {s τ} Z s, U s =1 {s τ} Z s, U s, ds dp a.s., akg =τ 6. yelds ha T Y τ = Y τ +η+ = Y τ +η + τ τ τ T 1 {s<τ} gs, Z s, U s ds 1 {s<γ} gs, Z s, U s ds T = ξ+η+ g γ s, Z s, U s ds τ T τ τ τ τ Z s db s Z s db s Z s db s τ,t τ,t τ,τ U s xñds, dx U s xñds, dx U s xñds, dx, P a.s. 6.3 Mullyg 1 { τ} o 6. ad mullyg 1 {<τ} o 6.3 leads o ha Y = 1 {<τ} Y +1 { τ} Y +η = ξ+η+ T g γ s, Z s, U s ds T Z s db s U s xñds, dx,,t P a.s. The rgh couy of rocess Y he shows ha P a.s. Y =ξ+η+ T g γ s, Z s, U s ds T Z s db s U s xñds, dx,, T.,T So Y, Z, U solves BSDJ ξ + η, g γ. By uqueess, oe has P { Y = Y ξ+η,gγ,, T } = 1. I follows ha τ,γξ+η=y g τ ξ+η,gγ =Y τ =Y τ +η =τ,γξ+η, g P a.s. Therefore, g5 holds. 6 Assume ha g s covex y, z, u ad le η L F γ, α, 1. We se Y 1, Z 1, U 1 := Y ξ,gγ, Z ξ,gγ, U ξ,gγ, Y, Z, U := Y η,gγ, Z η,gγ, U η,gγ ad Y, Z, U := αy αy, αz αz, αu αu. As g :=αg γ, Y 1, Z 1, U 1 +1 αg γ, Y, Z, U,, T s a F rogressvely measurable rocess, oe ca regard

18 g valuaos wh L Domas uder Jum Flrao 18 as a secal geeraor. I holds P a.s. ha Y = αy 1 +1 αy = αξ +1 αη+ T αz 1 s +1 αzs dbs = αξ +1 αη+ T g s ds T,T T Z s db s αgγ s, Y 1 s, Z 1 s, U 1 s +1 αg γ s, Y s, Z s, U s ds αu 1 s x+1 αu s x Ñ ds, dx,t U s xñds, dx,, T. Sce he covexy of g y, z, u shows ha P { g γ, Y, Z, U g,, T } = 1, a alcao of Theorem. wh g 1, Y 1, Z 1, U 1 = g γ, Y αξ+1 αη,gγ, Z αξ+1 αη,gγ, U αξ+1 αη,gγ ad g, Y, Z, U = g, Y, Z, U yelds ha P { Y αξ+1 αη,gγ Y, τ, γ } = 1. Hece, we oba τ,γ g αξ+1 αη,g αξ+1 αη = Y γ τ Y τ = ατ,γξ+ g 1 ατ,γη, g P a.s. 7 Nex, assume ha g s osvely homogeeous y, z, u. Le α, ad se Y, Z, U:= Y ξ,gγ, Z ξ,gγ, U ξ,gγ. I holds P a.s. ha αy = αξ+ = αξ+ T T αg γ s, Y s, Z s, U s ds T g γ s, αy s, αz s, αu s ds T αz s db s,t αz s db s,t αu s xñds, dx αu s xñds, dx,, T, whch shows ha αy, αz, αu S solves BSDJ αξ, g γ. Thus, P { Y αξ,gγ = αy,, T } = 1. I arcular, τ,γ αξ=y g τ αξ,gγ = αy τ = α τ,γξ, g P a.s. Proof of xamle 3.1: 1 Sce 1.6 ad Hölder s equaly mly ha u ± 1 x u ± x νdx u1 x u x 1 νdx ν q u 1 u L ν, u 1, u L ν, we see ha u u± xνdx s a couous fuco o L ν. I follows ha g Ξ ad g Ξ are wo P BR BR d B Lν /BR measurable mags. Clearly, g Ξ ad g Ξ sasfy A wh coeffces β, Λ, ad g Ξ,,, g Ξ,,,. To verfy A3 for g Ξ, we le, ω, y, z, u 1, u, T Ω R R d L ν L ν. As g Ξ, ω, y, z, u 1 g Ξ, ω, y, z, u = κ 1 u 1 x u x νdx+κ u + 1 x u + x νdx = κ1 u1 x u x + κ κ 1 u + 1 x u+ x νdx, 6.4 g Ξ sasfes.3 wh h κ, ω, y, z, u 1, u x:=κ 1 +1 {u1x u x}κ κ 1 u+ 1 x u+ x u 1x u x, x. Clearly, h κ, ω, y, z, u 1, u s a real-valued, F measurable fuco. Sce a + b + for ay a, b R wh a b, 6.5 we ca deduce from 1.6 ha κ 1 h κ, ω, y, z, u 1, u x = κ 1 +1 {u1x u x}κ κ 1 u+ 1 x u+ x u 1x u x κ, whch mles ha h κ, ω, y, z, u 1, u L q ν. I remas o show ha he mag h κ s P BR BR d BL ν BL ν/bl q ν measurable: le, ω, y, z, u 1, u, T Ω R R d L ν L ν, le λ> ad defe f λ α:= α+ λ 1, α R. For ay u L ν, 6.5 ad 1.6 show ha he fuco φ u1 λ u x:=f λ u1 x ux u + 1 x u+ x u 1x ux, x akes values, 1, so φu1 λ u Lq ν. We frs show ha φ u1 λ s a couous mag from L ν o L q ν. Fx ε> ad se δ =δλ, ε:= λε 3 ν 1 q. Le u, ũ L ν wh ũ u L ν <δ1/ 1 1 ε/ 1. Sce u+ 1 x u+ x u 1 x ux u+ 1 x ũ+ x u 1 x ũx = ũ+ x u + xu 1 x ũx+u + 1 x ũ+ xux ũx u 1 x uxu 1 x ũx ũx ux u 1x ũx + u 1 x ũx ux ũx = ux ũx u 1 x ux u 1 x ũx u 1 x ux, x

19 6. Proofs of Seco 3 19 ad sce he Lschz coeffce of f λ s o larger ha 1/λ, oe ca deduce from 1.6 ha for ay x { ũ u <δ} φ u1 λ u x φ u1 λ ũ x = f λ u1 ux u + 1 x u+ x u 1 x ux u+ 1 x ũ+ x + f λ u1 ux f λ u1 ũx u + 1 x ũ+ x u 1 x ũx u 1 x ũx u 1x ux + ux ũx λ u 1 x ux + 1 λ ux ũx 3 ux ũx < 3δ λ λ. I follows ha φ u1 λ u x φ u1 λ ũ x q νdx { ũ u <δ} or φ u 1 λ u φu1 λ ũ L q ν ε. Ths shows ha 3δ q 3δ qν νdx+ q q νdx + λ { ũ u δ} λ δ ũ u L <εq, ν he mag φ u1 λ s uformly couous from L ν o L q ν ad hus BL ν/bl q ν measurable. 6.6 For ay u L ν, we defe a fuco φ u1 u L q ν by φ u 1 u x := lm φ u 1 λ u u + 1 x=1 x u+ x {u1x ux>} λ u 1 x ux I lgh of he bouded covergece heorem, lm φ u1 λ u q u φu1 λ L = lm q ν λ Namely, φ u1 u s he lm of { φ u1 1, 1, x. φ u1 λ ux φu1 ux q νdx=. λ u} λ> Lq ν. I he follows from 6.6 ha he mag φ u1 s BL ν/bl q ν measurable. Defe φu 1 u u x := x u+ x {u1x ux<} u 1x ux 1, 1, u L ν, x. Oe ca smlarly show ha φ u1 s also a BL ν/bl q ν measurable mag. Cosequely, he mag u h κ, ω, y, z, u 1, u s aga BL ν/bl q ν measurable. Symmercally, he mag u h κ, ω, y, z, u, u s BL ν/bl q ν measurable. Pug hem ogeher yelds he execed measurably of h κ. 3 Smlar o 6., we see from 6.4 ha for ay, ω, y, z, u 1, u, T Ω R R d L ν L ν g Ξ, ω, y, z, u 1 g Ξ, ω, y, z, u = u x+u 1 x h κ, ω, y, z, u, u 1 xνdx. So g Ξ sasfes.3 wh h κ, ω, y, z, u 1, u x = h κ, ω, y, z, u, u 1 x κ 1, κ, x. Clearly, he mag h κ s also P BR BR d BL ν BL ν/bl q ν measurable. Therefore, g Ξ also sasfes A3. Proof of Prooso 3.1: Fx τ T, γ T τ ad ξ, η L F T. Se Y 1, Z 1, U 1 = Y ξ,gγ, Z ξ,gγ, U ξ,gγ, Y, Z, U = Y η,gγ, Z η,gγ, U η,gγ ad Y 3, Z 3, U 3 = Y ξ η,gξ γ, Z ξ η,g Ξ γ, U ξ η,g Ξ γ. The P BR BR d B L ν /BR measurably of g, he P measurably of rocess Y, he P measurably of rocess Z ad he P F measurably of radom feld U mly ha he mag g, ω, y, z, u := g, ω, y+y, ω, z+z, ω, u+u, ω g, ω, Y, ω, Z, ω, U, ω,, ω, y, z, u, T Ω R R d L ν s also P BR BR d B L ν /BR measurable. For Y, Z, U :=Y 1 Y, Z 1 Z, U 1 U S, holds P a.s. ha Y =ξ η+ T 1 {<γ} gs, Y 1 s, Z 1 s, U 1 s gs, Y s, Z s, U s ds T Z s db s U s xñds, dx,, T.,T Namely, Y, Z, U solves he BSDJ ξ η, g γ. We ca deduce from A ad A3 ha d dp a.s. g, Y, Z, U =g, Y 1, Z 1, U 1 g, Y, Z, U =g, Y 1, Z 1, U 1 g, Y, Z, U 1 +g, Y, Z, U 1 g, Y, Z, U β Y +Λ Z + U x h, Y, Z, U 1, U xνdx β Y +Λ Z +κ U + xνdx κ 1 U xνdx=g Ξ, Y, Z, U.

20 g valuaos wh L Domas uder Jum Flrao Sce g Ξ also sasfes A ad A3 by xamle 3.1, alyg Theorem. wh τ, γ =, T, Y 1, Z 1, U 1 = Y, Z, U ad Y, Z, U =Y 3, Z 3, U 3 yelds ha P { Y 1 Y =Y Y 3,, T } =1. I arcular, g τ,γξ g τ,γη=y 1 τ Y τ Y 3 τ = Ξ τ,γξ η, P a.s. 6.7 Mullyg 1 o BSDJ η ξ, gγ Ξ shows ha Y η ξ,g Ξ γ, Z η ξ,g Ξ γ, U η ξ,g Ξ γ s he uque soluo of BS- DJ } ξ η, g Ξ γ. So P { Y η ξ,gξ γ =Y ξ η,gξ γ,, T = 1, whch ogeher wh 6.7 mles ha g τ,γξ g τ,γη = g τ,γη g τ,γξ Ξ τ,γη ξ = Y η ξ,gξ γ τ = Y ξ η,gξ γ τ = Ξ τ,γξ η, P a.s. 6.3 Proofs of Seco 4 Proof of Lemma 4.1: Le < s +1 for some {1, 1} ad le ξ L F τ s. Se Y, Z, U := Y ξ,g τ s, Z ξ,gτ s, U ξ,gτ s ad Ỹ, Z, Ũ := Y ξ,gs, Z ξ,gs, U ξ,gs. Le, s. Sce {τ } = {τ +1 } c F F, ad sce Z r, U r = 1 {r τ s} Z r, U r, dr dp a.s. by Theorem.1, mullyg 1 {τ } ad 1 {τ +1} o BSDJ ξ, g τ s over erod, T resecvely yelds ha P a.s. ad ha P a.s. T T 1 {τ }Y = 1 {τ }ξ+ 1 {τ }1 {r<τ s} gr, Y r, Z r, U r dr 1 {τ }1 {r τ s} Z r db r 1 {τ }1 {r τ s} U r xñdr, dx=1 {τ }ξ, 6.8,T T T 1 {τ +1}Y =1 {τ +1}ξ+ 1 {τ +1}1 {r<τ s} gr, Y r, Z r, U r dr 1 {τ +1}1 {r τ s} Z r db r 1 {τ +1}1 {r τ s} U r xñdr, dx,t = 1 {τ +1}ξ+ s 1 {τ +1}gr, Y r, Z r, U r dr Also, a aalogy o 6.8 shows ha P a.s. 1 {τ }Ỹ =1 {τ }ξ+ s s 1 {τ }g r, Ỹr, Z s r, Ũr dr 1 {τ +1}Z r db r 1 {τ +1}U r xñdr, dx. 6.9,s 1 Z {τ } r db r 1 {τ }ŨrxÑdr, dx. 6.3,s Nex, se Y r, Z r, U r := 1 {τ }Ỹr, Z r, Ũr + 1{τ +1}Y r, Z r, U r, r, s. As Y L F, Theorem.1 shows ha he BSDJ Y, g adms a uque soluo Y, Z, U S. Defe Y r := 1 {r<} Y r +1 {r } Y r s ad Zr, U r :=1{r } Z r, U r +1 {<r s} Z r, U r, r, T. Oe ca deduce ha Y, Z, U belog o S. For ay, T, addg 6.9 o 6.3 yelds ha s Y = Y s =1 {τ }Ỹ s+1 {τ +1}Y s =ξ + = ξ + T g s r, Y r, Z r, U r dr T s Z r db r,t s gr, Y r, Z r, U r dr s Z r db r s,s U r xñdr, dx U r xñdr, dx, P a.s O he oher had, for ay,, as Theorem.1 shows ha Y =Y =Y T =Y, P a.s., we have Y Y = Y Y = = g r, Y r, Z r, U r dr g r, Y r, Z r, U r dr Z r db r Z r db r,, U r xñdr, dx, U r xñdr, dx P a.s.

21 6.3 Proofs of Seco 4 1 Takg = 6.31 yelds ha T Y = ξ+ T g s r, Y r, Z r, U r dr Z r db r U r xñdr, dx, P a.s. 6.3,T By he rgh-couy of Y, we see from 6.31 ad 6.3 ha P a.s. T T Y =ξ+ g s r, Y r, Z r, U r dr Z r db r U r xñdr, dx,, T,,T whch shows ha Y, Z, U solves BSDJ ξ, g s. I follows ha,sξ g = Y ξ,gs 6.8 wh =, we see from Theorem.1 aga ha P a.s. = Y = Y, P a.s. The alyg g τ,τ sξ=y τ =Y =1 {τ }Y + 1 {τ +1}Y =1 {τ }ξ+1 {τ +1}Y =1 {τ }ξ+1 {τ +1} g,sξ. Proof of Prooso 4.1: Le us oly cosder he g submargale case, as he oher cases ca be derved smlarly. 1 Assume frs ha γ akes values a fe se {= 1 < < =T }. If, T, 3.1 shows ha γ,γ g γ =Y γ,gγ γ argue ha for ay, T, =Y γ,gγ γ = g γ,γ γ = γ = γ, P a.s. The le us ducvely g γ,γ γ γ, P a.s Suose ha for some {,, }, 6.33 holds for each, T. Gve 1,, he g1, g3 roeres of g evaluaos ad 4.1 mly ha g γ,γ γ = g γ,γ g γ,γ γ g γ,γ γ =1 {γ 1} γ +1 {γ } g, γ, P a.s Sce {γ }={γ 1 } c F 1 F, he g4 of g evaluaos ad he g submargaly of show ha P a.s. 1 {γ } g, γ =1 {γ } g, 1 {γ } γ =1 {γ } g, 1 {γ } =1 {γ } g, 1 {γ } =1 {γ } γ. Pug back o 6.34 roves 6.33 for ay 1, T. Ths comlees he ducve se. Hece, 6.33 holds for ay, T. If s also fely valued, for examle {=s 1 < <s m =T }, he we see from 6.33 ha P a.s. τ,γ g γ =Yτ γ,gγ =Y γ,gγ γ τ = m j=1 1 {τ=sj}y γ,gγ γ s j = m m 1 {τ=sj}γ s g j,γ γ 1 {τ=sj} γ sj = γ τ = τ. j=1 Nex, assume ha s rgh-couous bu τ, γ are geeral sog mes. Se Y, Z, Y := Y γ,gγ, Z γ,gγ, U γ,gγ. For ay N, we se := T, =,, ad defe τ := =1 1 { 1 <τ } ad γ := =1 1 { 1 <γ } T. Le m, N wh m> ad se Y, Z, Y := Y γ,gγ, Z γ,gγ, U γ,gγ. Sce τm τ γ, Par 1 shows ha Yτ m =τ g m,γ γ τm, P a.s. As lm τ m =τ, he rgh couy of rocesses Y ad mles ha m Yτ = lm Y τ m m lm τ m = τ, P a.s m By Prooso.1, Yτ Y τ Y Y γ C D γ γ + g, Y, Z, U d Also, A1 A3, 6.1, 1.7 ad Hölder s equaly mles ha T g, Y, Z, U T d g,,, +β Y +Λ Z +κ ν 1 q U L ν d T + T 4 1 g,,, d Ĉ q T Y +Ĉ T Z d +κ ν T q U <. L d ν γ j=1