FULLY COUPLED FBSDE WITH BROWNIAN MOTION AND POISSON PROCESS IN STOPPING TIME DURATION

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1 J. Au. Mah. Soc. 74 (23), FULLY COUPLED FBSDE WITH BROWNIAN MOTION AND POISSON PROCESS IN STOPPING TIME DURATION HEN WU (Received 7 Ocober 2; revied 18 January 22) Communicaed by V. Sefanov Abrac We fir give he exience and uniquene reul and a comparion heorem for backward ochaic differenial equaion wih Brownian moion and Poion proce a he noie ource in opping ime (unbounded) duraion. Then we obain he exience and uniquene reul for fully coupled forwardbackward ochaic differenial equaion wih Brownian moion and Poion proce in opping ime (unbounded) duraion. We alo proved a comparion heorem for hi kind of equaion. 2 Mahemaic ubjec claificaion: primary 6H1, 6G4. Keyword and phrae: ochaic differenial equaion, opping ime, random meaure, Poion proce, comparion heorem. 1. Inroducion Nonlinear backward ochaic differenial equaion wih Brownian moion a noie ource (BSDE in hor) have been independenly inroduced by Pardoux and Peng [11] and Duffie and Epein [4]. I wa oon dicovered by Peng [13] ha, coupled wih a forward ochaic differenial equaion (SDE in hor), uch BSDE give a probabiliic inerpreaion for a large kind of econd order quailinear parial differenial equaion (PDE in hor). In hi paper Peng alo gave an exience and uniquene reul of BSDE in opping ime duraion which can ake infinie value. And hen Darling and Pardoux [3] proved an exience and uniquene reul for BSDE in opping ime under differen aumpion. They applied heir reul o conruc a coninuou vicoiy oluion for a cla of emilinear ellipic PDE. In [8], El Karoui, Peng and Thi work i uppored by Chinee Naional Naural Science Foundaion (Gran No. 1122), he Excellen Young Teacher Program and he Docoral Program Foundaion of MOE, P.R.C. c 23 Auralian Mahemaical Sociey /3 $A2: : 249

2 25 hen Wu [2] Quenez gave a comparion heorem o BSDE and ome applicaion in opimal conrol and financial mahemaic. Fully coupled forward-backward ochaic differenial equaion wih Brownian moion (FBSDE in hor) can be encounered in he opimizaion problem when applying ochaic maximum principle and mahemaical finance conidering large inveor in ecuriy marke. Anonelli [1] fir udied hi kind of equaion and obained he local exience and uniquene reul, ha i, he ime duraion on which he oluion exi (wihou exploion) ha o be ufficienly mall. He alo gave a counerexample o how ha he Lipchiz condiion i no enough for he exience of FBSDE in an arbirarily large ime duraion. Uing PDE mehod, Ma, Proer and Yong [9] uccefully obained he exience and uniquene reul for an arbirarily precribed ime duraion. Bu hey needed he forward SDE o be nondegenerae and he coefficien no o be randomly diurbed. Uing probabiliy mehod, Hu and Peng [6] obained he exience and uniquene reul when forward and backward equaion ake ame dimenion under ome monoone aumpion. Hamadène [5] weaken heir monoone aumpion and dicued he applicaion in ochaic differenial game. Peng and he auhor [17] exend heir reul o differen dimenional FBSDE and weaken he monoone aumpion o ha he reul can be ued widely. The main mehod i o inroduce an m n full rank marix G o overcome he difficuly of he differen dimenion. Yong [21] made he above mehod yemaic and called i coninuaion mehod. In [12], Pardoux and Tang alo gave he exience and uniquene reul for FBSDE under ome monoone condiion differen from [6] and [17]. Recenly, Peng and Shi [16] gave an exience and uniquene reul of FBSDE wih infinie horizon. Bu he oluion i in a quare inegrable pace, he infinie ime value of he oluion mu be zero. The BSDE wih Poion proce (BSDEP in hor) wa fir dicued by Tang and Li [19]. The ochaic proce in he equaion i diconinuou wih random jump. Afer hen Siu Rong [18] obained an exience and uniquene reul wih non-lipchiz coefficien for BSDEP. Uing hi kind of BSDEP Barle, Buckdahn and Pardoux [2] gave he probabiliic inerpreaion for a yem of parabolic inegroparial differenial equaion and proved ha here exi a unique vicoiy oluion for hi kind of PDE yem. In Secion 2 we udy he BSDEP in opping ime duraion, here he opping ime i unbounded and can ake infinie value. Under a Lipchiz condiion uiable for our cae, we ge he exience and uniquene reul for BSDEP uing fixed poin principle and oher echnique. Furher in Secion 2, we give a comparion heorem for BSDEP in opping ime. The concluion i imilar wih ha in [8]. We only need o conrol he heigh of he jump in BSDEP. In Secion 3, we conider fully coupled forward-backward ochaic differenial equaion wih Brownian moion and Poion proce (FBSDEP in hor) in opping ime duraion. Suiable for he cae ha he opping ime can be infinie, we prove

3 [3] Fully coupled FBSDE wih Brownian moion 251 an exience and uniquene reul under a Lipchiz and monoone aumpion, he infinie ime value of he oluion no necearily be required zero. In Secion 4, we give a comparion heorem for FBSDEP in opping ime. The idea in he proof i o ue dualiy echnique and opping ime echnique. The dualiy echnique i uually ued in opimal conrol heory o inroduce he adjoin equaion for proving he maximum principle (ee [14, 2]). Anoher echnique i o analyze he jump heigh under he limi aumpion. Thi kind of comparion heorem can be ued o connec FBSDEP wih a parabolic inegro-pde yem and udy he exience of he vicoiy oluion for hi PDE yem. The PDE yem form hould be a PDE combined by he algebra equaion. For no jump cae hi kind of PDE form can be een in [15]. Here he comparion heorem of FBSDEP i eablihed only a ime, we canno ge he reul in he whole random inerval. We alo give a counerexample o how hi poin. 2. BSDEP in opping ime duraion Le.; F ; {F } ; P/ be a ochaic bai uch ha F conain all P-null elemen of F and F = ž> F ž = F ;. We uppoe ha he filraion {F } i generaed by he following wo muually independen procee: a d-dimenional andard Brownian moion {B } and a Poion random meaure N on R, where R l i nonempy open e equipped wih i Borel field B. /, wih compenaor N.dz; d/ = n.dz/ d, uch ha Ñ.A [; ]/ =.N N /.A [; ]/ i a maringale for all A B. / aifying n.a/ <. n i aumed o be a -finie meaure on.; B. // and called he characeriic meaure. F = F.Le ={.!/} be F opping ime and ake value in [; ]. We inroduce he following noaion: S 2 = { v,,iaf adaped proce uch ha E[up v 2 ] < }, H 2 = { v,,iaf adaped proce uch ha E [ v 2 d ] < }, L 2 = { ¾, ¾ i a F meaurable random variable uch ha E ¾ 2 < }, F 2 N = { k. /,,iaf predicable proce uch ha E [ k.z/ 2 n.dz/ d ] < }. We conider he following BSDEP in opping ime duraion (2.1) p = ¾ f.; p ; q ; k / d q db k.z/ñ.dz d/; where, ¾ L 2 and f i a map from [; ] R m R m d R m ono R m which aifie

4 252 hen Wu [4] (H2.1) For every.p; q; k/ R mm dm, f. ; p; q; k/ i progreively meaurable and E ( f.; ; ; / d )2 <. (H2.2) There exi hree poiive deerminiic funcion u 1./, u 2./ and u 3./, uch ha.p i ; q i ; k i /, i = 1; 2, f.; p 1 ; q 1 ; k 1 / f.; p 2 ; q 2 ; k 2 / u 1./ p 1 p 2 u 2./ q 1 q 2 u 3./ k 1 k 2 ; ; and u 1./ d <, u 2./ d <, 2 u 2 3./ d <. Then we have THEOREM 2.1. Aume ¾ L 2 and f aifie (H2.1) (H2.2), hen here exi a unique oluion.p; q; k/ S 2 H 2 FN 2 aifying he BSDEP (2.1). PROOF. For he uniquene, le. p; q; k/ be anoher oluion, we e ˆp =. p p/, ˆq =.q q/, ˆk =.k k/. Uing Iô formula o ˆp 2, imilarly wih he proof in [11] for fixed ime T wihou jump excep he Lipchiz conan being replaced by u 1./, u 2./ and u 3./,, we can ge he concluion from he aumpion (H2.2) and Gronwall lemma. For he exience we wan o conruc one conracion map for (2.1) and ge he oluion. However, he opping ime duraion i unbounded and can be infinie, o we canno ge hi in one ep. We divide he proof ino wo ep. Fir ep. Aume ( 2 u 1./ d) u 2 2./ d u 2 3./ d < 1 15 : For every.p; q; k/ S 2 H 2 F 2,wehave N [ ] 2 ¾ f.; p ; q ; k / d and E E [ ¾ ( E E E ( f.; ; ; / u1./ p u 2./ q u 3./ k ) ] 2 d 2 ( u 1./ p d) ) 2 u 1./ d p. / 2 < ; S 2 ( 2 ( ) u 2./ q d) u 2./ d 2 q. / 2 < ; H 2 ( 2 ( u 3./ k d) u 2 3./ d ) k. / 2 F 2 N < :

5 [5] Fully coupled FBSDE wih Brownian moion 253 Then E [ ¾ f.; p ; q ; k / d ] F i a quare inegral maringale. From maringale repreenaion heorem, here exi.q ; K / aifying [ ] ¾ f.; p ; q ; k / d E = E [ ¾ F f.; p ; q ; k / d ] Q db K.z/Ñ.dz d/: We le P = E[¾ f.; p ; q ; k /d F ], hen P. / S 2 and.p; Q; K / i he oluion of he BSDEP (2.2) P = ¾ f.; p ; q ; k / d Q db K.z/Ñ.dz d/: Thi equaion inroduce he map 8 : S 2 H 2 F 2 N S 2 H 2 F 2 N by 8 :.p; q; k/.p; Q; K /. We ue he following mehod, which i imilar wih ha in [1], o ge he oluion of BSDE in L 1 pace wihin he fixed ime duraion, o prove he above map i a conracion. Le 8 :.p i ; q i ; k i /.P i ; Q i ; K i /, i = 1; 2, P = P 1 P 2, Q = Q 1 Q 2, K = K 1 K 2, ˆp = p 1 p 2, ˆq = q 1 q 2, ˆk = k 1 k 2, ˆ f = f.; p 1 ; q 1 ; k 1 / f.; p 2 ; q 2 ; k 2 /. From Doob inequaliy, ( P. / 2 = E up S 2 E ) 2 fˆ d F Q. / 2 K 2. / 2 = E ˆ H 2 FN 2 f d E We noe ha B 2 = S 2 H 2 F 2 N.So ( 4E fˆ d 2 ˆ f d) 2 ; ( E ˆ f d) 2 : 8.p 1 ; q 1 ; k 1 / 8.p 2 ; q 2 ; k 2 / 2 B 2 ( ) 2 5E fˆ d = P. / 2 Q. / 2 K. / 2 S 2 H 2 FN 2 [ ( 2 ] 15 u 1./ d) u 2./ d 2 u 2./ d 3 [ ] ˆp. / 2 S 2 ˆq. / 2 ˆk. / 2 : H 2 FN 2 From he aumpion ( u 1./ d )2 u 2./ d 2 u 2 3./ d < 1=15, hen 8 : B 2 B 2 i a ric conracion, BSDEP (2.1) ha one unique oluion. Second ep. Aume u 1./ d <, u 2./ d <, 2 u 2 3./ d <. Then here exi T >, uch ha ( u T 1./ d )2 T u2./ d 2 T u2 3./ d < 1=15. We le f 1.; p; q; k/ = I [T; ]./ f.; p; q; k/, hen f 1 aifie Lipchiz condiion

6 254 hen Wu [6] (H2.2) wih ū 1./ = I [T; ]./u 1./, ū 2./ = I [T; ]./u 2./, ū 3./ = I [T; ]./u 3./ and ( ū 1./ d )2 ū 2./ d 2 ū 2 3./ d < 1=15. So here exi oluion. p. /; q. /; k. // from he fir ep, uch ha, for T, (2.3) p = ¾ f 1.; p ; q ; k / d q db k.z/ñ.dz d/: Then we conider he following BSDEP, (2.4) p = p T T f.; p ; q ; k / d T q db T k.z/ñ.dz d/; [; T ]. From he reul in [1] or he reul for fixed ime in [19], which only need minor change uiable for our cae, here exi unique oluion. p; q; k/. Le u e p = I [;T ]./ p I.T ; ] p, q = I [;T ]./ q I.T ; ] q, k = I [;T ]./ k I.T ; ] k, i i eay o check ha hi i a oluion of BSDEP (2.1). The proof i compleed. Similarly o he comparion heorem of BSDE in [8], we will give hi kind of heorem for BSDEP in opping ime in he remaining par of hi ecion. Bu he appearance of jump proce need one new condiion o limi he heigh of he jump beide he Lipchiz condiion in (H2.2). We conider he following wo BSDEP in opping ime, here m = 1. (2.5) p i = ¾ i f i.; p i ; qi; ki/ d q i db k i.z/ñ.dz d/; where i = 1; 2, ¾ i L 2, f i aify (H2.1) and (H2.2). From Theorem 2.1, here exi.p i. /; q i. /; k i. // S 2 H 2 F 2 N which aify BSDEP (2.5) repecively. We alo aume (H2.3) ¾ 1 ¾ 2, f 1.; p 2 ; q 2 ; k 2 / f 2.; p 2 ; q 2 ; k 2 /,. (H2.4) c 2./ < ( f 1.; p 2 ; q 2 ; k 1 / f 2.; p 2 ; q 2 ; k 2 / ) =.k 1 k 2 /<c 1./, when k 1 k 2 =, c 1./ and c 2./ are wo poiive deerminiic funcion which aify c 1./ d <, c 2./ d < and c 2./ <1,. Then we have THEOREM 2.2. For every, p 1 p2. The proof i almo he ame a he proof of he comparion heorem [8, Theorem 2.2] for BSDE wihou jump. We omi i. When T <, we can ake u 1./, u 2./ and u 3./, T, o be conan, hen he reul of BSDEP in bounded ime duraion i he pecial cae of our reul in hi ecion.

7 [7] Fully coupled FBSDE wih Brownian moion Exience and uniquene of FBSDEP in opping ime duraion In hi ecion, we dicu he fully coupled FBSDEP in opping ime duraion. We conider (3.1) x = a p = 8.x / b.; x ; p ; q ; k / d.; x ; p ; q ; k / db g. ; x ; p ; q ; k.z/; z/ñ.dz d/; f.; x ; p ; q ; k / d k.z/ñ.dz d/: Here >,.x; p; q; k/ ake value in R m R m R m d R m, q db b : [; ] R m R m R m d R m R m ; : [; ] R m R m R m d R m R m d ; g : [; T ] R m R m R m d R m R m ; f : [; T ] R m R m R m d R m R m ; 8 : R m R m : We aume he following: (H3.1) For every.x; p; q; k/ R mmm dm, 8.x/ L 2, b,, g and f are progreively meaurable and ( 2 ( ) 2 E b.; ; ; ; / d) E f.; ; ; ; / d E.; ; ; ; / 2 d E g.; ; ; ; ; z/ 2 n.dz/ d < : (H3.2) There exi a poiive deerminiic bounded funcion u 1./, uch ha for every.x i ; p i ; q i ; k i / R mmm dm, i = 1; 2, l.; x 1 ; p 1 ; q 1 ; k 1 / l.; x 2 ; p 2 ; q 2 ; k 2 / u 1./ [ x 1 x 2 p 1 p 2 q 1 q 2 k 1 k 2 ] ; l = b; ; f; g repecively, and u 1./ d <, u 2 1./ d <. There exi a conan C > uch ha 8.x 1 / 8.x 2 / C x 1 x 2. We inroduce he noaion x f u = p q ; A.; u/ = b.; u/; k g

8 256 hen Wu [8] where =. 1 d /. We ue he uual inner produc and Euclidean norm in R m, R m d and aume he following monoone aumpion: (H3.3) For every u =.x; p; q; k/, ū =. x; p; q; k/, û =. ˆx; ˆp; ˆq; ˆk/ =.x x; p p; q q; k k/, { A.; u/ A.; ū/; û þ1 u 1./ ˆx 2 þ 2 u 1./. ˆp 2 ˆq 2 ˆk 2 /; 8.x/ 8. x/; x x ¼ 1 ˆx 2 ; where þ 1, þ 2 and ¼ 1 are given nonnegaive conan wih þ 1 þ 2 >, ¼ 1 þ 2 >. REMARK 3.1..i/ For noaional impliciy, we ake he ame funcion u 1./ in (H3.2) and (H3.3)..ii/ We only conider he ame dimenional cae of x and p. When x and p ake differen dimenion uch a x R n, p R m, we can inroduce a full rank m n marix and deal wih i uing he mehod in [17] o ge he ame reul a he following Theorem 3.1. THEOREM 3.1. We aume (H3.1), (H3.2) and (H3.3) hold, hen FBSDEP (3.1) ha a unique oluion.x. /; p. /; q. /; k. // S 2 S 2 H 2 F 2 N. PROOF. For he uniquene, le u =.x ; p ; q ; k / and ū =. x ; p ; q ; k / be wo oluion of (3.1). We e û =.x x; p p; q q; k k/ =. ˆx; ˆp; ˆq; ˆk/ and apply Iô formula o ˆx ; ˆp. Uing he ame echnique, which wa ued o prove he uniquene for FBSDE in [17], and he he uniquene reul for BSDEP and for ochaic differenial equaion wih jump in [7], we can eaily ge he concluion. To prove he exience, we can conider wo cae according o he ign of þ 1, þ 2 and ¼ 1, hi make he proof clear and eay o underand. Fir cae. þ 1 >, ¼ 1 > andþ 2. We conider he following family of FBSDEP paramerized by Þ [; 1]. x Þ = a [Þb.; u Þ / ] d [Þ.; u Þ / ] db [Þg. ; u Þ ; z/ ½.z/]Ñ.dz d/; (3.2) p Þ = Þ8.x Þ /.1 Þ/xÞ ¾ [.1 Þ/þ 1 u 1./x Þ Þ f.; uþ/ ] d q Þ db k Þ.z/Ñ.dz d/;

9 [9] Fully coupled FBSDE wih Brownian moion 257 where,, and ½ are given procee wih value in R m, R m d, R m repecively, ¾ L 2 and ( 2 ( 2 E d) E d) E 2 d E and R m ½.z/ 2 n.dz/ d < : Clearly, when Þ = 1, he exience of he oluion of (3.2) implie hi of (3.1). When Þ =, i i eay o ee ha here exi a oluion of (3.2). So we need he following lemma. LEMMA 3.2. We aume ha (H3.1), (H3.2) and (H3.3) hold. Then here exi a poiive conan Ž uch ha if, apriorily, for Þ [; 1/ here exi a oluion.x Þ ; p Þ ; q Þ ; k Þ / of (3.2), hen for each Ž [;Ž ] here exi a oluion.x ÞŽ ; p ÞŽ ; q ÞŽ ; k ÞŽ / S 2 S 2 H 2 F 2 of (3.2) for Þ = Þ N Ž. PROOF. Since for each,,, ½, Þ [; 1/, here exi a oluion of (3.2), hen, for each riple u =.x ; p ; q ; k / S 2 S 2 H 2 F 2 N ; x L 2 ; here exi a unique riple U =.X ; P ; Q ; K / S 2 S 2 H 2 F 2 N aifying he following FBSDEP X = a [Þ b.; U / Žb.; u / ] d [Þ.; U / Ž.; u / ] db [Þ g. ; U ; z/ Žg. ; u ; z/ ½ ]Ñ.dz d/ P = Þ 8.X /.1 Þ /X Ž.8.x / x / ¾ [ ].1 Þ /þ 1 u 1./X Þ f.; U /Ž. þ 1 u 1./x f.; u // d Q db K.z/Ñ.dz d/: We wan o prove ha he mapping defined by i a conracion. I ÞŽ.u x / = U X : S 2 S 2 H 2 F 2 N L2 S 2 S 2 H 2 F 2 N L2

10 258 hen Wu [1] We noe ha B 2 = S 2 S 2 H 2 F 2 and le ū =. x; p; q; N k/ B 2, U X = I ÞŽ.ū x /. Uing he ame noaion for û and Û a above and applying Iô formula o X ; P,wege (3.3) [Þ ¼ 1.1 Þ /]E X 2 þ 1 E ŽC 1 E ˆx 2 ŽC 1 E X 2 [ ( ŽC 1 u 2./ d 1 u 1./ X 2 d ) 2 ] ] u 1./ d [ Û. / 2 B 2 û. / 2 B : 2 Uing Iô formula o P 2 and hen Gronwall Lemma and he Burkholder-Davi- Gundy inequaliy, we ge P. / 2 Q. / 2 K. / 2 S 2 H 2 FN 2 ] u 1./ X 2 d E X 2 ŽC 2 E ˆx 2 C 2 [E [ ( ŽC 2 u 2./ d 1 ) 2 ] u 1./ d û. / 2 2: B Applying he uual echnique o he forward ochaic differenial equaion and combining wih (3.3), we ge Û. / 2 X B 2 2 ŽM [ L û. / 2 ˆx ] 2 B 2 2 L : 2 Here he conan C 1, C 2 and M depend on þ 1, ¼ 1 and C. We now chooe Ž = 1=.2M/. I i clear ha, for each fixed Ž [;Ž ],he mapping I ÞŽ i a conracion and ha a unique fixed poin U ÞŽ =.X ÞŽ ; P ÞŽ ; Q ÞŽ ; K ÞŽ / which i he oluion of (3.2) forþ = Þ Ž. The proof i complee. Second cae: þ 2 >, þ 1, ¼ 1. We need o conider he following family of FBSDEP paramerized by Þ [; 1]. (3.4) x Þ = a [Þb.; u Þ /.1 Þ/þ 2. u 1./p Þ / ] d p Þ = Þ8.x Þ / ¾ [Þ.; u Þ /.1 Þ/þ 2. u 1./q Þ / ] db [Þg. ; u Þ ; z/.1 Þ/þ 2. u 1./k Þ / ½.z/]Ñ.dz d/ [Þ f.; u Þ / ] d q Þ db k Þ.z/Ñ.dz d/;

11 [11] Fully coupled FBSDE wih Brownian moion 259 where,,, ½ and ¾ aify he ame aumpion a ha in (3.2). Similarly o Lemma 3.2, we can how he following reul. LEMMA 3.3. We aume (H3.1), (H3.2) and (H3.3) hold, hen here exi a poiive conan Ž uch ha if, apriorily, for an Þ [; 1/ here exi a oluion.x Þ ; p Þ ; q Þ ; k Þ / of (3.4), hen for each Ž [;Ž ] here exi a oluion.x ÞŽ ; p ÞŽ ; q ÞŽ ; k ÞŽ / S 2 S 2 H 2 F 2 N of (3.4) for Þ = Þ Ž. PROOF OF THEOREM 3.1 (EXISTENCE). From he aumpion (H3.3), we know ha eiher (i) þ 1 >, ¼ 1 >, þ 2 or (ii) þ 1, ¼ 1, þ 2 >. In he fir cae, we conider (3.2) andwhenþ =, (3.2) ha a unique oluion. I hen follow from Lemma 3.2 ha here exi a poiive conan Ž uch ha for each Ž [;Ž ], (3.2) ha a unique oluion for Þ = Þ Ž. We can repea hi proce N-ime wih 1 NŽ < 1 Ž. I hen follow ha, in paricular, for Þ = 1 wih =, =, =, ½ = and¾ = (3.2) ha a unique oluion. In he econd cae,we conider (3.4) and when Þ =, FBSDEP (3.4) ha a unique oluion. I hen follow from Lemma 3.3, by repeaing he ame proce a in he fir cae, ha we ge he deired concluion. The proof i compleed. REMARK 3.2. If we replace (H3.3) by he following (H3.4) For every u =.x; p; q; k/, ū =. x; p; q; k/, û =. ˆx; ˆp; ˆq; ˆk/ =.x x; p p; q q; k k/, { A.; u/ A.; ū/; û þ1 u 1./ ˆx 2 þ 2 u 1./. ˆp 2 ˆq 2 ˆk 2 / 8.x/ 8. x/; x x ¼ 1 ˆx 2 ; where þ 1, þ 2 and ¼ 1 are given nonnegaive conan wih þ 1 þ 2 >, ¼ 1 þ 2 >. Uing a imilar mehod a in Theorem 3.1, we can alo prove ha FBSDEP (3.1) ha he unique oluion. REMARK 3.3. When he opping ime T <, u 1./, T, can be replaced by he conan, hen he exience and uniquene reul of FBSDEP in bounded ime duraion i he pecial cae of Theorem The comparion heorem of FBSDEP in opping ime duraion In hi ecion, we give a comparion heorem o FBSDEP in opping ime. Thi heorem i one of imporan properie of FBSDEP. We conider he following wo

12 26 hen Wu [12] FBSDEP, (4.1) x i = a i b.; x i ; pi ; qi/ d.; x i ; pi ; qi/ db g. ; x i ; p i ; q i ; z/ñ.dz d/; i = 1; 2; p i = 8 i.x i / f i.; x i ; pi; qi; ki/ d q i db k i Ñ.dz d/: The coefficien of FBSDEP (4.1), i = 1; 2, boh aify (H3.1), (H3.2) and (H3.3), hen here exi he oluion.x i ; p i ; q i ; k i / S 2 S 2 H 2 FN 2 repecively. In he following par, we only conider m = 1, in fac we can alo deal wih he cae when x ake mulidimenional value uch a x R n. For ha cae, we need o inroduce a 1 n nonzero vecor G in he monoone aumpion o enure he exience and uniquene for differen dimenional FBSDEP he ame a ha in [17]. We aume (H4.1) For every x R,, { a1 a 2 ; 8 1.x/ 8 2.x/; a.. f 1.; x; p; q; k/ f 2.; x; p; q; k/; The inroducion of a random jump le he oluion x and p o be no coninuou, o we alo need he following condiion o conrol he jump heigh. (H4.2) 1 < f 1.; x 2; p2; q 2; k1/ f 1.; x 2; p2; q 2; k2/ k 1, k k2 1 k2 =, a.. Then we have THEOREM 4.1. p 1 p2. PROOF. For noaional convenience, we aume d = 1 and fir conider he following FBSDEP: x = a 1 b.; x ; p ; q / d.; x ; p ; q / db g. ; x ; p ; q ; z/ñ.dz d/; i = 1; 2; (4.2) p = 8 2. x / f 2.; x ; p ; q ; k / d q db k Ñ.dz d/: a..

13 [13] Fully coupled FBSDE wih Brownian moion 261 Obviouly, he above FBSDEP ha a unique oluion. x; p; q; k/. Weeˆx = x 1 x, ˆp = p 1 p, ˆq = q 1 q, ˆk = k 1 k, he quare. ˆx; ˆp; ˆq; ˆk/ aifie ˆx =.b 1 ˆx b 2 ˆp b 3 ˆq / d. 1 ˆx 2 ˆp 3 ˆq / db.g 1 ˆx g 2 ˆp g 3 ˆq /Ñ.dz d/ (4.3) ˆp = 8 ˆx 8 1. x / 8 2. x /. f 11 ˆx f 12 ˆp f 13 ˆq f 14 ˆk f / d ˆq db ˆk.z/Ñ.dz d/; where f = f 1.; x; p; q; k/ f 2.; x; p; q; k/, 8 1.x 1/ 81. x / 8 = x 1 x ; ˆx = ; ; oherwie; l.; x 1; p1; q 1/ l.; x ; p 1; q 1/ l 1 = x 1 x ; ˆx = ; ; oherwie; l.; x ; p 1; q 1/ l.; x ; p ; q 1/ l 2 = p 1 p ; ˆp = ; ; oherwie; l.; x ; p ; q 1/ l.; x ; p ; q / l 3 = q 1 q ; ˆq = ; ; oherwie l = b; ;g repecively. f 1.; x 1; p1; q 1; k1/ f 1.; x ; p 1; q 1; k1/ f 11 = x 1 x ; ˆx = ; ; oherwie; f 1.; x ; p 1; q 1; k1 / f 1.; x ; p ; q 1; k1 / f 12 = p 1 p ; ˆp = ; ; oherwie; f 1.; x ; p ; q 1; k1/ f 1.; x ; p ; q ; k 1/ f 13 = q 1 q ; ˆq = ; ; oherwie;

14 262 hen Wu [14] f 14 = f 1.; x ; p ; q ; k 1/ f 1.; x ; p ; q ; k / k 1 ; ˆk = ; k ; oherwie: I ieay o checkha (4.3) aifie (H3.1), (H3.2) and (H3.3), hu. ˆx; ˆp; ˆq; ˆk/ i he unique oluion of (4.3). We fir need o prove ha ˆp. We ue he dualiy echnique and inroduce he dual FBSDEP (4.4) M = 1 N = 8M. f 12 M b 2 N 2 U g 2 V / d. f 13 M b 3 N 3 U g 3 V / db f 14 U db M Ñ.dz d/;. f 1 M b 1 N 1 V.z/Ñ.dz d/: U g 1 V / d The dualiy echnique i uually ued o inroduce he adjoin equaion in opimal conrol heory when we wan o ge he maximum principle (ee [14] and [2]). From (4.3) aifying (H3.1), (H3.2) and (H3.3), we can verify ha (4.4) aifie (H3.1), (H3.2) and (H3.4). Then i follow from Remark 3.2 ha here exi a unique quare.m; N; U; V / which i he oluion of (4.4). Applying Iô formula o ˆx N ˆp M,wehave ˆp = E.8 1. x / 8 2. x //M E M f d: From (H4.1) andm = 1 >, if we can prove M, a.., hen ˆp. Le u define he following opping ime ¹ = inf{ > ; M } : So ¹, a.. and M ¹. In he fir equaion of (4.4), he nonconinuou par of M i only produced by random meaure N, from (H4.2), 1M ¹ M ¹ ; M ¹ = M ¹ 1M ¹ ; o M ¹ =, when ¹<. We can inroduce. M ; N ; U ; V /, [¹; ], which aifie

15 [15] Fully coupled FBSDE wih Brownian moion 263 he following FBSDEP M = (4.5). f 12 ¹ ¹ ¹. f 13 M b 2 N 2 U g 2 V / d f 14 N = 8 M U db M b 3 N 3 U g 3 V / db M Ñ.dz d/. f 1 M b 1 N 1 U g 1 V / d V.z/Ñ.dz d/: Then i i eay o ee ha. M ; N ; U ; V /.; ; ; / i he unique oluion. Now we le M = 1 [;¹]./M 1.¹; ]./ M ; N = 1 [;¹]./N 1.¹; ]./ N U = 1 [;¹]./U 1.¹; ]./ U ; V = 1 [;¹]./V 1.¹; ]./ V ; : I i eay o ee ha.m ; N ; U ; V / i a oluion of (4.4), from Remark 3.2, hi i he unique oluion. From M = M = 1 > andm ¹, obviouly M, a.., ha i, M. So we have p 1 p. Now we ry o compare p wih p 2, and hen ge he deired concluion. If a1 = a 2, from Theorem 3.1, p = p 2, hen p1 p2.ifa1 > a 2,wee andapplyiô formula o x p, x =. x x 2 /; p =. p p 2 /; ũ =.ū u 2 / q =. q q 2 /; k =. k k 2 /; E.8. x / 8.x 2 // x. p p 2 /.a1 a 2 / = E A.; ū / A.; u 2 /; ũ d: Here we ue he noaion from Secion 3 for u and A. From(H3.3), we have. p p 2 /.a 1 a 2 / ; o p p 2,andhenp1 p2. The proof i compleed. Now we give an example of FBSDEP o how he comparion heorem.

16 264 hen Wu [16] EXAMPLE 4.1. We conider he following wo FBSDEP, p 1 x 1 = 2 x 1 q1 q 1 d p1.1 / 2.1 / db 2 x 1 Ñ.dz d/;.1 / 2 (4.6) ( x 1 p 1 = x 1 5 p1 ) k1 1 d.1 / 2.1 / 2 q 1 db k 1.z/Ñ.dz d/; and (4.7) x 2 p 2 = 1 p 2 x 2 q2 d.1 / 2 x 2 q 2 p2.1 / 2 db Ñ.dz d/;.1 / 2 = x 2 x 2 p2 k2 d.1 / 2 q 2 db k 2.z/Ñ.dz d/; : I i eay o check ha (4.6) and (4.7) aify (H3.1), (H3.2) and (H3.3), o according o Theorem 3.1, here exi unique oluion.x 1 ; p 1 ; q 1 ; k 1 / and.x 2 ; p 2 ; q 2 ; k 2 / repecively. We can check ha he above wo FBSDEP aify (H4.1) and (H4.2), o from Theorem 4.1, we know ha p 1 p2. We noice ha he comparion Theorem 4.1 of FBSDEP, which hold only a ime =, i weaker han ha of BSDEP, ha i, Theorem 2.2. In he forward-backward cae, we canno eaily jump o a concluion like p 1 = 81.x 1 / 82.x 2 / = p2 from he aumpion ha 8 1.x/ 8 2.x/ becaue in he preen iuaion, he forward oluion x 1 and x 2 are differen if 81 and 8 2 are. Thu unlike he claical (pure backward) cae, no common comparion heorem can be made even a 1 = a 2 excep for =. We will give a counerexample o how hi poin. EXAMPLE 4.2. For impliciy, we conider he fixed ime duraion T >, he Lipchiz coefficien being conan, a one dimenional Brownian moion and udy he following wo FBSDEP, x 1 = a. p 1 q 1 / d.x 1 p 1 q 1 / db ; T; (4.8) T T T p 1 = x 1 T 2.x 1 q 1 2/ d q 1 db k 1.z/Ñ.dz d/;

17 [17] Fully coupled FBSDE wih Brownian moion 265 x 2 = a. p 2 q2/d.x 2 p2 q2/ db ; T; (4.9) T T T p 2 = x 2 T.x 2 q2/ d q 2 db k 2.z/Ñ.dz d/; From Theorem 3.1 and Remark 3.3, here exi a unique oluion.x 1 ; p 1 ; q 1 ; k 1 / for (4.8)and.x 2 ; p 2 ; q 2 ; k 2 / for (4.9) repecively. Then, from Theorem 4.1, p 2 p1. Now we ry o check hi concluion for hi example. Firly, i i eay o know ha.x 1 ; p 1 ; q 1 ; k 1 / i he unique oluion of (4.8), where p 1 = x 1 2, q 1 = x 1 1, k 1 = andx 1 i he oluion of he following ochaic differenial equaion: { dx 1 =. 2x 1 3/ d.x 1 1/ db ; (4.1) x 1 = a: Then we ge x 1 = ae 5=2 B e 5=2 B 4e 5=2B d e 5=2 B e 5=2B db and p 1 = x 1 2, T. We alo can ge.x 2 ; p 2 ; q 2 ; k 2 / i he unique oluion of (4.9), where p 2 = x 2, q2 = x 2, k2 = andx 2 aifie he following ochaic differenial equaion (4.11) { dx 2 = 2x 2 d x 2dB x 2 = a: Then p 2 = x 2 = ae 5=2 B.So p 1 p 2 = 2 e 5=2 B 4e 5=2B d e 5=2 B e 5=2B db : For =, p 1 p2 = 2 >, bu for any >, i can be boh poiive or negaive wih poiive probabiliy. Acknowledgemen and uggeion. The auhor hank he referee for hi many helpful commen Reference [1] F. Anonelli, Backward-forward ochaic differenial equaion, Ann. Appl. Probab. 3 (1993),

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