Local Strict Comparison Theorem and Converse Comparison Theorems for Reflected Backward Stochastic Differential Equations

Transcription

1 arxiv:mah/07002v [mah.pr] 3 Dec 2006 Local Sric Comparison Theorem and Converse Comparison Theorems for Refleced Backward Sochasic Differenial Equaions Juan Li and Shanjian Tang Absrac A local sric comparison heorem and some converse comparison heorems are proved for refleced backward sochasic differenial equaions under suiable condiions. AMS 2000 Subjec Classificaion: 60H0, 60H30 Keywords: Refleced Backward Sochasic Differenial Equaions, Comparison Theorem Inroducion The comparison heorem urns ou o be one classical resul for backward sochasic differenial equaions (BSDEs). I allows us o compare he soluions of wo real-valued BSDEs by comparing he erminal condiions and he generaors. In a converse way, Peng is concerned in 997 wih he following converse comparison propery for BSDEs: if he soluions of wo real-valued BSDEs are equal a he iniial ime for any idenical erminal condiion, heir generaors are idenical. For works on his problem, he reader is referred o among ohers: Chen [3], Briand e al. [], Coque e al. [2], and Jiang [9]. In heir argumens, he sric comparison heorem of BSDEs plays a crucial role. Parially suppored by he Naural Science Foundaion of China under grans , (disinguished youh foundaion, eniled wih Conrol Theory of Sochasic Sysems ), he Chang Jiang Scholars Programme, and he Science Foundaion of Chinese Minisry of Educaion under gran Insiue of Mahemaics, School of Mahemaical Sciences, Fudan Universiy, Shanghai , China; Key Laboraory of Mahemaics for Nonlinear Sciences (Fudan Universiy), Minisry of Educaion; & Deparmen of Mahemaics, Shandong Universiy a Weihai, Weihai , China. juanli@sdu.edu.cn. Deparmen of Finance and Conrol Sciences, School of Mahemaical Sciences, Fudan Universiy, Shanghai , China; & Key Laboraory of Mahemaics for Nonlinear Sciences (Fudan Universiy), Minisry of Educaion. sjang@fudan.edu.cn.

2 On he oher hand, he soluion Y of a refleced BSDE (RBSDE) characerizes he value process of an opimal sopping ime problem, and he price process {Y } 0 T of an American opion is a soluion of an RBSDE (See El Karoui e al. [5]): wih Y = (X T k) + Y S := (X k) +, [ry s + θz s ]ds + K T K [0, T]; 0 (Y S ) dk = 0. Z s db s, (.) Here θ := σ (µ r) is he premium of he marke risk, and {X } 0 T is he sock price process saisfying he following SDE: X = X µx s ds + 0 σx s db s, [0, T]. Define he sopping ime τ := inf { : Y = S }, which is he ime when he invesor would ake he acion o sell or buy he sock. The heory of RBSDEs exising in he lieraure only reveals how he price Y depends on he generaor g(y, z) := ry + θz, y lr, z lr and he srike price k (more generally speaking, he obsacle and he erminal value) as well. I is naural o ask how he premium θ (more generally speaking, he generaor g) can be obained from a family of American opions parameerized by he srike price k? Then, he relaion among he soluion, he generaor and he obsacle becomes ineresing. In his paper, we are concerned wih comparison heorems and converse comparison heorems for RBSDEs under suiable condiions, which reveal some monooniciy beween he soluion, and he generaor and he obsacle of a RBSDE. The res of he paper is organized as follows. In Secion 2, we provide some preliminary resuls on BSDEs and RBSDEs. In Secion 3, we firs illusrae ha quie differen from BSDEs, RBSDEs do no have he global comparison propery. Then we prove a local sric comparison heorem for RBSDEs. In Secions 4 and 5, we discuss converse comparison properies for RBSDEs when he obsacle is no previously given and when he obsacle is previously given, respecively. Some ineresing comparison heorems are obained in boh cases. 2 Preliminaries In his secion, we give some basic resuls on BSDEs and RBSDEs. They will be used in he subsequen secions. Le (Ω, F, P) be a probabiliy space and {B } 0 be a d-dimensional sandard Brownian moion on his space such ha B 0 = 0. Denoe by {F } 0 he filraion generaed by Brownian moion {B } 0 : F := σ{b s, s [0, ]} N, [0, T], where N is he se of all P-null subses. Le T > 0 be a given real number. For any posiive ineger n and z R n, z denoes he Euclidean norm. 2

3 Define he following wo spaces of processes: H 2 (0, T;R n ) := {{ψ } 0 T and is an R n -valued predicable process s.. E 0 ψ 2 d < + } S 2 (0, T;R) := {{ψ } 0 T is a predicable process s.. E[ sup ψ 2 ] < + }. 0 T Consider funcion g : Ω [0, T] R R d R such ha {g(, y, z)} [0,T] is progressively measurable for each (y, z) in R R d. We make he following assumpions on g hroughou he paper. (A) There exiss a consan K > 0 such ha a.s. for [0, T], g(, y, z ) g(, y 2, z 2 ) K( y y 2 + z z 2 ), [0, T], y, y 2 R, z, z 2 R d. (A2) The process g(, 0, 0) H 2 (0, T;R). (A3) g(, y, 0) = 0 a.s. for any (, y) [0, T] R. (A4) The mapping g(, y, z) is coninuous a.s. for any (y, z) R R d. Remark 2.. I is obvious ha Assumpion (A3) implies Assumpion (A2). I is by now well known (see Pardoux and Peng [0] for he proof) ha under Assumpions (A) and (A2), for any random variable ξ L 2 (Ø, F T, P), he BSDE y = ξ + g(s, y s, z s )ds z s db s, 0 T (2.) has a unique adaped soluion (y T,g,ξ, z T,g,ξ ) S 2 (0, T;R) H 2 (0, T;R d ). In he sequel, we always assume ha g saisfies (A) and (A2). We inroduce he following operaor ε g,t : for any ξ L 2 (Ø, F T, P), denoe by ε g,t [ξ] and ε g,t [ξ F ] he iniial a ime of he soluion o BSDE (2.), respecively. For a sopping ime τ, he operaor ε g,τ can be defined in an idenical way. We give some basic resuls of BSDEs, including Lemmas 2., 2.2 and 2.3, which can be found in Briand e al. [] or Peng [], El Karoui e al. [6], and Jiang [9], respecively. value y T,g,ξ 0 and he value y T,g,ξ Lemma 2.. (Comparison Theorem) Assume ha wo fields g and saisfy (A) and (A2). Consider ξ, ξ 2 L 2 (Ω, F T, P). We have (i) (Monooniciy) If ξ ξ 2 and g a.s., hen ε g,t[ξ ] ε g2,t[ξ 2 ], and ε g,t[ξ F ] ε g2,t[ξ 2 F ] a.s. for [0, T]. (ii)(sric Monooniciy) If ξ ξ 2, g, and P({ξ > ξ 2 }) > 0, hen P({ε g,t [ξ F ] > ε g,t [ξ 2 F ]}) > 0 for [0, T]. In paricular, ε g,t [ξ ] > ε g,t [ξ 2 ]. 3

4 Lemma 2.2. Assume ha he field g saisfies assumpions (A) and (A2). Consider he sopping ime τ T and ξ L 2 (Ω, F τ, P). Define g(, y, z) := g(, y, z) [0,τ] (), (, y, z) [0, T] R R d. Then ε g,τ [ξ] = ε g,t [ξ] and ε g,τ [ξ F ] = ε g,t [ξ F ] a.s. for [0, τ]. Lemma 2.3. Assume ha wo funcions g and saisfy assumpions (A), (A2) and (A4). Then he following wo asserions are equivalen: (i) ε g,τ[ξ] = ε g2,τ[ξ] for each sopping ime τ T and any ξ L 2 (Ω, F τ, P). (ii) g (, y, z) = (, y, z) a.s. for any (, y, z) [0, T] R R d. We inroduce he condiional g-expecaion (see Peng [, 2], Chen [3], Coque e al. [2, 4]). Suppose g saisfies (A), (A3) and (A4). We se, for any sopping ime τ aking values in [0, T], ε g [ξ F τ ] := y T,g,ξ τ (= ε g,t [ξ F τ ]). I can be shown ha ε g [ξ F τ ] is he unique F τ -measurable, square-inegrable random variable η such ha ε g [ A η] = ε g [ A ξ], A F τ. Therefore i is called he g-expecaion condiioned on F τ. Noice ha g-expecaion ε g is a paricular example of he nonlinear expecaion inroduced in [3, 4,, 2]. Now we borrow from [2] he converse comparison heorem for g-expecaion. Lemma 2.4. Suppose ha wo funcions g and saisfy assumpions (A), (A3) and (A4). Then he following wo asserions are equivalen: (i) ε g [ξ] ε g2 [ξ] for any ξ L 2 (Ω, F T, P). (ii) g (, y, z) (, y, z) a.s. for any (, y, z) [0, T] R R d. A refleced BSDE is associaed wih a erminal condiion ξ L 2 (Ω, F T, P), a generaor g, and an obsacle process {S } 0 T. We make he following assumpion: (A5) {S } 0 T is a coninuous process such ha {S } 0 T S 2 (0, T;R). The soluion of a RBSDE is a riple (Y, Z, K) of F -progressively measurable processes aking values in R R d R + and saisfying (i) Z H 2 (0, T;R d ), Y S 2 (0, T;R), and K T L 2 (Ω, F T, P); (ii) Y = ξ + g(s, Y s, Z s )ds + K T K Z s db s, [0, T]; (2.2) (iii) Y S a.s. for any [0, T]; (iv) {K } is coninuous and increasing, K 0 = 0 and (Y 0 S )dk = 0. The following wo lemmas are borrowed from El Karoui e al. [7]. 4

5 Lemma 2.5. Assume ha g saisfies (A) and (A2), ξ L 2 (Ω, F T, P), {S } 0 T saisfies (A5), and S T ξ a.s.. Then RBSDE (2.2) has a unique soluion (Y, Z, K). Remark 2.2. For simpliciy, a given riple (ξ, g, S) is said o saisfy he Sandard Assumpions if he generaor g saisfies (A) and (A2), he erminal value ξ L 2 (Ω, F T, P), he obsacle S saisfies (A5), and S T ξ a.s.. Lemma 2.6. (comparison heorem) Suppose ha wo riples (ξ, g, S ) and (ξ 2,, S 2 ) saisfy he Sandard Assumpions (in fac, i is sufficien for eiher g or o saisfy he Lipschiz condiion (A)). Furhermore, we make he following assumpions: (i) ξ ξ 2 a.s.; (ii) g (, y, z) (, y, z) a.s. for (, y, z) [0, T] R R d ; (iii) S S2 a.s. for [0, T]. Le (Y, Z, K ) and (Y 2, Z 2, K 2 ) be adaped soluions of RBSDEs (2.2) wih daa (ξ, g, S ) and (ξ 2,, S 2 ), respecively. Then Y Y 2 a.s. for [0, T]. Lemma 2.7. Assume ha (ξ, g, S) and (ξ 2,, S) saisfy he Sandard Assumpions. Furhermore, we make he following assumpions: (i) ξ ξ 2 a.s.; (ii) g (, y, z) (, y, z) a.s. for (, y, z) [0, T] R R d. Le (Y, Z, K ) and (Y 2, Z 2, K 2 ) be adaped soluions of RBSDEs (2.2) wih daa (ξ, g, S) and (ξ 2,, S), respecively. Then we have K K2 a.s. for [0, T], and K K2 is increasing in ime variable. (2.3) See Hamadène e al. [8, Proposiion 4.3] for he deailed proof of Lemma Local sric comparison heorem for RBSDEs In conras o BSDEs, he sric comparison heorem is no rue in general for RBSDEs. Here are wo counerexamples. Example 3. Take T =, g = 3, ξ = 3, ξ 2 = 2, and S = 2 + for [0, T]. Then he soluion (Y, Z, K ) of RBSDE (2.2) wih daa (ξ, g, S) is given by { { 2, if 0 Y 5 = 5 2, if < ; Z = 0, 0 ; K =, if 0 3 5, if < The soluion (Y 2, Z 2, K 2 ) of RBSDE (2.2) wih daa (ξ 2, g, S) is given by { 2, if 0 Y 2 = 0 5, if < ; Z 2 = 0, 0 ; K2 = { 5 3, if 0 0, if <. 6 0

6 Obviously, Y = Y 2 for [0, ] and Y 0 < Y 2 for (, ]. Moreover, 0 K K2 for [0, ]. The sric comparison heorem of RBSDE (2.2) does no hold. The following example shows ha even if he generaor is zero, i happens ha he sric comparison heorem of RBSDE (2.2) may no be rue. Example 3.2 Take T =, g = 0, ξ =, ξ 3 2 =, and S 2 = 2 + for [0, T]. Then he soluion (Y, Z, K ) of RBSDE (2.2) wih daa (ξ, g, S) is given by { 2, if 0 Y = 3, if < ; Z = 0, 0 ; K = 3 3 The soluion (Y 2, Z 2, K 2 ) of RBSDE (2.2) wih daa (ξ 2, g, S) is given by { 2, if 0 Y 2 = 4, if < ; Z 2 = 0, 0 ; K 2 = 2 4 { 2, if , if 3 <. { 2, if 0 4 2, if 4 <. Obviously, Y = Y 2 for [0, ] and Y 4 < Y 2 for (, ]. Moreover, 4 K K 2 for [0, ]. The sric comparison heorem of RBSDE (2.2) does no hold. However, we have he local sric comparison heorem. Theorem 3.. Suppose ha wo riples (ξ, g, S) and (ξ 2, g, S) saisfy he Sandard Assumpions. Moreover, assume ha ξ ξ 2 a.s. and P({ξ < ξ 2 }) > 0. Le (Y, Z, K ) and (Y 2, Z 2, K 2 ) be adaped soluions of RBSDE (2.2) wih daa (ξ, g, S) and (ξ 2, g, S), respecively. Then here exiss a sopping ime τ such ha τ < T almos surely and P({Y < Y 2, [τ, T]}) > 0. Proof. From Lemma 2.6, we have Y Y 2 a.s. for [0, T]. (3.) Define a sequence of sopping imes {τ k } k= in he following way: and τ := 0 τ k+ = inf{ τ k + 2 (T τ k) : Y = Y 2 } T, k =, 2,.... I is obvious ha he sequence {τ k } k= is boh bounded by T and non-decreasing. Therefore, i has an almos sure limi τ, which is sill a sopping ime saisfying τ T. Since τ k+ τ k + 2 (T τ k), k =, 2,..., we have by passing o he limi ha τ τ + (T τ), ha is, τ T. Hence, τ = T. 2 6

7 Furhermore, we asser ha here is some posiive ineger k 0 such ha P({τ k0 = T }) > 0. Oherwise, we have P({τ k < T }) =, k =, 2,.... This implies he following Y τ k = Y 2 τ k, k =, 2,.... Passing o he limi, we have YT = Y T 2 a.s.. Tha is, ξ = ξ 2, which conradics he assumpion ha P({ξ < ξ 2 }) > 0. Take he smalles ineger k among hose posiive inegers k 0 such ha P({τ k0 = T }) > 0. Then, we have k, P({τ k = T }) > 0, τ k < T a.s.. We asser ha he sopping ime τ := [τ k + 2 (T τ k )] < T is a desired one of he heorem. In fac, by definiion of τ k, we have Y < Y 2 on he inerval [ τ, T] whenever τ k = T. Therefore, The proof is complee. P({Y < Y 2, [ τ, T]}) P({τ k = T }) > 0. (3.) Corollary 3.. In Theorem 3., if g is eiher bounded from below by a nonnegaive consan C or saisfies (A3), ξ i (i =, 2) are bounded from below and he obsacle process S are bounded from above by a consan C 2, hen we have In paricular, we have Y 0 < Y 2 0. Y Y 2 a.s.,and P({Y < Y 2 }) > 0 for [0, T]. Proof. Consider he following BSDEs: Y i = ξ i + g(s, Y s i, Z i s )ds Z s i db s, 0 T, i =, 2. Obviously, i follows from Lemma 2. ha Y i C 2 S a.s., i =, 2, and P({Y < Y 2 }) > 0 for [0, T]. From Lemma 2.5, we have Y i = Y i, Z i = Z i, and K i = 0 a.s. for [0, T]. Therefore P({Y < Y 2 }) > 0 for [0, T]. Then he proof is complee. 7

8 4 A converse problem for RBSDEs In his secion, we consider he general converse comparison heorem for RBSDE (2.2). Coque e al. [2] prove he converse comparison for BSDEs: if g, saisfy (A), (A3) and (A4), and ε g,t[ξ] ε g2,t[ξ] for any ξ L 2 (Ω, F T, P), hen g (, y, z) (, y, z) a.s. for (, y, z) [0, T] R R d. Consider he converse comparison for RBSDE (2.2). I is ineresing since he sric comparison heorem is no rue for RBSDE (2.2) and several argumens developed for BSDEs have o be modified for RBSDEs. In he sequel, we always assume ha he daa (ξ, g, S) saisfies he Sandard Assumpions for RBSDEs. We inroduce he following operaor ε r g : denoe by εr g,t [ξ] and εr g,t [ξ F ] he iniial value Y T,g,ξ,S 0 and he value Y T,g,ξ,S a ime of he soluion of RBSDE (2.2) wih daa (ξ, g, S), respecively. Theorem 4.. Suppose ha wo funcions g and saisfy assumpions (A), (A3) and (A4). Then he following wo condiions are equivalen: (i) ε r g,t [ξ] εr,t [ξ] for any ξ L2 (Ω, F T, P) and any obsacle process (S ) 0 T saisfying (A5) and ξ S T a.s.. (ii) g (, y, z) (, y, z) a.s. for (, y, z) [0, T] R R d. Proof. Thanks o Lemma 2.6, i is obvious ha (ii) implies (i). I is sufficien o prove ha (i) implies (ii). For ξ L 2 (Ω, F T, P), consider he following BSDE: { dy i () = g i (, y i (), z i ())d z i ()db, [0, T], y i (T) = ξ, for i =, 2. In view of Lemma 2.4, i suffices o show ha ε g,t[ξ] ε g2,t[ξ]. Consider he following BSDE: Y = ξ + [ K Y s K Z s ]ds Z s db s, 0 T, where K = max(k, K 2 ) wih K and K 2 being he Lipschiz consans of g and, respecively. Then {Y } 0 T saisfies (A5) (see El Karoui e al. [6] for he deailed proof). Se S := Y. Since ξ S T, from assumpions (A), (A3) and Lemma 2., i follows ha y i () S a.s. for [0, T]. From Lemma 2.5, we see ha (y i, z i, 0) is he soluion of RBSDE (2.2) wih daa (ξ, g i, S) for i =, 2. In paricular, ε r g i,t [ξ] = yi (0) for i =, 2. Therefore, we ge from he assumpion ha ε g,t[ξ] = ε r g,t [ξ] εr,t [ξ] = ε,t[ξ]. The proof is complee. As an immediae consequence of Theorem 4., we have 8

9 Corollary 4.. Assume ha funcions g and saisfy assumpions (A), (A3) and (A4). Then he following wo condiions are equivalen: (i) ε r g,t [ξ] = εr,t [ξ] for any ξ L2 (Ω, F T, P) and any obsacle process {S } 0 T saisfying (A5) and ξ S T a.s.. (ii) g (, y, z) = (, y, z) a.s. for any (, y, z) [0, T] R R d. Remark 4.. The asserion of Theorem 4. is no rue if assumpion (A3) fails o be saisfied. To show his fac, consider he following example: g () = [0, T 2 )() + T 2 [ T 2,T]() and () = T 2 [0, T 2 )() + (T ) [ T 2,T](). Then, for any ξ L 2 (Ω, F T, P), any {S } 0 T saisfying (A5), and ξ S T a.s., i follows from El Karoui e al. [5, Proposiion 2.3] ha However, g. ε r g,t [ξ] εr,t [ξ]. Remark 4.2. If he obsacle process {S } 0 T is previously given and ε r g,t [ξ] εr,t[ξ] only for hose ξ L 2 (Ω, F T, P) such ha ξ S T a.s., hen Theorem 4. is no rue in general. I suffices o consider he following example: g (, y, z) = µ (y c ) z and (, y, z) = µ 2 (y c 2 ) z, wih c < c 2 and µ > µ 2. I is obvious ha g and saisfy assumpions (A), (A3), and (A4). Furhermore, ake S = c 2, [0, T]. Then for any ξ L 2 (Ω, F T, P) saisfying ξ S T a.s., we have ε r g,t[ξ] = ε r,t[ξ] and ε r g,t[ξ F ] = ε r,t[ξ F ] a.s. for (0, T]. However, we have neiher g nor g. When assumpion (A2) insead of (A3) is made on g, we have he following converse comparison resul. Theorem 4.2. Assume ha funcions g and saisfy assumpions (A), (A2) and (A4). Then he following wo condiions are equivalen: (i) ε r g,τ[ξ] = ε r,τ[ξ] for any sopping ime τ T, any ξ L 2 (Ω, F τ, P), and any obsacle {S } 0 T saisfying (A5) and ξ S τ a.s.. (ii) g (, y, z) = (, y, z) a.s. for any (, y, z) [0, T] R R d. Proof. Thanks o Lemma 2.6, i is obvious ha (ii) implies (i). I is sufficien o prove ha (i) implies (ii). For ξ L 2 (Ω, F τ, P), consider he following BSDE defined on he inerval [0, τ] : { dy i () = g i (, y i (), z i ())d z i ()db, y i (τ) = ξ, 9

10 for i =, 2. In view of Lemma 2.3, i suffices o show ha ε g,τ[ξ] = ε g2,τ[ξ]. Consider he obsacle process {S } 0 T which is defined o be ξ on [τ, T], and on [0, τ] is aken o be one componen of he soluion of he following BSDE: S = ξ + τ [g (s, 0, 0) (s, 0, 0) K S s K Z s ]ds τ Z s db s, where K = max(k, K 2 ), K and K 2 are he Lipschiz consans of g and, respecively. The above equaion admis a unique soluion {S } 0 T, which saisfies (A5). Since ξ S τ, i follows from assumpions (A), (A2), Lemmas 2. and 2.2 ha y i () S a.s. on he inerval [0, τ]. From Lemma 2.5, we ge ha (y i, z i, 0) is he soluion of RBSDE (2.2) wih daa (ξ, g i, S) on he inerval [0, τ] for i =, 2. In paricular, ε r g i,t [ξ] = yi (0) for i =, 2. Therefore, i follows from he assumpion ha The proof is complee. ε g,τ[ξ] = ε r g,τ [ξ] = εr,τ [ξ] = ε,τ[ξ]. Remark 4.3. From he example in Remark 4., we can ge ha if g only saisfies (A), (A2), and (A4), asserion (ii) of Theorem 4.2 fails o hold under he condiion (i) of Corollary Alernaive converse problem for RBSDEs wih he obsacle process {S } 0 T being given Remark 4.2 shows ha if he obsacle process {S } 0 T is previously given, i is impossible in general o compare he generaor g on he whole space Ø [0, T] R R d. In his secion we shall show ha we can sill have he local converse comparison heorem for RBSDEs on an upper semi-space Ω [0, T] [C, + ) R d, specified by he uniform upper bound C of he obsacle, which is acually he whole space if he generaor does no depend on he firs unknown variable y (see Theorem 5.2 below). Assume ha he daa (ξ, g, S) saisfies he Sandard Assumpion for RBSDEs. Bu o emphasize he dependence on he obsacle process {S } 0 T, denoe by g,t [ξ] and εr,s g,t [ξ F ] he iniial value Y T,g,ξ,S 0 and he value Y T,g,ξ,S a ime of he soluion of RBSDE (2.2) wih daa (ξ, g, S), respecively. Proposiion 5.. Suppose ha g saisfies (A) and (A2), and he obsacle process {S } 0 T saisfies (A5). For he sopping ime τ T, and he erminal value ξ L 2 (Ω, F τ, P) such ha ξ S τ a.s., hen we have g,τ [ξ] = εr,s g,t [ξ] where g(, y, z) := g(, y, z) [0,τ] () and S := S τ a.s. for (, y, z) [0, T] R R d. 0

11 Proof. Consider he soluion (Y τ,g,ξ,s, Z τ,g,ξ,s, K τ,g,ξ,s ) of RBSDE (2.2) wih daa (ξ, g, S) on he inerval [0, τ], and (Y T,g,ξ,S, Z T,g,ξ,S, K T,g,ξ,S ) of RBSDE (2.2) wih daa (ξ, g, S) on he inerval [0, T]. Obviously, g,τ[ξ] = Y τ,g,ξ,s 0 and T,g,ξ,S g,t [ξ] = Y 0. For simpliciy, denoe (Y τ,g,ξ,s, Z τ,g,ξ,s, K τ,g,ξ,s ) and (Y T,g,ξ,S, Z T,g,ξ,S, K T,g,ξ,S ) by (Y, Z, K) and (Y, Z, K), respecively. From Lemma 2.5, we have Y () = ξ, Z() = 0, K() = K(τ) on he inerval (τ, T]; Y () = Y (), Z() = Z(), K() = K() on he inerval [0, τ]. Therefore, g,τ[ξ] = g,t [ξ]. Theorem 5.. Assume ha wo funcions g and saisfy assumpions (A), (A3) and (A4), and he obsacle process {S } 0 T saisfies (A5). Moreover, assume ha here is a consan C such ha sup S C a.s.. 0 T If for each sopping ime τ T, we have hen we have g,τ[ξ],τ[ξ] for any ξ L 2 (Ω, F τ, P) such ha ξ S τ a.s., g (, y, z) (, y, z) a.s. for any (, y, z) [0, T] [C, + ) R d. Proof. For each δ > 0 and (y, z) (C, + ) R d, define he following sopping ime: τ δ = τ δ (y, z) = inf{ 0 : g (, y, z) (, y, z) δ} T. If he resul does no hold, hen here exiss δ > 0 and (y, z) (C, + ) R d such ha P({τ δ (y, z) < T }) > 0. For such a riple (δ, y, z), consider he following SDEs defined on he inerval [τ δ, T]: { dy () = g (, Y (), z)d zdb, Y (τ δ ) = y and { dy 2 () = (, Y 2 (), z)d zdb, Y 2 (τ δ ) = y. For i =, 2, he above equaions admi a unique soluion Y i S 2 (τ δ, T;R).

12 Now we define he following sopping imes: τ δ = inf{ τ δ : Y S } T, τ 2 δ = inf{ τ δ : Y 2 S } T, τ δ = inf{ τ δ : g (, Y (), z) (, Y 2 (), z) δ 2 } T. Noe ha τ δ = τ2 δ = τ δ = T, if τ δ = T. Obviously, {τ δ < τ δ } = {τ δ < τ 2 δ } = {τ δ < τ δ } = {τ δ < T }. Define τ 3 δ = τ δ τ2 δ τ δ. Hence P({τ δ < τδ 3 }) > 0. Moreover, we have Y > S and Y 2 > S on he inerval [τ δ, τδ 3). Then he soluion (Y i (), z, 0) is he soluion of RSBDE (2.2) wih daa (Y i (τδ 3), g i, S) on he inerval [τ δ, τδ 3 ] for i =, 2. We firs ge he following hree lemmas. Lemma 5.. g i,τ δ [y] = ε gi,τ δ [y] = y for i =, 2. Proof. Consider he following BSDE defined on he inerval [0, τ δ ]: { dy () = g (, y (), z ())d z ()db, y (τ δ ) = y. From assumpion (A3), we see ha y () = y and z () = 0 on he inerval [0, τ δ ]. Obviously, he riple (y, z, 0) is he soluion of RSBDE (2.2) wih daa (y, g, S) on he inerval [0, τ δ ]. Similarly, we have,τ δ [y] = ε g2,τ δ [y] = y. The proof is complee. Lemma 5.2. The sric inequaliy Y (τ 3 δ ) > Y 2 (τ 3 δ ) holds on {τ δ < τ 3 δ }. Proof. From he definiions of τ δ and Y i, we have Y (τ 3 δ ) Y 2 (τ 3 δ ) = τ3 δ on {τ δ < τδ 3 }. The proof is complee. Lemma 5.3. [Y (τ 3,τδ 3 δ )] = ε,τδ 3[Y (τδ 3)]. Proof. Consider he following BSDE: τ δ [ (s, Y 2 (s), z) g (s, Y (s), z)]ds δ 2 (τ3 δ τ δ) > 0, { d Ỹ 2 () = (, Ỹ 2 (), Z 2 ())d Z 2 ()db, [0, τ 3 δ ]; Ỹ 2 (τ 3 δ ) = Y (τ 3 δ ). 2

13 From he definiion of τ 3 δ and Lemma 5.2, we ge On he oher hand, we have Y (τδ 3 ) Y 2 (τδ 3 ) and P({Y (τδ 3 ) > Y 2 (τδ 3 )}) > 0. [Y 2 (τ 3,τδ 3 δ )] = ε,τδ 3[Y 2 (τδ 3 )] and εr,s [Y 2 (τ 3,τδ 3 δ ) F ] = ε g2,τδ 3[Y 2 (τδ 3 ) F ] on [0, τδ 3 ]. From Lemma 2., we ge Ỹ 2 () ε g2,τδ 3[Y 2 (τδ 3) F ] S() on [0, τδ 3]. Therefore (Ỹ 2, Z 2, 0) is he soluion of RBSDE (2.2) wih daa (Y (τδ 3),, S) on he inerval [0, τδ 3 ]. The proof is complee. Le us reurn o he proof of Theorem 5.. Thanks o Lemma 5., we have y = g,τ δ [y] = g,τ δ [ g [Y (τ 3,τδ ]] = g [Y (τ 3,τδ 3 δ )], and y =,τ δ [y] =,τ δ [ [Y 2 (τ 3,τδ 3 δ ) F τδ ]] = [Y 2 (τ 3,τδ 3 δ )]. On he oher hand, from he definiion of τ 3 δ and Lemma 5.3, i follows ha [Y 2 (τ 3,τδ 3 δ )] = ε,τδ 3[Y 2 (τδ 3 )] and [Y (τ 3,τδ 3 δ )] = ε g2,τδ 3[Y (τδ 3 )], respecively. Furhermore, from Lemma 2.2 we ge [Y i (τ 3,τδ 3 δ )] = ε,τδ 3[Y i (τδ 3 )] = ε,t[y i (τδ 3 )], i =, 2. Here, (, y, z) := (, y, z) [0,τ 3 δ ]() for æ [0, T] and any (y, z) R R d. From he definiion of τδ 3 and Lemma 5.2, i follows ha Therefore, in view of Lemma 2., we have Concluding he above, we ge Y (τδ 3 ) Y 2 (τδ 3 ) and P({Y (τδ 3 ) > Y 2 (τδ 3 )}) > 0. ε g2,t[y 2 (τδ 3 )] < ε,t[y (τδ 3 )]. (5.) y = [Y 2 (τ 3,τδ 3 δ )] = ε,τδ 3[Y 2 (τδ 3)] = ε,t[y 2 (τδ 3)] < ε,t[y (τδ 3)] = ε,τδ 3[Y (τδ 3)] =,τ 3 δ [Y (τδ 3 )] εr,s g [Y (τ 3,τδ 3 δ )] = y. This is a conradicion. The proof is complee. 3

14 Remark 5.. Consider he example given in Remark 4.2. Furhermore, assume ha µ 2 > 0. Immediaely, we have he following hree facs: (i) g (, y, ) = (, y, ) when y c 2 ; (ii) g (, y, z) < (, y, z) when c y < c 2 and z 0; and (iii) g (, y, z) > (, y, z) when y c z and z 0. On he oher hand, since ε r g,t [ξ] = εr,t [ξ] and εr g,t [ξ F ] = ε r,t [ξ F ] a.s. for (0, T] for any ξ L 2 (Ω, F T, P) saisfying ξ S T a.s., we deduce he above fac (i) from Theorem 5.. The oher wo facs (ii) and (iii) demonsrae ha he conclusion of Theorem 5. is he bes possible in he underlying example. In Theorem 5., he bound assumpion on he obsacle process appears o be very resricive. In wha follows, we show ha if he generaor of RBSDE (2.2) does no depend on he firs unknown variable y, we can ge he following global converse comparison resul wihou he bound assumpion. Theorem 5.2. Suppose ha wo fields g and saisfy assumpions (A), (A3) and (A4), and he obsacle process {S } 0 T saisfies (A5). Furhermore, assume ha g and do no depend on y. If for each sopping ime τ T, g,τ[ξ] εr,s,τ [ξ] for any ξ L2 (Ω, F τ, P) such ha ξ S τ a.s., (5.2) hen we have g (, z) (, z) a.s. for (, z) [0, T] R d. (5.3) Proof. Sep. If sup 0 T S is bounded from above, hen he desired asserion is immediae. Sep 2. For a large ineger n, define he sopping ime τ n := inf{ 0 : S n} T. Then 0 τ n T a.s.. Since S 0 is a deerminisic finie number and S is coninuous, we have τ n > 0 a.s. for any n > S 0 +. For every n > S 0 +, define g i (, z) := g i (, z) [0,τn]() and S = S τn for (, z) [0, T] R d wih i =, 2. Then for each sopping ime τ T and any ξ L 2 (Ω, F τ, P) such ha ξ S τ, if we have g,τ[ξ] εr,s,τ[ξ], (5.4) hen noing ha S n on [0, T] (in view of he definiion of τ n ), we have from Sep ha g (, z) (, z) a.s. for (, z) [0, T] R d. Tha is, g (, z) (, z) a.s. for (, z) [0, τ n ] R d. 4

15 Obviously, τ n T as n. Passing o limi, from assumpion (A4) we ge g (, z) (, z) a.s. for (, z) [0, T] R d. The proof is hen complee. Therefore we only need o prove inequaliy (5.4). Define g i (, z) = g i (, z) [0,τ] () and S = S τ for (, z) [0, T] R d wih i =, 2. I follows ha g i (, z) = g i (, z) [0,τ τn]() and S = S τ τn for (, z) [0, T] R d, i =, 2. From Proposiion 5., we have Therefore Similarly, g i,τ[ξ] = εr, S g i,t[ξ], i =, 2. On he oher hand, from he definiions of g (, z) and S, we have ε r, S g,t[ξ] = εr, S g,τ τ n [ε r, S g,t [ξ F τ τ n ]] = g,τ τ n [ε r, S g,t [ξ F τ τ n ]]. g,τ[ξ] = εr,s g,τ τ n [ε r, S g,t [ξ F τ τ n ]].,τ[ξ] = εr,s,τ τ n [ε r, S,T [ξ F τ τ n ]]. Also, hanks o he definiions of g (, z), (, z) and S, we ge ε r, S g,t [ξ F τ τ n ] = ε r, S,T [ξ F τ τ n ]. For simpliciy, we se η := ε r, S g,t [ξ F τ τ n ]. Obviously η L 2 (Ω, F τ τn, P) and η S τ τn. Then from he assumpion, i follows ha Now we end up wih he proof. g,τ[ξ] = εr,s g,τ τ n [η],τ τ n [η] =,τ [ξ]. Remark 5.2. Obviously, (5.2) and (5.3) in Theorem 5.2 are also equivalen. If assumpion (A3) is replaced wih assumpion (A2) in Theorem 5., hen we have Theorem 5.3. Assume ha wo random fields g and saisfy assumpions (A), (A2) and (A4), and he obsacle process {S } 0 T saisfies (A5). If for any wo sopping imes τ and σ such ha τ σ T, g,σ [ξ F τ],σ [ξ F τ] a.s. for ξ L 2 (Ω, F σ, P) such ha ξ S σ a.s., (5.5) hen for any coninuous process Y S 2 (0, T;R) such ha Y () S a.s. wih [0, T], we have g (, Y (), z) (, Y (), z) a.s. for (, z) [0, T] R d. (5.6) In paricular, g (, S(), z) (, S(), z) a.s. for (, z) [0, T] R d. (5.7) 5

16 Proof. In view of he coninuiy of g (, y, z) and (, y, z) in y, i is sufficien o prove (5.6) for any coninuous process Y S 2 (0, T;R) such ha Y () S + ǫ a.s. wih [0, T] for some consan ǫ > 0. We shall prove i by conradicion. Oherwise, here would exis δ > 0 and z R d such ha P({τ δ (z) < T }) > 0. Here for δ > 0 and z R d, we have defined he following sopping ime: τ δ = τ δ (z) := inf{ 0 : g (, Y (), z) (, Y (), z) δ} T. For such a pair (δ, z), analogous o he proof of Theorem 5., consider he following SDEs defined on he inerval [τ δ, T]: { dy () = g (, Y (), z)d zdb, Y (τ δ ) = Y (τ δ ) and { dy 2 () = (, Y 2 (), z)d zdb, Y 2 (τ δ ) = Y (τ δ ). The above SDEs admi unique soluions Y i S 2 (τ δ, T;R) wih i =, 2. Define he following sopping imes: τ δ = inf{ τ δ : Y S } T, and τ 2 δ = inf{ τ δ : Y 2 S } T, τ δ = inf{ τ δ : g (, Y (), z) (, Y 2 (), z) δ 2 } T. Noe ha τ δ = τ2 δ = τ δ = T, if τ δ = T. Obviously, {τ δ < τ δ } = {τ δ < τ 2 δ } = {τ δ < τ δ } = {τ δ < T }. We define τ 3 δ = τ δ τ2 δ τ δ. Hence P({τ δ < τδ 3 }) > 0. Moreover, we have Y > S and Y 2 > S on he inerval [τ δ, τδ 3). Therefore, he riple (Y i, z, 0) is he soluion of RSBDE (2.2) wih daa (Y i (τδ 3), g i, S) on he inerval [τ δ, τδ 3 ] for i =, 2. Consequenly, g [Y (τ 3,τδ ] = ε g,τδ 3[Y (τδ 3 ) F τ δ ] = Y (τ δ ) and [Y 2 (τ 3,τδ 3 δ ) F τδ ] = ε g2,τδ 3[Y 2 (τδ 3 ) F τδ ] = Y (τ δ ). Idenical o he proof of lemmas 5.2 and 5.3, we ge 6

17 Lemma 5.4. We have and Y (τ 3 δ ) > Y 2 (τ 3 δ ) on {τ δ < τ 3 δ } (5.8) [Y (τ 3,τδ ] = ε g2,τδ 3[Y (τδ 3 ) F τ δ ]. (5.9) From he definiion of τ 3 δ and (5.8), we have Y (τδ 3 ) Y 2 (τδ 3 ) a.s. and P({Y (τδ 3 }) > Y 2 (τδ 3 )) > 0. Then i follows from Lemmas 2. and (5.9) ha and [Y (τ 3,τδ ] = ε g2,τδ 3[Y (τδ 3 ) F τ δ ] ε g2,τδ 3[Y 2 (τδ 3 ) F τ δ ] = [Y 2 (τ 3,τδ ] a.s. P({ [Y (τ 3,τδ ] > [Y 2 (τ 3,τδ ]}) = P({ε g2,τδ 3[Y (τδ 3 ) F τ δ ] > ε g2,τδ 3[Y 2 (τδ 3 ) F τ δ ]}) > 0. The las relaion implies ha which conradics he assumpion ha P({ [Y (τ 3,τδ ] > Y (τ δ )}) > 0 [Y (τ 3,τδ ] g [Y (τ 3,τδ ] = Y (τ δ ) a.s.. The proof is complee. The following gives an immediae consequence of Theorem 5.3. Corollary 5.. Suppose ha wo generaors g, saisfy assumpions (A), (A2) and (A4), and he obsacle process {S } 0 T saisfies (A5). Furhermore, assume ha g and do no depend on he firs unknown variable y. If for each pair of sopping imes τ and σ such ha τ σ T, we have g,σ[ξ F τ ],σ[ξ F τ ] a.s. for any ξ L 2 (Ω, F σ, P) such ha ξ S σ a.s., hen we have g (, z) (, z) a.s. for any (, z) [0, T] R d. Acknowledgemens. Boh auhors would like o hank Professor Shige Peng for his helpful commens, and Guangyan Jia and Zhiyong Yu for deecing an error in he original proof of Theorem 3.. They are also very graeful o he Associae Edior and he anonymous referee for heir helpful suggesions and criicisms. 7

18 References [] P. Briand, F. Coque, Y. Hu, J. Mémin, and S. Peng, A converse comparison heorem for BSDEs and relaed properies of g-expecaion, Elecron. Comm. Probab., 5 (2000), pp [2] F. Coque, Y. Hu, J. Mémin, and S. Peng, A general converse comparison heorem for BSDEs, C. R. Acad. Sci. Paris, Série I, 333 (200), pp [3] Z. Chen, A propery of backward sochasic differenial equaions, C. R. Acad. Sci. Paris, Série I, 326(4) (998), pp [4] F. Coque, Y. Hu, J. Mémin, and S. Peng, Filraion consisen nonlinear expecaions and relaed g-expecaion, Probab. Theory Relaed Fields, 23 (2002), pp [5] N. El Karoui, E. Pardoux, and M. C. Quenez, Refleced Backward SDEs and American opions, in: Numerical mehods in finance, Publ. Newon Ins. Cambridge Univ. Press, Cambridge (997), pp [6] N. El Karoui, S. Peng, and M. C. Quenez, Backward sochasic differenial equaions in finance, Mah. Finance 7 () (997), pp [7] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez, Refleced soluions of backward SDE s, and relaed obsacle problems for PDE s, Annals. Probabiliy, 25 (2) (997), pp [8] S. Hamadène, J. P. Lepelier, and A. Maoussi, Double barrier backward SDEs wih coninuous coefficien. In: El Karoui N. and Mazliak L. (Eds.), Backward Sochasic Differenial Equaions. Piman Research Noes in Mahemaics Series, 364 (997), pp [9] Jiang L., Some resuls on he uniqueness of generaors of backward sochasic differenial equaions, C. R. Acad. Sci. Paris, Série I 338 (2004) [0] Pardoux E., Peng S., Adaped soluion of a backward sochasic differenial equaion, Sysems Conrol Le. 4 ( - 2) (990) [] Peng S., Backward Sochasic Differenial Equaions, Nonlinear Expecaions, Nonlinear Evaluaions and Risk Measures, Lecure Noes in Chinese Summer School in Mahemaics, Weihai, [2] Peng S., Backward SDE and relaed g-expecaion, in: Backward Sochasic Differenial Equaions, Paris, , Piman Res. Noes Mah. Ser., Vol. 364, Longman, Harlow, 997, pp