A DECOMPOSITION THEOREM OF g-MARTINGALES

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(1)SUT Journal of Mathematics Vol. 34, No. 2 (1998), 197{208. A DECOMPOSITION THEOREM OF g-MARTINGALES . Zengjing Chen and Shige Peng (Received May 11, 1998). Abstract. In this paper, we rst introduce the notions of nonlinear mathematical expectations and nonlinear martingales via backward stochastic dierential equations(BSDEs) introduced by Due & Epstein and Skiadas. And then, we prove a general nonlinear decomposition theorem. Our decomposition theorem generalizes Doob-Meyer decomposition theorem. AMS 1991 Mathematics Subject Classication . 60H10, 60H20, 60H99. Key words and phrases. backward stochastic dierentional equations, g{expectation, decomposition theorem of g{martingales.. 1. Introduction The famous Doob-Meyer decomposition theorem is an important theorem in stochastic calculus. For many years, various versions of decomposition theorems and their applications have been discussed by many researchers, such as optional decomposition theorem ( El.Karoui and Quneez(1995), D. Kramkov (1994)]) F{S decomposition theorem ( F ollmer and Schweizer(1991) Jacka's martingale representation theorem (Jacka (1992)) and Ansel and Stricker 's theorem ( Ansel and Stricker(1992)) etc.. In this paper, we rst introduce the notions of general expectations (in short g-expectations), (which usually are non-additive) and the corresponding g-martingales via a class of backward stochastic dierential equations (BSDEs) introduced by Due and Epstien and Skiadas, and then, we study a decomposition theorem for g-supermartingales. This result extends Doob-Meyer Theorem. Since g-supermartingale discussed in this paper usually is non-linear, thus the classical method is not valid. The method used in this paper is dierent from the classical method. This paper is organized as follows: In Section 2, we present the notions of g-expectations, conditional g-expectations and g-martingales via a class of BSDEs introduced by Due & Epstein and Skiadas. In Section 3, we study a decomposition theorem of g-supermartingales. This work was partially supported by the National Natural Science Foundation of China (79790130). . 197.

(2) 198. Z. CHEN AND S. PENG. 2. g-expectation and g-martingale introduced via BSDEs. In this section, we rst present a BSDE rst introduced by Due and Epstein 4], and then, we introduce the notions of g-expectations and gmartingales. Let ( F P ) be probability space endowed with the ltration fFt gt0 satisfying the \usual hypotheses". We suppose that F0 is trivial set and S F = F1 = ( Ft ): All processes mentioned in this paper are supposed t0 to be fFt g-adapted. For any given t 2 0 1], let us denote by L1 ( Ft P ) the set of all R-valued, Ft -measurable random variables such that E jj < +1: Let T > 0 be xed time horizon, we denote by L1 (0 T F P ) the set of all R-valued, fFt g-adapted processes ft g such that. E. T. Z. 0. js jds < +1:. We identify two processes 1 and 2 in L1 (0 T F P ), if. E. T. Z. 0. j1s ; 2s jds = 0:. Due and Epstein 4] introduced the following BSDE: . yt = E ( +. T. Z. t. . gs (ys)ds)jFt 0 t T. (2.1). Here 2 L1 ( F P ) is given and g : 0 T ] R ! R is B(0 T ]) F B(R)jB(R) measurable function satisfying the following conditions: 8 (i) g is uniformly Lipschitz with Lipschitz constant , i.e. there9 > > > > > > > > i < = exists a constant > 0, such that 8y 2 R (i = 1 2) > > jgt (y1 ) ; gt (y2 )j jy1 ; y2 j 8t 2 0 T ] a.s. > > > > > > :. 1 (ii) For any y 2 R fgt (y)g 2 L (0 T F P ): (H.1) The following existence and uniqueness theorem is a special case of 2]: Lemma 2.1. Assume (H.1) holds on g, if 2 L1 ( F P ), then (1) BSDE (2.1) has a unique RCLL adapted solution fyt g in L1 (0 T F P ). (2) Particularly, if we choose gt (y) := at y + Ht then the solution (yt ) of linear BSDE(2.1) can be given by. yt = E exp(. T. Z. t. as ds) +. Z. t. T. Z. Hs exp(. t. s. ar dr)ds=Ft ] 0 t T.

(3) A DECOMPOSITION THEOREM. 199. Where fat gt0 is a bounded process and fHt g 2 L1 (0 T F P ): The following Lemma is called comparison theorem which plays an important role in our main results. Lemma 2.2. Under the assumption of Lemma 2.1, let (yt ) be the solution of BSDE (2.1) and (yt ) be the solution of the following BSDE:. yt = E ( +. Z. T. gs ds)jFt ] 0 t T. t. (2:2). where 2 L1 ( F P ) and g 2 L1 (0 T F P ): (1) If , gt (yt ) g t a.s., 8t 2 0 T ) then yt yt 8t 2 0 T ): (2) (Strict comparison theorem) If > (i.e. a.s. and 6= ), then yt > yt t 2 0 T ]: Proof. Set. ybt := yt ; yt b := ; Ht := gt (yt ) ; gt ( gs (ys );gs (ys ) if y 6= y s s ys ;ys as := 0 otherwise. Obviously, for all t 2 0 T ] jat j for the reason that g satises uniform Lipschitz condition. With the above notations, (ybt ) can be viewed as the solution of the following linear BSDE: . ybt = E + b. T. Z. t. (as ybs + Hs )dsjFt. . (2.3). Solving linear BSDE (2.3), by Lemma 2.1(2), we can obtain . ybt = E (bexp(. Z. t. T. as ds) +. Z. t. T. Z. Hs exp(. t. s. . ar dr)ds) Ft :. (2.4). It then follows by b 0 and Ht 0 for all t 2 0 T ] that the proof of (1) is complete. Note that for all t 2 0 T ] jat j from Eq.(2.4), we can deduce that ybt e;(T ;t) E bjFt ] > 0: The proof of (2) is complete. We now introduce the generalized notions of g-expectation and g-martingale via BSDE(2.1)..

(4) 200. Z. CHEN AND S. PENG. Denition 2.3. Suppose 2 L1 ( Fg P ) and (H.1) holds on g. Let fyt g be the solution of BSDE (2.1), we call EtT ( ) denoted by g ( ) := y 0 t T EtT t. the conditional g-expectation of random variable on time interval t T ] generated by g, in short conditional g-expectation if t = 0 we call E0T ( ) thegexpectation of on time interval 0 T ]: Remark. (1) The above denition is based on the following observation: For any 2 L1 ( F P ) if we denote the conditional mathematical expectation of random variable by yt := E jFt ] then conditional mathematical expectation E jFt ] is the solution of BSDE(2.1) under g 0: Moreover, E = y0 : (2) If g is nonlinear, then ( conditional) g-expectation is nonlinear too, for this reason, we sometimes call g-expectation nonlinear mathematical expectation. If we further assume that g satises the following condition:. gt (0) = 0 8t 2 0 T ]. (H.2). then,by the above denitions, we have Lemma 2.4. Assume 2g L1( F P ), (H.1) and (H.2) hold on g, then for any r 2 0 T ] let := ErT ( ) then 2 L1 ( Fr P ) is the unique random variable such that. E0gT (1A ) = E0gr (1A ) 8A 2 Fr . (2.5). Proof. Let (yt ) be the solution of BSDE: . yt = E ( +. Z. t. T. . gs (ys )ds) Ft :. For any A 2 Fr , multiplying 1A on both sides of the above equation and then observing yt 1A on r T ]: From the assumption (H.2), we can deduce the following relation. gt (1A (!)y) = gt (y)1A (!) 8(t ! y) 2 0 T ] R: Note that 8t 2 r T ] yt 1A is Ft -adapted and yt 1A solving the following BSDE on r T ]: Z T yet = E (1A + gs (yes )ds)jFt t 2 0 T ]: t.

(5) A DECOMPOSITION THEOREM. 201. Let (yet ) be the solution of the above BSDE, immediately, by Lemma 2.1 ( the unicity of solution), we have. ys 1A = yes 8s 2 r T ]. (2.6). Set := yr , obviously 2 L1 ( Fr P ): According to the denition of E0gT (1A ), applying equality (2.6), we have. E0gT (1A ) = ye0 = E0gr (yer ) = E0gr (yr 1A) = E0gr ( 1A ): We now prove is unique. Assume that there exists another 2 L1 ( Fr P ) such that for any A 2 Fr ,. E0gr ( 1A ) = E0gr ( 1A). (2.7). but P ( 6= ) > 0: We can choose A := f 6= g obviously A 2 Fr , it then follows by Strict Comparison Theorem (Lemma 2.2(2)) that. E0gr ( 1A ) 6= E0gr ( 1A) which is in contradiction to (2.7). The proof is complete. The following counter-example is to show that equation (2.5) does not hold without the assumption of g(0) 0: Example. Suppose 2 L1 ( F P ) let g 1, we choose r = T2 > 0: Then there is no 2 L1 ( F T2 P ) satisfying(2.5). In fact, if there exists such satisfying (2.5),i.e.. E01T (1A ) = E01 T2 ( 1A ) 8A 2 F T2 : then by Denition 2.3, E01T ( 1A ) = E ( 1A +T ) and E01 T2 ( 1A ) = E ( 1A + T2 ). Thus E (1A + T ) = E ( 1A + T2 ) 8A 2 F T2 :. In particular, let A :=

(6) then from the above equality, we have T = T2 which is impossible. Remark. Similar to the classical mathematical expectation, we can call dened in (2.5) the condtional g-expectation of in the interval r T ]: However, motivated by Lemma 2.4 (ii), we can dene the more general conditional gexpectation without the assumption (H.2)..

(7) 202. Z. CHEN AND S. PENG. Using conditional g-expectation, we can naturally dene g-martingale just as in the classical case. To do this, we rst introduce the solution of BSDEs on a random variable interval. Assume that

(8) is a stopping time with value in 0 T ] in this paper, we denote the solution of the following BSDE on random variable interval 0

(9) ] : . yt = E ( + in the sense of . yt = E ( +. T. Z. t. Z. t. . . gs (ys )ds) Ft t 2 0

(10) ]. . 10 ] (s)gs (ys )ds) Ft t 2 0 T ]. g ( ) by Similar to Denition 2.5, we denote Et g ( ) := y : Et t Denition 2.5. A right-continuous adapted process fXt g is called g-martingale on 0 T ] ( resp. g-supermartingale, g-submartingale ), if for any t 2 0 T ], E jXt j < 1 and for any stopping times and

(11) , if 0

(12) < T , then g (X ) = X E ( resp X X ): A g-supermartingale fXt gt0 is said to be of class (DL), if fXt g is of class (DL). Remark. (1) For notational simplicity, we adopt the above (strong) denition of g-martingale, In fact, we can adopt the following ( weak ) denition similar to the denition of the classical martingale i.e. g (X ) = X (resp. X X ) 80 s t T: Est t s s s Chen and Peng have showed that the above denitions are equivalent under some assumptions on g (see 3]). (2) Obviously, if g 0, then a g-martingale is a classical martingale. (3) g-martingales usually are nonlinear, i.e. g-martingales usually are non-additive.. 3. Nonlinear Decomposition Theorem for g-martingales. In this section, we assume that fXt g is a right continuous g-supermartingale RT such that E 0 jXs jds < 1: thus, for each n 1, the following BSDE: ytn = E XT +. Z. T;. . gs (ysn) + n(Xs ; ysn)+ ds Ft t 2 0 T ]: (3.1) t has a unique solution (ytn ): Moreover, by Comparison Theorem (Lemma 2.2), for each t 2 0 T ] fytn g is increasing as n increases. Where X + := maxfX 0g no loss of generacity, in this section we assume gt (0) = 0: The following Lemma shows that fytn g is bounded by fXt g:.

(13) A DECOMPOSITION THEOREM. 203. Lemma 3.1. Let fXt gt0 be a right continuous g-supermartingale and fytng be the solution of BSDE (3.1), then for any n > 0, we have Moreover,. ytn Xt t 2 0 T ]:. (3.2). g (X )j: jytn j jXt j + jEtT T. (3.3). Proof. Obviously, (3.2) holds when t = T , we now prove (3.2) holds when t 2 0 T ): We argue by contradiction. If fXt g is not the case, since fXt g and ytn are right continuous, thus, there exist n 1 and > 0 such that the measure of f(! t) : ytn ; Xt g is a non-zero subset of 0 T ] we denote by the following stopping times:. := inf ft 0 ytn ; Xt g ^ T

(14) := inf ft ytn Xt g ^ T:. Since ytn ; Xt is right continuous, we have (i) yn X + onf < T g (ii) X yn : Obviously, 0

(15) T and P (

(16) > ) > 0 otherwise, if P (

(17) = ) = 1 then (i) is in contradiction to (ii). Furthermore, according to Comparison Theorem, from (ii), we can deduce that g (X ) E g (yn ): E Noting that ytn ; Xt 0 on

(18) ): Thus, from equation (3.1),. yn = E XT + = E yn +. Z. T. Z. . Z. (gs (ysn ) + n(Xs ; ysn )+ )dsjF ]. (gs (ysn ) + n(Xs ; ysn )+ )dsjF ]. = E yn + gs (ysn )dsjF ] g (yn ) = E g E (X ):.

(19) 204. Z. CHEN AND S. PENG. On the other hand, since fXt g is a g-supermartingale, thus g (X ): X E Consequently, g (yn ) yn : X E This is in contrary with (i). Hence, we obtain (3.2). From (3.2) and Lemma 2.2, we have g (X ) Xt ytn EtT T which implies (3.3). The proof is complete. Set Z t Ant := n (Xs ; ysn)+ ds 0 t T 0. (3:4). then, BSDE (3.1) can be rewritten as: ytn = E XT + AnT +. Z. T. . gs (ysn )dsjFt ; Ant 0 t T: t. (3:5). We have the following Lemma: Lemma 3.2. If fXt gt0 is a right-continuous g-supermartingale of class (DL), then fAnT gn>0 de ned in (3.4) is uniformly integrable in L1 ( F P ). Proof. see Appendix. Lemma 3.3. For the above (ytn) and AnT , we have (1) There exists a constant C , which is independent of n, such that EAnT C (2) For each t 2 0 T ] limn!1 ytn = Xt : Proof. (1) is from Lemma 3.2. Now let us prove (2) According to (3.2) and (3.4), applying the above result, we can obtain. E which implies. Z 0. T. jXs ; ysn jds = E. Z 0. T. (Xs ; ysn )ds. n C T = EA n n ! 0 as n ! 1.. lim yn = X. n!1. in L1 (0 T F P ), it then follows by the fact that (Xt ) (ytn ) are right-continuous in 0 T ) that we can prove (2). The following is adopted from 11]..

(20) A DECOMPOSITION THEOREM. 205. Lemma 3.4. Let X i ( ) be a family of RCLL adapted increasing on 0 T ] f. g. (i.e. for any t 0 T ] X i (t) X (t)) such that X (t) = b(t) A(t), here b( ) is a RCLL adapted process and A( ) is a increasing process with A(0) = 0 and EA(T ) < . then, X ( ) and A( ) are also RCLL processes. Proof. See Appendix. The following theorem is so-called nonlinear decomposition theorem. Theorem 3.5. If Xt is a right continuous g-supermartingale of class (DL), then there exists a unique RCLL increasing At , such that Xt satises the following BSDE: 2. ". ;. . . 1. . f. . g. f. . Xt = E XT + AT +. T. Z. gs (Xs )ds t jF. t. g. . ;. f. At . g. t 0 T ]: 2. Proof. For each n > 0, let (ytn ) be the solution of BSDE: T. Z. . ytn = E XT + AnT +. . gs (ysn )ds t F. t. . ;. Ant. Where Ant = n 0t (Xs ysn )+ ds: Obviously, for each n > 0, Ant is a continuous and increasing process. By Lemma 3.2 and Lemma 3.3(1) and the Dunford-Pettis compactness criterion ( Dunford & Schwartz (1963), P. 294), the set AnT n>0 is relatively compact in the weak topology of L1 ( P ). Thus, there exists AT L1 ( T P ) such that AnT weakly converges to AT in L1 ( P ). Moreover, it is easy to check that for each t 0 T ], E AnT t ] weakly converges to E AT t ] (see Problem 4.11, P.27 7]), Denote At by R. ;. f. g. f. g. F. 2. F. F. 2. jF. Z. . jF. T. . At := E XT + AT + gs (Xs )ds t Xt t 0 T ] (3.6) t We only need to check that, for each t 0 Ant also is weakly converges to At : R In fact, since E 0T ysn Xs ds 0 as n thus jF. ;. 2. . j. E. . T. Z. t. ;. j. !. gs (ysn)dsjFt ;! E. ! 1. T. Z. . gs (Xs )ds t weakly in L1 : jF. t. Applying Lemma 3.3(2), we have Ant = E XT + AnT +. Z. t. T. gs (ysn )ds t jF. . ;. ytn.

(21) 206. Z. CHEN AND S. PENG. weakly converges to . E XT + AT + That is. T. Z. gs (Xs )ds t jF. t. Xt = E XT + AT +. ;. Xt = At. T. Z. . . . gs (Xs )ds t At : t Obviously (At ) is an increasing process with A0 = 0 such that EAT < . From Lemma 3.4, (At ) is RCLL process. The proof is complete. Remark. Obviously, if g 0 let Mt := E (XT + AT ) t ] then Xt = Mt + At jF. ;. 1. . jF. which is Doob-Meyer decomposition theorem.. 4. Appendix. The proof of Lemma 3.2 is similar to the classical case: The proof of Lemma 3.2. Let c > 0 be xed, and set cn = inf t 0 Ant > c T 2nc = inf t 0 Ant > 2c T: . ^. . ^. then 0 Ancn c cn < T nc2 < T and (AnT Annc )1fcn <T g 2c : 2 Applying the classical optional stopping theorem to BSDE (3.5),. f. g f. g. yncn = E XT + AnT +. T. Z. . ;. . gs (ysn )ds cn n F. c. . ;. Ancn :. (A.1). Noting that AnT > c = cn < T applying inequality (3.2) and (3.3), we can obtain, from (A.1) f. g. f. g. . EAnT 1AnT >c] E yncn XT. ;. ;. E Xcn XT ;. . ;. ;. cn Z T c. ;. E Xcn XT + ;. !. gs (ysn )ds 1cn<T ] + cP (cn < T ). gs (ysn )ds 1cn<T ] + cP (cn < T ) n. E Xcn XT + ". T. Z. Z. cn Z T 0. !. T. + cP (cn < T ). ysn ds 1cn <T ] + cP (cn < T ). j. j. #. g (X ) )ds 1 n ( Xs + EsT T c <T ] j. j. j. j. (A.2).

(22) A DECOMPOSITION THEOREM. 207. On the other hand, from (A.2),. E ynnc 2. ;. XT. Z ;. T. n2c. gs (ysn )ds]1f nc2 <T g = E (AnT Annc )1f nc2 <T g ;. 2.

(23). E AnT ; Annc 2 c n 2 P (c < T ):. 1cn <T ]. Applying (3.2) and (3.3),. cP (cn < T ) 2E X n2c. ;. XT + . T. Z. g (X ) )ds]1 n ): ( Xs + EsT T c <T ] 2 j. 0. j. j. j. Consequently, from (A.2). + 2E X. nc 2. ;. T. Z. EAnT 1AnT >c] E Xcn ; XT + . j. 0. Z. XT + . g (X ) )ds]1 n ( Xs + EsT T c <T ]. 0. T. j. j. j. g (X ) )ds]1 n : ( Xs + EsT T f c <T g 2 (A.3) j. j. j. j. g (X ) Note that Xt is a g-supermartingale of class(DL), and Xt and EtT T belong to L1 (0 T P ). Thus, from BSDE (3.5), we have f. g. f. g. F. n. P (cn < T ) = P (AnT > c) = EAc T. = 1c E y0n XT ;. !. Similarly,. 1 E X. c. 0. 0. ;. Z ;. 0. T Z. T. g (X ) )ds] XT + ( Xs + EsT T 0 as c + (A.4) j. ". P ( n2c < T ). gs (ysn )ds] j. j. j. 1. 0 as c : Combining (A.3) with (A.4), we can now conclude that AnT n>0 is uniformly integrable. The proof of Lemma 3.4. Since the processes b( ) and A( ) have paths with left limits, so is X ( ), thus we only need to prove that X ( ) is right-continuous. !. " 1. f. . . . g. .

(24) 208. Z. CHEN AND S. PENG. Since for any t 0 T ), A(t+) A(t), thus 2. . X (t+) = b(t) A(t+) X (t): (A:5) On the other hand, for any > 0, there exists a positive integer j = j ( t) such that X (t) X j (t) + . but X j ( ) is RCLL, therefore, there exists a positive integer 0 = 0 (j t ) such that X j (t) X j (t + ) + , (0 0 ]. thus, for any (0 0 ], X (t) X j (t + ) + 2 X (t + ) + 2: in particular, X (t) X (t+) + 2 and X (t) X (t+). It then follows by (A.5) that we can obtain that X ( ) is right-continuous. ;. . 8. 2. 2. . Acknowledgments. The authors would like to thank the referee for his comments and advice.. References 1. J.P. Ansel and C. Stricker, Lois de martingale densit es et d ecomposition de F ollmer Schweizer, Ann.Inst.Henri Poincar e, 28(1992), 375-392. 2. F. Antonelli, Backward{forward stochastic dierential equations, Ann. Appl.Prob., 3(1993), 777-793. 3. Z. Chen and S. Peng,Continuous properties of g-martingales, (1996) Preprint . 4. D. Due and L. Epstein, Stochastic dierential utility, Econometrica, 60(1992), 353-394. 5. N. Dunford and J. Schwartz, Linear operations. Part I: General Theory. J. Wiley and Sons/ interscience, New York, 1963. 6. H. F ollmer and M. Schweier, Hedging of contingent claims under incomplete information, Appl. Stochastic Anal., 5(1991), 389-414. 7. Jacka, A martingale representation result and an application to incomplete nancial markets, Math. Finance, 2(1992), 239-250. 8. I. Karatzas and I. Shereve, Brownian motion and stochastic calculus, SpringVerlag, 1988. 9. N. Karoui and M. Quenez, Dynamic programming and pricing of contingent claims in an incomplete markets, SIAM Control and Optimization, 33(1995), 29-66. 10. D. Kramkov, Optional decomposition of supermartingales and hedging contingent claims incomplete security markets, to appear Prob. Theory & related Field, (1994). 11. S.Peng, BSDE and related g-expectation, Pitman research note series, 364(1997), 141-159. 12. S. Peng,Monatomic limit theorem of BSDE and its application to Doob-Meyer decomposition theorem, to appear Prob. Theory & related Field, 1998. 13. S. Peng, and Darling, A useful nonlinear version of conditional expectation, (1996), Preprint. Zengjing Chen and Shige Peng Department of Mathematics, Shandong University Jinan 250100, China.

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