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Reflected solutions of backward stochastic differential equations with continuous coefficient
7,>-~-S.--~7,
STATISTICS& PROBABILITY LETTERS
~",!jiII ELSEVIER
Statistics & Probability Letters 34 (1997) 347-354
Reflected solutions of backward stochastic differential equations with continuous coefficient Anis Matoussi Uni~ersit~ du Maine, Laboratoire Statistiques et Processus, BP 535, 72017 Le Mans cedex, France
Received May 1996~revised October 1996
Abstract
We prove the existence of a reflected solution of one-dimensional backward stochastic differential equations with continuous and linear growth coefficient and squarcd integrable terminal condition. Kevwords: Backwards SDEs
1. Introduction
Backward stochastic differential equations (BSDEs) have been first introduced by Pardoux and Peng (1990). It has been since widely acknowledged that they are applied in many problems in mathematical finance; see in particular Duffle and Epstein (1992) and El Karoui et al. (1995). They also are used for problems in stochastic control and differential games; see El Karoui et al. (1995) and Hamedene and Lepeltier (1995). The existence and uniqueness of a reflected and adapted solution of one dimensional backward SDEs under lipschitz hypothesis on the coefficient were proved by El Karoui et al. (1996). Moreover, Lepeltier and San Martin (1996) established the existence of a solution of one-dimensional BSDEs with continuous and linear growth coefficient. In this paper we will be inspired by the works of E1 Karoui et al. (1996) and Lepeltier and San Martin (19961 to establish existence of reflected solutions of one-dimensional BSDEs with continuous and linear growth coefficient.
2. Formulation
Let {Bt, O<~t <<,T} be a d-dimensional standard Brownian motion defined on a probability space (fL.T,P). Let {Ftt, 0 ~
A. Matoussi I Statistics & Probability Letters 34 (1997) 347--354
348
such that (i) V(y,z) E R × Ra,(og; t) ~ f(og, t, y,z) is Ft -progressively measurable, (ii) For fixed t, o9 : f ( t , o9.... ) is continuous, (iii) f has a linear growth i.e. there exists K > 0 such that Vt, o9, y,z, If(t, og, Y,Z)l<~g(l + lyl + Izl) • An obstacle {St, O<~t<~T}, which is a continuous and Ft-progressively measurable real valued process satisfying
(iv) E(o<~t<~rsup(S~) 2) < 2 . We shall always assume that Sr ~<¢ a.s, and denote by .~ the predictable a-field. We consider H2(Ra) = {X : [0, T] × ~ --, R~,X E .~, IlXll 2 = Efo r IXslZds < 2 } . O u r main result is Theorem 1. Let ( ~ , f , S ) be a triple satisfyin 9 the above assumptions, in particular (i)-(iv). Then there exists a Ft-progressively measurable triple {(Yt, Zt, Kt ); 0 <~t <~T} solution of the reflected backward stochastic differential equation ( RBSDE):
Yt=~+
l
f(s, og,¥s,Z,)ds+Kr-Kr-
It
ZsdBs; O ~ < t ~ r
(1)
such that
(v) E fr(Y~ + IZ, I)Z)dt < 2 , (vi) Yt>~S, O<~t<~T, (vii) {Kt, O<~t<~T} is a continuous and increasin9 process, Ko = O, and or(yt -
S,)dg,
= 0.
(2)
To prove Theorem 1, we need an important result which gives an approximation of continuous functions by Lipschitz functions (see Lepeltier and San Martin (1996) to appear for the proof). Lemma 1. Let f : RP---, R be a continuous function with linear 9rowth, that is, there exists a constant K < 2 such that Vx E Rp, If(x)l ~1K and satisfies (a) Linear 9rowth: Vx E R p, Ifn(x) I ~
3. Preliminary results and Proof of Theorem 1
Consider, for fixed (t, og), the sequence fn(t, o9,y,z) associated to f by Lemma 1. Then, fn is a ~ ® R a+l measurable function as well as a Lipschitz function. Moreover, since ~ E L2(f2,Fr, P) and {St, O<~t<~T} satisfy (iv), we get from El Karoui et al. (1996) that there is a unique triple {(Y~,Z~,KT) , O<~t<~T} of
A. Matoussi I Statistics & Probability Letters 34 (1997) 3 4 7 - 3 5 4
349
Ft progressively measurable processes taking values in R,Ra and R+, respectively, and satisfying
Y[' = ~ +
E
(
f.(s,o~, y~,Z' " " )ds + K rn - Kt -
/
Z'~ dBs,
(3)
O<~t<<.T,
for(] vTI2 + IZTI2) dt < oo,
Yt>~S,,
(4)
O<~t<~r,
{K7, O<~t<~T} is
a
(5)
continuous and increasing (in t) process, K~ = 0, and
or(Y7 - S,)dK 7 = 0.
(6)
Using the comparison theorem of RBSDE's in El Karoui et al. (1996), we obtain that
Vn>~m>~K,
yn>~yrn,
dt ® dP-a.s.
(7)
The idea of the proof of Theorem 1 is to establish that the limit of the sequence (Yn,Zn, K") is a solution of the RBSDE (1) with parameters ( f , ~ , S ) . We first give uniform in n a priori estimates on the sequence (Y",Z",K") of solutions of RBSDE (3). Lemma 2. There exists a constant C depending only on K, T,E(~ 2) and E(supo
gn>>.g,E
(
sup
\O<<.t<~T
1Y712+
IZfflZds+(gD 2 <~C.
Proof. Here and in the sequel, C > 0 denotes a real constant, whose value may vary from line to line. From It6's formula applied to the semimartingale (YT) 2, it follows that
(Yt")2 +
12~'12 ds = #z + 2 ={2+2
f,(s, Y~",2~')ds + 2
(
f.(s, Yi,g~')rl ds + 2
Ys"dK~' - 2
(
SsdK~- 2
Y~"Z~dB,
(
YIZ~ dBs,
where we have used the identity foT (Yin - S t ) d K 7 = 0. Taking expectation in both sides of the above equation, and using the fact that fo Y~nZ~dBs is uniformly integrable (by Proposition 2.1 and Corollary 2.3 of El Karoui et al., 1996), we deduce
EIYII2 + E
IZ~I2 ds = E42 + 2E
f,(s,Y~,Z~)Y[ ds + 2E
SsdK i.
We have that for each ~, using the uniform linear growth property of f , and the elementary inequality: 2ab <~a2/[3 + [3b2, V[3 > O,
ElYt l2 + E f r lZ~lZds ~ c (l + E f r ly~,2 ds) + l/3E f r lZ~12ds +
O~
( sup (s:);) + \O<.s<~T
/
-
350
A. Matott~si / Statistics & Probability Letters 34 (1997) 3 4 7 - 3 5 4
Now from (3), K~- - K 7 = Yt" - ~ - . ~ r f , ( s , V , Z ~ ) d s + J ,
E((K~.-Kt)e)<~C(E(yt)2~-E~2 + I +E
•l
n
Z,. dB,.
T f ((Y,~')e +IZ;~n~2)ds).
(8)
Choosing ~ = 1/3C, we obtain
~E(Y, l 2 " )2 + .gE
f
r IZs~12ds~
( I
It then follows from Gronwall's lemma that:
E
)
1 +E
(Y,~')2ds , s u p 0 ~ < t ~< r
E(Y~)z<~C(I +fTE(y~)gds).
(9)
El Yt]2~ C, and from (9) and the last inequality that
]Z~I2ds<~C;E(K~)2<~C.
The result of Lemma 2 then follows from Eq. (3), the above estimates and the Burkholder-Davis-Gundy inequality. Now, we have from (7) and the result of Lemma 2 (respectively),
Y~ <~Yt +I,
O <~t <~T, P-a.s.
and E sup ( ! Y t l 2 ) ~ C . O<~t<~T
Hence,
Y~ /~ Y,,
O <~t <~T, P-a.s.
and from Fatou's lemma, we have E(supo<~t<~r IY,12)~
E,
IY, - Yfll2dt ~ 0
(10)
as n -~ oc.. Now, we should prove that the sequence of processes Z n converge in HZ(R). For all n >~p>~no >~K, from lt6's formula for t = 0,
glYg-
YoPl2 ~-g
J0
lEt - Z / ' 1 2 d t = 2 E
f0' fo"
± 2E
(Y~ - YtP)(f.(t, Y 2 , Z ~ ) - fp(t, YP,gtP))dt ( Y~ - Y,P)(dK7 - dK,p).
Using the fact that for all n, Y[ >>-&, 0 <.%t <. T, we obtain
EIYg - Y0Pl2 + E
IZt - Z/'I2 d t = 2 E
+2e
(Y~ - YtP)(f,(t,Y~,ZT) - fp(t, YeP,ztP))dt
Io'( V - r , P ) d X t + 2 e fo (r,p - V ) O x p,
A. Matoussi / Statistics & Probability Letters 34 (1997) 347-354
'
351
/o"( Y t - S t ) d K T + 2 E // (YtP-St)dK p 2E L' ( • - Yp)(fn(t,Y~,ZT)- fp(t,Y,P, ZP))dt.
E L ]z~-zFI 2dt<<-2E
From the identity f : ( Y :
- S t ) d K 7 = 0, we have
E L r IZ," - Z:12dt ~2E .Lr(Yp - YtP)(f.(t, Yp,ZT)- fp(t, YtP,ZtP) ) dt
~<2(E L [Yp- Yt'12dt)112(E T
.
\1i2
LT
ifn(t,Yp,Z~)- fp(t,YtP,ZtP)i2dt) ,
where we have used the H61der inequality. Now, using the uniform linear growth condition on the sequence (fn) and the fact that II(Y",Z")[I is bounded, we obtain the existence of a constant C depending only on K,T,E¢ 2 and E(suPo<.tzr(S+) 2) such that
Vn, p>~no,
IIZ"-ZPlI<~CIIY"- YPII1/2
Then from (10), (Z ") is a Cauchy sequence in H2(Ra), and there exists a Ft progressively measurable process Z such that Z n ---+Z in HZ(Ra), as n ---+
E
([Y:-Y,PlZ+lzT-zflZ)dt--,o
(11)
as n, p---, ~o. Now, we prove that E(0~t~
T12: - / : 1 2 d s
= 2f
T( Y ~ - y f ) ( f , ( s , Ysn,Z~)-fp(s, YsP,ZsP))ds
+2 f r(r2- rf(dX2 -dKf)-2 f r (rs - r:)(Z~ -Zf)dB,. From the above proof, we have Vn>~p, ft r (Y~~ - YiP)(dK"" -dKp)~<0. Then,
it._ r:12.
yf)(f.(s,Y:,Zp)- fp(s, Yp,Zf))ds- 2
from which we deduce E(o~
.<
.
),j2
( Y : - Yfl)(Z~- Z.:)dBs,
A. MatoussiI Statistics & ProbabilityLetters 34 (1997) 347-354
352
Using again the uniform linear growth condition on the sequence(fn) and the fact that we deduce T
II(Y.,Z~)II
is bounded,
1,:2
(=/o ,s.~s,~,=,,_~,,s,,.s,,Zs,,,~,,s) ~.~.
(12)
Afterwards, from the Davis-Burkholder-Gundy inequality, we obtain
2E(ko<~t<~Tsup ,It f T ( Y n - Yf)(Zn-ZP)dBs ) <~2CE(fo T I t ; <~2CE
(
sup IYt -
Y,Pl
\ O<~t<~T
~
2
(
sup
IV-
O<~t<~T
Y, Pl 2
(z'
)
]Z; - Zf] 2 ds
+ CE
/o~IZ~ -
- r2121z: - Zpl:
~)1,,'2
J)
Zfl2 ~.
Hence, from (12) and the above inequality I/2
=C.o..,..,~u,,,.~_Y,,,~)..~(=1",,.:_,.:,~ds) + 21E(\o<~t~rsupIYF- Y,PI2) + CE £TIZff - ZFI2ds, and so 1,"2
= (oSUp~'','-',''~) -'~ (=/o"~-':'~s) += for Iz~'- zPI=ds). Then from (11), we have
e(\0.<,<.rSUp IV-r, Pl~) --,0
(13)
as n, p ~ oo, from which we deduce that P-almost surely, Y~ converges uniformly in t to Y and that Y is a continuous process. [] Now according to (3), we have for all
E (o<.,<~rSUp[Kt
-
n, p>~no>~K,
KP,2) <~E[Y~ - YoP[2 +E \o<~t<~r ( sup
[Ytn -
Y,P]2I
I"T
+CE Jo If.(s,Y=",Z~)- fp(s, YsP,ZsP)[2ds +E
sup
(Z~ - ZsP)dBs
.
(14)
O<~t<~T
We need to show that the sequence of processes ( f , ( . , Y.~,Z~))~ converges to f ( . , Y,Z.) in deduced from the following facts: • Y~/~ to Y in H2(R) and dt ® dP a.-s. • Since Z ~ ---* Z in H2(R) then there exists a process Z n ~ Z, dt ® dP-a.s.
Z'
in
H2(Ra)
H2(R).This
is
and a subsequence such that Vn, ]Znl ~
A. Matoussi I Statistics & Probability Letters 34 (1997) 347-354
353
Therefore, from (a) and (d) of Lemma 1 we get for almost surely all ~o,
fn(t, Yt, Zt) --~ f(t, Yt,Zt),
dt-a.s.
as n ~ ~ and If,(/,
YT,ZT)l ~
Thus, it follows by the dominated convergence theorem that
E
If.(s,Y~,Z~')-f(t,Y,,Z,)lZds~O
(15)
as n---, vc. Now from Burkholder-Davis-Gundy inequality and (13)-(15) we obtain
E(ko~
sup [K7 - K ' 1 2 ~
/
\ O<~t<~T
~0
(16)
as n ~ vo and then {Kt, 0 <~t <~T} is clearly an increasing (with K0 = 0) and a continuous process. Finally, taking limits in the RBSDE (3) we obtain that the triple {(Yt, Z, Kt), O<~t<~T} is a solution of the RBSDE (1). Now in order to finish the Proof of Theorem 1, it remains to check (v),(vi) and (2). From Lemma 2, we have E
([Yt"[2 +
[Ztl2)dt<~C.
Taking limits in the above inequality, we obtain (v)
E J~r(IY, I2 + IZ,12)dt<~C.
On the other hand, we have
Vn~K,
Y~>~St, Vt C [0, T].
Taking limits we have clearly (vi). From (13), (16) and the result of Saisho (1987, p. 465) we have
J0 [ (Y" - S')dK7 --'
fo
(Y, - S,) aK,
P-a.s. as n --+ ao. Using the identity
f r ( Y 7 - S,)dK 7 = 0, we obtain
f0r(yt - St) dKt = 0. The proof of Theorem 1 is complete.
Acknowledgements I would like to thank Lepeltier and Hamadene for the helpful conversations we had together and for comments they made in a previous version of this paper.
354
A. Matoussi I Statistics & Probability Letters 34 (1997) 347-354
References Duffie, D. and L. Epstein (1992), Stochastic differential utility, Econometrica 60, 353-394. El Karoui, N., C. Kapoudjian, E. Pardoux, S. Peng and M.C. Quenez (1996), Reflected solutions of BSDE's and related obstacle problems for PDE's, Prepublications 96-02, C.M.I. Universit~ de provence. El Karoui, N., S. Peng and M.C. Quenez (1995), BSDE, finance and optimization, preprint. Hamedene, S. and J.P. Lepeltier (1995), Zero-sum stochastic differential games and BSDEs, Systems Control Lett. 24, 259-263. Lepeltier, J.P. and J. San Martin (1996), Backward stochastic differential equations with continuous coefficient, to appear in: Statist. Probab. Lett. Pardoux, E. and S. Peng (1990), Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14, 55-61. Saisho, Y. (1987), SDE for multidimensional domains with reflecting boundary, Probab. Theory Related Fields 74, 455-477.
STATISTICS& PROBABILITY LETTERS
~",!jiII ELSEVIER
Statistics & Probability Letters 34 (1997) 347-354
Reflected solutions of backward stochastic differential equations with continuous coefficient Anis Matoussi Uni~ersit~ du Maine, Laboratoire Statistiques et Processus, BP 535, 72017 Le Mans cedex, France
Received May 1996~revised October 1996
Abstract
We prove the existence of a reflected solution of one-dimensional backward stochastic differential equations with continuous and linear growth coefficient and squarcd integrable terminal condition. Kevwords: Backwards SDEs
1. Introduction
Backward stochastic differential equations (BSDEs) have been first introduced by Pardoux and Peng (1990). It has been since widely acknowledged that they are applied in many problems in mathematical finance; see in particular Duffle and Epstein (1992) and El Karoui et al. (1995). They also are used for problems in stochastic control and differential games; see El Karoui et al. (1995) and Hamedene and Lepeltier (1995). The existence and uniqueness of a reflected and adapted solution of one dimensional backward SDEs under lipschitz hypothesis on the coefficient were proved by El Karoui et al. (1996). Moreover, Lepeltier and San Martin (1996) established the existence of a solution of one-dimensional BSDEs with continuous and linear growth coefficient. In this paper we will be inspired by the works of E1 Karoui et al. (1996) and Lepeltier and San Martin (19961 to establish existence of reflected solutions of one-dimensional BSDEs with continuous and linear growth coefficient.
2. Formulation
Let {Bt, O<~t <<,T} be a d-dimensional standard Brownian motion defined on a probability space (fL.T,P). Let {Ftt, 0 ~
A. Matoussi I Statistics & Probability Letters 34 (1997) 347--354
348
such that (i) V(y,z) E R × Ra,(og; t) ~ f(og, t, y,z) is Ft -progressively measurable, (ii) For fixed t, o9 : f ( t , o9.... ) is continuous, (iii) f has a linear growth i.e. there exists K > 0 such that Vt, o9, y,z, If(t, og, Y,Z)l<~g(l + lyl + Izl) • An obstacle {St, O<~t<~T}, which is a continuous and Ft-progressively measurable real valued process satisfying
(iv) E(o<~t<~rsup(S~) 2) < 2 . We shall always assume that Sr ~<¢ a.s, and denote by .~ the predictable a-field. We consider H2(Ra) = {X : [0, T] × ~ --, R~,X E .~, IlXll 2 = Efo r IXslZds < 2 } . O u r main result is Theorem 1. Let ( ~ , f , S ) be a triple satisfyin 9 the above assumptions, in particular (i)-(iv). Then there exists a Ft-progressively measurable triple {(Yt, Zt, Kt ); 0 <~t <~T} solution of the reflected backward stochastic differential equation ( RBSDE):
Yt=~+
l
f(s, og,¥s,Z,)ds+Kr-Kr-
It
ZsdBs; O ~ < t ~ r
(1)
such that
(v) E fr(Y~ + IZ, I)Z)dt < 2 , (vi) Yt>~S, O<~t<~T, (vii) {Kt, O<~t<~T} is a continuous and increasin9 process, Ko = O, and or(yt -
S,)dg,
= 0.
(2)
To prove Theorem 1, we need an important result which gives an approximation of continuous functions by Lipschitz functions (see Lepeltier and San Martin (1996) to appear for the proof). Lemma 1. Let f : RP---, R be a continuous function with linear 9rowth, that is, there exists a constant K < 2 such that Vx E Rp, If(x)l ~
3. Preliminary results and Proof of Theorem 1
Consider, for fixed (t, og), the sequence fn(t, o9,y,z) associated to f by Lemma 1. Then, fn is a ~ ® R a+l measurable function as well as a Lipschitz function. Moreover, since ~ E L2(f2,Fr, P) and {St, O<~t<~T} satisfy (iv), we get from El Karoui et al. (1996) that there is a unique triple {(Y~,Z~,KT) , O<~t<~T} of
A. Matoussi I Statistics & Probability Letters 34 (1997) 3 4 7 - 3 5 4
349
Ft progressively measurable processes taking values in R,Ra and R+, respectively, and satisfying
Y[' = ~ +
E
(
f.(s,o~, y~,Z' " " )ds + K rn - Kt -
/
Z'~ dBs,
(3)
O<~t<<.T,
for(] vTI2 + IZTI2) dt < oo,
Yt>~S,,
(4)
O<~t<~r,
{K7, O<~t<~T} is
a
(5)
continuous and increasing (in t) process, K~ = 0, and
or(Y7 - S,)dK 7 = 0.
(6)
Using the comparison theorem of RBSDE's in El Karoui et al. (1996), we obtain that
Vn>~m>~K,
yn>~yrn,
dt ® dP-a.s.
(7)
The idea of the proof of Theorem 1 is to establish that the limit of the sequence (Yn,Zn, K") is a solution of the RBSDE (1) with parameters ( f , ~ , S ) . We first give uniform in n a priori estimates on the sequence (Y",Z",K") of solutions of RBSDE (3). Lemma 2. There exists a constant C depending only on K, T,E(~ 2) and E(supo
gn>>.g,E
(
sup
\O<<.t<~T
1Y712+
IZfflZds+(gD 2 <~C.
Proof. Here and in the sequel, C > 0 denotes a real constant, whose value may vary from line to line. From It6's formula applied to the semimartingale (YT) 2, it follows that
(Yt")2 +
12~'12 ds = #z + 2 ={2+2
f,(s, Y~",2~')ds + 2
(
f.(s, Yi,g~')rl ds + 2
Ys"dK~' - 2
(
SsdK~- 2
Y~"Z~dB,
(
YIZ~ dBs,
where we have used the identity foT (Yin - S t ) d K 7 = 0. Taking expectation in both sides of the above equation, and using the fact that fo Y~nZ~dBs is uniformly integrable (by Proposition 2.1 and Corollary 2.3 of El Karoui et al., 1996), we deduce
EIYII2 + E
IZ~I2 ds = E42 + 2E
f,(s,Y~,Z~)Y[ ds + 2E
SsdK i.
We have that for each ~, using the uniform linear growth property of f , and the elementary inequality: 2ab <~a2/[3 + [3b2, V[3 > O,
ElYt l2 + E f r lZ~lZds ~ c (l + E f r ly~,2 ds) + l/3E f r lZ~12ds +
O~
( sup (s:);) + \O<.s<~T
/
-
350
A. Matott~si / Statistics & Probability Letters 34 (1997) 3 4 7 - 3 5 4
Now from (3), K~- - K 7 = Yt" - ~ - . ~ r f , ( s , V , Z ~ ) d s + J ,
E((K~.-Kt)e)<~C(E(yt)2~-E~2 + I +E
•l
n
Z,. dB,.
T f ((Y,~')e +IZ;~n~2)ds).
(8)
Choosing ~ = 1/3C, we obtain
~E(Y, l 2 " )2 + .gE
f
r IZs~12ds~
( I
It then follows from Gronwall's lemma that:
E
)
1 +E
(Y,~')2ds , s u p 0 ~ < t ~< r
E(Y~)z<~C(I +fTE(y~)gds).
(9)
El Yt]2~ C, and from (9) and the last inequality that
]Z~I2ds<~C;E(K~)2<~C.
The result of Lemma 2 then follows from Eq. (3), the above estimates and the Burkholder-Davis-Gundy inequality. Now, we have from (7) and the result of Lemma 2 (respectively),
Y~ <~Yt +I,
O <~t <~T, P-a.s.
and E sup ( ! Y t l 2 ) ~ C . O<~t<~T
Hence,
Y~ /~ Y,,
O <~t <~T, P-a.s.
and from Fatou's lemma, we have E(supo<~t<~r IY,12)~
E,
IY, - Yfll2dt ~ 0
(10)
as n -~ oc.. Now, we should prove that the sequence of processes Z n converge in HZ(R). For all n >~p>~no >~K, from lt6's formula for t = 0,
glYg-
YoPl2 ~-g
J0
lEt - Z / ' 1 2 d t = 2 E
f0' fo"
± 2E
(Y~ - YtP)(f.(t, Y 2 , Z ~ ) - fp(t, YP,gtP))dt ( Y~ - Y,P)(dK7 - dK,p).
Using the fact that for all n, Y[ >>-&, 0 <.%t <. T, we obtain
EIYg - Y0Pl2 + E
IZt - Z/'I2 d t = 2 E
+2e
(Y~ - YtP)(f,(t,Y~,ZT) - fp(t, YeP,ztP))dt
Io'( V - r , P ) d X t + 2 e fo (r,p - V ) O x p,
A. Matoussi / Statistics & Probability Letters 34 (1997) 347-354
'
351
/o"( Y t - S t ) d K T + 2 E // (YtP-St)dK p 2E L' ( • - Yp)(fn(t,Y~,ZT)- fp(t,Y,P, ZP))dt.
E L ]z~-zFI 2dt<<-2E
From the identity f : ( Y :
- S t ) d K 7 = 0, we have
E L r IZ," - Z:12dt ~2E .Lr(Yp - YtP)(f.(t, Yp,ZT)- fp(t, YtP,ZtP) ) dt
~<2(E L [Yp- Yt'12dt)112(E T
.
\1i2
LT
ifn(t,Yp,Z~)- fp(t,YtP,ZtP)i2dt) ,
where we have used the H61der inequality. Now, using the uniform linear growth condition on the sequence (fn) and the fact that II(Y",Z")[I is bounded, we obtain the existence of a constant C depending only on K,T,E¢ 2 and E(suPo<.tzr(S+) 2) such that
Vn, p>~no,
IIZ"-ZPlI<~CIIY"- YPII1/2
Then from (10), (Z ") is a Cauchy sequence in H2(Ra), and there exists a Ft progressively measurable process Z such that Z n ---+Z in HZ(Ra), as n ---+
E
([Y:-Y,PlZ+lzT-zflZ)dt--,o
(11)
as n, p---, ~o. Now, we prove that E(0~t~
T12: - / : 1 2 d s
= 2f
T( Y ~ - y f ) ( f , ( s , Ysn,Z~)-fp(s, YsP,ZsP))ds
+2 f r(r2- rf(dX2 -dKf)-2 f r (rs - r:)(Z~ -Zf)dB,. From the above proof, we have Vn>~p, ft r (Y~~ - YiP)(dK"" -dKp)~<0. Then,
it._ r:12.
yf)(f.(s,Y:,Zp)- fp(s, Yp,Zf))ds- 2
from which we deduce E(o~
.<
.
),j2
( Y : - Yfl)(Z~- Z.:)dBs,
A. MatoussiI Statistics & ProbabilityLetters 34 (1997) 347-354
352
Using again the uniform linear growth condition on the sequence(fn) and the fact that we deduce T
II(Y.,Z~)II
is bounded,
1,:2
(=/o ,s.~s,~,=,,_~,,s,,.s,,Zs,,,~,,s) ~.~.
(12)
Afterwards, from the Davis-Burkholder-Gundy inequality, we obtain
2E(ko<~t<~Tsup ,It f T ( Y n - Yf)(Zn-ZP)dBs ) <~2CE(fo T I t ; <~2CE
(
sup IYt -
Y,Pl
\ O<~t<~T
~
2
(
sup
IV-
O<~t<~T
Y, Pl 2
(z'
)
]Z; - Zf] 2 ds
+ CE
/o~IZ~ -
- r2121z: - Zpl:
~)1,,'2
J)
Zfl2 ~.
Hence, from (12) and the above inequality I/2
=C.o..,..,~u,,,.~_Y,,,~)..~(=1",,.:_,.:,~ds) + 21E(\o<~t~rsupIYF- Y,PI2) + CE £TIZff - ZFI2ds, and so 1,"2
= (oSUp~'','-',''~) -'~ (=/o"~-':'~s) += for Iz~'- zPI=ds). Then from (11), we have
e(\0.<,<.rSUp IV-r, Pl~) --,0
(13)
as n, p ~ oo, from which we deduce that P-almost surely, Y~ converges uniformly in t to Y and that Y is a continuous process. [] Now according to (3), we have for all
E (o<.,<~rSUp[Kt
-
n, p>~no>~K,
KP,2) <~E[Y~ - YoP[2 +E \o<~t<~r ( sup
[Ytn -
Y,P]2I
I"T
+CE Jo If.(s,Y=",Z~)- fp(s, YsP,ZsP)[2ds +E
sup
(Z~ - ZsP)dBs
.
(14)
O<~t<~T
We need to show that the sequence of processes ( f , ( . , Y.~,Z~))~ converges to f ( . , Y,Z.) in deduced from the following facts: • Y~/~ to Y in H2(R) and dt ® dP a.-s. • Since Z ~ ---* Z in H2(R) then there exists a process Z n ~ Z, dt ® dP-a.s.
Z'
in
H2(Ra)
H2(R).This
is
and a subsequence such that Vn, ]Znl ~
A. Matoussi I Statistics & Probability Letters 34 (1997) 347-354
353
Therefore, from (a) and (d) of Lemma 1 we get for almost surely all ~o,
fn(t, Yt, Zt) --~ f(t, Yt,Zt),
dt-a.s.
as n ~ ~ and If,(/,
YT,ZT)l ~
Thus, it follows by the dominated convergence theorem that
E
If.(s,Y~,Z~')-f(t,Y,,Z,)lZds~O
(15)
as n---, vc. Now from Burkholder-Davis-Gundy inequality and (13)-(15) we obtain
E(ko~
sup [K7 - K ' 1 2 ~
/
\ O<~t<~T
~0
(16)
as n ~ vo and then {Kt, 0 <~t <~T} is clearly an increasing (with K0 = 0) and a continuous process. Finally, taking limits in the RBSDE (3) we obtain that the triple {(Yt, Z, Kt), O<~t<~T} is a solution of the RBSDE (1). Now in order to finish the Proof of Theorem 1, it remains to check (v),(vi) and (2). From Lemma 2, we have E
([Yt"[2 +
[Ztl2)dt<~C.
Taking limits in the above inequality, we obtain (v)
E J~r(IY, I2 + IZ,12)dt<~C.
On the other hand, we have
Vn~K,
Y~>~St, Vt C [0, T].
Taking limits we have clearly (vi). From (13), (16) and the result of Saisho (1987, p. 465) we have
J0 [ (Y" - S')dK7 --'
fo
(Y, - S,) aK,
P-a.s. as n --+ ao. Using the identity
f r ( Y 7 - S,)dK 7 = 0, we obtain
f0r(yt - St) dKt = 0. The proof of Theorem 1 is complete.
Acknowledgements I would like to thank Lepeltier and Hamadene for the helpful conversations we had together and for comments they made in a previous version of this paper.
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A. Matoussi I Statistics & Probability Letters 34 (1997) 347-354
References Duffie, D. and L. Epstein (1992), Stochastic differential utility, Econometrica 60, 353-394. El Karoui, N., C. Kapoudjian, E. Pardoux, S. Peng and M.C. Quenez (1996), Reflected solutions of BSDE's and related obstacle problems for PDE's, Prepublications 96-02, C.M.I. Universit~ de provence. El Karoui, N., S. Peng and M.C. Quenez (1995), BSDE, finance and optimization, preprint. Hamedene, S. and J.P. Lepeltier (1995), Zero-sum stochastic differential games and BSDEs, Systems Control Lett. 24, 259-263. Lepeltier, J.P. and J. San Martin (1996), Backward stochastic differential equations with continuous coefficient, to appear in: Statist. Probab. Lett. Pardoux, E. and S. Peng (1990), Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14, 55-61. Saisho, Y. (1987), SDE for multidimensional domains with reflecting boundary, Probab. Theory Related Fields 74, 455-477.