Stochastic Analysis and Applications, 23: 97–110, 2005 Copyright © Taylor & Francis, Inc. ISSN 0736-2994 print/1532-9356 online DOI: 10.1081/SAP-200044444
Comparison Theorems of Backward Doubly Stochastic Differential Equations and Applications Yufeng Shi School of Mathematics and System Sciences, Shandong University, Jinan, China
Yanling Gu School of Economics, Fudan University, Shanghai, China
Kai Liu Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK
Abstract: A class of backward doubly stochastic differential equations (BDSDEs for short) are studied. We obtain a comparison theorem of these BDSDEs. As one of its applications, we derive the existence of solutions for BDSDEs with continuous coefficients. Keywords: Backward doubly stochastic differential equations; Backward stochastic integral; Comparison theorem. Mathematics Subject Classification: 60H10.
1. INTRODUCTION Since the fundamental paper on backward stochastic differential equations (BSDEs for short) was published by Pardoux and Peng [13] Received August 28, 2003, Accepted March 10, 2004 Address correspondence to Yufeng Shi, School of Mathematics and System Sciences, Shandong University, Jinan 250100, China; E-mail:
[email protected]
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in 1990, BSDEs have drawn the attentions of more and more researchers. For instance, the theory of BSDEs has been found to be a powerful tool for mathematical economics (cf. [1, 3]), and, particularly, for mathematical finance (cf. [6] and its references). For instance, the theory of BSDEs has been found to be a powerful tool for the theory of contingent claims valuation in both complete and incomplete markets (cf. [7] and [16]). Another main reason is due to their enormous range of applications in such diverse fields as partial differential equations (cf. [4, 14, 17, 18]), variational inequalities and obstacle problems (cf. [5]), stochastic partial differential equations (cf. [15]), stochastic control (cf. [19]), stochastic differential games (cf. [8]), nonlinear mathematical expectations (cf. [2, 20, 21]), and so on. A class of backward doubly stochastic differential equations (BDSDEs for short) were introduced by Pardoux and Peng [15] in 1994, with two different directions of stochastic integrals, i.e., the equations involve both a standard (forward) stochastic integral dWt and a backward stochastic integral dBt . They have proved the existence and uniqueness of solutions for BDSDEs under uniformly Lipschitz conditions. However, they have not showed a comparison theorem for BDSDEs. As we know, the comparison theorem is a very useful result in the theory of BDSDEs, for example, it can be used to study viscosity solutions of stochastic partial differential equations. In this paper, we shall prove the comparison theorem of BDSDEs. Then we study BDSDEs with continuous coefficients as an application of the comparison theorem of BDSDEs. Some efforts have been made in relaxing the Lipschitz hypothesis of BSDEs (cf. [9, 10, 11]). So far, however, there is little similar work for BDSDEs. In this paper, we shall do so for BDSDEs. For one-dimensional BDSDEs, we will weaken the usual Lipschitz assumptions to linear growth conditions by virtue of the comparison theorem. We show the existence of the minimal solution of BDSDEs. This method is due to [10].
2. PRELIMINARIES: THE EXISTENCE AND UNIQUENESS OF BDSDEs Notation. The Euclidean norm of a vector x √ ∈ Rk will be denoted by x, and for a d × k matrix A, we define A = Tr AA∗ , where A∗ is the transpose of A. Let P be a probability space, and T > 0 be an arbitrarily fixed constant throughout this paper. Let Wt 0 ≤ t ≤ T and Bt 0 ≤ t ≤ T be two mutually independent standard Brownian Motions with
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values in Rd and Rl , respectively, defined on P. Let denote the class of P-null sets of . For each t ∈ 0 T , we define B t = tW ∨ tT where for any process t st = r − s s ≤ r ≤ t ∨ t = 0t . Note that the collection t t ∈ 0 T is neither increasing nor decreasing, so it does not constitute a filtration. For any n ∈ N , let M 2 0 T Rn denote the set of (classes of dP ⊗ dt a.e. equal) n-dimensional jointly measurable stochastic processes t t ∈ 0 T , which satisfy:
T (i) 2M 2 = E 0 t 2 dt < ; (ii) t is t -measurable, for any t ∈ 0 T . Similarly, we denote by S 2 0 T Rn the set of n-dimensional continuous stochastic processes, which satisfy: (iii) 2S2 = Esup0≤t≤T t 2 < ; (iv) t is t -measurable, for any t ∈ 0 T . Let f × 0 T × Rk × Rk×d → Rk g × 0 T × Rk × Rk×d → Rk×l be jointly measurable such that for any y z ∈ Rk × Rk×d , f· y z ∈ M 2 0 T Rk g· y z ∈ M 2 0 T Rk×l Moreover, we assume that there exist constants C > 0 and 0 < < 1 such that for any t ∈ × 0 T , y1 z1 y2 z2 ∈ Rk × Rk×d , ft y1 z1 − ft y2 z2 2 ≤ Cy1 − y2 2 + z1 − z2 2 (H1) gt y1 z1 − gt y2 z2 2 ≤ Cy1 − y2 2 + z1 − z2 2 Given ∈ L2 T P Rk , we shall consider the following BDSDE: Yt = +
T t
fs Ys Zs ds +
T t
gs Ys Zs dBs −
T t
Zs dWs 0 ≤ t ≤ T (1)
Note that the integral, with respect to Bt , is a “backward Itô integral” and the integral with respect to Wt is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Sokorohod integral, see [12].
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Definition 2.1. A pair of process y z × 0 T → Rk × Rk×d is called a solution of BDSDE Eq. (1), if y z ∈ S 2 0 T Rk × M 2 0 T Rk×d , and satisfies BDSDE Eq. (1). Proposition 2.1. Under the above conditions, in particular (H1), BDSDE, Eq. (1) has a unique solution Y Z ∈ S 2 0 T Rk × M 2 0 T Rk×d This proposition was derived in [15].
3. COMPARISON THEOREMS In this paper, we only consider one-dimensional BDSDEs, i.e., k = 1. We consider the following BDSDEs: 0 ≤ t ≤ T T T T Yt1 = 1 + f 1 s Ys1 Zs1 ds + gs Ys1 Zs1 dBs − Zs1 dWs (2) t
Yt2 = 2 +
t
T t
f 2 s Ys2 Zs2 ds +
t
T t
gs Ys2 Zs2 dBs −
T t
Zs2 dWs (3)
where BDSDEs Eq. (2) and Eq. (3) satisfy the conditions of Proposition 2.2. Then there exist two pairs of measurable processes Y 1 Z1 and Y 2 Z2 satisfying BDSDEs Eq. (2) and Eq. (3), respectively. Assume 1 ≥ 2 as f 1 t y z ≥ f 2 t y z as ∀t y z ∈ 0 T × R × Rd (H2) Then we have the following comparison theorem. Theorem 3.1. Assume BDSDEs Eq. (2) and Eq. (3) satisfy the conditions of Proposition 2.2, let Y 1 Z1 and Y 2 Z2 be solutions of BDSDEs Eq. (2) and Eq. (3), respectively. If (H2) holds, then Yt1 ≥ Yt2 , a.s. ∀t ∈ 0 T . Proof. For notational simplicity, we assume l = d = 1. Then Yt1 − Yt2 , Zt1 − Zt2 satisfies the following BDSDE. T Yt1 − Yt2 = 1 − 2 + f 1 s Ys1 Zs1 − f 2 s Ys2 Zs2 ds + −
t
T t T t
g s Ys1 Zs1 − g s Ys2 Zs2 dBs Zs1 − Zs2 dWs 0 ≤ t ≤ T
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Applying Itô’s formula to Ys1 − Ys2 − 2 , we get Yt1 − Yt2 − 2 = 1 − 2 − 2 − 2 −2 +
T t
T t
Ys1 − Ys2 − f 1 s Ys1 Zs1 − f 2 s Ys2 Zs2 ds
− Ys1 − Ys2 g s Ys1 Zs1 − g s Ys2 Zs2 dBs
t
+2
T
2 1Ys1 ≤Ys2 g s Ys1 Zs1 − g s Ys2 Zs2 ds T
Ys1 − Ys2
t
Zs1 − Zs2 dWs −
−
T t
2 1Ys1 ≤Ys2 Zs1 − Zs2 ds (4)
From (H2), we have − ≥ 0, so 1
2
E1 − 2 − 2 = 0
(5)
Since Y 1 Z1 and Y 2 Z2 are in S 2 0 T R × M 2 0 T Rd , it easily follows that T Ys1 − Ys2 − Zs1 − Zs2 dWs = 0 (6) E t
T
E t
Ys1 − Ys2 − gs Ys1 Zs1 − gs Ys2 Zs2 dBs = 0
Let
= −2
T
− 1 f s Ys1 Zs1 − f 2 s Ys2 Zs2 ds
T
− 1 f s Ys1 Zs1 − f 1 s Ys2 Zs2 ds
T
− 1 f s Ys2 Zs2 − f 2 s Ys2 Zs2 ds
Ys1 − Ys2
t
= −2
Ys1 − Ys2
t
−2
Ys1 − Ys2
t
(7)
= 1 + 2 where 1 = −2 2 = −2
T
T
t
t
Ys1 − Ys2 Ys1 − Ys2
− 1 f s Ys1 Zs1 − f 1 s Ys2 Zs2 ds − 1 f s Ys2 Zs2 − f 2 s Ys2 Zs2 ds ≤ 0
From (H1) and Young’s inequality, it follows that T − Ys1 − Ys2 Ys1 − Ys2 + Zs1 − Zs2 ds ≤ 1 ≤ 2c t
T T 1 c2 Y − Y 2 − 2 ds + 1 − 1Y 1 ≤Y 2 Z1 − Z2 2 ds ≤ 2c + s s s s s s 1− t t (8)
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where c > 0 only depends on the Lipschitz constant C in (H1). Using the assumption (H1), again, we deduce T 2 1Ys1 ≤Ys2 g s Ys1 Zs1 − g s Ys2 Zs2 ds t T 2 2 ≤ 1Ys1 ≤Ys2 C Ys1 − Ys2 + Zs1 − Zs2 ds t T T 2 Y 1 − Y 2 − 2 ds + =C 1Ys1 ≤Ys2 Zs1 − Zs2 ds (9) s s t
t
Taking expectation on both sides of Eq. (4) and noting Eq. (5–9), we get
T − 2 − 2 c2 E Ys1 − Ys2 ds E Yt1 − Yt2 ≤ C + 2c + 1− t By Gronwall’s inequality, it follows that − 2 E Yt1 − Yt2 = 0 ∀t ∈ 0 T That is, Yt1 ≥ Yt2 , a.s., ∀t ∈ 0 T .
Remark. Let Y 1 Z1 and Y 2 Z2 be solutions of BDSDEs, Eq. (2) B and Eq. (3), respectively. If 1 2 ∈ L2 TT P R f 1 f 2 g are B tT -measurable and satisfy (H1). From the existence and uniqueness theorem of SDEs, it easily follows that Z1 ≡ 0 and Z2 ≡ 0, i.e., Y 1 and Y 2 are the solutions of the following SDEs, respectively, T T f 1 s Ys1 0 ds + g s Ys1 0 dBs 0 ≤ t ≤ T Yt1 = 1 + t
Yt2 = 2 +
t
T t
f s Ys2 0 ds + 2
T t
g s Ys2 0 dBs 0 ≤ t ≤ T
In particular, it is worth noting that Theorem 3.1, in this case, degenerates into a counterpart of the classical comparison theorem of forward stochastic differential equations. 4. BDSDEs WITH CONTINUOUS COEFFICIENTS This section is devoted to the study of BDSDEs with continuous coefficients as an application of BDSDE’s comparison theorem obtained in Section 3. We divide it into several steps. Our main result is: Theorem 4.1. Assume f × 0 T × R × Rd → R and g × 0 T × R × Rd → Rl are jointly measurable functions and satisfy (1) Linear growth: ∃0 < K < , such that f t y z ≤ K1 + y + z ∀ t y z ∈ × 0 T × R × Rd
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(2) For fixed and t f t · · is continuous. (3) There exist constants C > 0 and 0 < < 1 such that g t y1 z1 − g t y2 z2 2 ≤ Cy1 − y2 2 + z1 − z2 2 ∀ t ∈ × 0 T y1 z1 ∈ R × Rd y2 z2 ∈ R × Rd Then, if ∈ L2 T P, BDSDE Eq. (1) has a solution Y Z ∈ S 2 0 T R × M 2 0 T Rd . Moreover, there is a minimal solution Y Z of BDSDE Eq. (1) in the sense that, for any other solution Y Z of Eq. (1), we have Y ≤ Y . For notational simplicity, we still assume that l = d = 1. For fixed t, we define, as in Lemma 1 of [10], the sequence fn t y z associated to f , f t y z + ny − y + z − z fn t y z = inf y z ∈Q
then, for n ≥ K fn is jointly measurable and uniformly linear growth in y z with constant K. We also define the function. F t y z = K1 + y + z Given ∈ L2 T P, by Proposition 2.2, there exist two pairs of processes Y n Zn and U V, which are the solutions to the following BDSDEs Eq. (10) and Eq. (11), respectively, Ytn = +
Ut = +
T t
fn s Ysn Zsn ds + T
t
Fs Us Vs ds +
T t T
t
g s Ysn Zsn dBs − gs Us Vs dBs −
T t
Zsn dWs
(10)
T t
Vs dWs
(11)
From Theorem 3.1 and Lemma 1 of [10], we get ∀n ≥ m ≥ K Y m ≤ Y n ≤ U dt ⊗ dP-a.s.
(12)
Lemma 4.2. There exists a constant A > 0 depending only on K C T , and , such that ∀n ≥ K Y n S2 ≤ A Zn M 2 ≤ A U S2 ≤ A V M 2 ≤ A Proof. First of all, we prove that U S2 and V M 2 are all bounded. Clearly, from Eq. (12), there exists a constant B depending only on K C T and , such that T
21 T
21 n 2 2 E ≤ B E ≤ B V M 2 ≤ B Ys ds Us ds 0
0
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Applying Itô’s formula to Ut 2 , we have Ut 2 = 2 + 2 −2
t
T
T t
Us · Fs Us Vs ds + 2
Us · Vs dWs +
T t
T t
Us · gs Us Vs dBs
gs Us Vs 2 ds −
T t
Vs 2 ds
(13)
From (H1), for all < < 1, there exists a constant C > 0 such that gt u v2 ≤ C u2 + gt 0 02 + v2
(14)
From Eq. (13) and Eq. (14), it follows that Ut 2 +
T t
Vs 2 ds ≤ 2 + 2K + C +2
t
T
t
T t
T
Us 1 + Us + Vs ds
Us 2 + gs 0 02 ds +
Us · gs Us Vs dBs − 2
≤ 2 + K 2 T − t + C
t
T
T t
T t
Vs 2 ds
Us · Vs dWs
gs 0 02 ds
1 + T Vs 2 ds 2 t
2K 2 T + 1 + 2K + C + Us 2 ds 1 − t T T +2 Us · gs Us Vs dBs − 2 Us · Vs dWs +
t
t
Taking supremum and expectation, we get by Young’s inequality, U 2S2
T 1 − 2 2 2 2 V M 2 ≤ E + K T + C + gs 0 0 ds 2 0
T 2K 2 + 1 + 2K + C + E Us 2 ds 1 − 0 T + 2E sup Us · gs Us Vs dBs t 0≤t≤T T + 2E sup (15) Us · Vs dWs 0≤t≤T
t
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By Burkholder-Davis-Gundy’s inequality, we deduce T
Us · gs Us Vs dBs E sup t
0≤t≤T
≤ cE ≤ cE
T
0
Us · gs Us Vs ds 2
2
sup Ut 2
21 0
0≤t≤T
≤ 2c2 C E
0
T
T
21
gs Us Vs 2 ds
Us 2 ds +
T 0
21
gs 0 02 ds
1 + U 2S2 + 2c2 V 2M 2 8
(16)
In the same, way, we have T
1 Us · Vs dWs ≤ U 2S2 + 2c2 V 2M 2 E sup 8 t 0≤t≤T
(17)
From Eqs. (16), (17) and (15), it follows that 1 − U 2S2 + V 2M 2 2 ≤ 2 E2 + K 2 T + C 1 + 4c2 E
T
0
gs 0 0 ds 2
T 2K 2 2 + 2 1 + 2K + + C 1 + 4c E Us 2 ds 1 − 0 + 8c2 1 + V 2M 2 ≤ 2 E2 + K 2 T + C 1 + 4c2 E
0
T
gs 0 02 ds
2K 2 2 2 + 2 1 + 2K + + C 1 + 4c + 4c 1 + B2 1 −
=
1 − 2 B 2
that is U S2 ≤ B V M 2 ≤ B From Eq. (12), it easily follows that Y n S2 ≤ B
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Next, we prove that boundedness of Zn M 2 . Applying Itô’s formula to Ytn 2 , it follows that T T n 2 n n n Y = 2 + 2 s Y ds + 2 Y · f Z Ysn · g s Ysn Zsn dBs n t s s s −2
t
t
T
Ysn · Zsn dWs +
t
T t
g s Y n Zn 2 ds − s
Taking expectation, we have T 2 Zn 2 ds = E2 + 2E E Ytn + E s t
+E
T
s
T t
T
n 2 Z ds s
t
Ysn · fn s Ysn Zsn ds
g s Y n Zn 2 ds
t
s
s
From the well-known Young’s inequality, it follows that T 2 Zn 2 ds E Ytn + E s t
T n 2 Y ds + 1 − E Zn 2 ds s s 2 t t T + K 2 T − t + C E gs 0 02 ds + E
≤ E2 + C E
T
t
t
T
n 2 Z ds s
2
2K and we know 0 < < 1 from where C = 1 + 2K + C + 1− , Eq. (14). Then
T 2 2 2 2 2 Zn 2M 2 ≤ C TB + K T + E + C E gs 0 0 ds 1 − 0
= A2
The proof is completed. 2 2 Lemma 4.3. Y n Zn + n=1 converges in S 0 T R × M 0 T R.
Proof. Let n0 ≥ K. Since Y n is increasing and bounded in S 2 0 T R, we deduce from the dominated convergence theorem that Y n converges in S 2 0 T R. We shall denote by Y the limit of Y n . Applying Itô’s formula to Ytn − Ytm 2 , we get for n m ≥ n0 , T 2 Zn − Zm 2 ds E Y0n − Y0m + E s s = 2E
0
+E
T
0
0
T
Ysn − Ysm fn s Ysn Zsn − fm s Ysm Zsm ds
g s Y n Zn − g s Y m Zm 2 ds s
s
s
s
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≤2 E +E
T
0
T
0
n Y − Y m 2 ds s
s
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21 E
T
fn s Y n Zn − fm s Y m Zm 2 ds s
0
s
s
21
s
2 2 n C Ys − Ysm + Zsn − Zsm ds
Since fn and fm are uniformly linear growth and Y n Zn is bounded, similarly to Lemma 4.2, there exists a constant K > 0 depending only on K C T and , such that T 2 Zn − Zm 2 ds E Y0n − Y0m + E s s ≤E
0
T
0
2 2 n K Y − Y m + Zn − Zm ds s
s
s
s
So KT Y n − Y m 2S2 1−
Zn − Zm 2M 2 ≤
Thus Zn is a Cauchy sequence in M 2 0 T R, from which the result follows. Proof of Theorem 41. For all n ≥ n0 ≥ K, we have Y n0 ≤ Y n ≤ U , and Y n converges in S 2 0 T R dt ⊗ dP-a.s. to Y ∈ S 2 0 T R. On the other hand, since Zn converges in M 2 0 T R to Z, we can assume, choosing a subsequence if needed, that Zn → Z dt ⊗ dP-a.s. and G = supn Zn is dt ⊗ dP integrable. Therefore, from (i) and (iv) of Lemma 1 in [10], we get for almost all , fn t Ytn Ztn −→ ft Yt Zt n → dt-ae fn t Y n Zn ≤ K 1 + sup Y n + sup Zn t t t t
n
n
= K 1 + sup Ytn + Gt ∈ L1 0 T dt n
Thus, for almost all and uniformly in t, it holds that T T fn s Ysn Zsn ds −→ fs Ys Zs ds n → t
t
From the continuity properties of the stochastic integral, it follows that T n T sup t Zs dWs − t Zs dWs −→ 0 in probability, 0≤t≤T
T T sup t g s Ysn Zsn dBs − t gs Ys Zs dBs −→ 0 in probability. 0≤t≤T
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Choosing, again, a subsequence, we can assume that the above convergence is P-a.s. Finally, n Y − Y m ≤ t t
T
fn s Y n Zn − fm s Y m Zm ds s
t
s
s
s
T T n n m m + g s Ys Zs dBs − g s Ys Zs dBs t t T T + Zsn dWs − Zsm dWs t
t
and taking limits on m and supremum over t, we get sup Ytn − Yt ≤
o≤t≤T
0
T
fn s Y n Zn − fs Ys Zs ds s
s
T T n n + sup g s Ys Zs dBs − gs Ys Zs dBs t t 0≤t≤T T T + sup Zsn dWs − Zs dWs P-a.s. 0≤t≤T
t
t
from which it follows that Y n converges uniformly in t to Y (in particular, Y is a continuous process). Note that Y n is monotone; therefore, we actually have the uniform convergence for the entire sequence and not just for a subsequence. Taking limits in Eq. (10), we deduce that Y Z is a solution of Eq. (1). Let Y Z ∈ S 2 0 T R × M 2 0 T R be any solution of Eq. (1). Y , ∀n ∈ N and therefore Y ≤ Y From Theorem 3.1, we get that Y n ≤ proving that Y is the minimal solution. 5. ACKNOWLEDGMENTS The authors thank Professor Shige Peng for his helpful comments. Shi’s work is partially supported by Foundation for University Key Teacher by Ministry of Education of China, National Natural Science Foundation of China grant 10201018, and Doctor Promotional Foundation of Shandong grant 02BS127. Liu’s work is partially supported by EPSRC grant No. GR/R 37227.
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