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Backward Stochastic Differential Equation with

Monotone and Continuous Coefficient

Xiaoqin Huang

1

_{,Xiaojie Liu}

2 ∗

Abstract—In this paper, we consider a backward stochastic differential equation(BSDE) with mono-tone and continuous coefficient and obtain the exis-tence and uniqueness of solution.

Keywords: backward stochastic differential equation; Yosida approximations; monotone and continuous co-efficients

1

Introduction

In 1973, Bismut first introduced the adapted solution for a linear BSDE which is the adjoint process for a stochastic control problem. Later Pardoux and Peng [6] obtained the existence and uniqueness of solution for the following nonlinear BSDE with Lipschitz coefficientf

yt=ξ+ Z T

t

f(s, ys, zs)ds− Z T

t

zsdWs (1)

where (Ws)0≤s≤T is a standard d-dimensional Brownian motion defined on a complete probability space (Ω,F, P), Ftis the natural filtration. Ftcontains all P-null sets of

F. ξis a givenFT measurable random vector. Since then, many researchers have devoted to obtaining its existence and uniqueness of solution under weaker assumptions on f. For example, Lepeltier [4] obtained the existence for one-dimensional BSDE under the continuous assumption, and Mao [5] obtained the existence and uniqueness under the non-Lipschitz assumption. In this paper, using the Yosida approximations (see for example Da Prato and Zabczyk [1,2] or Hu [3]), we also obtain the existence and uniqueness of solution for BSDE (1).

∗_{Manuscript received May 6, 2009. This work is supported} by the science foundation of Hebei University of Science and Technology (XL200824),Dr. fund research of Hebei Univer-sity of Science and Technology(QD200950) and Hebei province science and technology projects (No.09213508d; No.09213558). Xiaoqin Huang is with the College of Sciences, Hebei Univer-sity of Science and Technology, Shijiazhuang, Hebei 050018, China(e-mail:[email protected]). Xiaojie Liu is with the Institute of Mathematics, Shijiazhuang No.2 middle school, Shijiazhuang 050000,China(e-mail: [email protected]).

2

Preliminary

Let M2_{(0, T;R}n_{) denote the set of all R}n_{-valued F} t -progressively measurable processesv(·) satisfyingERT

0 |

v(s)|2_{ds <+∞.}

In this paper, (·) denotes the usual inner product inRn_; We use the usual Euclidean norm inRn_{. Forz∈R}n×d_, its Euclidean norm is defined by | z |= tr(zzT₎12 _and its inner product is ((z1_{, z}2_{)) = tr(z}1_(z2₎T_{). For u}1 ₌

(y1, z1) ∈ Rn×Rn×d, u2 = (y2, z2) ∈ Rn×Rn×d, we denote [u1, u2] = (y1, y2) + ((z1, z2)) and |u1 |2_{=| y}1 _|2

+|z1|2_.

Definition 2.1The solution of Eq.(1) is a couple (y, z) which belongs to M2_{(0, T; R}n _×Rn×d_{) and satisfies} Eq.(1).

Theorem 2.2[6] Iff0: Ω×[0, T]×Rn_×Rn×d_→Rn _is a progressively measurable function and satisfies

(i)f0(t,0,0)t∈[0,T] belongs toM2(0, T;Rn);

(ii) There exists a constant K ≥ 0 s.t.P-a.s., for all y1,

y2 ∈ Rn, z1, z2 ∈ Rn×d, |f0(t, y1, z1)−f0(t, y2, z2)| ≤

K(|y1−y2|+|z1−z2|).

Thenyt=ξ+ RT

t f

0_{(s, y}

s, zs)ds− RT

t zsdWshas a unique adapted solution (y(.), z(.))∈M2_{(0, T;R}n_×Rn×d_).

Theorem 2.3 [6] Iff0 : Ω×[0, T]×Rn_×Rn×d _{→ R}n andg0 : Ω×[0, T]×Rn_×Rn×d_→Rn×d_{are progressively} measurable functions, and there exist constants λ > 0 andα >0 such that

|f0(t, y1, z1)−f0(t, y2, z2)|+|g0(t, y1, z1)−g0(t, y2, z2)|

≤λ(|y1−y2|+|z1−z2|)

|g0(t, y, z1)−g0(t, y, z2)|≥α|z1−z2|

for all y1, y2 ∈ Rn, z1, z2 ∈ Rn×d. Then the following

equation

yt=ξ+ Z T

t

f0(s, ys, zs)ds− Z T

t

g0(s, ys, zs)dws

In this paper, we consider the following BSDE

yt=ξ+ Z T

t

f(s, ys, zs)ds− Z T

t

zsdWs

Assumption 2.4f is continuous in (y, z) for almost all (t, ω). f(., y, z)∈M2_{(0, T;R}n_{). For allu= (y, z)∈R}n_× Rn×d_{, there exist constants} 3

4 ≤C ≤1 andC1 >0 such

that P-a.s.,a.e.

(H1) |f(t, u)|≤|f(t,0)|+C1|u|

(H2) (f(t, y1, z1)−f(t, y2, z2), y1−y2)

≤(1−C)|z1−z2|2_−C|y1_−y2_|2

We denote g(t, y, z)≡z. Then the equation (1) is equal to

yt=ξ+ Z T

t

f(s, ys, zs)ds− Z T

t

g(s, ys, zs)dWs (10)

Let F(t, u) = F(t, y, z) = (f(t, y, z),−g(t, y, z)) = (f(t, u),−g(t, u)). Then by the Assumption 2.4, F(t, .) is also continuous, and (H2) is equal to

(H20) [F(t, u1)−F(t, u2), u1−u2]≤ −C|u1−u2|2

Lemma2.5[3]. Let Φ : Rn _{−→ R}n _{be a continuous} function, and there exists a constant c > 0 such that (Φ(x1_)−Φ(x2_{), x}1_−x2_{)≤ −c|x}1_−x2_|2_{, ∀x}1_{, x}2_∈Rn_. Then for the Yosida approximations Φα_{of Φ, α >0, we} have

(i) (Φα(x1)−Φα(x2), x1−x2)≤ −c|x1−x2|2

|Φα(x1)−Φα(x2)|≤(_α2 +c)|x1−x2| |Φα_{(x)|≤|Φ(x)|+2c|x|}

(ii) For anyα, β >0, we have

(Φα_(x1_)−Φβ_(x2_{), x}1_−x2_{)≤(α+β)(|Φ(x}1_)|+|Φ(x2_)|

+c|x1_|+c|x2_|)2_−c|x1_−x2_|2

(iii) For any {xα_}

α>0⊂Rn, x∈Rn, if limα→0xα=x,

then

limα→0Φα(xα) = Φ(x)

3

Main results

Theorem 3.1Let Assumption 2.4 hold, then there exists a unique adapted solution (y(.), z(.)) ∈ M2(0, T;Rn ×

Rn×d) for Eq.(1).

Proof. First, we prove the existence. We divide the proof into four steps.

Step 1. There exists a unique adapted solution for the approximating BSDE.

For arbitrary α > 0, we consider the approximating BSDE of (1’)

yα_t =ξ+ Z T

t

fα(s, y_sα, zα_s)ds−

Z T

t

gα(s, yα_s, z_sα)dws(2)

where Fα_{(t, y}α

t, ztα) = (fα(t, ytα, ztα),−gα(t, ytα, ztα)) is the Yosida approximation ofF(t, yα

t, ztα).

Letv1_{= (y}1_{, z}1_{) and v}2 _{= (y}2_{, z}2_{), then by Lemma 2.5,}

we have

|Fα(t, v1)−Fα(t, v2)|2_≤(2

α+C)

2_|v1_−v2_|2

hence

2|fα(t, v1)−fα(t, v2)|2_+2|gα_{(t, v}1_)−gα_{(t, v}2_)|2

≤ 2(2 α+C)

2₍

|y1−y2|2+|z1−z2|2)

(|fα(t, v1)−fα(t, v2)|+|gα(t, v1)−gα(t, v2)|)2

≤ 2(2 α+C)

2_(|y1_−y2_|2_+|z1_−z2_|2₎

therefore

|fα(t, y1, z1)−fα(t, y2, z2)|+|gα(t, y1, z1)−gα(t, y2, z2)|

≤ √2(2

α+C)(|y

1

−y2|2+|z1−z2|2)12

≤ √2(2

α+C)(|y

1

−y2|+|z1−z2|).(3)

Let w1 _{= (y, z}1_{) andw}2 _{= (y, z}2_{), then by Lemma 2.5,}

we have

(Fα(t, w1)−Fα(t, w2), w1−w2)≤ −C|w1−w2|2

so (fα_{(t, y, z}1_{) − f}α_{(t, y, z}2_{), y − y) + (g}α_{(t, y, z}2_{) −}

gα_{(t, y, z}1_{), z}1_−z2_{)≤ −C|z}1_−z2_|2

thus

|gα(t, y, z1)−gα(t, y, z2)| · |z1−z2|≥C|z1−z2|2

Step 2. There exists a constant L, such thatERT

0 |u

α_|2

dt≤L.

Applying the Itˆoformula to|yα

t |2 and taking the expec-tation, we get

Eξ2=E|y₀α|2−E Z T

0

2(yα_t, fα(t, y_tα, zα_t))dt

+E Z T

0

|gα(t, yαt, z α t)|

2 dt so E Z T 0

2(yα_t, fα(t, y_tα, zα_t)−fα(t,0,0))dt

−E Z T

0

2((z_tα, gα(t, yα_t, z_tα)−gα(t,0,0)))dt+Eξ2

=−E Z T

0

2(y_tα, fα(t,0,0))dt+E Z T

0

|gα(t, y_tα, z_tα)|2_dt

+E |y₀α|2_−E

Z T

0

2((z_tα, gα(t, y_tα, z_tα)−gα(t,0,0)))dt hence

Eξ2−E Z T

0

2C|uα|2_dt

≥ E|y₀α|2+E Z T

0

|gα(t, y_tα, z_tα)|2dt

−E Z T

0

2(yα_t, fα(t,0,0))dt

−E Z T

0

2((z_tα, gα(t, y_tα, z_tα)−gα(t,0,0)))dt

then

E|y₀α|2+E Z T

0

2C|uα|2dt

≤ E Z T

0

2(y_tα, fα(t,0,0))dt

+E Z T

0

2((z_tα, gα(t, y_tα, z_tα)−gα(t,0,0)))dt

+Eξ2−E Z T

0

|gα(t, yαt, ztα)|2dt

≤ Eξ2+E Z T

0

|y_tα|2_dt

+E Z T

0

|fα(t,0,0)|2dt−E Z T

0

|gα(t, ytα, z α t)|

2

dt

+E Z T

0

2((zα_t, gα(t, yα_t, z_tα)))dt−E Z T

0

|zα_t |2_dt

−E Z T

0

2((zα_t, gα(t,0,0)))dt+E Z T

0

|z_tα|2_dt

≤ E Z T

0

|fα(t,0,0)|2dt−E Z T

0

2((z_tα, gα(t,0,0)))dt

+E Z T

0

|z_tα|2_dt+Eξ2_+E

Z T

0

|yα_t |2_dt

= E

Z T

0

|fα(t,0,0)|2_dt−EZ

T

0

2((z_tα, gα(t,0,0)))dt

−2E Z T

0

|gα(t,0,0)|2_dt−1

2E Z T

0

|ztα|2dt

+2E Z T

0

|gα(t,0,0)|2_dt+1

2E Z T

0

|z_tα|2_dt

+E Z T

0

|z_tα|2dt+Eξ2+E Z T

0

|yα_t |2dt

By | Fα(x) |≤|F(x) | +2C | x |, we have | Fα(0) |2_≤|

F(0) |2_{. Thus | f}α_{(t,0,0) |}2 _{+ | g}α_{(t,0,0) |}2_≤|

f(t,0,0) |2 _{+| g(t,0,0) |}2_{=| f(t,0,0) |}2_{. Therefore we}

get

E|yα₀ |2_+E

Z T

0

2C|uα|2_dt

≤ Eξ2+E Z T

0

|y_tα|2_{dt+ 2E}Z

T

0

|f(t,0,0)|2_dt

+3 2E

Z T

0

|zαt |2dt

≤ Eξ2+3 2E

Z T

0

|uα|2_{dt+ 2E}

Z T

0

|f(t,0,0)|2_dt

so

E|y₀α|2_+E

Z T

0

(2C−3

2)|u α_|2_dt

≤ Eξ2+ 2E Z T

0

|f(t,0,0)|2dt=M

Because ₄3 ≤C ≤1, then letL= _4C2M₋₃, we have ERT

0 |

uα_|2_dt≤L.

Step 3. uα_{= (y}α_{, z}α_{) converges inM}2_{(0, T;R}n_×Rn×d_). Let α > 0 and β > 0, applying the Itˆo formula to |

ytα−y β

t |2 and taking the expectation, we get 0 =E

Z T

0

2(y_tα−yβt, fβ(t, y β t, z

β

t)−fα(t, yαt, z α t))dt

+E Z T

0

|gα(t, yα_t, z_tα)−gβ(t, yβt, z β t)|2dt +E|y₀α−yβ₀ |2

hence

E Z T

0

−E Z T

0

2((z_tα−z_tβ, gα(t, yα_t, z_tα)−gβ(t, yβ_t, zβ_t)))dt +E|y₀α−yβ₀ |2

= E

Z T

0

2(yα_t −y_tβ, fα(t, y_tα, zα_t)−fβ(t, y_tβ, z_tβ))dt

−E Z T

0

2((zαt −z β

t, gα(t, yαt, ztα)−gβ(t, y β t, z

β t)))dt

= E

Z T

0

2[Fα(t, uα)−Fβ(t, uβ), uα−uβ]dt

≤ −2CE Z T

0

|uα−uβ|2dt+

2(α+β)ERT

0 (| F(t, u

α_{) | + | F(t, u}β_{) | +C(| u}α _{| + |} uβ_|))2_dt

therefore

E |y₀α−yβ₀ |2+2CE Z T

0

|uα−uβ |2dt

≤2(α+β)ERT

0 (|F(t, u

α_{)|+|F(t, u}β_)|+C|uα_|+C| uβ_|)2_dt

+E Z T

0

2((z_tα−zβ_t, gα(t, yα_t, z_tα)−gβ(t, y_tβ, z_tβ)))dt

−E Z T

0

|ztα−z β

t |2dt+E Z T

0

|ztα−z β t |2dt

−E Z T

0

|gα(t, yα_t, z_tα)−gβ(t, y_tβ, zβ_t)|2_dt

≤8(α+β)E Z T

0

(|F(t, uα)|2_{+|F(t, u}β_)|2

+C2|uα|2_+C2_|uβ_|2_)dt

+E Z T

0

|zα_t −z_tβ|2_dt+E

Z T

0

|yα_t −yβ_t |2_dt

then

E|y₀α−yβ₀ |2_+(2C−1)EZ

T

0

|uα−uβ|2_dt

≤ 8(α+β)E Z T

0

(|F(t, uα)|2_{+|F(t, u}β_)|2

+C2|uα|2_+C2_|uβ _|2_)dt

Because | F(t, u) |2_{=| f(t, u) |}2 _{+ | z |}2_{, | f(t, u) |≤|}

f(t,0)|+C1|u|andER

T

0 |u|

2_{dt≤L, we deduce that}

there exists a constantk >0, such that

E|yα₀ −y₀β|2_+(2C−1)EZ

T

0

|uα−uβ |2_{dt≤k(α+β)}

Because 3₄ ≤ C ≤ 1, then {uα, α > 0} is a Cauchy se-quence in M2(0, T;Rn×Rn×d). We denote its limit by u= (y, z)∈M2_{(0, T;R}n_×Rn×d_).

Step 4. Taking weak limits in the approximating equa-tions (2).

From Lemma 2.5 and Assumption (H1), there exist con-stantsl andmsuch that

|Fα(t, uα)|2 _{≤ (|F(t, u}α_)|+2C|uα_|)2

≤ l|f(t,0)|2_+m|uα_|2

So, there exists a constant C2 > 0 such that E

RT

0 |

Fα(t, uα) |2 _{dt ≤ C}

2. Therefore there exists a

subse-quences of {Fα_{(., u}α_{), α > 0} converge weakly to limits} G= (H,−B) in the spaceM2_{(0, T;R}n_×Rn×d_{). Taking} weak limits in the approximating equations (2) yields

yt = ξ+ Z T

t

H(s)ds−

Z T

t

B(s)dws

Similar to the proof of Hu[3], we can prove that G = (H,−B) =F(t, yt, zt) = (f(t, yt, zt),−g(t, yt, zt)). Therefore

yt=ξ+ Z T

t

f(s, ys, zs)ds− Z T

t

zsdWs

We deduce that (y, z) is an adapted solution of Eq.(1). The existence is proved.

Next, we prove the uniqueness of solution of Eq.(1).

Let u1 = (yt1, zt1) and u2 = (y2t, zt2) be two solutions of Eq.(1). ˆyt=yt1−y2t and ˆzt=zt1−zt2, then we have

dˆyt= (f(t, y2t, z

2

t)−f(t, y

1

t, z

1

t))dt+ (z

1

t −z

2

t)dwt

Applying the Itˆoformula to|yˆt|2and taking the expec-tation, we get

0 =E|yˆ0|2+E

Z T

0

2(ˆyt, f(t, yt2, zt2)−f(t, y1t, zt1))dt

+E Z T

0

|z_t1−z_t2|2_dt

Hence by Assumption 2.4, we get

E|yˆ0|2−E

Z T

0

|z_t1−z2_t |2_dt

= E

Z T

0

2(ˆyt, f(t, y1t, z

1

t)−f(t, y

2

t, z

2

t))dt

−2E Z T

0

|z_t1−z2_t |2_dt

≤ −2CE Z T

0

Then 2CERT

0 (| zˆt | 2 _{+ | yˆ}

t |2)dt−ER T

0 | zˆt |

2 _{dt ≤ 0.}

Because 3

4 ≤C≤1, then we haveE

RT

0 |yˆt|

2_{dt= 0 and}

ER₀T |zˆt|2dt= 0. Sou1=u2.

Thus there exists a unique adapted solution (y(.), z(.)) for Eq.(1) inM2(0, T; Rn×Rn×d). The proof is completed.

References

[1] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.

[2] G. Da Prato and J. Zabczyk,Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge, 1996.

[3] Hu,Y., “On the solution of forward-backward SDEs with monotone and continuous coeffi-cients,”Nonlinear analysis, v42,pp.1-12,2000

[4] Lepeltier,J.P., and Martin,J.S. , “Backward stochas-tic differential equations with continuous coef-ficient,”Statistics Probability Letters,v32, pp.425-430,1997

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