MuandWu Advances in Difference Equations (2015) 2015:265 DOI 10.1186/s13662-015-0607-3

0 Advances in Difference Equations

a SpringerOpen Journal

One kind of multiple dimensional Markovian BSDEs with stochastic linear growth generators

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Rui Mu and Zhen Wu*

Correspondence: wuzhen@sdu.edu.cn Schoolof Mathematics, Shandong University, Jinan, 250100, P.R. China

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Abstract

In this article, we deal with a multiple dimensional coupled Markovian BSDEs system with stochastic linear growth generators with respect to volatility processes. An existence result is provided by virtue of approximation techniques.

MSC: 60H10; 39A05

Keywords: backward stochastic differential equations; stochastic differential equations; approximation techniques

1 Introduction

Backward stochastic differential equations (BSDEs) were proposed firstly by Bismut [1] in linear case to solve the optimal control problems. Later this notion was generalized by Pardoux and Peng [2] into the general nonlinear form, and the existence and uniqueness results were proved under the classical Lipschitz condition. A class of BSDEs was also introduced by Duffie and Epstein [3] in point of view of recursive utility in economics. During the past twenty years, BSDEs theory has attracted many researchers' interest and has been fully developed into various directions. Among the abundant literature, we refer readers to the florilegium book edited by El-Karoui and Mazliak [4] for the early works before 1996. Surveys on BSDEs theory also include [5] which is written by El-Karoui, Hamadene and Matoussi collected in book [6] (see Chapter 8) and the book by Yong and Zhou [7] (see Chapter 7). Some applications on optimization problems can be found in [5]. About other applications, such as in the field of economics, we refer to El-Karoui, Peng and Quenez [8]. Recently, a complete review on BSDEs theory as well as some new results on nonlinear expectation were introduced in a survey paper by Peng [9].

One possible extension to the pioneering work of [2] is to relax as much as possible the uniform Lipschitz condition on the coefficient. Mao [10] provided an existence and uniqueness result under a weaker condition than the Lipschitz one. Hamadene introduced in [11] a one-dimensional BSDE with local Lipschitz generator. Later Lepeltier and San Martin [12] provided an existence result of minimum solution for one-dimensional BSDE where the generator function/ is continuous and of linear growth in terms of (y, z). When f is uniformly continuous in z with respect to («, t) and independent of y, a uniqueness result was obtained by Jia [13]. BSDEs with polynomial growth generator were studied by

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Briand in [14]. The case of one-dimensional BSDEs with coefficient which is monotonic in y and non-Lipschitz on z is shown in work [15]. About the BSDE with continuous and quadratic growth driver, a classical research should be the one by Kobylanski [16] which investigated a one-dimensional BSDE with driver \f (t,y, z) < C(1 + |y| + |z|2) and bounded terminal value. This result was generated by Briand and Hu into the unbounded terminal value case in [17].

There are plenty of works on one-dimensional BSDE. However, limited results have been obtained about the multi-dimensional case. We refer to Hamadene, Lepeltier and Peng [18] for an existence result on BSDEs system of Markovian case where the driver is of linear growth on (y, z) and of polynomial growth on the state process. See Bahlali [19, 20] for high-dimensional BSDE with local Lipschitz coefficient.

In the present article, we consider a high-dimensional BSDE under Markovian framework as follows:

Yl = gi(Xr)+ f Hi(s,X„ Y^,...,Ysn,Z],...,Zn) ds -/ ZlsdBs Jt Jt

for i = 1,2,..., n, with process X as a solution of a stochastic differential equation (SDE for short). For each i = 1,2,...,n, the coefficient Hi is continuous on (y1,...,yn, zL,...,zn) and satisfies

\Hi(t,x,y\...,yn,z\...,z?)\ < C( 1+ |x|)|zi| + C( 1+ |x|Y + |/|), y >0,

which means that Hi is of stochastic linear growth on Zi, or in other words, it is of linear growth w by w. Similar situation was considered in [21] in the background of nonzero-sum stochastic differential game problem. However, in [21], the generator Hi is independent of (y\...,yn). According to our knowledge, this general form of high-dimensional coupled BSDEs system with stochastic linear growth generator has not been considered in literature. This is the main motivation of the present work.

The rest of this article is organized as follows. In Section 2, we give some notations and assumptions on the coefficient. The properties of the forward SDE are also provided. The main existence result of BSDEs is proved in Section 3 where a measure domination result plays an important role. This domination result holds true when we assume that the diffusion process of the SDE satisfies the uniform elliptic condition. For the proof of the main result, we adopt an approximation scheme following the well-known mollify technique. The irregular coefficients are approximated by a sequence of Lipschitz functions. Then, we obtain the uniform estimates of the sequence of solutions as well as the convergence result in some appropriate spaces. Finally, we verify that the limit of the solutions is exactly the solution to the original BSDE, which completes the proof.

2 Notations and assumptions

In this section, we give some basic notations, the preliminary assumptions throughout this paper, as well as some useful results. Let (fi, F, P) be a probability space on which we define an m-dimensional Brownian motion B = (Bt)0<t<T with integer m > 1. Let us denote by F = {Ft, 0 < t < T} for fixed T >0 the natural filtration generated by process B and augmented by NP the P-null sets, i.e., Ft = a{Bs,s < t} vNP.

Let P be the a-algebra on [0, T] x ^ of Ft-progressively measurable sets. Letp e [1, to) be a real constant and t e [0, T] be fixed. We then define the following spaces: Lp = (f Ft-measurable and Rm-valued random variable s.t. E[|f |p] < to}; SpT = (y = (%)t<s<T P-measurable and Rm-valued s.t. E[supse[tT] |%|p] < to} and %pj = (y = (%)t<s<T P-measurable and Rm-valued s.t. E[(/tT |^s |2 ds)2] < to}. Hereafter, S0,T and Hp T are simply denoted by STp and HPT.

The following assumptions are in force throughout this paper. Let a be the function defined as

a : [0, T] x Rm Rmxm which satisfies the following assumption. Assumption 2.1

(i) a is uniformly Lipschitz w.r.t. x, i.e., there exists a constant C\ such that, Vt e [0, T], Vx,X e Rm, |a(t,x)-a(t,x')| < C^x-x'|.

(ii) a is invertible and bounded and its inverse is bounded, i.e., there exists a constant Ca such that V(t,x) e [0, T] x Rm, |a(t,x)| + |a-1(t,x)| < Ca.

Remark 2.1 (Uniform elliptic condition) Under Assumption 2.1, we can verify that there exists a real constant e >0 such that for any (t,x) e [0, T] x Rm,

e.1 < a(t,x).aT(t,x) < e-1.1, (.1)

where 1 is the identity matrix of dimension m.

Suppose that we have a system whose dynamic is described by a stochastic differential equation as follows: for (t,x) e [0, T] x Rm,

= x + fts a(u,XUx) dBu, s e [t, T]; (2 2)

Xlx = x, s e [0, t]. .

The solution Xtix = (Xstx)s<T exists and is unique under Assumption 2.1 (cf. Karatzas and Shreve [22], p.289). We recall a well-known result associated to integrability of the solution. For any fixed (t,x) e [0, T] x Rm, p > 2, it holds that, P-a.s.,

sup |Xff < C(l+|x|p), (.3)

L0<s<T -I

where the constant C only depends on the Lipschitz coefficient and the bound of a.

For integer n > l, we first present the following Borelian function as the terminal coefficient of the n-dimensional BSDE that we are considering:

gi : Rm R, i = l,2,...,n,

which satisfies the following.

Assumption 2.2 The function gi, i = 1,2,..., n, is of polynomial growth with respect to x, i.e., there exist a constant Cg and y > 0 such that

|gi(x)| < Cg (1+ |x|Y), Vx e Rm, fori = 1,2,...,n.

Now, we consider Borelian functions Hi, i = 1,2,..., n, from [0, T] x Rm x Rn x Rnm into R as follows:

Hi(t, x, y\...,yn, z1,...,zn), i = 1,2,..., n,

which satisfy the following hypothesis.

Assumption 2.3

(i) For each (t, x, y\...,yn, z\...,zn) e [0, T] x Rm x Rn x Rnm, there exist constants C2, Ch and y >0 such that, for each i = 1,2,..., n,

(ii) themapping(y1,...,yn,z\...,zn) e Rn x Rnm Hi(t,x,y\...,yn,z1,...,zn) e Ris continuous for any fixed (t, x) e [0, T] x Rm.

For a fixed constant a e Rm, and i = 1,2,..., n, let us consider the following BSDE:

From Assumptions 2.2 and 2.3, we know that this is a multiple dimensional coupled BSDEs system under Markovian framework with unbounded terminal value.

3 Existence of solutions for the multiple dimensional coupled BSDEs system

In this section, we provide an existence result of BSDEs (2.5) when n = 2 as an example. Actually, the case for n >2 can be dealt with in the same way without any difficulties.

3.1 Measure domination

Before we state our main theorem, let us first recall a result related to measure domination.

Definition 3.1 (Lq-domination condition) Let q e (1, to) be fixed. For given ti e [0, T], a family of probability measures {v1(s,dx),s e [t1, T]} defined on Rm is said to be Lq-dominated by another family of probability measures {v0(s, dx),s e [t1, T]} if for any S e (0, T -11], there exists an application : [t1 + S, T] x Rm ^ R+ such that:

(i) v1(s,dx) ds = «(s,x)v0(s,dx) ds on [t1 + S, T] x Rm.

(ii) Vk > 1, (s,x) e Lq([t1 + S, T] x [-k, k]m; V0(s, dx) ds).

Lemma 3.1 Let a e Rm, (t,x) e [0, T] x Rm, s e (t, T] and fi(t,x; s, dy) the law ofXf, i.e.,

|Hi(t,x,y1,...,yn,zV..,zn)| < C2(1+ |x|)z\ + Ch(1 + |x|Y + |yi|); (2.4)

Yi = gi(X°0a) + j THi(s, X0,a, Yl,...,Y:, Z1,...,ZD ds

t e [0, T].

VA e B(R^, Mi,x;s, A) = P(Xf e A).

Under Assumption 2.1 on a ,foranyq e (1, to), the family of laws [¡x(t, x; s, dy), s e [t, T]} is Lq-dominated by [^(0, a; s, dy), s e [t, T ]} for fixed a e Rm.

Proof See [21], Lemma 4.3 and Corollary 4.4, pp.14-15. □

3.2 High-dimensional coupled BSDEs system

Our main result in this section is the following theorem.

Theorem 3.1 Let a e Rm befixed. Then under Assumptions 2.1,2.2 and 2.3, there exist two pairs of P-measurable processes (Yi, Zi) with values in R1+m, i = 1,2, and two deterministic functions gi (t, x) which are of polynomial growth, i.e., |g i(t, x)| < C(1 + \x\y) with y > 0, i = 1,2, such that

P-a.s., Vt < T, Y¡: = gi(t,X°'a) and Zi is dt-square integrable P-a.s.; Y1= g1(x0,a) + ftTH1(s,X'.Y], Y2, Z\, Z2) ds - ftTZ¡ dBs; (3.1)

Y2 = g2(x0,a) + ft H2(s,X°a, Yj1, Y2, Z1,Z2) ds - ftTZ2 dBs.

The result holds true as well for the case i > 2 following the same way.

Proof The structure of this proof is as follows. We first use the mollify technique on the generator Hi to construct a sequence of BSDEs with generators which are Lipschitz continuous. Then, we provide uniform estimates of the solutions as well as the convergence property. Finally, we verify that the limits of the sequences are exactly the solutions for BSDE (3.1). Step 1. Approximation.

Let f be an element of CTO(R2+2m, R) with compact support and satisfy í f (y1, y2, z1, z2) dy1 dy2 dz1 dz2 = 1.

jR2+2m

For (t,x,y1,y2,z1, z2) e [0, T] x Rm x R2+2m, we set H1n(t, x, y1, y2, z1, z2)

n4H^ s, ^n(x), p1, p2, q1, q2)

x f (n(y1 -p1),n(y2 -p2), n(z1- q1), n(z2 - q2)) dp1 dp2 dq1 dq2,

where the truncation function y„(x) = ((x¡ v (-n)) A n)j=1,2,...,m for x = (x¡);=1,2,...,m e Rm. We next define f e CTO(R2+2m, R) by

f(yy1, y2, z1, z2) =

1, ly112 + |y2 |2 + |z1|2 + |z2|2 < 1, 0, |y112 + |y2|2 + |z1|2 + |z2|2 > 4.

Then we define the measurable function sequence (H\f)n>\ as follows: V(i,x,y1,y2, z1, z2) e [0, T] x Rm x R2+2m,

Hm(t,x,y1,y2,z1,z2) = f(y-,y-, —, Hm(t,x,y1,y2,z1,z2). nnnn

We have the following properties:

(a) H1n is uniformly Lipschitz w.r.t. (y1,y2, z1,z2);

(b) |H1n(t,x,y1,y2,z1,z2)| < C2(1 + |^n(x)|)|z1| + Ch(1 + ten(x)|Y + |y1|);

(c) |H1n(t,x,y1,y2,z1,z2)| < cn for any (t,x,y1,y2,z1,z2);

(d) For any (t, x) e [0, T] x Rm, and K a compact subset of R2+2m, sup(r1,y2,z1,z2)eK |H1n(t,x,y1,y2,z1,z2) -H^t,x,y1,y2,z1,z2)| ^ 0, as n ^ to.

The construction of the approximating sequence (H2[)n->1 is carried out in the same way. For each n > 1 and (t,x) e [0, T] x Rm, since H1n and H2n are uniformly Lipschitz w.r.t. (y1,y2, z1, z2), by the result of Pardoux-Peng (see [2]), we know that there exist two pairs of processes (Yin;(tx), Zin;(tx)) e <S2T(R) x H2T(Rm), i = 1,2, which satisfy, for s e [t, T],

Yln;(tx) = g1(XtTx) + // H1n(r,X*. Yrin;(t,x), Y2n;(t,x),Z1n;(t,x),Zr2n;(tx)) dr

1n;(t,x) v2n;(t,x) ry1n;(t,x) ry2n;(t,x)\

- fsT Z1n;(t,x) dB,

yW = g2(XTx) + fsTH2n(r,X*. Yrln;(tx), Y2n;(tx),Z1n;(tx),Zr2n;(tx)) dr

T 72n;(t,x)

Meanwhile, properties (3.2)(a), (c) and the result of El-Karoui etal. (ref. [8]) yield that there exist two sequences of deterministic measurable applications g1n (resp. g2n) : [0, T] x Rm ^ R and z1n (resp. z2n): [0, T] x Rm ^ Rm such that for any s e [t, T],

Yin;(tx) = g 1n(s,Xtx) (resp. Y2n;(tx) = g2n(s,X^ (3.4)

Z1n;(tx) = z1^s, Xf) (resp. Zs2n;(tx) = 32n(s, Xf)).

Besides, we have the following deterministic expression: for i = 1,2 and n > 1,

g in(t, x) = E where

gi(XtTx) +j T Fin (s, Xf) ds

, V(t, x) e [0, T] x Rm,

FUs,x) = Hm(s,x, g 1n(s,x), g2n(s,x),31n(s,x),32n(s,x)).

Step 2. Uniform integrability of (Y1n;(tx),Z1n;(t^))n>1 for fixed (t,x) e [0, T] x Rm. For each n > 1, let us first consider the following BSDE:

Yln = g1(XtTc) + jT C2(1 + ^(xr) |) |Z1n |

+ Ch(1 + Mx;x)|y + |Yrln|) dr -j T ZlndBr. (3.6)

For any x e Rm and n > 1, the mapping (y1,z1) e R x Rm ^ C2(l + \yn(Xt;x)\)\z1\ + Ch(l + \^n(X£x)\Y + y^) is Lipschitz continuous; therefore, the solution (r1n,Z1n) e S2T(R) x %2T(Rm) exists and is unique. Moreover, it follows from the result of El-Karoui et al. (see [8]) that T1n can be characterized through a deterministic measurable function g1n : [0, T] x Rm ^ R, that is, for any 5 e [t, T],

Y¡" = g ln(s, Xf).

Next let us consider the process

Bn = Bs - i'1[t,T](r)C2(1 + MXtx) |) sign(^1n) dr, 0 < 5 < T,

which is, thanks to Girsanov's theorem, a Brownian motion under the probability Pn on (fi,F) whose density with respect to P is ET := ET(/0TC2(1 + \^n(Xstx)\) sign(Z5ln) x 1[t,T] (5) dBs), where for any z = (z0i=1,...,d e Rm, sign(z) = (1[\z<\=0] )i=1,...,d and Et(■) is defined by

£(M) := (exp{Mt - {M)t/2})t

for any (Ft, P)-continuous local martingale M = (Mt)t<T. Then (3.6) becomes

Y]n = /(XT*) + / TCh(1+ |^n(xt^)|Y + |Y;ln|) dr - fT ~Z]ndBnr, 5 e [t, T]. (3.9)

Applying Ito-Meyer's formula to (eCht Ytln)+, t < T,we know

(eChtnn)+ + jt TdL0 = (eChY(xTx))+-^ Td(eChYn)

= (eChy(xT?))+ + fTCheChs(l + \Vn(X¡*)\y)ds- f

ds - / eChSZlndBns,

where L0 is the local time of the continuous semimartingale eChtYY}n at time 0 which is an increasing process. Therefore, the term ff dL0s is positive. Considering (3.7), we have

eChtgln(t,x) < (eChtgln(t,x))+

(eChTgl(xT,*))+ + JT CheChs(l + \^n(Xf) \Y)

where En is the expectation under the probability Pn. Taking the expectation on both sides under the probability Pn and taking account of gln is deterministic and f+ < \ | for any integrable function f, we obtain

(t, x) < e-ChtEn

\g1(XtTx)\+f CheChs(l+\Vn(xtx)\y)ds

Since g1 is of polynomial growth, and e Cht < 1 for t e [0, T ], we infer that, V(t, x) e [0, T ] x Rm,

g 1n(t, x) < CEn sup (1 + |Xf|Y) < CE Et ■ sup (1 + Xf^)

where the constant C depends only on T, Ch and Cg. Next, by a result of Haussmann ([23], p.14, see also [21], Lemma 3.1), there exist somep0 e (1,2) and a constant C which is independent of n such that E[|ET |p0] < C uniformly. As a result of Young's inequality and estimate (2.3), we have

g 1n(t, x) < Cp0\ E[|Et |p0 ] + e\ sup (1 + |xf|pp00-1 )! },

1 Lt<s<T JJ

which yields

g 1n(t,x) < C(1 + |x|X) with X = P0Y/(P0 -1) > 2.

(3.10)

Next, by the comparison theorem of BSDEs and property (3.2)(b), we actually have, for any s e [t, T ],

Yin;(tx) = g 1n(s,Xf) < Y}n = g 1n(s,Xtx),

and by choosing s = t,wegetthatg1n(t,x) < C(1+ |x|X),(t,x) e [0, T] x Rm.Inasimilarway, we can also show that for any (t,x) e [0, T] x Rm, g 1n(t,x) > -C(1 + |x|X). As a conclusion, g1n is of polynomial growth on (t, x) uniformly in n, i.e., there exist a constant C, which is independent of n, and X >2 such that

|g 1n(t,x)| < C(1 + |x|X). (.11)

Combining (3.11) and (2.3), we deduce that, for any a > 1, i = 1,2,

■t<s<T

Yin;(t,x) |

< C. (3.2)

On the other hand, by applying Ito's formula to (Yin;(tx))2 and considering the uniform estimate (3.12), we can infer in a regular way that, for any t e [0, T], i = 1,2,

/■ T

\Zisn;(t,x)\2 ds

< C. (3.13)

Step 3. For fixed a e Rm, there exists a subsequence of ((Ysln;(0,a), Zjln;(0,a))0<j<T)n>1 which converges in space S|(R) x H|(Rm) respectivelyto (Y/, Z])0<s<T, a solution of BSDE (3.1). Let us recall expression (3.5) for case i = 1 and apply property (3.2)(b) combined with the uniform estimates (3.12), (3.13), (2.3) and Young's inequality to show that, for 1 < q <2,

i F^s,X0a)|qds

iT (1 + |^n(Xs°,a) |)q|Z1n;(0,a) |q + (1+ MXs°,a)|Y q + |Ysln;(0,a)|q) ds J0

^Z-MM!2 ds

sup |Ysln;(0,a)|q 0<s<T s

Therefore, there exists a subsequence (nk} (for notation simplification, we still denote it by (n}) and a B([0, T]) <g> B(Rm)-measurable deterministic function F1(s,y) such that

F1n — F1 weakly in Lq{[0, T] x Rm; ¡(0, a;s, dy) ds).

(3.15)

Next we aim to prove that (g 1n(t,x))n>1 is a Cauchy sequence for each (t,x) e [0, T] x Rm. Now let (t,x) be fixed, n >0, k, n and m > 1 be integers. From (3.5), we have

g 1n(t,x)-g 1m(t,x)| = E j Fm(s,Xf) -F1m(s,Xf)ds

/- t+n

|Fm(s,Xl,x) - F1m(s,X^) | d

/ (Fm(s,Xt,x) -F1m(s,Xf))

fT (F1n(s, Xf) - F1m(s, Xf ))1( Xtx | >k} ds

-J t+n

where on the right-hand side, noticing (3.14), we obtain

/- t+n

|Fm(s,Xl,x) - F^s,XT) | d

< n q i e

< Cn q .

r^s,Xf) -F1m(s,X^ds

At the same time, Lemma 3.1 associated with the Lq-1 -domination property implies

fT (F1n(s, Xf) - F1m(s, Xf ))1{ Xt,x | <k} ds] -Jt+n J

= (F1n(s, n)-F1m(s, n))1( | n | <k}l(t, x; s, dn) ds

JRm Jt+n

= (F1n(s, n)-F1m(s, n))1( | n | <k}$t,x,a(s, n)l(0, a; s, dn) ds .

JRm Jt+n

Since 4>t,x,a(s, n) e Lq-1([t + n, T] x [-k, k]m; ¡1.(0, a; s, dn) ds),for k > 1, it follows from (3.15) that for each (t, x) e [0, T] x Rm, we have

fT (F1n(s, Xf) - F1m(s, Xf ))1{ |xf | <k} ds

— 0 as n, m — to.

Finally,

fT (Fln(s, Xf) - Flm(s, Xf))l{

{\xl^\>k}

L{\xr\>k}'

iT|Fi„(s,Xf) -Fim(s,Xf)|qds

< Ck~ q .

Therefore, for each (t,x) e [0, T] x Rm, (gln(t,x))n> is a Cauchy sequence, and then there exists a Borelian application gl on [0, T] x Rm such that for each (t, x) e [0, T] x Rm, limn^TO gln(t,x) = gl(t,x), which indicates that for t e [0, T], limn^TO Ytln,(0,a)(«) = gl(t,Xt0,a), P-a.s. Taking account of (3.l2) and Lebesgue's dominated convergence the-

orem, we obtain that the sequence ((Yt1"'(0,a))0<t<T)„>l converges to Y1 = (gl(t,Xtu,a))0<t<T

in LP ([0, T] x Rm) for any p > l, that is,

|Ytln;(M- YHPdt

^ 0, as n ^to.

Next, we will show that for any p > l, Zln;(0a) = ((zln(t,Xt0,a))0<t<T)n>l has a limit in HT(Rm). Besides, (Yln;(0,a))n>l is convergent in 5|(R) as well. We now focus on the first claim. For n, m > l and 0 < t < T, using Ito's formula with (Ytln - Ytlm)2 (we omit the subscript (0, a) for convenience) and considering (3.2)(b), we get

|| ln lm || 2 \1t Yt +

|Zln - Zlm|2 ds

= 2 j (Yln - Yslm)(Hln(s,Xs0'a, Ysln, Y2n,Zsln,Z2sn)

- Hlm (s, Xs0'a, Yslm, Y2m, Zlm, Zs2m)) ds -2 j (Yln - Yslm)(Zsln - Zslm) dBs

< cj |Yln - Yslm|[(|Zsln| + |Zlm|)(l +

+ |Ysln| + | Yslm| + (l + XMf ] ds -2 jfT (Yln - Yslm)(Zsln - Zslm) dBs.

Since for any x,y,z e R, \xyz\ < a \x\a + b \y\b + l\z\c with a + b + \ = l, then for any e > 0 we have

|| ln lm || 2 Yt - Yt +

|Zln - Zlm|2 ds

e2 fT o e4 rT

e ' 'Z^U^D2 ds + e

jt (l + |X?1)4

4? j,' lY'" " Y'"l* ds + 211 (

2 /r (l + iX0'a!)2' ds+2e jf № - Ys

2 jf T(Ysln - Yslm)(Zsln -Zslm) dBs.

Y m | + | Yl

lm || 2

Taking now t = 0 in (3.17), expectation on both sides and the limit w.r.t. n and m, we deduce that

limsup E

\Zln - Z-m|2 ds

, s2 s -

< a — + — + -

" 2 4 2

(3.18)

due to (3.13), (2.3) and the convergence of (3.16). As - is arbitrary, then the sequence (Z1n)n>L is convergent in HT to a process Z1.

Now, returning to inequality (3.17), taking the supremum over [0, T] and using BDG's inequality, we obtain that

sup | Y-n - Y-m |2 + / |Z-n - Z-m |2 ds

0<t<T Jo

. — s4 -1 1 < — + — + -\ + - E " 2 4 2 I 4

sup |Ytln - Ytlm|2

L0<t<T

|Z1n - Z1m|2 ds

which implies that

limsup E

sup |Ytln - Ytlm|2

0<t<T t t

since - is arbitrary and (3.18). Thus, the sequence of (Y1n)n>L converges to Y1 in Sj., which is a continuous process.

Finally, repeating the procedure for player i = 2, we have also the convergence of (Z2n)n>L (resp. (Y2n)n>L) in Hj (resp. Sj) to Z2 (resp. Y2 = ^2(.,X0x)). Step 4. The limit process (Ylt, Zlt)0<t<T, i = 1,2, is the solution of BSDE (3.1). Indeed, we need to show that (for case i = 1)

F- (t,X°°,a) = H- (s,X0a, Y-, Y2,Z-,Z2S) dt ® dP-a.e. For k > 1, we have

/Vn(s, Xs0,a, Ysln, Y2n, Z-n, Zs2n) - H-(s, Xs0a, Ys1, Y2, Z1, Zs2) | ds

fTHn (s,Xs0a, Ysln, Y2n, Z-n, Zs2n)

- H-(s, X°'a, Ysln, Ys2n, Zs1n, Zs2n) | • 1{|Ysln \+\Y}n |+|Z1n |+|Z2n |<k} ds

+ e\ f Hm(s,X0a, Ysln, Y2n,Z-n,Zs2n) -J0

- H-(s, Xs0,a, Ysln, Ys2n, Zs1n, Zs2^ | • 1[|Y-n|+|Y■2n|+|z1n|+|z2sn|->k} ds

/v (s,X0,a, Ysln, Y2n, Z1n, Zs2n) - H-(s,X0a, Ys1, Y2, Z1, Zs2)|

:= ¡I + In2 + I3n.

The sequence ¡H, n > 1, converges to 0. On the one hand, for n > 1, property (3.2)(b) in Step 1 implies that

|Hm(s,X0a, Y}ln, Y2n,Z]n,Z2n) -Hi(s,X0a, Yln, Y2n,Zsln,Zs2n) |

' 1{|Ysln|+|YS!n|+|Zi1n|+|z2n|<i)

— C2(1 + |X0-"+ Ch(1+ |Xs0,a|y + k).

On the other hand, considering property (3.2)(d), we obtain that |H1n(s,Xs0,a, Y}ln, Y2n, Z1n, Zs2n) - H1(s,Xs°,a, Y}ln, Y2n, Zs1n, Z2n) |

' '1{|Y}n|+|Y}n|+|Z}n|+|Z2n|<k)

— sup |Hn(}, X}0,a, y1, y2, z1, z2) - H1(}, X0,a, y1, y2, z1, z2 )|

(ylyizlzi) |y11+ |y21+ |z! |+ |z! |<k

as n ^ to. Thanks to Lebesgue's dominated convergence theorem, the sequence ¡1 of (3.19) converges to 0 in H^.

The sequence ¡1 in (3.19) is bounded by k2(?C1)/? with q e (1,2). Actually, from property (3.2)(b) and Markov's inequality, for q e (1,2), we get

¡2n — C^ E

fT (1 + |Xs0'a|)q|Zs1n|q + (1 + X'Y + | Y^fds

/ 1{|Y/n |+| Y2n|+|Z1n |+|Z2n |>k) 0

Z1n|2 ds

/ |Y1n|2 ds

f (1+ |Xs0,a|)

fT (1+Kiy

{E[/0T |Y}ln|2 + | Y2n|2 + |Z1n|2 + |Zs2n|2d s])^

(k2) ?

— 2(q-1) •

The last inequality is a straightforward result of estimates (2.3), (3.12) and (3.13).

The third sequence ¡31, n > 1, in (3.19) also converges to 0, at least for a subsequence. Actually, since the sequence (Z1n)n>i converges to Z1 in HT, then there exists a subsequence (Z1nk)k>1 such that it converges to Z1, dt ® dP-a.e.; and furthermore, supk>1 |Z1nk («)| e HT. On the other hand, (Y1nk )k>1 converges to Y1, dt <g> dP-a.e. Thus, by the continuity of function H1(t,x,y1,y2, z1, z2) w.r.t. (y1,y2, z1, z2), we obtain that

H (t X0,a Ylnk Y2nk Zlnk Z2nk) 1 \ , t ,t ,t ,t ,t /

> H1 (t,Xt0'a, Y), Y2,Z\,Z2t) dt <g> dP-a.e.

In addition, considering that

supHi(t,X0a, Y-nk, Y2nk,Z-nk,Z]nk) | e HqT(R) for 1 < q <2,

which follows from (3.14). Finally, by the dominated convergence theorem, one can get that

H-(t,X°'a, Y-nk, Y2nk,Z-nk,Z2nk) H-(t,X00*, Y— Yt2,Z-,Zt2) in HqT,

which yields the convergence of ¡n in (3.19) to 0.

It follows that the sequence (H-n(t,Xt0,a, Ytln, Y2n,Z-n,Z2n)0<t<T)n>-i converges to (H-(t, X°0,a, Y-, Y2,Z-,Zt2))0<t<T in £-([0, T] x dt<g>dP) andthenF-(t,X°0,a)=Hi(t,X°0,a, Y— Y2, Z-, Z2), dt < dP-a.e. In the same way, we have Fz(t,X°0,x) = H2(t,X°0,a, Y— Y2, Z-, Z2), dt < dP-a.e. Thus, the processes (Yi, Zi), i = 1,2, are the solutions of the backward equation (3.1). □

4 Generalizations

As we can see from Assumption 2.3(i), this model deals with BSDE (2.5) with stochastic linear growth generators. However, when x takes value of X0,a, the coefficient of the component yi in (2.4) is deterministic. Therefore, one possible generalization is to improve this model by considering the following assumption instead of Assumption 2.3.

Assumption 4.1

(i) For each (t, x, y1,...,yn, z1,...,zn) e [0, T] x Rm x Rn x Rnm, there exist a constant C and 0 < y <2 such that, for each i = 1,2,..., n,

|Hi(t,x,y-,...,yn,z1,...,zn)| < C(l+|x|)|zt| + C(l + |x|Y)|yi|; (4.1)

(ii) themapping(y-,...,yn,z-,...,zn) e Rn x Rnm Hi(t,x,y-,...,yn,z-,...,zn) e Ris continuous for any fixed (t, x) e [0, T] x Rm.

Then we will obtain a similar conclusion below. However, notice that we can only conveniently deal with the case when y is strictly smaller than 2.

Theorem 4.1 Leta e Rm befixed. Then, under Assumptions 2.1,2.2 and 4.1, there existtwo pairs of P-measurable processes (Yi, Zi) with values in R1+m, i = 1,2, and two deterministic functions g i(t, x) with the growth property as follows, i(t, x)| < eC(1+|x|Y) with 0 < y <2, i = 1,2, such that equations (3.1) hold true.

Sketch of the proof One can follow the same method of the proof of Theorem 3.1. We only point out here some difference in Step 2 related to the growth property of the deterministic function g1n.

In Step 2, we consider the following BSDE sequence: Yln = g-(XTx) + JT C(1 + ^nX*) |) |^-n |

+ C^+^X^)|Yrln|dr- iTZ-ndBr. (4.2)

After changing probability following the same line as (3.8), we obtain

Y1n = g1 (Xf) + J TC (1 + ^ (Xrtx) |Y) | Yrln| dr -J T Z1ndBn, s e [t, T]. (4.3) Considering Y1n = g1n (s,Xls,x) for the deterministic function g1n, we obviously have

g 1n(t, x) = En [g1 (XTVtTC(1+^n(Xrt,X)|Y) sign(Yrln)dr]

— En [eC SUP0—s—T (1+|Xst,X |Y) j = E[eC suP0—s—T (1+|Xst,X |Y) Et j

— eC (1+|x|Y).

This inequality is obtained from the integrability of ET in Lp0 for some p0 e (1,2) and the fact that (1 + |x|)< ex+|x| for any x. Therefore, by comparison theorem, we know gln(t,x) — eC(1+|x|Y). in a similar way, we can also show that g1n(t,x) > e-C(1+|x|Y).

Combining this growth property and the fact that Ysln:(t,x) = gln (s, Xsi,x), we conclude that, for any a >1 and 0 < y <2,

e\ sup |r;n;(t,x)N — C, t—s—T -I

E[ sup eC(1+|Xt,x|Y) 1 — C (4.4)

t—s—T -I

is true only for positive y strictly smaller than 2. When y touches 2 or bigger than 2, then estimate (4.4) is not necessarily true for arbitrary T > 0. In this case, (4.4) may explode, and this is the reason why we only consider y e (0,2) in Assumption 4.1. However, for Y > 2, we can still expect that (4.4) holds true for T small enough.

The rest of the proof will have no practical difficulties, therefore, we omit it.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

RM carried out the model formulation, the proof and the edit work. ZW reviewed and edited the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the anonymous reviewers for their useful comments and suggestions. The first author was supported in part by the Natural Science Foundation for Young Scientists of Jiangsu Province, P.R. China (No. BK20140299).The second author was supported in part by National Natural Science Foundation of China (11221061 and 61174092), 111 project (B12023), the National Natural Science Fund for Distinguished Young Scholars of China (11125102) and the Chang Jiang Scholar Program of Chinese Education Ministry.

Received: 8 January 2015 Accepted: 12 August 2015 Published online: 27 August 2015 References

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