Representation theorem for generators of BSDEs with monotonic and polynomial-growth generators in the space of processes

In this paper, on the basis of some recent works of Fan, Jiang and Jia, we establish a representation theorem in the space of processes for generators of BSDEs with monotonic and polynomial-growth generators, which generalizes the corresponding results in Fan (2006, 2007), and Fan and Hu (2008).


Introduction
By Pardoux and Peng (1990), we know that there exists a unique square-integrable and adapted solution to a backward stochastic differential equation (BSDE for short in the remainder of this paper) of the type y s = ξ + T s g (u, y u , z u provided that g is Lipschitz in both variables y and z and that ξ and (g(t, 0, 0)) t∈ [0,T ] are square integrable. The g is called the generator of BSDE (1), ξ the terminal data and the triple (ξ, T, g) the parameters of BSDE (1). We denote the unique solution by ( y ξ,T,g s , z ξ,T,g s ) s∈ [0,T ] , and often denote y ξ,T,g t by g t,T [ξ] for every t ∈ [0, T ]. One of the achievements of BSDE theory is the comparison theorem. Recently, many papers have been devoted to studying the converse comparison theorem. For studying the converse comparison theorem, Briand et al. (2000) established the following representation theorem of generators for BSDEs in the space of random variables: For every (t, y, z) ∈ [0, T [×R 1+d , lim n→∞ n{ g t,t+1/n [ y + z · (B t+1/n − B t )] − y} = g (t, y, z) (2) holds true in (the space of random variables) L 2 when g satisfies two additional assumptions that E sup 0≤t≤T |g(t, 0, 0)| 2 < ∞ and (g(t, y, z)) t∈ [0,T ] is continuous in t for every ( y, z). Since then, much effort has been made to weaken and eliminate these two assumptions mentioned above. For instance, after weakening these two assumptions step by step in Jiang (2005a, b, c), under the most elementary conditions that g is Lipschitz in both variables y and z and that ξ and (g(t, 0, 0)) t∈ [0,T ] are square-integrable, Jiang (2006Jiang ( , 2008 finally proved that (2) holds true in (the space of random variables) L p (1 ≤ p < 2) for almost every t ∈ [0, T ). Furthermore, under a continuity condition in t on stochastic differential equations (SDEs in short), Jiang (2005d) generalized this result to the case where the terminal data of BSDEs are solutions of the SDEs.
On the other hand, from the point of view of Fan and Hu (2008), it seems to be more appropriate for this kind of representation theorem to be investigated in the space of processes rather than in the space of random variables, that is to say, without fixing t, we are to investigate whether (2) holds in some kinds of spaces of processes. Accordingly, Fan (2006Fan ( , 2007 and Fan and Hu (2008) investigated this kind of representation theorem in the space of processes and eliminated the above continuity condition in t on SDEs used in Jiang (2005d).
Furthermore, Mao (1995) established an existence and uniqueness result of solutions for BSDE (1) where g satisfies a non-Lipschitz condition in y, the corresponding representation theorem in L p (1 ≤ p < 2) was established in Liu and Jiang (2008). Lepeltier and San Martin (1997) proved the existence and uniqueness of the minimal and maximal solutions for BSDE (1) where (g(ω, t, 0, 0)) t∈[0,T ] is a bounded process and g is continuous with linear growth in ( y, z), the corresponding representation theorem in L p (1 ≤ p < 2) was obtained in Jia (2008). Very recently, Fan and Jiang (2010a) extended the existence and uniqueness result in Lepeltier and San Martin (1997) by eliminating the condition that (g(ω, t, 0, 0)) t∈[0,T ] is a bounded process, the corresponding representation theorem in the space of processes has also been established in Fan and Jiang (2010b).
It should be noted that all these representation results dealt with the case that the generator g is of linear growth in y. In this paper, we are the first time to consider the case that g is of polynomial growth in y. More precisely, on basis of the existence and uniqueness result of the minimal and maximal solutions for BSDE (1) obtained in Briand et al. (2007), we establish a new representation theorem in the space of processes, where the generator g is continuous in ( y, z) and monotonic in y, it has a polynomial growth in y and a linear growth in z, and the terminal data are solutions of SDEs (see Theorem 1 in Section 2). This representation theorem further generalizes the corresponding results in Fan (2006Fan ( , 2007 and Fan and Hu (2008).
Finally, we would like to mention that the representation theorem has been playing an important role in investigating properties of generators of BSDEs by virtue of solutions of BSDEs. In fact, a lot of problems in BSDE theory and nonlinear mathematical expectation theory are related to the above representation theorem. For example, it was just with the help of the representation theorem that many important results have been obtained in Briand et al. (2000), Chen et al. (2003), Jiang and Chen (2004), Jiang (2004Jiang ( , 2005aJiang ( , b, c, d, 2006Jiang ( , 2008, Fan (2006Fan ( , 2007, Fan and Hu (2008) and Fan and Jiang (2010b).
This paper is organized as follows: In section 2, after introducing some notations and assumptions, we put forward our main result-Theorem 1. Section 3 is devoted to the proof of the main result. Finally, some applications are given in Section 4.

Notations, assumptions and the main result
Let (Ω, , P) be a probability space carrying a standard d-dimensional Brownian motion (B t ) t≥0 , and let ( t ) t≥0 be the σ-algebra generated by B augmented by the P-null sets of . Then ( t ) t≥0 is right continuous and complete. Let T > 0 be a given real number. In this paper, we always work in the space (Ω, T , P), and only consider processes indexed by t ∈ [0, T ]. For every n ∈ N, let |z| denote the Euclidean norm of z ∈ R n . R m×d is identified with the space of real matrices with m rows and d columns, and if z ∈ R m×d , we have |z| 2 = trace(zz * ). For every p ∈ [1, 2] and 0 ≤ t 1 ≤ t 2 ≤ T , we define the following space of processes: It is well known that n p (t 1 , t 2 ) is a Banach space endowed with the norm · p . For simplicity, n p (0, T ) is also denoted by n p . Let b (ω, t, x) : σ(ω, t, x) : Ω × [0, T ] × R m → R m×d be two functions such that for any x ∈ R m , b(·, x) and σ(·, x) are ( t )-progressively measurable. Let b and σ also satisfy the following hypotheses (H1) and (H2): (H1) There exists a constant K 1 ≥ 0 such that dP × dt − a.e., (H2) There exists a constant K 2 ≥ 0 such that dP × dt − a.e., Given (t, x) ∈ [0, T ] × R m , by classical SDE theory, the following SDE: has a unique s-continuous solution, denoted by (X t,x s ) s∈ [0,T ] , with the properties that (X t,x s ) s∈[0,T ] is ( s )-adapted and for every β ≥ 1, where the constant C β depends on x, β, K 1 , K 2 , T .
In this paper, the generator g of a BSDE is a function g (ω, t, y, z) : The following Proposition 1 comes from Theorem 4.1 in Briand et al. (2007).
Proposition 1 Let the generator g satisfy the following assumptions: (A3') There exists a constant A ≥ 0, a nonnegative continuous process (g t ) t∈[0,T ] which belongs to 1 β for some β > 1 and a nondecreasing continuous function ϕ : Then, for every ξ ∈ L β (Ω, T , P), the BSDE with parameters (ξ, T, g) has a unique minimal solution Remark 1 Theorem 4.1 in Briand et al. (2007) pointed out that there also exists a unique maximal In the remainder of this paper, for notational simplicity, for each t ∈ [0, T ] and n ∈ N, we denote (t + 1/n) ∧ T by t n , and (t + 1/n k ) ∧ T by t n k . Furthermore, we fix a constant α ≥ 1 and always assume that the g satisfies (A1), (A2) and the following assumption (A3): (A3) There exists a constant C ≥ 0 and a nonnegative continuous process ( f t ) t∈[0,T ] which belongs to 1 2α such that dP × dt − a.e., Let g satisfy (A1), (A2) and (A3). Given (x, y, q) ∈ R m+1+m . For every t ∈ [0, T ] and n ∈ N, in view of (4) and the fact that 2α ≥ 2, it follows from Proposition 1 with β = 2 that the following BSDE: has a unique minimal solution in the space 1 Moreover, it follows from (4) and Proposition 1 with β = 2α that this solution also belongs to the Remark 2 In view of the definition of t n , we know that for every n ∈ N, the random variable X [0,T ] are both well defined. This is exactly why we let t n = (t + 1/n) ∧ T .
With respect to the above sequence of processes, we have the following conclusion which is the main result of this paper.
Theorem 1 (Representation Theorem I) Let (A1), (A2) and (A3) hold true for the generator g; let (H1) and (H2) hold true for b and σ. Then for every (x, y, q) ∈ R m+1+m and every p ∈ [1, 2), the following equality holds true in the space of processes Moreover, if the process then (6) holds true in the space of processes 1 2 and (7) also holds true.
By letting m = d, b ≡ 0, σ ≡ 1 and q = z in Theorem 1, the following Theorem 2 follows immediately.
Theorem 2 (Representation Theorem II) Let (A1), (A2) and (A3) hold true for the generator g. Then for every ( y, z) ∈ R 1+d and every p ∈ [1, 2), the equality holds true in the space of processes 1 p . And, there exists a subsequence {n k } ∞ k=1 such that dP × dt − a.e., lim Moreover, if (8) is also satisfied, then (9) holds true in the space of processes 1 2 and (10) also holds true.

Proof of the main result
This section aims at giving a proof of our main result-Theorem 1. let us first introduce some Lemmas which will play important roles in the proof of Theorem 1. The following Lemma 1 is a direct corollary of Proposition 3.2 in Briand et al. (2003).
Lemma 3 Assume that f (·) : R k → R with k ∈ N is a continuous function with polynomial growth, i.e., there exists constantsK 1 ,K 2 ≥ 0 and β ≥ 1 such that Let f n be the function defined as follows: Then the sequence of functions f n is well defined for every n ≥ 1, and it satisfies: (16) does not contain the constantK 2 in (15). This fact will be made full use of in the proof of the following Proposition 2, which explains why we use (15) rather than the usual expression (i.e., | f (x)| ≤ K(1 + |x| β ), ∀ x) although they are equivalent.

Remark 5
The case of β = 1 in Lemma 3 has been proved in Lepeltier and San Martin (1997). In addition, in view of the continuity of the f , we can use Q k instead of R k in (16).

Hence, (iii) follows and the proof of Lemma 3 is complete.
With Lemma 3 in hand, we can establish the following proposition which will play a key role in the proof of Theorem 1.
Furthermore, it follows from Lebesgue's dominated convergence theorem that the above limit also holds true in the process space 1 2α . On the other hand, it is clear that for every n ∈ N and (ȳ,z) ∈ R 1+d , which is the desired result. The proof of Proposition 2 is complete. Now we are in a position to prove our main result-Theorem 1.
The Proof of Theorem 1. Given (x, y, q) ∈ R m+1+m and p ∈ [1, 2). For notational simplicity, we denote the unique solution of SDE (3) by (X t s ) s∈ [0,T ] for every t ∈ [0, T ], and denote the minimal solution of BSDE (5) Let By letting s = t in (18) and then taking the conditional expectation with respect to t , it follows Thus, in view of the relation between the moment convergence and almost sure convergence, for completing the proof of Theorem 1 it suffices to prove that the right hand side of equality (19) tends to 0 in the space of process 1 p as n → ∞, and that if (8) also holds true, then the right hand side of equality (19) tends to 0 in 1 2 as n → ∞. First, it should be noted that the following statement has been proved in Fan and Hu (2008)(see (3.11) in Fan and Hu (2008)): Second, it follows from (3.16) and (3.19) in Fan and Hu (2008) that and (22) Since g satisfies (A3) and σ satisfies (H2), it is not difficult to verify that the process (g t, y, σ * (t, x)q ) t∈[0,T ] belongs to 1 2α and then 1 2 . Then, it follows from the absolute continuity of integral that the second term of the right hand side of (22) tends to zero as n → ∞. Applying (12) with φ(t) = g t, y, σ * (t, x)q yields that the first term of the right hand side of (22) also tends to zero as n → ∞. Thus, we have and then the second term of the right hand side of (21) tends to zero as n → ∞. Furthermore, applying (13) with φ(t) = g t, y, σ * (t, x)q yields that the first term of the right hand side of (21) also tends to zero as n → ∞. Consequently, we can conclude that Third, let us prove that It follows from Proposition 2 that there exists a non-negative process sequence {(ψ k (t)) t∈[0,T ] } ∞ k=1 in 1 2α depending on (x, y, q) such that lim k→∞ ψ k (t) 2α = 0 and for every k ∈ N, dP × dt − a.e., P t,n u := g u, Y t,n u + y + q · (X t u − x), Z t,n u + σ * (u, X t u )q − g u, y, σ * (u, where the constantC = C(1 + |q|K 2 ) and the constant C 1 depends only on (α, q,C). Note that we also have lim k→∞ ψ k (t) 2 = 0. By Fubini's Theorem, Jensen's inequality and Hölder's inequality, we can deduce that then, it follows from (26) that there exists a constant C 2 > 0 depending only on C 1 such that for every k ∈ N, Furthermore, it follows from (4) and (H2) , and from (18) that it solves the following BSDE: where for every (ω, u,ỹ,z) ∈ g(u,ỹ,z) .

Some Applications
In this section, we will give some applications relating to Theorem 1 and Theorem 2. The following Theorem 3 gives a converse comparison theorem for generators of BSDEs with monotonic and polynomial-growth generators.
Like the representation theorem for generators of BSDEs with Lipschitz generators, Theorem 2 can be used to investigate properties of generators of BSDEs with monotonic and polynomial-growth generators by virtue of their solutions. The following Theorem 4 and Theorem 5 are two specific examples which are both direct corollaries of Theorem 2. Some further results can be obtained like Section 2.3.2 in Jia (2008).

Remark 6
It is easy to see that if the minimal solutions of BSDEs are replaced by the maximal solutions (see Remark 1), Theorems 1-5 also hold true.