Infinite time interval backward stochastic differential equations with continuous coefficients

In this paper, we study the existence theorem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document}Lp \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1<p<2)$$\end{document}(1<p<2) solutions to a class of 1-dimensional infinite time interval backward stochastic differential equations (BSDEs) under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document}Lp \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1<p<2)$$\end{document}(1<p<2) solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Theorem 3.1 in Zong (Turkish J Math 37:704–718, 2013).

was continuous, it had linear growth, and the terminal condition was square integrable. They also obtained the existence of a minimal solution. Chen and Wang (2000) obtained the existence and uniqueness theorem for L 2 solutions of BSDEs with non-uniformly Lipschitz coefficients when T ≡ ∞, by the martingale representation theorem and fixed point theorem. In fact, such a problem has been investigated by Peng (1990), Pardoux (1997), Darling and Pardoux (1997) and other researchers under the assumption that terminal value ξ = 0 or E[e pρT |ξ | p ] < ∞ for some constant ρ and random terminal time T (i.e., T is a stopping time). But in L p (1 < p < 2), there is no the martingale representation theorem. Zong (2013) studied L p solutions to infinite time interval BSDEs with non-uniformly Lipschitz coefficients. She gave a new a priori estimate. By using this a priori estimate, she got the existence and uniqueness of L p solutions to infinite time interval BSDEs.
In this paper, we study the existence theorem for L p (1 < p < 2) solutions to a class of 1-dimensional infinite time interval BSDEs under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem for L p (1 < p < 2) solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Theorem 3.1 in Zong (2013).
This paper is organized as follows. In "Preliminaries" section, we introduce some notations, assumptions and lemmas. In "Main results and proofs" section, we give our main results including the proofs.

Preliminaries
In this section, we shall present some notations, assumptions and lemmas that are used in this paper.
Notation. The Euclidean norm of a vector x ∈ R k will be denoted by |x|, and for a k × d matrix A, we define ||A|| = √ TrAA * , where A * is the transpose of A. Let (�, F, P) be a completed probability space, (W t ) t≥0 be a d-dimensional standard Brownian motion defined on this space and (F t ) t≥0 be the natural filtration generated by Brownian motion (W t ) t≥0 , that is where N is the set of all P-null subsets. Furthermore, we define F := σ t≥0 F t .
We consider the following spaces: In the sequel, we assume that 1 < p < 2.

Consider the following infinite time interval BSDE
Let such that for any (y, z) ∈ R k × R k×d , g(·, y, z) is F t -progressively measurable. We make the following assumptions: (A.1) There exist two positive non-random functions α(t) and β(t), such that for all There exist two positive non-random functions α(t) and β(t), such that for all y 1 , y 2 ∈ R k , z 1 , z 2 ∈ R k×d , where α(t) and β(t) satisfy that 3) Linear growth: There exists a positive non-random function γ (t) such that where γ (t) satisfies that ∞ 0 γ (t)dt < ∞, ∞ 0 γ 2 (t)dt < ∞; (A.4) For fixed ω and t, g(ω, t, ·, ·) is continuous.

Main results and proofs
In this section, first we study the existence and uniqueness theorem for L p solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Lemma 1.

Lemma 3 Suppose that (A.1) holds for g. Furthermore, each
Then there exists a positive constant C p depending only on p such that, for any τ ∈ [0, T ], It follows that where c p is a positive constant depending only on p. By the Burkholder-Davis-Gundy inequality, we get where d p is a positive constant depending only on p. From (8) and (10) where D p is a positive constant depending only on p. From the Lipschitz assumption (A.1) on g, we have where M p is a positive constant depending only on p. From (14) and (15), we have where C ′ is a positive constant depending only on p.
Combining (11) with (16), we get where C p is a positive constant depending only on p. The proof of Lemma 3 is complete.
Theorem 4 (Comparison Theorem) Assume that k = 1. We make the same assumptions on ξ, g and ξ , g as in Theorem 2. Let Y , Z be a solution of BSDE If we suppose that: then where W i t is the ith components of W t . Let us consider the following BSDEs  (24), we can obtain Ŷ t ≤ 0, a.s. and if ξ = ξ a.s., g(t, Y t , Z t ) = g(t, Y t , Z t ) a.s., then Y t = Y t a.s.. Choosing t = 0 in (24) and from the strict monotonicity of E[·], we can obtain that if Y 0 = Y 0 , then ξ = ξ a.s., g(t, Y t , Z t ) = g(t, Y t , Z t ) a.s.. The proof of Theorem 4 is complete. Now we prove the existence theorem for L p solutions of 1-dimensional infinite time interval BDSDEs which generalizes Theorem 1 in Lepeltier and San Martin (1997).
In order to prove Theorem 5, we need the following lemmas. (ω, t, y, z)
We also define the function For each given ξ ∈ L p (�, F, P, R), by Theorem 2, there exist two pair of processes (Y n , Z n ) and (U, V), which are the solutions to the following BSDEs respectively. From Theorem 4 and Lemma 6, we get Lemma 7 There exists a constant A > 0 independent of n, such that Proof Since (U, V) is the solution of BSDE (26), there exists a constant B > 0 independent of n, such that From Inequality (27), we can obtain that for each n ∈ N, g n (ω, t, y, z) := inf y ′ ,z ′ ∈Q g(ω, t, y ′ , z ′ ) + nγ (t) y − y ′ + z − z ′ , g n (ω, t, y, z) − g n (ω, t, y ′ , z ′ ) ≤ nγ (t) y − y ′ + z − z ′ ; g n (ω, t, y n , z n ) → g(ω, t, y, z), as n → ∞.