Properties of Solutions of BSDEs with Integrable Parameters

Abstract

The main result of this study is to obtain, using the localization method in Briand et al. [3] . Levi, Fatou and Lebesgue type theorems for the solutions of certain one-dimensional backward stochastic differential equation (BSDEs) with integrable parameters with respect to the terminal condition. PreliminariesLet (Ω, F , P ) be a probability space carrying a standard d-dimensional Brownian motionFor t ∈ [0, T ] and real p > 0, let L p (Ω, F t , P ) denote the set of all F t -measurable random variable ξ such that E|ξ| p < +∞. Let S p denote the set of R R R-valued, F t -adapted and càdlàg process {X t , t ∈ [0, T ]} such thatand let S ∞ denote the set of predictable bouned processes. Let M p denote the set of F t -In this paper, we are concerned with the following one-dimensional BSDE:where the random function g (ω, t, y, z) :termed the generator of BSDE(1), and ξ is an F T -measurable random variable termed the terminal condition. (ξ, g) is called the parameters of BSDE(1). A solution Manuscript

L p (p>1) solutions for one-dimensional BSDEs with linear-growth generators

“…A Lebesgue type theorem on L 1 solutions is also obtained in this section (see Theorem 4). We mention that Theorems 3 and 4 improve, in some sense, the main results in Fan [8].…”

“…In the last section, by virtue of Theorem 1 in Fan and Jiang [9], we establish a general comparison theorem on L 1 solutions when the generator g is weakly monotonic in y and uniformly continuous in z as well as it has a stronger sublinear growth in z (see Theorem 9), which improves the corresponding results in Fan and Liu [11] and Xiao, Li and Fan [22]. As a byproduct, we also obtain a general existence and unique theorem on L 1 solutions when g is stronger continuous in (y, z), monotonic in y and uniformly continuous in z as well as it has a general growth in y and a stronger sublinear growth in z (see Theorem 10), which also extends, in some sense, the corresponding results in Fan and Liu [11], Xiao, Li and Fan [22] and Briand, Delyon, Hu, Pardoux and Stoica [4] (see Remark 11).…”

“…We would like to mention that, to our knowledge, (H2), (H3) together with (H4'), and (H4") are, respectively, put forward at the first time in Peng [20], Briand, Delyon, Hu, Pardoux and Stoica [4] and Fan and Liu [11]. But, (H1) and (H4) are new.…”

“…For this purpose, similar to Theorem 4.1 in Briand, Lepeltier and San Martin [6] we construct a sequence of g n to approach the generator g. However, we would like to mention that the sequence g n is obtained by "the infinite evolution" made between g and |z| α with 0 < α < 1, but not between g and |z| as usual (see Proposition 1 and Remark 2). At the same time, in order to deal with the general growth of g in y, we use two stopping time sequences {τ k } and {σ m } different from not only those in Theorem 2.1 of Fan [8] but also those in Theorem 4.1 of Briand, Lepeltier and San Martin [6]. The use of these two stopping time sequences allow us to eliminate the additional continuity assumptions employed in two results quoted before.…”

Jensen's inequality for filtration consistent nonlinear expectation without domination condition

“…Moreover, from Theorem 2.1 in [9], we can also know that this nonlinear expectation E[·] satisfies the continuous assump-…”