Existence, uniqueness and approximation for L p solutions of reflected BSDEs with generators of one-sided Osgood type

Abstract

We establish a general existence and uniqueness result of L 1 solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator g satisfying a one-sided Osgood condition as well as a general growth condition in y, and a Lipschitz condition together with a sublinear growth condition in z, which improves some existing results. In particular, we put forward and prove a stability theorem of the L 1 solutions for the first time. A new type of L 1 solution is also investigated. Some delicate techniques involved in the relationship between convergence in L 1 and in probability and dividing appropriately the time interval play crucial roles in our proofs.

$L^p$ solutions of doubly reflected BSDEs under general assumptions

“…Enlightened by these works, particularly by Klimsiak [38] and Fan [16], in this paper we will establish several general existence and uniqueness results for L p (p > 1) solutions of reflected BSDEs with two continuous barriers under weaker assumptions, which improves considerably some corresponding works.…”

“…Finally, we would like to mention that the results obtained in this paper can be regarded as a generalization of Klimsiak [38] in the sense that the conditions required for the generator are greatly relaxed, and also as a generalization of Fan [16] from reflected BSDEs with one continuous barrier to those with two continuous barriers. 2…”

“…Preprint submitted to ArXiv September 5, 2018 RBSDEs with data that are in L p (p > 1) and L 1 , see [6], [7], [9], [12], [15], [33] and [44] for non-reflected BSDEs, and [1], [2], [28], [37], [38] and [55] for reflected BSDEs; and more papers focused their attention on weakening the linear growth condition and Lipschitz-continuity condition of the generator g with respect to variables y and z, see [5], [6], [7], [12], [15], [17], [18], [19], [20], [31], [33], [34], [35], [44], [45] and [47] for non-reflected BSDEs, and [1], [2], [3], [13], [14], [16], [24], [26], [29], [32], [36], [37], [38], [39], [40], [41], [46], [55],…”

$L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

“…During the evolution of BSDE theory, many papers have also been interested in the existence and uniqueness of the L 1 solutions for non-reflected BSDEs and RBSDEs with only integrability data, see, for example, El Karoui et al [11], Briand et al [3], Briand and Hu [4], Fan and Liu [24], Fan [17], Klimsiak [36], Rozkosz and S lomiński [52], Fan [16], Bayraktar and Yao [2], and Hu and Tang [32] for details, where generally speaking (except [32]), an additional sub-linear growth condition in z (see assumption (H2)(ii) or (H2')(ii) in Subsection 2.3) needs to be satisfied by the generator g.…”

Existence, uniqueness, comparison theorem and stability theorem for unbounded solutions of scalar BSDEs with sub-quadratic generators

“…(1.2) with α ∈ (0, 1) is satisfied for g. It follows from Briand et al [3] that for (ξ, g 0 ) ∈ L 1 × L 1 , BSDE(ξ, g) admits a solution (Y • , Z • ) such that Y • belongs to class (D), and the solution is unique when g further satisfies the uniformly Lipschitz condition in (y, z). See for example Briand and Hu [5] and Fan [13,14] for more details on this topic.…”