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

Abstract

We establish several existence and uniqueness results for L p (p > 1) solutions of reflected BSDEs with two continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. To get the existence a necessary and sufficient condition relating the growth of generator with that of barriers is given. We also prove that the L p solutions can be approximated by the penalization method and by some sequences of the L p solutions of reflected BSDEs with two barriers.

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

“…The rest proof of this proposition is divided into 3 steps, some of ideas among them are lent from the proof of Proposition 5.1 in Fan [18].…”

“…In the existence and uniqueness results for L 2 solutions or L p (p > 1) solutions established respectively in Fan and Jiang [21], Fan [12], Fan [15], Fan [13] and Fan [18], the monotonicity condition of g in y was further weakened to the one-sided Osgood condition (see assumption (H1)(i) in Subsection 2.3) and the weak monotonicity condition, which both unify the monotonicity condition, the Mao's non-Lipschitz condition (see Mao [42]) and the usual Osgood condition (see Fan et al [23]).…”

“…[5], Xu [54], Fan [15] and Fan [18], besides the (weak) monotonic condition in y and the continuity condition in (y, z), a general growth in y and a linear growth in z (see assumption (HH) with α = 1 in Subsection 2.3) is the only requirement for the generator g, and in Lepeltier and San Martin [39],…”

“…Recently, several generalized Mokobodzki conditions, see (ii) of assumptions (H3), (H3L) and (H3U) in Subsection 2.3 for the case of L 1 solution, were put forward and proved to be sufficient and necessary to ensure existence of an L p (p > 1) or L 1 solution for a RBSDE with one or two barriers when the generator g has a general growth in y, see Klimsiak [36], Klimsiak [37], Fan [15] and Fan [18] for more details. Many efforts in this direct can also be found in Lepeltier et al [38], Xu [53], Xu [54], Rozkosz and S lomiński [52], Li and Shi [40] and references therein.…”

“…g) in (y, z) used in Theorem 4.3, Theorem 5.3, Theorem 5.4, Theorem 6.3 and Theorem 6.4 (resp. Corollary 5.6) can be relaxed to the left-continuity and lower semi-continuity condition in case of the minimal L 1 solution and the right-continuity and upper semi-continuity condition in case of the maximal L 1 solution, with a similar argument as in Fan and Jiang [20], Fan [14] and Fan [18]. The same is in Corollary 4.4, Corollary 5.5 and Corollary 6.5; 8) Since the associated assumptions are more general, the results of this paper strengthen some known corresponding works on the L 1 solutions obtained, for example, in 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] and Bayraktar and Yao [2]; 9) Under assumptions (H1)(i), (HH) and (H3L) (resp.…”