L1 solution to scalar BSDEs with logarithmic sub-linear growth generators ✩

Shengjun Fan (1) , Ying Hu (2) , Shanjian Tang (3)

1 CUMT - China University of Mining and Technology
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Fudan University [Shanghai]

By developing the test function method and combining the localization technique, we prove existence of an L 1 solution to a one-dimensional backward stochastic differential equation (BSDE for short) with L 1 terminal condition when the generator g has a one-sided linear growth in the first unknown variable y and a logarithmic sub-linear growth in the second unknown variable z, which improves some existing results. A new idea to study existence of an adapted solution to a BSDE is given. When the generator g additionally satisfies a one-sided Osgood condition in y and a logarithmic uniform continuity condition in z, we further establish a comparison theorem for the L 1 solutions to the above BSDEs, which yields immediately the uniqueness of the solution.

Mots clés: Backward stochastic differential equation; L1 solution; Existence and uniqueness; Logarithmic sub-linear growth generator; Comparison theorem

Domaines: Probabilités [math.PR]

Shengjun Fan, Ying Hu, Shanjian Tang. L1 solution to scalar BSDEs with logarithmic sub-linear growth generators ✩. Systems & Control Letters, 2023, 177, pp.article n° 105553. ⟨10.1016/j.sysconle.2023.105553⟩. ⟨hal-03681198⟩