CIMPA Research School - Algeria

Khaled Bahlali

Abstract: BSDEs (backward stochastic differential equations), which are equations with prescribed terminal data, are related to semilinear and quasilinear PDEs (partial differential equations), financial mathematics, stochastic control and game theory.
Due to difficulties in locating solutions, few results have been obtained on the solvability of multidimensional BSDEs under local assumptions on the generator. However, many results have been obtained in dimension one using comparison methods.
Recently, an important result was obtained (in [1]) on the solvability of one-dimensional BSDEs reflected between two continuous barriers. This result expresses that if we can pass a continuous semimartingale between two given barriers, then we can pass at least one solution of a quadratic reflected BSDE.
The purpose of this course is to show how one can obtain (from [1]) almost all the previous results on one-dimensional BSDEs (without barriers), and go beyond. This includes BSDEs with linear, logarithmic, and also quadratic growths.

[1] Essaky, E. H.; Hassani, M. Generalized BSDE with 2-reflecting barriers and stochastic quadratic growth. J. Differential Equations, 254 (2013), no. 3, 1500--1528.

Lecture notes: first part, second part, third part, and fourth part.