General Existence Results for Reflected BSDE and BSDE
Provides theoretical foundations for BSDEs used in finance, control, and game theory, but the results are incremental extensions of existing work.
The paper proves existence of solutions for generalized reflected and non-reflected BSDEs with generators having general growth in y and stochastic quadratic growth in z, for both bounded and unbounded terminal conditions and barriers.
In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator $fds + gdA_s$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. We deal with the case of a bounded terminal condition $ξ$ and a bounded barrier $L$ as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in \cite{EH}. The work is suggested by the interest the results might have in finance, control and game theory.