Multidimensional quadratic and subquadratic BSDEs with special structure

Abstract

We study multidimensional BSDEs of the form $$ Y_t = \xi + \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s $$ with bounded terminal conditions $\xi$ and drivers $f$ that grow at most quadratically in $Z_s$. We consider three different cases. In the first one the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver $f$ in $Z_s$ is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending the solution recursively.