Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the form $Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$ with bounded terminal conditions $ξ$ 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.

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