Two algorithms for the discrete time approximation of markovian backward stochastic differential equations under local conditions

Abstract

Two discretizations of a class of locally Lipschitz Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes Z, and the second a novel scheme based on Malliavin weights which approximates the marginals of the process Z directly. Extending the representation theorem of Ma and Zhang [24] leads to advanced a priori estimates and stability results for this class of BSDEs. These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in the time-grid. The class of BSDEs considered includes Lipschitz BSDEs with fractionally smooth terminal condition, thus extending the results of [15], quadratic BSDEs with bounded, Hölder continuous terminal condition (for bounded, differentiable volatility), and BSDEs related to proxy methods in numerical analysis.