Error Criteria for Numerical Solutions of Backward SDEs

Many option pricing and portfolio selection problems in mathematical finance can be reformulated in terms of backward SDEs (BSDEs). As the corresponding BSDE can rarely be solved in closed form, simulation of BSDEs is of prime importance. However, the quality of the simulated solution depends on the interplay of different error sources, such as the time discretization error, the simulation error, and e.g. the choice of basis functions (if the conditional expectations are estimated by least squares Monte Carlo). In this paper we suggest error criteria which can be calculated explicitly in terms of the simulated solutions. Under suitable conditions the convergence behaviour of these observable error criteria can be linked to the approximation error between the simulated solution and the unknown true solution. Finally, we illustrate how the error criteria can be applied to judge the quality of simulated solutions for a non-linear option pricing problem. We also suggest some kind of non-linear control variates which reduces the numerical error considerably in our simulation results