Solving BSDEs based on novel multi-step schemes and multilevel Monte Carlo

Qiang Han, Shaolin Ji

Abstract: For backward stochastic differential equations (BSDEs), we construct a fully explicit multi-step time-discretization scheme and prove its stability and convergence rates. To approximate conditional expectations in our scheme, we design a new algorithm based on the multilevel Monte Carlo (MLMC) method. This estimator can achieve a prescribed error ϵ<math><mi is="true">ϵ</mi></math> with a computational effort of order ϵ−2<math><msup is="true"><mrow is="true"><mi is="true">ϵ</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">2</mn></mrow></msup></math>. Numerical experiments are given to illustrate the theoretical results of the proposed methods.