Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression
arXiv:1601.01186 · doi:10.3150/14-BEJ667
Abstract
We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the -component by (Ann. Appl. Probab. 12 (2002) 1390-1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.
arXiv commentPublished at http://dx.doi.org/10.3150/14-BEJ667 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
References in corpus (3)
Cited by in corpus (18)
- Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
- Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
- On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
- Deep splitting method for parabolic PDEs
- An overview on deep learning-based approximation methods for partial differential equations
- Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs
- Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
- Numerical methods for backward stochastic differential equations: A survey
- Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
- Generalised multilevel Picard approximations
- Quasi-Regression Monte-Carlo scheme for semi-linear PDEs and BSDEs with large scale parallelization on GPUs
- Numerical approximation of singular Forward-Backward SDEs
- Novel multi-step predictor-corrector schemes for backward stochastic differential equations
- Full-Projection explicit FBSDE scheme for parabolic PDEs with superlinear nonlinearities
- Quantization-based approximation of reflected BSDEs with extended upper bounds for recursive quantization
- A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs
- Stability of backward stochastic differential equations: the general case
- Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators