A Nonintrusive Stratified Resampler for Regression Monte Carlo: Application to Solving Nonlinear Equations

Our goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that the model for X is not known in full detail and only a root sample X1, . . . , XM of such process is available. By a stratification of the space and a suitable choice of a probability measure ν, we design a new resampling scheme that allows to compute local regressions (on basis functions) in each stratum. The combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation (possibly in large dimensions) using only a relatively small set of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates in L2(ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root paths.

[1]  Robert Denk,et al.  A forward scheme for backward SDEs , 2007 .

[2]  John N. Tsitsiklis,et al.  Regression methods for pricing complex American-style options , 2001, IEEE Trans. Neural Networks.

[3]  Cornelis W. Oosterlee,et al.  Pricing high-dimensional Bermudan options using the stochastic grid method , 2012, Int. J. Comput. Math..

[4]  E. Gobet,et al.  Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations , 2006 .

[5]  Dan Crisan,et al.  RUNGE-KUTTA SCHEMES FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS , 2014 .

[6]  Robert B. Gramacy,et al.  Sequential Design for Optimal Stopping Problems , 2013, SIAM J. Financial Math..

[7]  Emmanuel Gobet,et al.  Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions , 2015, Math. Comput..

[8]  Denis Belomestny,et al.  Pricing Bermudan options using nonparametric regression: optimal rates of convergence for lower estimates , 2009, 0907.5599.

[9]  E. Gobet,et al.  A regression-based Monte Carlo method to solve backward stochastic differential equations , 2005, math/0508491.

[10]  Carlos Vázquez,et al.  Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs , 2016, SIAM J. Sci. Comput..

[11]  Denis Belomestny,et al.  Regression Methods for Stochastic Control Problems and Their Convergence Analysis , 2009, SIAM J. Control. Optim..

[12]  N. Barton,et al.  Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects , 2011, The American Naturalist.

[13]  S. Ritchie,et al.  Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission , 2011, Nature.

[14]  Jean-Philippe Bouchaud,et al.  Hedged Monte-Carlo: low variance derivative pricing with objective probabilities , 2000 .

[15]  Anis Matoussi,et al.  Empirical Regression Method for Backward Doubly Stochastic Differential Equations , 2016, SIAM/ASA J. Uncertain. Quantification.

[16]  R. Fisher THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .