Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs

We extend a recently developed method to solve semi-linear PDEs to the case of a degenerated diffusion. Being a pure Monte Carlo method it does not suffer from the so called curse of dimensionality and it can be used to solve problems that were out of reach so far. We give some results of convergence and show numerically that it is effective. Besides we numerically show that the new scheme developed can be used to solve some full non linear PDEs. At last we provide an effective algorithm to implement the scheme.

[1]  X. Warin.  Variations on branching methods for non linear PDEs , 2017, 1701.07660.

[2]  Thomas Kruse,et al.  Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities , 2017, SIAM J. Numer. Anal..

[3]  N. Touzi.  Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE , 2012 .

[4]  Christian Kahl,et al.  Fast strong approximation Monte Carlo schemes for stochastic volatility models , 2006 .

[5]  Nadia Oudjane,et al.  Unbiased Monte Carlo estimate of stochastic differential equations expectations , 2016, 1601.03139.

[6]  E Weinan,et al.  Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning , 2017, ArXiv.

[7]  E Weinan,et al.  Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations , 2017, Communications in Mathematics and Statistics.

[8]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[9]  E Weinan,et al.  On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations , 2017, Journal of Scientific Computing.

[10]  Pierre-Louis Lions,et al.  Applications of Malliavin calculus to Monte Carlo methods in finance , 1999, Finance Stochastics.

[11]  Xavier Warin,et al.  Nesting Monte Carlo for high-dimensional non-linear PDEs , 2018, Monte Carlo Methods Appl..

[12]  H. Soner,et al.  Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs , 2005, math/0509295.

[13]  Thaleia Zariphopoulou,et al.  A solution approach to valuation with unhedgeable risks , 2001, Finance Stochastics.

[14]  Nizar Touzi,et al.  A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs , 2009, 0905.1863.

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

[16]  B. Bouchard,et al.  Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations , 2004 .

[17]  Nadia Oudjane,et al.  Branching diffusion representation of semilinear PDEs and Monte Carlo approximation , 2016, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[18]  E Weinan,et al.  On full history recursive multilevel Picard approximations and numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations , 2016 .

[19]  E Weinan,et al.  Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations , 2017, J. Nonlinear Sci..

[20]  Xiaolu Tan,et al.  A splitting method for fully nonlinear degenerate parabolic PDEs , 2013 .

[21]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

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