Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with non-regular drift

We consider an Euler-Maruyama type approximation method for a stochastic differential equation (SDE) with a non-regular drift and regular diffusion coefficient. The method regu-larizes the drift coefficient within a certain class of functions and then the Euler-Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors. The first one is the error of regularization of the drift coefficient within a given class of parametrized functions. The second one is the error of the regularized Euler-Maruyama scheme. After an optimization procedure with respect to the parameters we obtain various rates, which improve other known results.

[1]  S. Menozzi,et al.  On Some non Asymptotic Bounds for the Euler Scheme , 2010, 1001.1347.

[2]  S. Kusuoka.  Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations , 2013, 1310.4600.

[3]  Rémi Munos,et al.  Sensitivity Analysis Using It[o-circumflex]--Malliavin Calculus and Martingales, and Application to Stochastic Optimal Control , 2005, SIAM J. Control. Optim..

[4]  S. Menozzi,et al.  WEAK ERROR FOR THE EULER SCHEME APPROXIMATION OF DIFFUSIONS WITH NON-SMOOTH COEFFICIENTS * , 2016, 1604.00771.

[5]  G. Strang,et al.  A Fourier Analysis of the Finite Element Variational Method , 2011 .

[6]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[7]  Gunther Leobacher,et al.  A numerical method for SDEs with discontinuous drift , 2015, BIT Numerical Mathematics.

[8]  G. Kyriazis.  Approximation from shift-invariant spaces , 1995 .

[9]  S. Geiss,et al.  Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces , 2007, 0711.1439.

[10]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[11]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[12]  George Kyriazis.  Wavelet-Type Decompositions and Approximations from Shift-Invariant Spaces , 1997 .

[13]  D. Talay,et al.  The law of the Euler scheme for stochastic differential equations , 1996 .

[14]  Nikolaos Halidias,et al.  A note on the Euler–Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient , 2008 .

[15]  L. Devroye,et al.  Nonparametric density estimation : the L[1] view , 1987 .

[16]  Denis Talay,et al.  The law of the Euler scheme for stochastic differential equations , 1996, Monte Carlo Methods Appl..

[17]  Rainer Avikainen.  On irregular functionals of SDEs and the Euler scheme , 2009, Finance Stochastics.

[18]  R. Bass.  Diffusions and Elliptic Operators , 1997 .

[19]  C. Kenig,et al.  Examples of singular parabolic measures and singular transition probability densities , 1981 .

[20]  Emmanuel Gobet,et al.  Fractional Smoothness and Applications in Finance , 2010, 1004.3577.

[21]  Arturo Kohatsu-Higa,et al.  Weak Approximations. A Malliavin Calculus Approach , 1999, Math. Comput..

[22]  Rainer Janßen.  Difference methods for stochastic differential equations with discontinuous coefficients , 1984 .

[23]  Luc Devroye,et al.  Nonparametric Density Estimation , 1985 .

[24]  Michael Frazier,et al.  Decomposition of Besov Spaces , 2009 .

[25]  H. Triebel.  Theory of Function Spaces III , 2008 .

[26]  Kung-Sik Chan,et al.  Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients , 1998 .

[27]  David Dereudre,et al.  An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers , 2016, Monte Carlo Methods Appl..

[28]  A. I. Stepanets.  Methods of Approximation Theory , 2005 .

[29]  Dai Taguchi,et al.  Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients , 2013, Math. Comput..

[30]  Hoang-Long Ngo,et al.  Approximation for non-smooth functionals of stochastic differential equations with irregular drift , 2015, 1505.03600.

[31]  N. V. Krylov.  An Inequality in the Theory of Stochastic Integrals , 1971 .

[32]  Exact sampling of diffusions with a discontinuity in the drift , 2015, Advances in Applied Probability.

[33]  Nira Dyn,et al.  Interpolation and Approximation of Piecewise Smooth Functions , 2005, SIAM J. Numer. Anal..

[34]  G. Weiss,et al.  Littlewood-Paley Theory and the Study of Function Spaces , 1991 .

[35]  Liqing Yan.  The Euler scheme with irregular coefficients , 2002 .

[36]  S. Attanasio.  STOCHASTIC FLOWS OF DIFFEOMORPHISMS FOR ONE-DIMENSIONAL SDE WITH DISCONTINUOUS DRIFT , 2010 .

[37]  D. Talay,et al.  One-dimensional parabolic diffraction equations: pointwise estimates and discretization of related stochastic differential equations with weighted local times , 2012 .

[38]  Miguel Martinez,et al.  Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift , 2013, 1301.3019.

[39]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[40]  E. Gobet.  SENSITIVITY ANALYSIS USING ITÔ – MALLIAVIN CALCULUS AND , 2002 .

[41]  Lena Schwartz,et al.  Theory Of Function Spaces Ii , 2016 .

[42]  A. Veretennikov.  ON STRONG SOLUTIONS AND EXPLICIT FORMULAS FOR SOLUTIONS OF STOCHASTIC INTEGRAL EQUATIONS , 1981 .

[43]  A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient , 2006 .

[44]  Alina Semrau.  Discrete Approximations of Strong Solutions of Reflecting SDEs with Discontinuous Coefficients , 2009 .

[45]  R. Mikulevicius,et al.  Rate of Convergence of the Euler Approximation for Diffusion Processes , 1991 .

[46]  Mireille Bossy,et al.  ON MEAN NUMBERS OF PASSAGE TIMES IN SMALL BALLS OF DISCRETIZED ITÔ PROCESSES , 2009 .

[47]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[48]  Pawel Przybylowicz,et al.  Optimality of Euler-type algorithms for approximation of stochastic differential equations with discontinuous coefficients , 2014, Int. J. Comput. Math..

[49]  R. Mikulevicius,et al.  Weak Euler Approximation for Itô Diffusion and Jump Processes , 2015 .

[50]  Emmanuel Gobet,et al.  Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition , 2011, 1103.0371.