Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation

We consider stochastic differential equations driven by a general Lévy processes (SDEs) with infinite activity and the related, via the Feynman–Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (1) we approximate small jumps by a diffusion; (2) we use restricted jump-adaptive time-stepping; and (3) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. Results of some numerical experiments, including pricing of barrier basket currency options, are presented.

[1]  Jean Jacod,et al.  The approximate Euler method for Lévy driven stochastic differential equations , 2005 .

[2]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[3]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[4]  M. V. Tretyakov,et al.  The Simplest Random Walks for the Dirichlet Problem , 2003 .

[5]  Antonio Castagna.  FX Options and Smile Risk , 2010 .

[6]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[7]  Changyong Zhang,et al.  Weak Euler Scheme for Lévy-Driven Stochastic Differential Equations , 2018 .

[8]  Luc Devroye,et al.  Non-Uniform Random Variate Generation , 1986 .

[9]  Ernesto Mordecki,et al.  Adaptive Weak Approximation of Diffusions with Jumps , 2008, SIAM J. Numer. Anal..

[10]  A. Kohatsu-Higa,et al.  Jump-adapted discretization schemes for Lévy-driven SDEs , 2010 .

[11]  Niels Jacob,et al.  PSEUDO DIFFERENTIAL OPERATORS AND MARKOV PROCESSES, VOLUME III: MARKOV PROCESSES AND APPLICATIONS , 2005 .

[12]  Maria Giovanna Garroni,et al.  Green Functions for Second Order Parabolic Integro-Differential Problems , 1993 .

[13]  Eckhard Platen,et al.  Time Discrete Taylor Approximations for Itǒ Processes with Jump Component , 1988 .

[14]  Sylvain Rubenthaler,et al.  Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process , 2003 .

[15]  X. Q. Liu,et al.  Weak Approximations and Extrapolations of Stochastic Differential Equations with Jumps , 2000, SIAM J. Numer. Anal..

[16]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[17]  Nicola Bruti-Liberati.  Numerical Solution of Stochastic Differential Equations with Jumps in Finance , 2010 .

[18]  On Cauchy-Dirichlet Problem in Half-Space for Linear Integro-Differential Equations in Weighted Hoelder Spaces , 2005 .

[19]  Arturo Kohatsu-Higa,et al.  Optimal simulation schemes for Lévy driven stochastic differential equations , 2012, Math. Comput..

[20]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[21]  Constance de Koning,et al.  Editors , 2003, Annals of Emergency Medicine.

[22]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[23]  Philip Protter,et al.  The Euler scheme for Lévy driven stochastic differential equations , 1997 .

[24]  O. Barndorff-Nielsen,et al.  Lévy processes : theory and applications , 2001 .

[25]  Uwe Wystup,et al.  FX Options and Structured Products , 2007 .