Strong approximations of stochastic differential equations with jumps

This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs. Strong approximations based on jump-adapted time discretizations, which produce no discretization error in the case of pure jump processes, are analyzed. The computational complexity of these approximations is proportional to the jump intensity. By exploiting a stochastic expansion for pure jump processes, higher order discrete time approximations, whose computational complexity is not dependent on the jump intensity, are proposed. For the specific case of pure jump SDEs, the strong order of convergence of strong Taylor schemes is established under weaker conditions than those currently known in the literature.

[1]  E. Platen.  An introduction to numerical methods for stochastic differential equations , 1999, Acta Numerica.

[2]  R. Jarrow,et al.  A Markov Model for the Term Structure of Credit Risk Spreads , 1997 .

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

[4]  A. Gardon.  The Order of Approximations for Solutions of Itô-Type Stochastic Differential Equations with Jumps , 2004 .

[5]  C. W. Li,et al.  Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions , 2000 .

[6]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[7]  Eckhard Platen,et al.  Rate of Weak Convergence of the Euler Approximation for Diffusion Processes with Jumps , 2002, Monte Carlo Methods Appl..

[8]  Y. Maghsoodi,et al.  In-Probability Approximation and Simulation of Nonlinear Jump-Diffusion Stochastic Differential Equations , 1987 .

[9]  P. Protter,et al.  Asymptotic error distributions for the Euler method for stochastic differential equations , 1998 .

[10]  Paul Glasserman,et al.  Numerical solution of jump-diffusion LIBOR market models , 2003, Finance Stochastics.

[11]  P. Kloeden,et al.  CONVERGENCE AND STABILITY OF IMPLICIT METHODS FOR JUMP-DIFFUSION SYSTEMS , 2005 .

[12]  Nicola Bruti-Liberati,et al.  On the Strong Approximation of Jump-Diffusion Processes , 2005 .

[13]  The Euler Scheme for L?evy Driven Stochastic Difierential Equations: Limit Theorems , 2004, math/0410118.

[14]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[15]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

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

[17]  Paul Glasserman,et al.  Convergence of a discretization scheme for jump-diffusion processes with state–dependent intensities , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[19]  Kevin Burrage,et al.  Stochastic approaches for modelling in vivo reactions , 2004, Comput. Biol. Chem..

[20]  Desmond J. Higham,et al.  Numerical methods for nonlinear stochastic differential equations with jumps , 2005, Numerische Mathematik.

[21]  S. Rachev.  Handbook of heavy tailed distributions in finance , 2003 .

[22]  David Douglas Engel,et al.  The Multiple Stochastic Integral , 1982 .

[23]  P. Protter.  Stochastic integration and differential equations , 1990 .

[24]  B. Øksendal,et al.  Applied Stochastic Control of Jump Diffusions , 2004, Universitext.

[25]  David J. Wright.  Digital simulation of Poisson stochastic differential equations , 1980 .

[26]  Y. Maghsoodi,et al.  Exact solutions and doubly efficient approximations of jump-diffusion itô equations , 1998 .

[27]  D. Gillespie.  Approximate accelerated stochastic simulation of chemically reacting systems , 2001 .

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

[29]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[30]  Almost Sure Convergence of Stochastic Differential Equations of Jump-Diffusion Type , 1995 .