Runge-Kutta methods for jump-diffusion differential equations

In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.

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

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

[3]  G. N. Milstein,et al.  Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises , 1997, SIAM J. Sci. Comput..

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

[5]  Floyd B. Hanson,et al.  Applied stochastic processes and control for jump-diffusions - modeling, analysis, and computation , 2007, Advances in design and control.

[6]  Desmond J. Higham,et al.  Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems , 2007 .

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

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

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

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

[11]  J. C. Jimenez,et al.  Weak local linear discretizations for stochastic differential equations , 2006 .

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

[13]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[14]  Jerzy Zabczyk,et al.  Stochastic Partial Differential Equations with Lévy Noise: References , 2007 .

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

[16]  Evelyn Buckwar,et al.  Stochastic Runge--Kutta Methods for It[o-circumflex] SODEs with Small Noise , 2010, SIAM J. Sci. Comput..

[17]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

[18]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

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

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

[21]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[22]  Werner Römisch,et al.  Stepsize Control for Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noise , 2006, SIAM J. Sci. Comput..

[23]  Nicola Bruti-Liberati,et al.  Strong approximations of stochastic differential equations with jumps , 2007 .

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

[25]  Evelyn Buckwar,et al.  Multi-Step Maruyama Methods for Stochastic Delay Differential Equations , 2007 .

[26]  K. Dekker,et al.  Error bounds for the solution to the algebraic equations in Runge-Kutta methods , 1984 .

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

[28]  John Waldron,et al.  The Langevin Equation , 2004 .

[29]  R. Winkler.  Stochastic differential algebraic equations of index 1 and applications in circuit simulation , 2003 .

[30]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

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

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

[33]  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.

[34]  Evelyn Buckwar,et al.  Multistep methods for SDEs and their application to problems with small noise , 2006, SIAM J. Numer. Anal..

[35]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

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

[37]  T. Shardlow.  Numerical simulation of stochastic PDEs for excitable media , 2005 .

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

[39]  Desmond J. Higham,et al.  Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes , 2007 .

[40]  Werner Römisch,et al.  Stochastic DAEs in Circuit Simulation , 2003 .

[41]  Donald L. Snyder,et al.  Random Point Processes in Time and Space , 1991 .

[42]  A. Gardon.  The Order 1.5 Approximation for Solutions of Jump-Diffusion Equations , 2006 .

[43]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[44]  Evelyn Buckwar,et al.  NUMERICAL ANALYSIS OF EXPLICIT ONE-STEP METHODS FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS , 1975 .

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

[46]  Werner Römisch,et al.  Simultaneous Step-Size and Path Control for Efficient Transient Noise Analysis , 2010 .

[47]  Werner Römisch,et al.  Efficient transient noise analysis in circuit simulation , 2006 .

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

[49]  Stefan Schäffler,et al.  Adams methods for the efficient solution of stochastic differential equations with additive noise , 2007, Computing.