Boundary Preserving Semianalytic Numerical Algorithms for Stochastic Differential Equations

Construction of splitting-step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations (SDEs) of Ito type are discussed. As the crucial assumption, we oppose conditions such that one can decompose the original system of SDEs into subsystems for which one knows either the exact solution or its conditional transition probability. We present convergence proofs for a newly designed splitting-step algorithm and simulation studies for numerous well-known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance (Cox-Ingersoll-Ross (CIR) and constant elasticity of variance (CEV) models) to measure-valued diffusion and super-Brownian motion (stochastic PDEs (SPDEs)) as met in biology and physics.

[1]  W. Feller.  THE PARABOLIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED SEMI-GROUPS OF TRANSFORMATIONS , 1952 .

[2]  Henri Schurz,et al.  Numerical Regularization for SDEs: Construction of Nonnegative Solutions , 1995 .

[3]  Donald A. Dawson,et al.  Measure-Valued processes and renormalization of branching particle systems , 1999 .

[4]  Khachik Sargsyan,et al.  A numerical method for some stochastic differential equations with multiplicative noise , 2005 .

[5]  Carl Mueller,et al.  A phase transition for a stochastic PDE related to the contact process , 1994 .

[6]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[7]  E. Platen,et al.  Balanced Implicit Methods for Stiff Stochastic Systems , 1998 .

[8]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[9]  René Carmona,et al.  Stochastic Partial Differential Equations: Six Perspectives , 1998 .

[10]  Andrew M. Stuart,et al.  Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations , 2002, SIAM J. Numer. Anal..

[11]  Rick Durrett,et al.  Rescaled contact processes converge to super-Brownian motion in two or more dimensions , 1999 .

[12]  W. Feller.  Diffusion processes in one dimension , 1954 .

[13]  J. Gaines.  Stochastic Partial Differential Equations: Numerical experiments with S(P)DE's , 1995 .

[14]  E. Dynkin.  Superprocesses and Partial Differential Equations , 1993 .

[15]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

[16]  H. Hinrichsen.  Non-equilibrium critical phenomena and phase transitions into absorbing states , 2000, cond-mat/0001070.

[17]  Christian Kahl,et al.  Balanced Milstein Methods for Ordinary SDEs , 2006, Monte Carlo Methods Appl..

[18]  P. Kloeden,et al.  Numerical Solution of Sde Through Computer Experiments , 1993 .

[19]  Angewandte Mathematik,et al.  Solving Dirichlet problems numerically using the Feynman-Kac representation , 2002 .

[20]  Herbert Levine,et al.  Interfacial velocity corrections due to multiplicative noise , 1999 .

[21]  Hans-Jürgen Zimmermann,et al.  Fuzzy Set Theory - and Its Applications , 1985 .

[22]  E. Moro.  Numerical schemes for continuum models of reaction-diffusion systems subject to internal noise. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[24]  A. Siegel.  The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity , 1979 .

[25]  William Feller,et al.  The General Diffusion Operator and Positivity Preserving Semi-Groups in One Dimension , 1954 .

[26]  Raul Tempone,et al.  Adaptive Monte Carlo Algorithms for Stopped Diffusion , 2005 .

[27]  Fabienne Castell,et al.  The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations , 1996 .

[28]  Eckhard Platen,et al.  Approximation of itô integral equations , 1980 .

[29]  Roger Pettersson,et al.  Penalization schemes for reflecting stochastic differential equations , 1997 .

[30]  V. Bogachëv,et al.  Liens entre équations différentielles stochastiques et ordinaires en dimension infinie , 1995 .

[31]  H. Schurz.  Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications , 1997 .

[32]  Tetsuya Misawa,et al.  A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations , 2001, SIAM J. Sci. Comput..

[33]  E. Gobet.  Weak approximation of killed diffusion using Euler schemes , 2000 .

[34]  D. Higham,et al.  Convergence of Monte Carlo Simulations involving the Mean-Reverting Square Root Process , 2005 .

[35]  S. Kanagawa,et al.  Strong Approximation of Reflecting Brownian Motion Using Penalty Method and its Application to Cumputer Simulation , 2000, Monte Carlo Methods Appl..

[36]  Henri Schurz,et al.  AN AXIOMATIC APPROACH TO NUMERICAL APPROXIMATIONS OF STOCHASTIC PROCESSES , 2005 .

[37]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[38]  Denis Talay,et al.  Simulation of stochastic differential systems , 1995 .

[39]  Henri Schurz.  GENERAL THEOREMS FOR NUMERICAL APPROXIMATION OF STOCHASTIC PROCESSES ON THE HILBERT SPACEH2((0;T );; IR d ) , 2003 .

[40]  Stability , 1973 .

[41]  D. Lépingle,et al.  Un schéma d'Euler pour équations différentielles stochastiques réfléchies , 1993 .

[42]  G. Milstein.  Numerical Integration of Stochastic Differential Equations , 1994 .

[43]  Gordon Slade.  Scaling limits and super-Brownian motion , 2002 .

[44]  E. Gobet,et al.  Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme , 2004 .

[45]  Mireille Bossy,et al.  A symmetrized Euler scheme for an efficient approximation of reflected diffusions , 2004, Journal of Applied Probability.

[46]  J. Runnenburg.  PROBABILITY THEORY AND ITS APPLICATIONS , 1985 .

[47]  Roger Tribe,et al.  Stochastic p.d.e.'s arising from the long range contact and long range voter processes , 1995 .

[48]  M. A. Muñoz,et al.  Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions. , 2004, Physical Review Letters.