A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients

The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a Skew Brownian Motion.

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

[2]  K. Semra,et al.  Three dimensional groundwater quality modelling in heterogeneous media , 1970 .

[3]  David Aldous,et al.  Stopping Times and Tightness. II , 1978 .

[4]  V. Zhikov,et al.  AVERAGING AND G-CONVERGENCE OF DIFFERENTIAL OPERATORS , 1979 .

[5]  N. Portenko,et al.  Diffusion Processes with Generalized Drift Coefficients , 1979 .

[6]  N. Portenko,et al.  Stochastic Differential Equations with Generalized Drift Vector , 1980 .

[7]  J. Harrison,et al.  On Skew Brownian Motion , 1981 .

[8]  V. Zhikov,et al.  G-convergence of parabolic operators , 1981 .

[9]  J. Gall,et al.  One — dimensional stochastic differential equations involving the local times of the unknown process , 1984 .

[10]  D. Stroock,et al.  Upper bounds for symmetric Markov transition functions , 1986 .

[11]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[12]  Masami Okada,et al.  Second order differential operators and Dirichlet integrals with singular coefficients. I: Functional calculus of one-dimensional operators , 1987 .

[13]  Daniel W. Stroock,et al.  Diffusion semigroups corresponding to uniformly elliptic divergence form operators , 1988 .

[14]  Y. Ouknine,et al.  “Skew-Brownian Motion” and Derived Processes , 1991 .

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

[16]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[17]  M. Freidlin,et al.  Necessary and Sufficient Conditions for Weak Convergence of One-Dimensional Markov Processes , 1994 .

[18]  Dominique Lépingle,et al.  Euler scheme for reflected stochastic differential equations , 1995 .

[19]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[20]  A. Rozkosz.  Weak convergence of diffusions corresponding to divergence form operators , 1996 .

[21]  A. Rozkosz.  Stochastic representation of diffusions corresponding to divergence form operators , 1996 .

[22]  The Limits of Stochastic Integrals of Differential Forms , 1999 .

[23]  R S Cantrell,et al.  Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design. , 1999, Theoretical population biology.

[24]  Michael V. Tretyakov,et al.  SIMULATION OF A SPACE-TIME BOUNDED DIFFUSION , 1999 .

[25]  Paul Seignourel.  Processus en milieux aleatoires ou irreguliers , 1999 .

[26]  A. Lejay,et al.  Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence : cas linéaires et semi-linéaires , 2000 .

[27]  Ming Zhang,et al.  Calculation of Diffusive Shock Acceleration of Charged Particles by Skew Brownian Motion , 2000 .

[28]  Antoine Lejay,et al.  On the decomposition of excursions measures of processes whose generators have diffusion coefficients discontinuous at one point , 2002 .

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

[30]  Antoine Lejay,et al.  Simulating a diffusion on a graph. Application to reservoir engineering , 2003, Monte Carlo Methods Appl..

[31]  J. Steele,et al.  ITÔ CALCULUS , 2003 .

[32]  Miguel Martinez,et al.  Interprétations probabilistes d'opérateurs sous forme divergence et analyse de méthodes numériques probabilistes associées , 2004 .

[33]  Ann De Schepper,et al.  Applications of δ-function perturbation to the pricing of derivative securities , 2004 .

[34]  R. Bass,et al.  One-Dimensional Stochastic Differential Equations with Singular and Degenerate Coefficients , 2005 .

[35]  P. Etoré.  On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients , 2006 .