×

zbMATH — the first resource for mathematics

A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. (English) Zbl 1094.60056
The aim of this paper is to study a new type of scheme for the simulation of diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients (as classical Euler scheme, Milstein scheme) appearing in the differential operator \(L\), which generates the diffusion process \(X\), but uses instead an exact description of the behaviour of their trajectories when they reach the points of discontinuity. The differential operator \(L\) that generates the diffusion process \(X\) depends on three piecewise smooth functions, called the characteristic coefficients of \(L\). These smooth functions may have an infinite number of discontinuities on a countable set of points. The basic idea for the new simulation scheme proceeds as follows: First, the differential operator \(L\) is replaced by another one whose coefficients are piecewise constant, which provides good approximations of the solutions of the elliptic and parabolic partial differential equations involving \(L\).
Second, the behaviour of the stochastic process generated by the approximation of \(L\) at a given time, around discontinuity points of the characteristic coefficients of \(L\), can be described as a skew Brownian motion. The advantage of using a random walk method is that the time step is incremented with a constant value and not with a random variable, while the algorithm can be implemented locally around the points where discontinuities hold. In the regions where the coefficients are smooth, one may use more efficient algorithms based on classical schemes. Detailed description of the algorithm, its numerical simulation, and local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion are provided. Two examples illustrate the new approach: (a) the doubly skew Brownian motion, and (b) diffusion with a coefficient discontinuous at one point (which is compared with the Euler scheme and with a deterministic scheme).

MSC:
60J60 Diffusion processes
60J65 Brownian motion
65C05 Monte Carlo methods
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335–340. · Zbl 0391.60007 · doi:10.1214/aop/1176995579
[2] Aronson, D. G. (1968). Nonnegative solutions of linear parabolic equation. Ann. Scuola Norm. Sup. Pisa 22 607–693. · Zbl 0182.13802 · numdam:ASNSP_1968_3_22_4_607_0 · eudml:83474
[3] Bass, R. and Chen, Z.-Q. (2005). One-dimensional stochastic differential equations with singular and degenerate coefficients. Sankhyā 67 19–45. · Zbl 1192.60081 · sankhya.isical.ac.in
[4] Blumenthal, R. M. (1992). Excursions of Markov Processes . Birkhäuser, Boston. · Zbl 0983.60504
[5] Breiman, L. (1981). Probability . Addison–Wesley, Reading, MA. · Zbl 0174.48801
[6] Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae . Birkhäuser, Basel. · Zbl 0859.60001
[7] Cantrell, R. S. and Cosner, C. (1999). Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design. Theoretical Population Biology 55 2 189–207. · Zbl 0958.92028 · doi:10.1006/tpbi.1998.1397
[8] Carlen, E. A., Kusuoka, S. and Stroock, D. W. (1987). Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 2 245–287. · Zbl 0634.60066 · numdam:AIHPB_1987__23_S2_245_0 · eudml:77309
[9] Decamps, M., De Schepper, A. and Goovaerts, M. (2004). Applications of \(\delta\)-function perturbation to the pricing of derivative securities. Phys. A 342 3–4 677–692. · doi:10.1016/j.physa.2004.05.035
[10] Devroye, L. (1986). Non-Uniform Random Variate Generation . Springer, New York. · Zbl 0593.65005
[11] Étoré, P. (2003). Équations différentielles stochastiques à dérive singulière. “D.E.A.” dissertation, IECN / INRIA Lorraine / Univ. Toulouse III.
[12] Étoré, P. (2005). On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. Preprint, Institut Élie Cartan, Nancy, France. Electron. J. Probab.
[13] Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Process . de Gruyter, Berlin. · Zbl 0838.31001
[14] Freidlin, M. and Wentzell, A. D. (1994). Necessary and sufficient conditions for weak convergence of one-dimensional Markov processes. In Festschrift Dedicated to 70th Birthday of Professor E. B. Dynkin (M. Freidlin, ed.) 95–109. Birkhäuser, Boston. · Zbl 0820.60058
[15] Gaveau, B., Okada, M. and Okada, T. (1987). Second order differential operators and Dirichlet integrals with singular coefficients. I. Functional calculus of one-dimensional operators. Tôhoku Math. J. (2) 39 4 465–504. · Zbl 0653.35034
[16] Harrison, J. M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Probab. 9 309–313. JSTOR: · Zbl 0462.60076 · doi:10.1214/aop/1176994472 · links.jstor.org
[17] Itô, K. and McKean, H. P. (1974). Diffusion and Their Sample Paths , 2nd ed. Springer, New York. · Zbl 0837.60001
[18] Ladyženskaja, O. A., Rivkind, V. Ja. and Ural’ceva, N. N. (1966). Solvability of diffraction problems in the classical sense. Trudy Mat. Inst. Steklov. 92 116–146. · Zbl 0165.11802
[19] Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1967). Linear and Quasilinear Equations of Parabolic Type . Amer. Math. Soc., Providence, RI. · Zbl 0164.12302
[20] Lejay, A. (2000). Méthodes probabilistes pour l’homogénéisation des opérateurs sous forme-divergence: Cas linéaires et semi-linéaires. Ph.D. thesis, Univ. de Provence, Marseille, France.
[21] Lejay, A. (2001). On the decomposition of excursions measures of processes whose generators have diffusion coefficients discontinuous at one point. Markov Process. Related Fields 8 117–126. · Zbl 0997.60085
[22] Lejay, A. (2003). Simulating a diffusion on a graph. Application to reservoir engineering. Monte Carlo Methods Appl. 9 241–255. · Zbl 1072.76062 · doi:10.1163/156939603322729003
[23] Le Gall, J.-F. (1985). One-dimensional stochastic differential equations involving the local times of the unknown process. Stochastic Analysis and Applications . Lecture Notes in Math. 1095 51–82. Springer, Berlin. · Zbl 0551.60059
[24] Lépingle, D. (1993). Un schéma d’Euler pour équations différentielles stochastiques réfléchies. C. R. Acad. Sci. Paris Sér. I Math. 316 601–605. · Zbl 0771.60046
[25] Lépingle, D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comput. Simulation 38 119–126. · Zbl 0824.60062 · doi:10.1016/0378-4754(93)E0074-F
[26] Lyons, T. J. and Stoica, L. (1999). The limits of stochastic integrals of differential forms. Ann. Probab. 27 1–49. · Zbl 0969.60078 · doi:10.1214/aop/1022677253
[27] Martinez, M. (2004). Interprétations probabilistes d’opérateurs sous forme divergence et analyse de méthodes numériques probabilistes associées. Ph.D. thesis, Univ. de Provence, France.
[28] Milstein, G. N. and Tretyakov, M. V. (1999). Simulation of a space-time bounded diffusion. Ann. Appl. Probab. 9 732–779. · Zbl 0964.60065 · doi:10.1214/aoap/1029962812
[29] Ouknine, Y. (1990). “Skew-Brownian motion” and derived processes. Theory Probab. Appl. 35 163–169. · Zbl 0724.60087 · doi:10.1137/1135018
[30] Portenko, N. I. (1979). Diffusion process with generalized drift coefficients. Theory Probab. Appl. 24 62–78. · Zbl 0432.60094 · doi:10.1137/1124005
[31] Portenko, N. I. (1979). Stochastic differential equations with generalized drift vector. Theory Probab. Appl. 24 338–353. · Zbl 0434.60062 · doi:10.1137/1124038
[32] Rozkosz, A. (1996). Stochastic representation of diffusions corresponding to divergence form operators. Stoch. Process. Appl. 63 11–33. · Zbl 0870.60073 · doi:10.1016/0304-4149(96)00059-2
[33] Rozkosz, A. (1996). Weak convergence of diffusions corresponding to divergence form operator. Stochastics Stochastics Rep. 57 129–157. · Zbl 0885.60067
[34] Rogers, L. C. G. and Williams, D. (2000). Itô Calculus , 2nd ed. Cambridge Univ. Press. · Zbl 0977.60005
[35] Seignourel, P. (1999). Processus en milieux aléatoires ou irréguliers. Ph.D. thesis, École Polytechnique, France.
[36] Semra, K., Ackerer, Ph. and Mosé, R. (1993). Three dimensional groundwater quality modelling in heterogeneous media. In Water Pollution II Modelling, Measuring and Prediction (L. C. Wrobel and C. A. Brebbia, eds.) 3–11. Comput. Mech., Billerica, MA.
[37] Stroock, D. W. (1988). Diffusion semigroups corresponding to uniformly elliptic divergence form operator. Séminaire de Probabilités XXII . Lecture Notes in Math. 1321 316–347. Springer, New York. · Zbl 0651.47031 · numdam:SPS_1988__22__316_0 · eudml:113641
[38] Walsh, J. B. (1978). A diffusion with a discontinuous local time. In Temps Locaux . Astérisque 52 – 53 37–45. Soc. Math. de France, Paris. · Zbl 0385.60063
[39] Yan, L. (2002). The Euler scheme with irregular coefficients. Ann. Probab. 30 1172–1194. · Zbl 1020.60054 · doi:10.1214/aop/1029867124
[40] Zhang, M. (2000). Calculation of diffusive shock acceleration of charged particles by skew Brownian motion. The Astrophysical Journal 541 428–435.
[41] Zhikov, V. V., Kozlov, S. M., Oleinik, O. A. and T’en Ngoan, K. (1979). Averaging and \(G\)-convergence of differential operators. Russian Math. Survey 34 69–147. · Zbl 0445.35096 · doi:10.1070/RM1979v034n05ABEH003898
[42] Zhikov, V. V., Kozlov, S. M. and Oleinik, O. A. (1981). \(G\)-convergence of parabolic operators. Russian Math. Survey 36 9–60. · Zbl 0479.35047 · doi:10.1070/RM1981v036n01ABEH002540
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.