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A Comparison of Higher-Order Weak Numerical Schemes for Stopped Stochastic Differential Equations

Part of: Stochastic processes Probability theory and stochastic processes Numerical analysis

Published online by Cambridge University Press:  31 August 2016

Francisco Bernal  and
Juan A. Acebrón
Francisco Bernal*
Affiliation:
INESC-ID\IST, TU Lisbon. Rua Alves Redol 9, 1000-029 Lisbon, Portugal Center for Mathematics and its Applications, Department of Mathematics, Instituto Superior Técnico. Av. Rovisco Pais 1049-001 Lisbon, Portugal
Juan A. Acebrón*
Affiliation:
INESC-ID\IST, TU Lisbon. Rua Alves Redol 9, 1000-029 Lisbon, Portugal ISCTE - Instituto Universitário de Lisboa Departamento de Ciências e Tecnologias de Informação. Av. das Forças Armadas 1649-026 Lisbon, Portugal
*
*Corresponding author. Email addresses:francisco.bernal@ist.utl.pt (F. Bernal), juan.acebron@ist.utl.pt (J. A. Acebrón)
*Corresponding author. Email addresses:francisco.bernal@ist.utl.pt (F. Bernal), juan.acebron@ist.utl.pt (J. A. Acebrón)
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Abstract

We review, implement, and compare numerical integration schemes for spatially bounded diffusions stopped at the boundary which possess a convergence rate of the discretization error with respect to the timestep h higher than . We address specific implementation issues of the most general-purpose of such schemes. They have been coded into a single Matlab program and compared, according to their accuracy and computational cost, on a wide range of problems in up to ℝ48. The paper is self-contained and the code will be made freely downloadable.

Keywords

Weak convergenceFeynman-Kacstochastic differential equationbounded diffusionfirst-exit problem

MSC classification

Secondary: 60G50: Sums of independent random variables; random walks 60-08: Computational methods (not classified at a more specific level) 65-02: Research exposition (monographs, survey articles)
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 3 , September 2016 , pp. 703 - 732
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Acebrón, J.A., Busico, M.P., Lanucara, P., and Spigler, R., Domain decomposition solution of elliptic boundary-value problems via Monte Carlo and quasi-Monte Carlo methods. SIAM J. Sci. Comp. 27(2), 440–457 (2005).CrossRefGoogle Scholar
[2] Baldi, P., Exact asymptotics for the probability of exit from a domain and applications to simulation. Ann. Probab. 23(4), 1644–1670 (1995).CrossRefGoogle Scholar
[3] Bayer, C., Szepessy, A., and Tempone, R., Adaptive weak approximation of reflected and stopped diffusions. Monte Carlo Methods and Applications 16, 1–67 (2010).CrossRefGoogle Scholar
[4] Bernal, F., Acebrón, J.A., and Anjam, I., A stochastic algorithm based on fast marching for automatic capacitance extraction in non-Manhattan geometries. SIAM J. Imag. Sci. 7(4), 2657–2674 (2014).CrossRefGoogle Scholar
[5] Buchmann, F.M. and Petersen, W.P., An exit probability approach to solving high dimensional Dirichlet problems. SIAM J. Sci. Comput. 28, 1153–1166 (2006).CrossRefGoogle Scholar
[6] Buchmann, F.M., Simulation of stopped diffusions. J. Comput. Phys. 202(2), 446–462 (2005).CrossRefGoogle Scholar
[7] Constantini, C., Pacchiarotti, B., and Sartoretto, F., Numerical approximation for functionals of reflecting diffusion processes. SIAM J. Appl. Math. 58, 73–102 (1998).Google Scholar
[8] Deaconu, M. and Lejay, A., A Random Walk on Rectangles algorithm. Methodol. Comput. Appl. Probab. 8, 135–151 (2006).CrossRefGoogle Scholar
[9] Freidlin, M., Functional Integration and Partial Differential Equations. Princeton University Press (1985).Google Scholar
[10] Giles, M.B., Multi-levelMonte Carlo path simulation. Operations Research, 56(3), 607–617 (2008).CrossRefGoogle Scholar
[11] Giles, M.B. and Bernal, F., Multilevel simulations of expected exit times and other functionals of stopped diffusions. In preparation. (Unpublished)Google Scholar
[12] Giraudo, M.T. and Sacerdote, L., An improved technique for the simulation of first passage times for diffusion processes. Comm. Statist. Simulation Comput. 28(4), 1135–1163 (1999).CrossRefGoogle Scholar
[13] Glasserman, P., Monte Carlo Methods in Financial Engineering. Springer (2003).CrossRefGoogle Scholar
[14] Gobet, E. and Menozzi, S., Stopped diffusion processes: Overshoots and boundary correction. Stoch. Proc. Appl., 120, 130–162, (2010).CrossRefGoogle Scholar
[15] Gobet, E., Euler schemes for the weak approximation of killed diffusion. Stoch. Process. Appl. 87, 167–197 (2000).CrossRefGoogle Scholar
[16] Gobet, E., Euler schemes and half-space approximation for the simulation of diffusions in a domain. ESAIM: Probab. Statist. 5, 261–297 (2001).CrossRefGoogle Scholar
[17] Higham, D.J., Mao, X., Roj, M., Song, Q., and Yin, G.. Mean exit times and the multilevel Monte Carlo method. SIAM J. Uncertainty Quantification, 1(1), 2–18 (2013).CrossRefGoogle Scholar
[18] Iyer, G., Novikov, A., Ryzhik, L., and Zlatos, A., Exit times for diffusions with incompressible drift. SIAM J. Math. Anal. 42, 2484–2498 (2010).CrossRefGoogle Scholar
[19] Jansons, K.M. and Lythe, G.D., Efficient numerical solution of stochastic differential equations using exponential timestepping. J. Statist. Phys. 100(5-6), 1097–1109 (2000).CrossRefGoogle Scholar
[20] Kloeden, P.E. and Platten, E., Numerical Solution of Stochastic Differential Equations. Springer (1999).Google Scholar
[21] Lemaire, P. and Pagés, G., Multilevel Richardson-Romberg extrapolation. To appear in Bernoulli, (2015).CrossRefGoogle Scholar
[22] Mairé, S. and Tanré, E., Some new simulation schemes for the evaluation of Feynman-Kac representations. Monte Carlo Methods and Applications, 14(1), 29–51 (2008).CrossRefGoogle Scholar
[23] Mairé, S. and Simon, M., A partially reflecting random walk on spheres algorithm for electrical impedance tomography. Preprint in arXiv:1502.04318 (2015). (Unpublished)CrossRefGoogle Scholar
[24] Mancini, S., Bernal, F., and Acebrón, J.A., An efficient algorithm for accelerating Monte Carlo approximations of the solution to boundary value problems. J. Sci. Comput. 66(2), 577–597 (2016).CrossRefGoogle Scholar
[25] Mannella, R., Absorbing boundaries and optimal stopping in a stochastic differential equation. Phys. Lett. A 254(5), 257–262 (1999).CrossRefGoogle Scholar
[26] Mascagni, M. and Simonov, N.A., Monte Carlo methods for calculating some physical properties of large molecules, SIAM J. Sci. Comput. 26, 339–357 (2004).CrossRefGoogle Scholar
[27] Michael, J.R., Schucany, W.R., and Haas, R.W., Generating random variates using transformations with multiple roots. Am. Statist. 30, 88–90 (1976).Google Scholar
[28] Milstein, G.N., Weak approximation of a diffusion process in a bounded domain. Stoch. Stoch. Rep. 62, 147–200 (1997).CrossRefGoogle Scholar
[29] Milstein, G.N. and Tretyakov, M.V., Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004).CrossRefGoogle Scholar
[30] Milstein, G.N. and Tretyakov, M.V., The simplest random walks for the Dirichlet problem. Teor. Veroyatnost. I Primenen. 47(1), 39–58 (2002).Google Scholar
[31] Miranda, C., Partial Differential Equations of Elliptic Type. Springer (1970).Google Scholar
[32] Muller, M.E., Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Statist. 27, 569–589 (1956).CrossRefGoogle Scholar
[33] Redner, S., A Guide to First-Passage Processes. Cambridge University Press (2001).CrossRefGoogle Scholar
[34] Schwabedal, J.T.C. and Pikovsky, A., Phase description of stochastic oscillations, Phys. Rev. Lett. 110(20), 204102 (2013).CrossRefGoogle ScholarPubMed
[35] Tamborrino, M., Sacerdote, L., and Jacobsen, M., Weak convergence of marked point processes generated by crossings of multivariate jump processes. Application to neural network modeling. Physica D, 288, 45–52 (2014).CrossRefGoogle Scholar
[36] Talay, D. and Tubaro, L., Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. App., 8(4), 94–120 (1990).Google Scholar