# American Institute of Mathematical Sciences

May  2013, 18(3): 667-679. doi: 10.3934/dcdsb.2013.18.667

## The spectral collocation method for stochastic differential equations

 1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States 2 Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

Received  January 2012 Revised  September 2012 Published  December 2012

In this paper, we use the Chebyshev spectral collocation method to solve a certain type of stochastic differential equations (SDEs). We also use this method to estimate parameters of stochastic differential equations from discrete observations by maximum likelihood technique and Kessler technique. Our numerical tests shows that the spectral method gives better results than the Euler's method and the Shoji-Ozaki method.
Citation: Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
##### References:
 [1] E. J. Allen, S. J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Rep., 64 (1998), 117-142. [2] I. Babuska and K.-M. Liu, On solving stochastic initial-value differential equations, Math. Models. Meth. Appl. Sci., 13 (2003), 715-745. doi: 10.1142/S0218202503002696. [3] N. Bruti-Liberati and E. Platen, "On the Strong Approximation of Jump-Diffusion Process," Technical Report, Quantitative Finance Research Paper, 157, University of Technology, Sydney, 2005. [4] N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, J. Comp. Appl. Math., 205 (2007), 982-1001. doi: 10.1016/j.cam.2006.03.040. [5] K. Burrage and P. M. Burrage, High strong order explicit Runge-Kutta methods for stochastic differenital equations, Appl. Numer. Math., 22 (1996), 81-101. doi: 10.1016/S0168-9274(96)00027-X. [6] K. Burrage and P. M. Burrage, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38 (2000), 1626-1646. doi: 10.1137/S0036142999363206. [7] K. Burrage, P. M. Burrage and T. Tian, Numerical methods for strong solutions of stochastidc differential equations: An overview, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 460 (2004), 373-402. doi: 10.1098/rspa.2003.1247. [8] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. Zang, "Spectral Mehtods in Fluid Dynamics," Springer Series in Computational Physics, Springer-Verlag, New York, 1988. [9] C. C. Chang, Numerical solution of stochastic differential equations with constant diffusion coefficients, Math. Comp., 49 (1987), 523-542. doi: 10.2307/2008326. [10] J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, in "Stochastic Differential Systems" (Proc. IFIP-WG 7/1 Working Conf. Vilnius, 1978), Lecture Notes in Control and Inform. Sc., 25, Springer, Berlin-New York, (1980), 162-171. [11] A. Gardoń, The order of approximations for solutions of Itó-type stochastic differential equations with jumps, Stoch. Anal Appl., 22 (2004), 679-699. doi: 10.1081/SAP-120030451. [12] A. Gardoń, The order 1.5 approximation for solution of jump-diffusion equations, Stoch. Anal. Appl., 24 (2006), 1147-1168. doi: 10.1080/07362990600958838. [13] R. G. Ghanem and P. D. Spanos, "Stochastic Finite Elements: A Spectral Approach," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3094-6. [14] D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications," CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, 1977. [15] D. Higham, An algorithm introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [16] M. Kessler, Simple and explicit estimating functions for a discretely observed diffusion process, Scan. J. Stat., 27 (2000), 65-82. doi: 10.1111/1467-9469.00179. [17] M. Kleiber and T. D. Hien, "The Stochastic Finite Element Method," John Wiley & Sons, Ltd., Chichester, 1992. [18] P. E. Kloeden, S. Cyganowski and J. Ombach, "From Elementary Probability to Stochastic Differential Equations with MAPLE®," Universitext, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-642-56144-3. [19] P. E. Kloeden and S. E. Platen, "Numerical Solutions of Stochastic Differential Equations," Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [20] P. E. Kloeden, E. Platen and H. Schurz, "Numerical Solution of SDE Through Computer Experiments," With 1 IBM-PC floppy disk (3.5 inch; HD), Universitext, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4. [21] I. V. Krasovsky, Asymptotic distribution of zeros of polynomials satisfying difference equations, J. Comp. Appl. Math., 150 (2003), 56-70. doi: 10.1016/S0377-0427(02)00564-2. [22] S. M. Lacus, "Simulation and Inference for Stochastic Differential Equations," Springer, 2007. [23] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability, 64, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-13694-8. [24] I. Shoji and T. Ozaki, Estimation for nonlinear stochastic differential equations by a local linearization method, Stoch. Anal. Appl., 16 (1998), 733-752. doi: 10.1080/07362999808809559. [25] L. N. Trefethen, "Spectral Methods in MATLAB," Software, Environments, and Tools, 10, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [26] X. Wan, D. Xiu and G. E. Karniadakis, Stochastic solutions for the two-dimensional advection-diffusion equation, SIAM J. Sci. Comp., 26 (2004), 578-590. doi: 10.1137/S106482750342684X. [27] D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp., 24 (2002), 619-644. doi: 10.1137/S1064827501387826.

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##### References:
 [1] E. J. Allen, S. J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Rep., 64 (1998), 117-142. [2] I. Babuska and K.-M. Liu, On solving stochastic initial-value differential equations, Math. Models. Meth. Appl. Sci., 13 (2003), 715-745. doi: 10.1142/S0218202503002696. [3] N. Bruti-Liberati and E. Platen, "On the Strong Approximation of Jump-Diffusion Process," Technical Report, Quantitative Finance Research Paper, 157, University of Technology, Sydney, 2005. [4] N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, J. Comp. Appl. Math., 205 (2007), 982-1001. doi: 10.1016/j.cam.2006.03.040. [5] K. Burrage and P. M. Burrage, High strong order explicit Runge-Kutta methods for stochastic differenital equations, Appl. Numer. Math., 22 (1996), 81-101. doi: 10.1016/S0168-9274(96)00027-X. [6] K. Burrage and P. M. Burrage, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38 (2000), 1626-1646. doi: 10.1137/S0036142999363206. [7] K. Burrage, P. M. Burrage and T. Tian, Numerical methods for strong solutions of stochastidc differential equations: An overview, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 460 (2004), 373-402. doi: 10.1098/rspa.2003.1247. [8] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. Zang, "Spectral Mehtods in Fluid Dynamics," Springer Series in Computational Physics, Springer-Verlag, New York, 1988. [9] C. C. Chang, Numerical solution of stochastic differential equations with constant diffusion coefficients, Math. Comp., 49 (1987), 523-542. doi: 10.2307/2008326. [10] J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, in "Stochastic Differential Systems" (Proc. IFIP-WG 7/1 Working Conf. Vilnius, 1978), Lecture Notes in Control and Inform. Sc., 25, Springer, Berlin-New York, (1980), 162-171. [11] A. Gardoń, The order of approximations for solutions of Itó-type stochastic differential equations with jumps, Stoch. Anal Appl., 22 (2004), 679-699. doi: 10.1081/SAP-120030451. [12] A. Gardoń, The order 1.5 approximation for solution of jump-diffusion equations, Stoch. Anal. Appl., 24 (2006), 1147-1168. doi: 10.1080/07362990600958838. [13] R. G. Ghanem and P. D. Spanos, "Stochastic Finite Elements: A Spectral Approach," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3094-6. [14] D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications," CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, 1977. [15] D. Higham, An algorithm introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [16] M. Kessler, Simple and explicit estimating functions for a discretely observed diffusion process, Scan. J. Stat., 27 (2000), 65-82. doi: 10.1111/1467-9469.00179. [17] M. Kleiber and T. D. Hien, "The Stochastic Finite Element Method," John Wiley & Sons, Ltd., Chichester, 1992. [18] P. E. Kloeden, S. Cyganowski and J. Ombach, "From Elementary Probability to Stochastic Differential Equations with MAPLE®," Universitext, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-642-56144-3. [19] P. E. Kloeden and S. E. Platen, "Numerical Solutions of Stochastic Differential Equations," Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [20] P. E. Kloeden, E. Platen and H. Schurz, "Numerical Solution of SDE Through Computer Experiments," With 1 IBM-PC floppy disk (3.5 inch; HD), Universitext, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4. [21] I. V. Krasovsky, Asymptotic distribution of zeros of polynomials satisfying difference equations, J. Comp. Appl. Math., 150 (2003), 56-70. doi: 10.1016/S0377-0427(02)00564-2. [22] S. M. Lacus, "Simulation and Inference for Stochastic Differential Equations," Springer, 2007. [23] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability, 64, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-13694-8. [24] I. Shoji and T. Ozaki, Estimation for nonlinear stochastic differential equations by a local linearization method, Stoch. Anal. Appl., 16 (1998), 733-752. doi: 10.1080/07362999808809559. [25] L. N. Trefethen, "Spectral Methods in MATLAB," Software, Environments, and Tools, 10, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [26] X. Wan, D. Xiu and G. E. Karniadakis, Stochastic solutions for the two-dimensional advection-diffusion equation, SIAM J. Sci. Comp., 26 (2004), 578-590. doi: 10.1137/S106482750342684X. [27] D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp., 24 (2002), 619-644. doi: 10.1137/S1064827501387826.
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