ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE

doi:10.4208/jcm.1612-m2016-0681

In this article, we propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. We compare the complexity with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012), and find, by means of Fourier analysis, that the MIMC method:(i) has suboptimal complexity of O(ε^-2|log ε|³) for a root mean square error (RMSE) ε if the same spatial discretisation as in the MLMC method is used; (ii) has a better complexity of O(ε^-2|log ε|) if a carefully adapted discretisation is used; (iii) has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.

Key words: Parabolic stochastic partial differential equations, Multilevel Monte Carlo, Multi-index Monte Carlo, Stochastic finite differences, Zakai equation

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[1] E. Buckwar and T. Sickenberger. A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods. Mathematics and Computers in Simulation, 81:6(2011), 1110-1127.

[2] K. Bujok and C. Reisinger. Numerical valuation of basket credit derivatives in structural jumpdiffusion models. Journal of Computational Finance, 15:4(2012), 115-158.

[3] N. Bush, B.M. Hambly, H. Haworth, L. Jin, and C. Reisinger. Stochastic evolution equations in portfolio credit modelling. SIAM Journal on Financial Mathematics, 2:1(2011), 627-664.

[4] R. Carter, and M. B. Giles. Sharp error estimates for discretizations of the 1D convectiondiffusion equation with Dirac initial data. IMA Journal of Numerical Analysis, 27:2(2007), 406-425.

[5] A.M. Davie and J. Gaines. Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Mathematics of Computation, 70:233(2001), 121-134.

[6] M.B. Giles. Multilevel Monte Carlo path simulation. Operations Research, 56:3(2008), 607-617.

[7] M.B. Giles. Multilevel Monte Carlo methods. Acta Numerica, 24(2015), 259-328.

[8] M.B. Giles and C. Reisinger. Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIAM Journal on Financial Mathematics, 3:1(2012), 572-592.

[9] E. Gobet, G. Pages, H. Pham, and J. Printems. Discretization and simulation of the Zakai equation. SIAM Journal on Numerical Analysis, 44:6(2006), 2505-2538.

[10] M. Griebel and H. Harbrecht. On the convergence of the combination technique. Sparse Grids and Applications, volume 97 of Lecture Notes in Computational Science and Engineering, Springer, 2014, 55-74.

[11] I. Gyöngy. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I. Potential Analysis, 1:9(1998), 1-25.

[12] I. Gyöngy. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise Ⅱ. Potential Analysis, 11:1(1999), 1-37.

[13] I. Gyöngy and D. Nualart. Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Analysis, 7:4(1997), 725-757.

[14] A.L. Haji-Ali, F. Nobile, and R. Tempone. Multi-index Monte Carlo:when sparsity meets sampling. Numerische Mathematik, 132:4(2015), 767-806.

[15] D.J. Higham. Mean-square and asymptotic stability of the stochastic theta method. SIAM Journal on Numerical Analysis, 38:3(2000), 753-769.

[16] A. Jentzen and P. Kloeden. The numerical approximation of stochastic partial differential equations. Milan Journal of Mathematics, 77:1(2009), 205-244.

[17] A. Jentzen and P. Kloeden. Taylor expansions of solutions of stochastic partial differential equations with additive noise. The Annals of Probability, 38:2(2010), 532-569.

[18] A. Kebaier. Statistical Romberg extrapolation:a new variance reduction method and applications to option pricing. The Annals of Applied Probability, 15:4(2005), 2681-2705.

[19] R. Kruse. Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA Journal of Numerical Analysis, 34:1(2014), 217-251.

[20] N.V. Krylov and B.L. Rozovskii. Stochastic evolution equations. Journal of Soviet Mathematics, 16:4(1981), 1233-1277.

[21] T.G. Kurtz and J. Xiong. Particle representations for a class of nonlinear SPDEs. Stochastic Processes and their Applications, 83:1(1999), 103-126.

[22] S. Ledger. Sharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients. Stochastic Partial Differential Equations:Analysis and Computations 2:1(2014), 1-26.

[23] C. Pflaum and A. Zhou. Error analysis of the combination technique. Numerische Mathematik, 84:2(1999), 327-350. 1999.

[24] C. Reisinger. Analysis of linear difference schemes in the sparse grid combination technique. IMA Journal of Numerical Analysis, 33:2(2013), 544-581.

[25] C. Reisinger. Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative. International Journal of Computer Mathematics, 89:18(2012), 2562-2575.

[26] G. Strang and K. Aarikka. Introduction to Applied Mathematics, volume 16. W ellesley-Cambridge Press Wellesley, MA, 1986.

[27] L. Szpruch. Numerical Approximations of Nonlinear Stochastic Systems. PhD Thesis, University of Strathclyde, 2010.

[28] J. B. Walsh. Finite element methods for parabolic stochastic PDE's. Potential Analysis, 23:1(2005), 1-43.

[29] C. Zenger. Sparse grids. Parallel Algorithms for Partial Differential Equations (W. Hackbusch, ed.), Vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg, Braunschweig/Wiesbaden, 1991.

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ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE

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