Book contents
- Frontmatter
- Contents
- Preface
- Contributors
- 1 The State of the Art in Smale's 7th Problem
- 2 The Shape of Data
- 3 Upwinding in Finite Element Systems of Differential Forms
- 4 On the Complexity of Computing Quadrature Formulas for SDEs
- 5 The Quantum Walk of F. Riesz
- 6 Modulated Fourier Expansions for Continuous and Discrete Oscillatory Systems
- 7 The Dual Role of Convection in 3D Navier-Stokes Equations
- 8 Algebraic and Differential Invariants
- 9 Through the Kaleidoscope: Symmetries, Groups and Chebyshev-Approximations from a Computational Point of View
- 10 Sage: Creating a Viable Free Open Source Alternative to Magma, Maple, Mathematica, and MATLAB
- References
4 - On the Complexity of Computing Quadrature Formulas for SDEs
Published online by Cambridge University Press: 05 December 2012
Book contents
- Frontmatter
- Contents
- Preface
- Contributors
- 1 The State of the Art in Smale's 7th Problem
- 2 The Shape of Data
- 3 Upwinding in Finite Element Systems of Differential Forms
- 4 On the Complexity of Computing Quadrature Formulas for SDEs
- 5 The Quantum Walk of F. Riesz
- 6 Modulated Fourier Expansions for Continuous and Discrete Oscillatory Systems
- 7 The Dual Role of Convection in 3D Navier-Stokes Equations
- 8 Algebraic and Differential Invariants
- 9 Through the Kaleidoscope: Symmetries, Groups and Chebyshev-Approximations from a Computational Point of View
- 10 Sage: Creating a Viable Free Open Source Alternative to Magma, Maple, Mathematica, and MATLAB
- References
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Foundations of Computational Mathematics, Budapest 2011 , pp. 72 - 92Publisher: Cambridge University PressPrint publication year: 2012
References
[1] and , The law of the Euler scheme for stochastic differential equations (I): convergence rate of the distribution function. Prob. Theory and Related Fields, 104, 43–60, 1995.Google Scholar
[2] , , and , Infinite-dimensional quadrature and approximation of distributions. Found. Comput. Math., 9, 391–429, 2009.Google Scholar
[3] and , On the convergence rates of a general class of weak approximation of SDEs. in: Stochastic Differential Equations, Theory and Applications, , , eds., 221–249, World Scientific, Singapore, 2007.
[4] , High resolution coding of stochastic processes and small ball probabilities. Ph. D. Thesis, Department of Mathematics, 2003, FU Berlin.
[5] , The coding complexity of diffusion processes under supremum norm distortion. Stochastic Processes Appl., 118, 917–937, 2008.Google Scholar
[6] , The coding complexity of diffusion processes under Lp [0, 1]-norm distortion. Stochastic Processes Appl., 118, 938–951, 2008.Google Scholar
[7] , Asymptotic formulae for coding problems and intermediate optimization problems: a review. in: Trends in Stochastic Analysis, , , , eds., 187–232, Cambridge Univ. Press, Cambridge, 2009.
[8] and , High-resolution quantization and entropy coding for fractional Brownian motion. Electron. J. Probab., 11, 700–722, 2006.Google Scholar
[9] , and , Constructive quantization: approximation by empirical measures. Preprint, arXiv:1108.5346, 2011.
[10] and , Foundations of Quantization for Probability Distributions. Lect. Notes in Math. 1730, Springer-Verlag, Berlin, 2000.
[11] , and , The optimal discretization of stochastic differential equations. J. Complexity, 17, 117–153, 2001.Google Scholar
[12] , and , Linear vs. standard information for scalar stochastic differential equations. J. Complexity, 18, 394–414, 2002.Google Scholar
[14] , Approximation of expectation of diffusion process and mathematical finance. Adv. Stud. Pure Math., 31, 147–165, 2001.Google Scholar
[15] , Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus. Adv. Math. Econ., 6, 69–83, 2004.Google Scholar
[16] and , High order recombination and an application to cubature on the Wiener space. Preprint. http://arxiv.org/pdf/1008.4942v1, 2010.
[17] and , Sharp asymptotics of the functional quantization problem for Gaussian processes. Ann. Appl. Prob., 32, 1574–1599, 2004.Google Scholar
[18] and , Functional quantization of a class of Brownian diffusion: a constructive approach. Stochastic Processes Appl., 116, 310–336, 2006.Google Scholar
[19] and , Functional quantization rate and mean regularity of processes with an application to Lévy processes. Ann. Appl. Prob., 18, 427–469, 2008.Google Scholar
[20] , and , Asymptotically optimal quantization schemes for Gaussian processes. ESAIM Probab. Stat., 14, 93–116, 2008.Google Scholar
[21] and , Cubature on Wiener space. Proc. Royal Soc. Lond., 460, 169–198, 2004.
[22] , Optimal uniform approximation of systems of stochastic differential equations. Ann. Appl. Prob., 12, 664–690, 2002.Google Scholar
[23] , Optimal pointwise approximation of SDEs based on Brownian motion at discrete points. Ann. Appl. Prob., 14, 1605–1642, 2004.Google Scholar
[24] and , A local refinement strategy for constructive quantization of scalar SDEs. Preprint 72, DFG SPP 1324, 2010.
[25] , and , A derandomization of the Euler scheme for scalar stochastic differential equations. To appear in J. Complexity.
[27] and , http://www.quantize.maths-fi.co m/, 2005.
[28] and , Optimal quantization for finance: from random vectors to stochastic processes. in: Mathematical Modelling and Numerical Methods in Finance, Handbook of Numerical Analysis, Vol. XV, , , eds., 595–648, North-Holland, Amsterdam, 2008.
[29] and , Convergence of multi-dimensional quantized SDE's. Preprint, 2010.
[30] and , On the complexity of parabolic initial value problems with variable drift. J. Complexity, 22, 118–145, 2006.Google Scholar
[31] , Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester, 1991.
[32] , Konstruktive Quantisierung skalarer Diffusionsprozesse. Diploma Thesis, Department of Mathematics, 2008, TU Darmstadt.
[33] , and , Information-Based Complexity. Academic Press, New York, 1988.
[34] and , Complexity of weighted approximation over ℝ. J. Approx. Theory, 103, 223–251, 2000.Google Scholar
[35] and , Complexity of weighted approximation over ℝd. J. Complexity, 17, 722–740, 2001.Google Scholar
[36] , Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions. Ann. Appl. Prob., 11, 470–487, 2001.Google Scholar
- 1
- Cited by