Stochastic differential equations

Itō and friends

Placeholder. A time-indexed, causal, measure-valued stochastic process. As seen in state filters, optimal control, financial mathematics etc.

Useful tools: infinitesimal generators, martingales.

One difficulty in this field is that many references take SDEs to be synonymous with Itō processes, whose driving noise is Brownian. In full generality, e.g. (Kallenberg 2002) they are a lot more general than that.

Applebaum, David, and Markus Riedle. 2010. “Cylindrical Levy Processes in Banach Spaces.” Proceedings of the London Mathematical Society 101 (3): 697–726. https://doi.org/10.1112/plms/pdq004.

Ariffin, Noor Amalina Nisa, and Norhayati Rosli. 2017. “Stochastic Taylor Expansion of Derivative-Free Method for Stochastic Differential Equations.” Malaysian Journal of Fundamental and Applied Sciences 13 (3). https://doi.org/10.11113/mjfas.v13n3.633.

Baudoin, Fabrice. 2014. Diffusion Processes and Stochastic Calculus. EMS Textbooks in Mathematics. Zurich, Switzerland: European Mathematical Society.

Baudoin, Fabrice, and Alice Vatamanelu. n.d. “Stochastic Calculus,” 114. https://www.academia.edu/2974879/Stochastic_Calculus.

Bertoin, Jean, Marc Yor, and others. 2001. “On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable.” Electron. Comm. Probab 6 (95): 106. http://www.emis.ams.org/journals/EJP-ECP/_ejpecp/ECP/include/getdoc73f8.pdf?id=3485&article=1622&mode=pdf.

Bruti-Liberati, Nicola, and Eckhard Platen. 2007. “Strong Approximations of Stochastic Differential Equations with Jumps.” Journal of Computational and Applied Mathematics, Special issue on evolutionary problems, 205 (2): 982–1001. https://doi.org/10.1016/j.cam.2006.03.040.

Coulaud, Benjamin, and Frédéric JP Richard. 2018. “A Consistent Framework for a Statistical Analysis of Surfaces Based on Generalized Stochastic Processes.” https://hal.archives-ouvertes.fr/hal-01863312.

Davis, Mark H. A., Xin Guo, and Guoliang Wu. 2009. “Impulse Control of Multidimensional Jump Diffusions,” December. http://arxiv.org/abs/0912.3297.

Hairer, Martin. 2009. “An Introduction to Stochastic PDEs.” http://www.hairer.org/notes/SPDEs.pdf.

Hanson, Floyd B. 2007. “Stochastic Processes and Control for Jump-Diffusions.” SSRN Scholarly Paper ID 1023497. Rochester, NY: Social Science Research Network. https://doi.org/10.2139/ssrn.1023497.

Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag. https://doi.org/10.1007/978-1-4757-4015-8.

Karczewska, Anna. 2007. “Convolution Type Stochastic Volterra Equations,” December. http://arxiv.org/abs/0712.4357.

Kloeden, P. E., and E. Platen. 1991. “Stratonovich and Ito Stochastic Taylor Expansions.” Mathematische Nachrichten 151 (1): 33–50. https://doi.org/10.1002/mana.19911510103.

Kloeden, P. E., E. Platen, and I. W. Wright. 1992. “The Approximation of Multiple Stochastic Integrals.” Stochastic Analysis and Applications 10 (4): 431–41. https://doi.org/10.1080/07362999208809281.

Kloeden, Peter E, and Eckhard Platen. 1992. Numerical Solution of Stochastic Differential Equations. Berlin, Heidelberg: Springer Berlin Heidelberg. https://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=3099793.

Kloeden, Peter E., and Eckhard Platen. 1992. “Stochastic Taylor Expansions.” In Numerical Solution of Stochastic Differential Equations, edited by Peter E. Kloeden and Eckhard Platen, 161–226. Applications of Mathematics. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-12616-5_5.

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Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (4): 423–98. https://doi.org/10.1111/j.1467-9868.2011.00777.x.

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Mikosch, Thomas, and Rimas Norvaiša. 2000. “Stochastic Integral Equations Without Probability.” Bernoulli 6 (3): 401–34. https://doi.org/10.2307/3318668.

Mohammed, Salah-Eldin A., and Michael K. R. Scheutzow. 1997. “Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies.” The Annals of Probability 25 (3): 1210–40. https://doi.org/10.1214/aop/1024404511.

Papaspiliopoulos, Omiros, Yvo Pokern, Gareth O. Roberts, and Andrew M. Stuart. 2012. “Nonparametric Estimation of Diffusions: A Differential Equations Approach.” Biometrika 99 (3): 511–31. https://doi.org/10.1093/biomet/ass034.

Rackauckas, Christopher, Yingbo Ma, Vaibhav Dixit, Xingjian Guo, Mike Innes, Jarrett Revels, Joakim Nyberg, and Vijay Ivaturi. 2018. “A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions,” December. http://arxiv.org/abs/1812.01892.

Revuz, Daniel, and Marc Yor. 2005. Continuous Martingales and Brownian Motion. Berlin; Heidelberg: Springer.

Rogers, L. C. G., and D. Williams. 2000. Diffusions, Markov Processes, and Martingales. 2nd ed. Cambridge Mathematical Library. Cambridge, U.K. ; New York: Cambridge University Press.

Rößler, Andreas. 2004. “Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes.” Stochastic Analysis and Applications 22 (6): 1553–76. https://doi.org/10.1081/SAP-200029495.

Särkkä, Simo, and Arno Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10. Cambridge ; New York, NY: Cambridge University Press.

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Solin, Arno. 2016. “Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.” Aalto University. https://aaltodoc.aalto.fi:443/handle/123456789/19842.

Yaglom, A. M. 1987. Correlation Theory of Stationary and Related Random Functions: Supplementary Notes and References. Springer Series in Statistics. New York, NY: Springer Science & Business Media.