Stochastic differential equations

September 19, 2019 — June 22, 2021

dynamical systems
Lévy processes
probability
sciml
SDEs
signal processing
stochastic processes
time series

By analogy with differential equations, which use vanilla calculus to define deterministic dynamics, we can define stochastic differential equations, which use stochastic calculus to define random dynamics.

SDEs are time-indexed, causal stochastic processes which notionally integrate an ordinary differential equation over some driving noise. SPDEs are to SDEs as PDEs are to ODEs.

Useful in state filters, optimal control, financial mathematics etc.

Usually we talk about differential equations, but the broader and I think more common class of ODEs is naturally defined through integral equations rather than differential equations, in the sense that the driving noise process is an integrator. When you differentiate the noise process, it leads, AFAICT to Malliavin calculus, or something? I am not sure about that theory.

Warning: beware the terminology problem that some references take SDEs to be synonymous with Itô processes, whose driving noise is Brownian. Some writers, when they really want to be clear that they are not assuming Brownian motion, but some SDEs driven by Lévy noise, use the term sparse stochastic processes.

1 Pathwise solutions

Random ODEs is the highly ambiguous phrasing of Bongers and Mooij (2018) when referring to a certain class of referring to smooth SDEs; better might be “noise-driven ODEs”. In practice this is useful for the kind of smooth systems I encounter often.

AFAICT we can consider these in the context of Wong-Zakai approximation in classical SDE context, wrt Stratonovich integrals. However, the cleanest introduction I have is in the context of rough paths so let’s look at it in that context instead. For a classical setting, see .

See SPDEs.

4 References

Adler. 2010. The Geometry of Random Fields.
Adler, and Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115.
Allen, Allen, Arciniega, et al. 2008. Stochastic Analysis and Applications.
Applebaum, and Riedle. 2010. Proceedings of the London Mathematical Society.
Arfken, Weber, and Harris. 2013. Mathematical Methods for Physicists: A Comprehensive Guide.
Ariffin, and Rosli. 2017. Malaysian Journal of Fundamental and Applied Sciences.
Arnold, and Kliemann. 1983. In Probabilistic Analysis and Related Topics.
Bain, and Crisan. 2008. Fundamentals of Stochastic Filtering.
Baudoin. 2014. Diffusion Processes and Stochastic Calculus. EMS Textbooks in Mathematics.
Baudoin, and Vatamanelu. n.d.
Bertoin, Yor, and others. 2001. Electron. Comm. Probab.
Bion-Nadal, and Talay. 2019. The Annals of Applied Probability.
Bongers, and Mooij. 2018. arXiv:1803.08784 [Cs, Stat].
Bosq. 2002. Statistical Inference for Stochastic Processes.
Bruti-Liberati, and Platen. 2007. Journal of Computational and Applied Mathematics, Special issue on evolutionary problems,.
Coulaud, and Richard. 2018.
Curtain, and Falb. 1971. Journal of Differential Equations.
Davis, Guo, and Wu. 2009. arXiv:0912.3297 [Math].
Eguchi, and Uehara. n.d. Scandinavian Journal of Statistics.
Goldys, and Peszat. 2021.
Hanson. 2007. SSRN Scholarly Paper ID 1023497.
Hassler. 2016. Stochastic Processes and Calculus. Springer Texts in Business and Economics.
Hodgkinson, Roosta, and Mahoney. 2021. “Stochastic Continuous Normalizing Flows: Training SDEs as ODEs.” Uncertainty in Artificial Intelligence.
Holden, Øksendal, Ubøe, et al. 1996. Stochastic Partial Differential Equations.
Inchiosa, and Bulsara. 1996. Physical Review E.
Jacod, and Shiryaev. 1987. In Limit Theorems for Stochastic Processes. Grundlehren Der Mathematischen Wissenschaften.
Kallenberg. 2002. Foundations of Modern Probability. Probability and Its Applications.
Karatzas, and Ruf. 2016. Annales de l’Institut Henri Poincaré, Probabilités Et Statistiques.
Karczewska. 2007. arXiv:0712.4357 [Math].
Kelly. 2016. The Annals of Applied Probability.
Kelly, and Melbourne. 2014a.
Klebaner. 1999. Introduction to Stochastic Calculus With Applications.
Kloeden, P. E., and Platen. 1991. Mathematische Nachrichten.
Kloeden, Peter E., and Platen. 1992. In Numerical Solution of Stochastic Differential Equations. Applications of Mathematics.
Kloeden, P. E., Platen, and Wright. 1992. Stochastic Analysis and Applications.
Korzeniowski. 1989. Statistics & Probability Letters.
Kotelenez. 2007. Stochastic Ordinary and Stochastic Partial Differential Equations: Transition From Microscopic to Macroscopic Equations.
Kushner, and DiMasi. 1978. Journal of Mathematical Analysis and Applications.
Lindgren, Rue, and Lindström. 2011. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Liu, Yeo, and Lu. 2020. Journal of the American Statistical Association.
Loerincz, Gingl, and Kiss. 1996. Physics Letters A.
Ma, Chen, and Fox. 2015. In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2. NIPS’15.
Mamporia. 2017. Transactions of A. Razmadze Mathematical Institute.
Matheron. 1973. Advances in Applied Probability.
Meidan. 1980. Journal of Mathematical Analysis and Applications.
Mikosch. 2004. Non-Life Insurance Mathematics: An Introduction With Stochastic Processes.
Mikosch, and Norvaiša. 2000. Bernoulli.
Miller, Glennie, and Seaton. 2020. Journal of Agricultural, Biological and Environmental Statistics.
Mohammed, and Scheutzow. 1997. The Annals of Probability.
Papanicolaou. 2019. arXiv:1504.05309 [Math, q-Fin].
Papapantoleon, and Siopacha. 2010. arXiv:0906.5581 [Math, q-Fin].
Papaspiliopoulos, Pokern, Roberts, et al. 2012. Biometrika.
Pavliotis. 2014. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Texts in Applied Mathematics.
Privault. n.d. Notes on Stochastic Finance.
Protter. 2005. Stochastic Integration and Differential Equations.
Pugachev, and Sinit︠s︡yn. 2001. Stochastic systems: theory and applications.
Rackauckas. 2019. The Winnower.
Rackauckas, Ma, Dixit, et al. 2018. arXiv:1812.01892 [Cs].
Rackauckas, Ma, Martensen, et al. 2020. arXiv.org.
Revuz, and Yor. 2004. Continuous Martingales and Brownian Motion.
Risken. 1996. The Fokker-Planck Equation: Methods of Solution and Applications. Springer Series in Synergetics, v. 18.
Rogers, and Williams. 1987. Diffusions, Markov Processes and Martingales 2.
Rogers, and Williams. 2000. Diffusions, Markov Processes, and Martingales. Cambridge Mathematical Library.
Rößler. 2004. Stochastic Analysis and Applications.
Rubenstein, Bongers, Schölkopf, et al. 2018. In Uncertainty in Artificial Intelligence.
Särkkä, and Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10.
Schoutens. 2000. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics.
Schoutens, Leuven, and Studer. 2001.
Sigrist, Künsch, and Stahel. 2015. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Şimşekli, Sener, Deligiannidis, et al. 2020. CoRR.
Sussmann. 1978. The Annals of Probability.
Szehr, Azzimonti, and Azzimonti. 2020. arXiv:2011.06848 [Math, Stat].
Tautu. 2014. Stochastic Spatial Processes.
Teye. 2010. “Stochastic Invariance via Wong-Zakai Theorem.”
Twardowska. 1996. “Wong-Zakai Approximations for Stochastic Differential Equations.” Acta Applicandae Mathematica.
Tzen, and Raginsky. 2019a. In Proceedings of the Thirty-Second Conference on Learning Theory.
van Kampen. 1976. Physics Reports.
Wedig. 1984. Nuclear Engineering and Design.
Wolpert, and Brown. 2021. arXiv:2105.14591 [Math].
Yaglom. 1987. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics.
Zozor, and Amblard. 2003. IEEE Transactions on Signal Processing.