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.
Useful tools: infinitesimal generators, martingales, Dale Robert’s cheat sheet, Itô-Taylor expansions…
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.
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 (Teye 2010; Wedig 1984).
Infinite dimensional
See SPDEs.
References
Adler, Robert J. 2010. The Geometry of Random Fields. SIAM ed. Philadelphia: Society for Industrial and Applied Mathematics.
Adler, Robert J., and Jonathan E. Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115. New York: Springer.
Allen, Edward J., Linda J. S. Allen, Armando Arciniega, and Priscilla E. Greenwood. 2008. “Construction of Equivalent Stochastic Differential Equation Models.” Stochastic Analysis and Applications 26 (2): 274–97.
Applebaum, David, and Markus Riedle. 2010. “Cylindrical Levy Processes in Banach Spaces.” Proceedings of the London Mathematical Society 101 (3): 697–726.
Arfken, George B., Hans-Jurgen Weber, and Frank E. Harris. 2013. Mathematical Methods for Physicists: A Comprehensive Guide. 7th ed. Amsterdam ; Boston: Elsevier.
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).
Arnold, Ludwig, and Wolfgang Kliemann. 1983. “Qualitative Theory of Stochastic Systems.” In Probabilistic Analysis and Related Topics, edited by A. T. Bharucha-reid, 1–79. Academic Press.
Bain, Alan, and Dan Crisan. 2008. Fundamentals of Stochastic Filtering. Springer.
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.
Bertoin, Jean, Marc Yor, et al. 2001. “On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable.” Electron. Comm. Probab 6 (95): 106.
Bion-Nadal, Jocelyne, and Denis Talay. 2019. “On a Wasserstein-Type Distance Between Solutions to Stochastic Differential Equations.” The Annals of Applied Probability 29 (3).
Bongers, Stephan, and Joris M. Mooij. 2018. “From Random Differential Equations to Structural Causal Models: The Stochastic Case.” arXiv:1803.08784 [Cs, Stat], March.
Bosq, Denis. 2002. “Estimation of Mean and Covariance Operator of Autoregressive Processes in Banach Spaces.” Statistical Inference for Stochastic Processes 5 (3): 287–306.
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.
Coulaud, Benjamin, and Frédéric JP Richard. 2018. “A Consistent Framework for a Statistical Analysis of Surfaces Based on Generalized Stochastic Processes.”
Curtain, Ruth F, and Peter L Falb. 1971. “Stochastic Differential Equations in Hilbert Space.” Journal of Differential Equations 10 (3): 412–30.
Davis, Mark H. A., Xin Guo, and Guoliang Wu. 2009. “Impulse Control of Multidimensional Jump Diffusions.” arXiv:0912.3297 [Math], December.
Eguchi, Shoichi, and Yuma Uehara. n.d. “Schwartz-Type Model Selection for Ergodic Stochastic Differential Equation Models.” Scandinavian Journal of Statistics n/a (n/a).
Goldys, Beniamin, and Szymon Peszat. 2021. “On Linear Stochastic Flows,” May.
Hanson, Floyd B. 2007. “Stochastic Processes and Control for Jump-Diffusions.” SSRN Scholarly Paper ID 1023497. Rochester, NY: Social Science Research Network.
Hassler, Uwe. 2016. Stochastic Processes and Calculus. Springer Texts in Business and Economics. Cham: Springer International Publishing.
Hodgkinson, Liam, Fred Roosta, and Michael W Mahoney. 2021. “Stochastic Continuous Normalizing Flows: Training SDEs as ODEs.” Uncertainty in Artificial Intelligence 37 (July): 11.
Holden, Helge, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang. 1996. Stochastic Partial Differential Equations. Boston, MA: Birkhäuser Boston.
Inchiosa, M. E., and A. R. Bulsara. 1996. “Signal Detection Statistics of Stochastic Resonators.” Physical Review E 53 (3): R2021–24.
Jacod, Jean, and Albert N. Shiryaev. 1987. “The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.” In Limit Theorems for Stochastic Processes, edited by Jean Jacod and Albert N. Shiryaev, 1–63. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.
Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag.
Karatzas, Ioannis, and Johannes Ruf. 2016. “Pathwise Solvability of Stochastic Integral Equations with Generalized Drift and Non-Smooth Dispersion Functions.” Annales de l’Institut Henri Poincaré, Probabilités Et Statistiques 52 (2): 915–38.
Karczewska, Anna. 2007. “Convolution Type Stochastic Volterra Equations.” arXiv:0712.4357 [Math], December.
Kelly, David. 2016. “Rough Path Recursions and Diffusion Approximations.” The Annals of Applied Probability 26 (1).
Kelly, David, and Ian Melbourne. 2014a. “Smooth Approximation of Stochastic Differential Equations,” March.
———. 2014b. “Deterministic Homogenization for Fast-Slow Systems with Chaotic Noise,” September.
Klebaner, Fima C. 1999. Introduction to Stochastic Calculus With Applications. Imperial College Press.
Kloeden, P. E., and E. Platen. 1991. “Stratonovich and Ito Stochastic Taylor Expansions.” Mathematische Nachrichten 151 (1): 33–50.
Kloeden, P. E., E. Platen, and I. W. Wright. 1992. “The Approximation of Multiple Stochastic Integrals.” Stochastic Analysis and Applications 10 (4): 431–41.
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.
———. 2010. Numerical Solution of Stochastic Differential Equations. Berlin, Heidelberg: Springer Berlin Heidelberg.
Korzeniowski, Andrzej. 1989. “On Diffusions That Cannot Escape from a Convex Set.” Statistics & Probability Letters 8 (3): 229–34.
Kotelenez, Peter. 2007. Stochastic Ordinary and Stochastic Partial Differential Equations: Transition From Microscopic to Macroscopic Equations. Springer Science & Business Media.
Kushner, Harold J, and Giovanni DiMasi. 1978. “Approximations for Functionals and Optimal Control Problems on Jump Diffusion Processes.” Journal of Mathematical Analysis and Applications 63 (3): 772–800.
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.
Liu, Xiao, Kyongmin Yeo, and Siyuan Lu. 2020. “Statistical Modeling for Spatio-Temporal Data From Stochastic Convection-Diffusion Processes.” Journal of the American Statistical Association 0 (0): 1–18.
Loerincz, K., Z. Gingl, and L. B. Kiss. 1996. “A Stochastic Resonator Is Able to Greatly Improve Signal-to-Noise Ratio.” Physics Letters A 224 (1): 63–67.
Ma, Yi-An, Tianqi Chen, and Emily B. Fox. 2015. “A Complete Recipe for Stochastic Gradient MCMC.” In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2, 2917–25. NIPS’15. Cambridge, MA, USA: MIT Press.
Mamporia, Badri. 2017. “Stochastic Differential Equations in a Banach Space Driven by the Cylindrical Wiener Process.” Transactions of A. Razmadze Mathematical Institute 171 (1): 76–89.
Matheron, G. 1973. “The Intrinsic Random Functions and Their Applications.” Advances in Applied Probability 5 (3): 439–68.
Meidan, R. 1980. “On the Connection Between Ordinary and Generalized Stochastic Processes.” Journal of Mathematical Analysis and Applications 76 (1): 124–33.
Mikosch, Thomas. 2004. Non-Life Insurance Mathematics: An Introduction With Stochastic Processes. Kluwer Academic Publishers.
Mikosch, Thomas, and Rimas Norvaiša. 2000. “Stochastic Integral Equations Without Probability.” Bernoulli 6 (3): 401–34.
Miller, David L., Richard Glennie, and Andrew E. Seaton. 2020. “Understanding the Stochastic Partial Differential Equation Approach to Smoothing.” Journal of Agricultural, Biological and Environmental Statistics 25 (1): 1–16.
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.
Øksendal, Bernt. 1985. Stochastic Differential Equations: An Introduction With Applications. Springer.
Papanicolaou, Andrew. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.” arXiv:1504.05309 [Math, q-Fin], January.
Papapantoleon, Antonis, and Maria Siopacha. 2010. “Strong Taylor Approximation of Stochastic Differential Equations and Application to the Lévy LIBOR Model.” arXiv:0906.5581 [Math, q-Fin], October.
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.
Pavliotis, Grigorios A. 2014. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Texts in Applied Mathematics. New York: Springer-Verlag.
Privault, Nicolas. n.d. Notes on Stochastic Finance.
Protter, Philip. 2005. Stochastic Integration and Differential Equations. Springer.
Pugachev, V. S., and I. N. Sinit︠s︡yn. 2001. Stochastic systems: theory and applications. River Edge, NJ: World Scientific.
Rackauckas, Christopher. 2019. “Neural Jump SDEs (Jump Diffusions) and Neural PDEs.” The Winnower, June.
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.” arXiv:1812.01892 [Cs], December.
Rackauckas, Christopher, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. 2020. “Universal Differential Equations for Scientific Machine Learning.” arXiv:2001.04385 [Cs, Math, q-Bio, Stat], August.
Revuz, Daniel, and Marc Yor. 2004. Continuous Martingales and Brownian Motion. Springer Science & Business Media.
Risken, H. 1996. The Fokker-Planck Equation: Methods of Solution and Applications. 2nd ed. Springer Series in Synergetics, v. 18. New York: Springer-Verlag.
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.
Rogers, L. C. G., and David Williams. 1987. Diffusions, Markov Processes and Martingales 2. 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.
Rubenstein, Paul K., Stephan Bongers, Bernhard Schölkopf, and Joris M. Mooij. 2018. “From Deterministic ODEs to Dynamic Structural Causal Models.” In Uncertainty in Artificial Intelligence.
Särkkä, Simo, and Arno Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10. Cambridge ; New York, NY: Cambridge University Press.
Schoutens, Wim. 2000. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics. New York: Springer-Verlag.
Schoutens, Wim, K U Leuven, and Michael Studer. 2001. “Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management,” February, 24.
Sigrist, Fabio, Hans R. Künsch, and Werner A. Stahel. 2015. “Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (1): 3–33.
Şimşekli, Umut, Ozan Sener, George Deligiannidis, and Murat A. Erdogdu. 2020. “Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks.” CoRR abs/2006.09313.
Solin, Arno. 2016. “Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.” Aalto University.
Sussmann, Hector J. 1978. “On the Gap Between Deterministic and Stochastic Ordinary Differential Equations.” The Annals of Probability 6 (1): 19–41.
Szehr, Oleg, Dario Azzimonti, and Laura Azzimonti. 2020. “An Exact Kernel Framework for Spatio-Temporal Dynamics.” arXiv:2011.06848 [Math, Stat], November.
Tautu, Petre. 2014. Stochastic Spatial Processes. Springer.
Teye, Alfred Larm. 2010. “Stochastic Invariance via Wong-Zakai Theorem.” PhD Thesis, University of Amsterdam.
Twardowska, Krystyna. 1996. “Wong-Zakai Approximations for Stochastic Differential Equations.” Acta Applicandae Mathematica 43 (3): 317–59.
Van Kampen, N. G. 1976. “Stochastic Differential Equations.” Physics Reports 24 (3): 171–228.
Wedig, W. 1984. “A Critical Review of Methods in Stochastic Structural Dynamics.” Nuclear Engineering and Design 79 (3): 281–87.
Wolpert, Robert L., and Lawrence D. Brown. 2021. “Markov Infinitely-Divisible Stationary Time-Reversible Integer-Valued Processes.” arXiv:2105.14591 [Math], May.
Xiu, Dongbin. 2010. Numerical Methods for Stochastic Computations: A Spectral Method Approach. USA: Princeton University Press.
Yaglom, A. M. 1987. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics. New York, NY: Springer Science & Business Media.
Zozor, S., and P.-Amblard. 2003. “Stochastic Resonance in Locally Optimal Detectors.” IEEE Transactions on Signal Processing 51 (12): 3177–81.
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