# 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.

Cosma’s explanation of SDEs looks good for cannibalising for parts when I write my own.

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.

One confusion is that Itō’s formula, which is an important tool here, is applicable more broadly than to Brownian-type diffusions

Let $$X=\left(X^{1}, \ldots, X^{n}\right)$$ be an n-tuple of semimartingales and let $$f: \mathbb{R}^{n} \rightarrow$$ R have continuous second order partial derivatives. Then $$f(X)$$ is again a semimartingale and the following formula holds:

\begin{aligned} f\left(X_{t}\right)=& f\left(X_{0}\right)+\sum_{i=1}^{n} \int_{0+}^{t} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) d X_{s}^{i} \\ &+\frac{1}{2} \sum_{1 \leq i, j \leq n} \int_{0+}^{t} \frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}\left(X_{s-}\right) d\left[X^{i}, X^{j}\right]_{s}^{c} \\ &+\sum_{0<s \leq t}\left(f\left(X_{s}\right)-f\left(X_{s-}\right)-\sum_{i=1}^{n} \frac{\partial f}{\partial x_{i}}\left(X_{s-}\right) \Delta X_{s}^{i}\right) \end{aligned}

This does get messy for non-Brownian processes, however. Schoutens, Leuven, and Studer (2001) give a tractable example for Poisson processes.

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.

Korzeniowski, Andrzej. 1989. “On Diffusions That Cannot Escape from a Convex Set.” Statistics & Probability Letters 8 (3): 229–34. https://doi.org/10.1016/0167-7152(89)90127-2.

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. https://doi.org/10.1016/0022-247X(78)90072-0.

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.

Matheron, G. 1973. “The Intrinsic Random Functions and Their Applications.” Advances in Applied Probability 5 (3): 439–68. https://doi.org/10.2307/1425829.

Meidan, R. 1980. “On the Connection Between Ordinary and Generalized Stochastic Processes.” Journal of Mathematical Analysis and Applications 76 (1): 124–33. https://doi.org/10.1016/0022-247X(80)90066-9.

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.

Papapantoleon, Antonis, and Maria Siopacha. 2010. “Strong Taylor Approximation of Stochastic Differential Equations and Application to the L\’evy LIBOR Model,” October. http://arxiv.org/abs/0906.5581.

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.

Rackauckas, Christopher, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, and Ali Ramadhan. 2020. “Universal Differential Equations for Scientific Machine Learning,” January. https://arxiv.org/abs/2001.04385v1.

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.

Schilling, René L. 2016. “An Introduction to Lévy and Feller Processes. Advanced Courses in Mathematics - CRM Barcelona 2014,” March. http://arxiv.org/abs/1603.00251.

Schoutens, Wim, K U Leuven, and Michael Studer. 2001. “Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management,” February, 24. https://www.eurandom.tue.nl/reports/2001/005-report.pdf.

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

Şimşekli, Umut, Ozan Sener, George Deligiannidis, and Murat A. Erdogdu. 2020. “Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks,” June. http://arxiv.org/abs/2006.09313.

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.