Stochastic differential equations

Itō and friends


SDEs are time-indexed, causal stochastic processes which notionally integrate an ordinary differential equation over some driving noise. As seen in state filters, optimal control, financial mathematics etc.

Terminology problem: when people talk about these they really mean stochastic integral equations, in the sense that the driving noise process is integrated. When you differentiate the noise process, it leads, AFAICT to Malliavin calculus.

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

Useful tools: infinitesimal generators, martingales, Dale Robert’s cheat sheet, Itō-Taylor expansions

Terminology problem: 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.
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).
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, and others. 2001. “On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable.” Electron. Comm. Probab 6 (95): 106.
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.”
Davis, Mark H. A., Xin Guo, and Guoliang Wu. 2009. “Impulse Control of Multidimensional Jump Diffusions.” December 16, 2009.
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).
Hairer, Martin. 2009. “An Introduction to Stochastic PDEs.”
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.
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.
Karczewska, Anna. 2007. “Convolution Type Stochastic Volterra Equations.” December 28, 2007.
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.
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, and Rimas Norvaiša. 2000. “Stochastic Integral Equations Without Probability.” Bernoulli 6 (3): 401–34.
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.
Papanicolaou, Andrew. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.” January 2, 2019.
Papapantoleon, Antonis, and Maria Siopacha. 2010. “Strong Taylor Approximation of Stochastic Differential Equations and Application to the Lévy LIBOR Model.” October 4, 2010.
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.
Privault, Nicolas. n.d. Notes on Stochastic Finance.
Protter, Philip. 2005. Stochastic Integration and Differential Equations. Springer.
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.” December 5, 2018.
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 13, 2020.
Revuz, Daniel, and Marc Yor. 2004. Continuous Martingales and Brownian Motion. Springer Science & Business Media.
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.
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.
Solin, Arno. 2016. “Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.” Aalto University.
Szehr, Oleg, Dario Azzimonti, and Laura Azzimonti. 2020. “An Exact Kernel Framework for Spatio-Temporal Dynamics.” November 13, 2020.
Şimşekli, Umut, Ozan Sener, George Deligiannidis, and Murat A. Erdogdu. 2020. “Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks.” June 16, 2020.
Tautu, Petre. 2014. Stochastic Spatial Processes. Springer.
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.
Øksendal, Bernt. 1985. Stochastic Differential Equations: An Introduction With Applications. Springer.
———. n.d. An Introduction to Maliavin Calculus with Applications to Economics.