Stochastic Taylor expansion

Polynomial approximations of small randomnesses



Placeholder, for discussing the Taylor expansion equivalent for an SDE.

Let \(f\) denote a smooth function. Then, using Itô’s lemma, we may construct a local approximation by \[ f\left(X_{t}\right)=f\left(X_{0}\right)+\int_{s=0}^{t} L^{0} f\left(X_{s}\right) d s+\int_{s=0}^{t} L^{1} f\left(X_{s}\right) d B_{s} \] where the operators \(L^{0}\) and \(L^{1}\) are defined by \[ L^{0}=a(x) \frac{\partial}{\partial x}+\frac{1}{2} b(x)^{2} \frac{\partial^{2}}{\partial x^{2}} \quad \text { and } \quad L^{1}=b(x) \frac{\partial}{\partial x} \] We may notionally repeat this procedure arbitrarily many times. Each repetition produces a higher order of Itô-Taylor expansion. In practice this seems to get ugly really fast in any problem that you would actually like to use it in.

We may also generalise it to other noises than Brownian noise, including, say, arbitrary Lévy noises.

In practice, we tend to prefer other methods of solving stochastic differential equations than starting from this guy. But I will keep him around for reference

TBD: Relationship to Malliavin calculus and infinitesimal generators, other methods of approximating the distribution of a transformed RV

References

Aït-Sahalia, Yacine, Lars Peter Hansen, and José A. Scheinkman. 2010. “Operator Methods for Continuous-Time Markov Processes.” In Handbook of Financial Econometrics: Tools and Techniques, 1–66. Elsevier. https://doi.org/10.1016/B978-0-444-50897-3.50004-3.
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.
Jacob, Niels, and René L. Schilling. 2001. “Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser. https://doi.org/10.1007/978-1-4612-0197-7_7.
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. “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.
———. 2010. Numerical Solution of Stochastic Differential Equations. Berlin, Heidelberg: Springer Berlin Heidelberg. https://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=3099793.
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. http://arxiv.org/abs/0906.5581.
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.
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.

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