Itō-Taylor expansion

Polynomial approximations of small randomnesses


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

Let \(f\) denote a smooth function. Then from Itō’s lemma, \[ f\left(X_{t}\right)=f\left(X_{0}\right)+\int_{s=0}^{t} L^{0} f\left(X_{s}\right) d s+\int_{s=0}^{t} L^{1} f\left(X_{s}\right) d B_{s} \] 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 repeat this procedure arbitrarily many times. Each repetition produces a higher order of Itō-Taylor expansion.

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.
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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.” October 4, 2010. 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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