# 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

## References

Aït-Sahalia, Yacine, Lars Peter Hansen, and José A. Scheinkman. 2010. In Handbook of Financial Econometrics: Tools and Techniques, 1–66. Elsevier.
Ariffin, Noor Amalina Nisa, and Norhayati Rosli. 2017. Malaysian Journal of Fundamental and Applied Sciences 13 (3).
Jacob, Niels, and René L. Schilling. 2001. In Lévy Processes: Theory and Applications, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser.
Kloeden, P. E., and E. Platen. 1991. Mathematische Nachrichten 151 (1): 33–50.
Kloeden, P. E., E. Platen, and I. W. Wright. 1992. Stochastic Analysis and Applications 10 (4): 431–41.
Kloeden, Peter E., and Eckhard Platen. 1992. 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.
Papapantoleon, Antonis, and Maria Siopacha. 2010. arXiv:0906.5581 [Math, q-Fin], October.
Rößler, Andreas. 2004. Stochastic Analysis and Applications 22 (6): 1553–76.
Schoutens, Wim, K U Leuven, and Michael Studer. 2001. February, 24.

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