Stochastic Taylor expansion

Polynomial approximations of small randomnesses, Itô’s lemma

October 15, 2020 — December 24, 2023

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

Figure 1

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 to take into account higher-order derivatives of the function \(f\). 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, but stuff can get weird.

In practice, we tend to prefer other methods of solving stochastic differential equations than starting from this guy. TODO: worked example showing how tedious this gets. 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

1 Incoming

  • Taylor expansion with integral remainder

    The Carr–Madan formula is really just a special case of a Taylor expansion. For completeness, let’s rederive the Taylor expansion with an integral remainder.

2 References

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Ariffin, and Rosli. 2017. Stochastic Taylor Expansion of Derivative-Free Method for Stochastic Differential Equations.” Malaysian Journal of Fundamental and Applied Sciences.
Jacob, and Schilling. 2001. Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications.
Kloeden, P. E., and Platen. 1991. Stratonovich and Ito Stochastic Taylor Expansions.” Mathematische Nachrichten.
Kloeden, Peter E., and Platen. 1992. Stochastic Taylor Expansions.” In Numerical Solution of Stochastic Differential Equations. Applications of Mathematics.
———. 2010. Numerical Solution of Stochastic Differential Equations.
Kloeden, P. E., Platen, and Wright. 1992. The Approximation of Multiple Stochastic Integrals.” Stochastic Analysis and Applications.
Papapantoleon, and Siopacha. 2010. Strong Taylor Approximation of Stochastic Differential Equations and Application to the Lévy LIBOR Model.” arXiv:0906.5581 [Math, q-Fin].
Rößler. 2004. Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes.” Stochastic Analysis and Applications.
Sadr. 2009. Appendix A: Taylor Series Expansion.” In Interest Rate Swaps and Their Derivatives: A Practitioner’s Guide.
Schoutens, Leuven, and Studer. 2001. Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management.”