# 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.

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

*Handbook of Financial Econometrics: Tools and Techniques*.

*Malaysian Journal of Fundamental and Applied Sciences*.

*Lévy Processes: Theory and Applications*.

*Mathematische Nachrichten*.

*Numerical Solution of Stochastic Differential Equations*. Applications of Mathematics.

*Stochastic Analysis and Applications*.

*arXiv:0906.5581 [Math, q-Fin]*.

*Stochastic Analysis and Applications*.

*Interest Rate Swaps and Their Derivatives: A Practitioner’s Guide*.