# 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.

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