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.

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