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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Schoutens, Wim, K U Leuven, and Michael Studer. 2001. “Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management,” February, 24. https://www.eurandom.tue.nl/reports/2001/005-report.pdf.