Stochastic Taylor expansion

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.

In practice, we tend to prefer other methods of solving stochastic differential equations than starting from this guy. 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…