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
We may notionally repeat this procedure arbitrarily many times to take into account higher-order derivatives of the function
We may also generalize 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🚧 clarify 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.