Ito's formula via rough paths

Stochastic Analysis Seminar Series

Non-geometric rough paths arise when one encounters stochastic integrals for which the classical integration by parts formula does not hold. We will introduce two notions of non-geometric rough paths - one old (branched rough paths) and one new (quasi geometric rough paths). The former (due to Gubinelli) assumes one knows nothing about products of integrals, instead those products must be postulated as new components of the rough path. The latter assumes one knows a bit about products, namely that they satisfy a natural generalisation of the "Ito" integration by parts formula. We will show why they are both reasonable frameworks for a large class of integrals. Moreover, we will show that Ito's formula can be derived in either framework and that this derivation is completely algebraic. Finally, we will show that both types of non-geometric rough path can be re-written as geometric rough paths living above an extended version of the original path. This means that every non-geometric rough differential equation can be re-written as a geometric rough differential equation, hence generalising the Ito-Stratonovich correction formula.  


David Kelly (University of Miami)

Monday, April 22, 2013 - 14:15
to 15:15