Notes on the Itô Calculus

Itô Calculus

This entry for the Encyclopedia of Actuarial Sciences provides an introduction to the Itô Calculus that emphasizes the definition of the Itô integral and the description of Itô’s Formula, the most widely used result in the Itô Calculus.

Stochastic Calculus: Consequences of the Itô Formula

Proof. The stochastic differential equation (2) follows directly from the (multivariate) Itô theorem, using the fact that Zt has the form Zt = u(Mt , [M ]t ). Every Itô integral process V ·M is a local martingale provided that M is a local martingale and V 2 L§(M), so for each n <1 the stopped process Z øn is a local martingale, where øn is the first time t such that [M ]t = n, and so it follow...

Conditional Stein approximation for Itô and Skorohod integrals

We derive conditional Edgeworth-type expansions for Skorohod and Itô integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain conditional Stein approximation bounds for multiple stochastic integrals and quadratic Brownian functionals.

Stein approximation for Itô and Skorohod integrals by Edgeworth type expansions

We derive Edgeworth-type expansions for Skorohod and Itô integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which apply to SDE solutions and to multiple stochastic integrals.

Flows and stochastic Taylor series in Itô calculus