The Ito Formula

Abstract

A process M _t = (M _t ¹, ... ,M _t ^d) taking values in R ^d is a martingale with respect to the increasing σ-field family (F _t ) if (M _t ⁱ, F _t ) is a martingale for each i = 1, ... ,d, or equivalently, if (θ,M _t ,) is a real-valued martingale with respect to (F _t ) for every θ, ∈ R ^d . Here we use (,) to denote inner product in R ^d. (M _t , F _t ) with M ₀ = 0 (a.s.) is a d-dimensional, continuous L ²-martingale if for every θ ∈ R ^d, (θ,M _t ) is a continuous L ²-martingale with respect toF _t . It is then easy to verify the existence of a unique d × d-matrix—valued process A _t , = (A _t ^ij) with the following properties:

• a. Each A _t ^ij is F _t -measurable.

• b. A ₀ = 0 and A _t (ω) is continuous in t for almost all ω.

• c. For θ ∈ R ^d, (A _t θ,θ) is the (continuous) increasing process associated with (θ,M _t ).