Stochastic Filtering Theory pp 77-93 | Cite as

# The Ito Formula

Chapter

## 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 }).

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## Notes

- The proof of Theorem 4.2.1 is due to Meyer [44]. The variant of Ito’s formula given in Theorem 4.2.2 is due to Neveu [73].Google Scholar

## Copyright information

© Springer Science+Business Media New York 1980