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Introduction to Stochastic Analysis: Integrals and Differential Equations by Vigirdas Mackevičius

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Chapter 5

Itô’s Formula

 

 

For finite-variation continuous processes X and functions FC1(images), recall the main formula of integration1

image

From example 4.10 we see that it does not hold for stochastic integrals. Its stochastic counterpart is the Itô formula which is proved below. For this, we need the Taylor2 formula with the remainder in some special Peano-type3 form.

LEMMA 5.1 (Taylor formula).– Let FC2(images) have a uniformly continuous second derivative (for example, F has a bounded third derivative F'"). Then

image

where, the remainder term satisfies the estimate |R(x, y)| images r(|y − x|)(y − x)2 with an increasing function r: imagesimages such that limh↓0r(h) = 0.

Proof. We use the Taylor formula with the well-known Lagrange remainder term

where the point ξ = ξ(x, y) is between x and y (x < ξ < y or y < ξ < ...

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