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

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Chapter 7

Itô Processes

 

 

DEFINITION 7.1.Adapted random process M = {Mt, t image 0} is called a martingale if

[7.1] images

for all bounded random variables images and s image t image 0.

REMARKS.- 1. Random variables Z1 and Z2 are said to be orthogonal if E(Z1Z2) = 0. Therefore, property [7.1] can be interpreted as follows: a random process M is a martingale if its increments Ms − Mt, s image t image 0, are orthogonal to the past images(i.e. to all bounded random variables images). A Brownian motion B is a martingale: recall that its increments are not only orthogonal to but, moreover, independent of the past. Many properties of Brownian motion ...

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