Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 7

Itô Processes

 

 

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

[7.1]

for all bounded random variables and s t 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 M_s − M_t, s t 0, are orthogonal to the past (i.e. to all bounded random variables ). 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 ...