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Option Pricing and Estimation of Financial Models with R by Stefano M. Iacus

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Chapter 3

Stochastic processes

When there is the need to model some form of dependence in a sample of observations, stochastic processes arise quite naturally. Dependence is a generic term which must be specified case by case in statistics and probability, but the most common situations include time and/or spatial dependence. In our context, the most interesting form of dependence is time dependence. Still, time dependency can be modelled in a variety of forms as we will discuss in this chapter.

3.1 Definition and First Properties

We assume to have a probability space images/c03_I0001.gif. A real-valued, one-dimensional, stochastic process is a family of random variables images/c03_I0002.gif defined on images/c03_I0004.gif taking values in images/c03_I0005.gif. The set images/c03_I0006.gif may be any abstract set, but we will restrict our attention to particular cases. For each images/c03_I0007.gif, the random variable is a measurable map . For a given fixed value of , say , the map , seen as a function of ...

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