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# 160a_ho0 - Pstat160a Intro Class An Example of Stochastic...

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Unformatted text preview: Pstat160a Intro Class An Example of Stochastic Process Modeling: the Stock Market As an introduction to the topic covered in this class (and Pstat160B), we look at a a simple example of modeling using a stochastic process. Suppose we are interested in historical values (past data) of the Dow Jones Industrial average. A plot of this index since its creation is plotted bellow in figure 1. Figure 1. Dow Jones index, 1920-2010 We want to give a Mathematical description of this index. That is, describe its statistical properties. In order to do so, let X n the value of the Dow Jones index at time n , where n stand for the number of days this index has been tracked. From this graph and our understanding of the stock market, X n is random and fluctuate with time: it is a stochastic process (more precisely, what we observe is the realization of a stochastic process. That is X 1 , ··· ,X n , ··· are random variables (positive!). In order to describe the properties of the process X n ,n = 1 , 2 , ··· and try to match the actual data, denote by L n the log returns: L n = log( X n +1 )- log( X n ) (1) ≈ X n +1- X n X n (2) A plot of the log returns is given below Figure 2. Dow Jones log-returns 1 2 Note that using (1) we can recover...
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160a_ho0 - Pstat160a Intro Class An Example of Stochastic...

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