Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
10.<br />
<strong>R<strong>and</strong>om</strong> <strong>Walk</strong><br />
The <strong>R<strong>and</strong>om</strong> <strong>Walk</strong> <strong>of</strong> small particles in fluid is named after<br />
Brown, who first observed it, Brownian Motion. It models<br />
other processes, such as the fluctuations <strong>of</strong> a stock price.<br />
In a volume <strong>of</strong> fluid, the path <strong>of</strong> a particle is in any direction<br />
in the volume, <strong>and</strong> <strong>of</strong> variable size<br />
10.1 The Bernoulli <strong>R<strong>and</strong>om</strong> Variables <strong>of</strong> the <strong>Walk</strong><br />
We restrict the <strong>Walk</strong> here to the line, in uniform<br />
infinitesimal size steps dx :<br />
To the left, with probability<br />
1<br />
p = ,<br />
2<br />
36
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