Monday, September 2, 2013

Stoch Lesson J88 when to add scaling factor sqrt(t)

See Lesson 05 for a backgrounder on h.
See Lesson 15 for a backgrounder on paths and realizations.
In the formulas, one fine point easy to missed out is whether to include or remove sqrt(t) in front of dW. As repeated many times, notation is extremely important here. Before addressing the question, we must spend a few paragraphs on notations.
It's instructive to use examples at this juncture. Suppose we adopt (h=) 16-sec intervals, and generate 9999 realizations of the canonical Wiener process. The 9999 "realized" stepsize values form a histogram. It should be bell-shaped with mean 0 and variance 16.0, stdev 4.0. If we next adopt (h=) 0.09-sec intervals, and generate 8888 realizations of the same process, then the resulting 8888 stepsize values should show variance 0.09, stdev 0.3.
That's the canonical Wiener variable. So dW is defined as the stepsize as h -> 0. So dW has a Gaussian distribution with variance -> 0. Therefore dW is not customized and has well-known standard properties, including the sqrt(t) feature.
The simplest, purest, canonical Wiener variable already shows the sqrt(t) feature. Therefore, we should never put sqrt() in front of dW.
In fact, sqrt(t) scaling factor is only used with epsilon (or Z), a random variable representing the standard normal noisegen, with a fixed variance = 1.0

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