Monday, September 2, 2013

Stoch Lesson J101 - W(t) isn't a traditional function-of-time

See lesson 05 for a backgrounder on Level, steps
See Lesson 33 for a backgrounder on the canonical Wiener variable W

Let's look at the notation W(t). This suggests the Level of W is a function of t. Suppose i = 55, I'd prefer the notation W_55 or Level_55, i.e. the level AFTER step_55. This level depends on i (i.e. 55), depends on t (i.e. 55 intervals after last-observation), and also depends on the 55 queries on the noisegen. Along one particular path W may be seen as a traditional function of t, but it's misleading to think of W as a function t. Across all paths, at time t_55, W is W_55 and includes all the 9999 realized values after step_55 and all the "unrealized" values.

In other words, W at time t_55 refers to the "distribution" of all these possible values. W at time t_55 is a cross section of the 9999+ paths. The symbol W(t) means the "Distribution of W's likely values at a future time t seconds after last observation". Since W isn't a traditional function of t, dW/dt is a freak. As illustrated elsewhere on this blog, the canonical Wiener variable W is not differentiable.

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