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in BUS 4028F - Financial Economics by


Just to confirm, from the notes, that limit is given after the definition of a time inverted Wiener Process. Does this mean that W(t) = t*W(1/t)

by (200 points)
edited by
[edited my comment and posted it as a solution]

1 Answer

+1 vote
by (200 points)

To be clear, we define a new process from the original Wiener process Wt.
It can then be shown that that new process (call it Wt_bar) is  itself a Wiener process.

I'd caution on equating the two processes as such, since the original process W_t is a stochastic process, it is not "guaranteed" that it's value at t will meet the above definition - i.e. W(t) need not equal t*W(1/t) - i.e. the value it realised at 1/t. However, in the probabilistic sense this is correct and thus to not confuse between realised values and the process, I'd keep the Wt_bar to show the new process,since the statement has said that Wt_bar is a Wiener process.

In fact, the following processes can be shown to be SBM (or Wiener):


by (950 points)

Interesting addition to this that a pulled off a website 

“The following comments are adapted from Stochastic Calculus and Fi- nancial Applications by J. Michael Steele. Springer, New York, 2001, page 40. These laws tie the Wiener process to three important groups of transfor- mations on [0, ∞), and a basic lesson from the theory of differential equations is that such symmetries can be extremely useful. On a second level, the laws also capture the somewhat magical fractal nature of the Wiener process. The scaling law tells us that if we had even one-billionth of a second of a Wiener process path, we could expand it to a billions years’ worth of an equally valid Wiener process path! The translation symmetry is not quite so startling, it merely says that Wiener process can be restarted anywhere. That is, any part of a Wiener process captures the same behavior as at the origin. The inversion law is perhaps most impressive, it tells us that the first second of the life of a Wiener process path is rich enough to capture the behavior of a Wiener process path from the end of the first second until the end of time.”

Taken from: https://www.math.unl.edu/~sdunbar1/MathematicalFinance/Lessons/BrownianMotion/Transformations/transformations.pdf