Wiener Process
So a Wiener process can be modeled by the following difference equation:
W(t) = W(t-1) + N(0,1) where N(0,1) = normal distribution with mean 0 and unit variance 1. Somewhere I saw someone use the following closed form for it: W(t)= sqrt(t) * N(0,1) Is the closed form an approximation for the difference equation? If not can someone show the steps to reach to the close form from the difference equation? |
Re: Wiener Process
If W(0) = 0, then W(1) = X_1, where X_1 ~ N(0,1), and W(t) = X_1 + ... + X_t where X_i ~ N(0,1). The sum of t iid normal RVs is normal with mean 0 and variance t, so W(t) ~ N(0,t) which is the same as sqrt(t) * N(0,1)
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Re: Wiener Process
Thanks. Can you explain how you get from N(0,t) ~ sqrt(t) * N(0,1)
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