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Proof in the paper by Feller: http://www.jstor.org/stable/1969318

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Try modelling samples every 20,000 ticks, instead of 2 hours (or any such number like that). Markets are often less fat tailed in terms of the trade- or volume-clock. See http://www.amazon.ca/Introduction-High-Frequency-Finance-Ramazan-Gen%C3%A7ay/dp/0122796713 and http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2034858

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I believe that the process you postulate has a Beta conditional distribution. If my memory serves me well, I have encountered it in the book by Liptser and Shiryayev "Statistics of Random Processes" as the evolution of the conditional probability in a HMM. This was 10+ years ago, therefore I might be well off. In that case you should be sampling from Beta ...

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You are not going to get a process which stays within bounds if your increments are normal random variables, which have an unbounded distribution. You probably want to look at some kind of random walk, where the increment is a discrete distribution. In other words you have a finite list of values which you add or subtract, each with a positive probability. ...

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The first process is a BM. The second does not exist in continuous time. The variance goes down too slowly with dt and the process blows up at the limit. You can break the (0,1) interval into 1, 100, 1000, 1000000 steps and see that happening. Variance of a martingale has to scale with dt: if it is too fast then the process dies, if it is too slow then ...

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The first process $$B_{t+dt} = B_t + Z$$ where $Z$ is independent of $(B_s)_{s \le t}$ and follows a Gaussian distribution with mean $0$ and varince $dt$ is a standard Brownian motion (thus the variance of $B_t$ is $t$). For the second process let us recall the definition from your link: $$E[B^H_t B^H_s] = \frac12 ( t^{2H} + s^{2H} - |t-s|^{2H}),$$ thus ...

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the answer is simple: look at key differences between these two models. GBM is diffusion, OU is mean-reversion

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Given efficient markets, asset prices should be unpredictable in the sense that any upcoming returns are uncorrelated with current or past returns. Hence for traded assets the price should follow something more similar to a GBM than an O-U process. However, many financial metrics are not prices; for example interest rates or volatility. O-U processes may ...

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