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5

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 ...


4

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 ...


3

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 ...


1

To answer this I sum up a paragraph of "Interest rate models - An Introduction" by A.Cairns: For $i=1,\ldots,d$ consider the OU-processes $$ dX^i_t = -\frac 12 \alpha X^i_t dt + \sqrt{\alpha} dW^i_t. $$ Looking at the squared radius $R_t = \sum_{i=1}^d (X^i_t)^2 $ (in $\mathbb{R}^d$) of this process we get by Ito: $$ dR_t = \sum_{i=1}^d (2 X^i_t dX^i_t) + d ...



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