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Sample Wiener process constrained to open (initial), high (max), low (min), close (final)

You can do this (within some small $\epsilon$) using the reflection principle. Let's take the simpler case, where you only want to constrain the maximum. Adding the minimum is a fairly simple ...
Brian B's user avatar
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3 votes

Geometric brownian motion vs. Ornstein Uhlenbeck

A more abstract yet simple way of looking at this may help. Consider a generic diffusion $dY = (a_t - b_t Y_t) dt + \sigma_t dW_t$, where $Y_t$ is either the modelled quantity itself or $Y_t = \log{...
achirikhin's user avatar
2 votes
Accepted

What are $\mu$ and $a$ in $ \mu = a + \frac{\sigma^2}{2} $

The above stochastic exponential is merely a solution of $dS/S = \mu dt + \sigma dW_t$ which is equivalent to, by Ito $d \log{S} = ( {\mu - \sigma^2/2 }) dt + \sigma dW_t$, which can be immediately ...
achirikhin's user avatar
0 votes

Identifying stochastic process from data

To see if it's lognormal, check if the log follows a normal distribution (statistical test, there are many). To see if it is normal, check if itself follows a normal distribution. Maybe you want a ...
Arshdeep's user avatar
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5 votes
Accepted

Differentiating Wiener process

Let $\text dX_t=\mu_t\text dt+\sigma_t\text dW_t$ be an Itô process. Itô's Lemma tells us $$\text df(t,X_t)=\left(f_t+\mu_tf_x+\frac{1}{2}\sigma_t^2f_{xx}\right)\text dt+\sigma_tf_x\text dW_t.$$ You'...
Kevin's user avatar
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