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Summary
Table below summarized what we have learned so far.
|
Model |
Brownian Motion |
ABM |
GBM |
| Initial Value |
\( 0 \) |
\( x_0 \) |
\( x_0 \) |
| SDE Model |
None |
\( dX_t=m dt + s dw_t \) |
\( dX_t=\mu X_t dt + \sigma X_t dw_t \) |
| Simulation Model |
\( w_{t+1}=w_t + N(0,1) \) |
\( X_t=x_0+mt+sw_t \)
|
\( X_t=x_0 e^{\tilde{\mu} t + \sigma w_t } \) Where \( \tilde{\mu} = \mu - \frac{1}{2}\sigma^2 \) |
| Parameters |
None |
\( m,s \) |
\( \mu, \sigma \) |
| Distribution |
Normal |
Normal |
Log-Normal |
| Mean |
0 |
\( x_0 + mt \) |
\( x_0 e^{\tilde{\mu}t} \) |
| Variance |
1 |
\( s^2 t \) |
\( x_{0}^{2} e^{2\tilde{\mu}t} (e^{\sigma^2t}-1) \) |
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These tutorial is copyrighted .
Preferable reference for this tutorial is
Teknomo, Kardi. (2017) Stochastic Process Tutorial .
http://people.revoledu.com/kardi/tutorial/StochasticProcess/