By Kardi Teknomo, PhD.

stochastic process

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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/