Learning Notes

# Stochastic – Incremental Process

## Independent Incremental Process

For a stochastic process $X_T$ with sample space $t_1, t_2, \dots, t_n \in T$ such that $t_1 < t_2 < \dots < t_n$ then

$\displaystyle X_{t2} - X_{t1} , X_{t3} - X_{t2} , \dots , X_{tn} - X_{tn-1}$

are independent.

## Stationary Independent Incremental Process

For an independent incremental process $X_T$, it is said to be stationary if it satisfy:

$\displaystyle \{X_{t_i + h} - X_{t_{i-1} + h}\} = \{X_{t_i} - X_{t_{i-1}}\}$

A stationary independent incremental process is also called Levy Process