Definition A stochastic process iXt, tET is said to be a martingales process if Definition A stochastic process IXt, tET is said to be an innovation process if A stochastic process (Xt, tET) is said to be a Markov process if Definition: A stochastic process (Xt, tET) is said to be a Brownian motion process if 3.2 Parametric stochastic processes 3.2.1(Weakly) Stationary Process Definition A stochastic process (Xt, tET is said to be a autoregressive of order one (AR(1)) if it satisfies the stochastic difference equation, X where is a constant and ut is a white-noise process. We first consider the index setT*={0.,±1,±2,} and assume that X-r→0asT→∞ Define a lag -operator L by LXt≡Xt-1 then the ar(1) process can be rewritten as (1-OLXL when o| 1, it can be inverted as X2=(1-oD)-ut=(1+oL+2L2+…,)u=t2+0u-1+c2u1-2Definition: A stochastic process {Xt , t ∈ T } is said to be a martingales process if... Definition: A stochastic process {Xt , t ∈ T } is said to be an innovation process if.... Definition: A stochastic process {Xt , t ∈ T } is said to be a Markov process if.... Definition: A stochastic process {Xt , t ∈ T } is said to be a Brownian motion process if... 3.2 Parametric stochastic processes 3.2.1 (Weakly) Stationary Process Definition: A stochastic process {Xt , t ∈ T } is said to be a autoregressive of order one (AR(1)) if it satisfies the stochastic difference equation, Xt = φXt−1 + ut where φ is a constant and ut is a white-noise process. We first consider the index set T ∗ = {0, ±1, ±2, ...} and assume that X−T → 0 as T → ∞. Define a lag − operator L by LXt ≡ Xt−1, then the AR(1) process can be rewritten as (1 − φL)Xt = ut , when |φ| < 1, it can be inverted as Xt = (1 − φL) −1 ut = (1 + φL + φ 2L 2 + ....)ut = ut + φut−1 + φ 2 ut−2 + ..... = X∞ i=0 φ iut−i , 10