Ornstein - Uhlenbeck Process

Also, a process {Yt : t ≥ 0} is said to have independent increments if, for all t0 < t1 < . . . < tn, the n random variables Yt1 − Yt0 , Yt2 − Yt1 , ..., Ytn − Ytn−1 are independent. This condition implies that {Yt : t ≥ 0} is Markovian, but not conversely. The increments are further said to be stationary if, for any t > s and h > 0, the distribution of Yt+h− Ys+h is the same as the distribution of Yt− Ys. This additional provision is needed for the following definition. A stochastic process {Wt : t ≥ 0} is a Wiener-Lévy process or Brownian motion if it has stationary independent increments, ifWt is normally distributed and E(Wt) = 0 for each t > 0, and if W0 = 0. It follows immediately that {Wt : t > 0} is Gaussian and that Cov(Ws,Wt) = θ min{s, t}, where the variance parameter θ is a positive constant. For concreteness’ sake, we henceforth assume that θ = 1. Almost all sample paths of Brownian motion are everywhere continuous but nowhere differentiable. One technical stipulation is required for the following. A stochastic process {Yt : t ≥ 0} is continuous in probability if, for all u ∈ R and ε > 0, P (|Yv − Yu| ≥ ε)→ 0 as v → u. This holds if Cov(Ys, Yt) is continuous over R × R. Note that this is a statement about distributions, not sample paths. Having dispensed with preliminaries, we turn to the central topic. A stochastic process {Xt : t ≥ 0} is an Ornstein-Uhlenbeck process or a Gauss-Markov