2021/05/31 by Robert L. Wolpert, Wolpert, Robert L
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2106.00087
openalex publication_date 2021/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each λ>0 and every square-integrable infinitely-divisible (ID) distribution there exists at least one stationary stochastic process t↦ Xt with the specified distribution for X1 and with first-order autoregressive (AR(1)) structure in the sense that the autocorrelation of Xs and Xt is exp(-λ|s-t|) for all indices s,t. For the special case of the standard Normal distribution, the process Xt is unique -- namely, the first-order autoregressive Ornstein-Uhlenbeck velocity process. The process Xt is also uniquely determined if X1 is accorded the unit rate Poisson distribution. For the Gamma distribution, however, Xt is not determined uniquely. In these lecture notes we describe six distinct processes with the same univariate marginal distributions and AR(1) autocorrelation function. We explore a few of their properties and describe methods of simulating their sample paths.