On the Exact Discretization of a Continuous Time AR(1) Model driven by either Long Memory or Antipersistent Innovations: A Fractional Algebra Approach

Exact discretization formulae are established for a first-order stochastic differential equation driven by a fractional noise of either long memory or antipersistent type. We assume that the underlying process is sampled at non-unit equispaced observational intervals. Using fractional algebra techniques the exact discretization formulae are derived in terms of confluent hypergeometric and incomplete gamma functions which admit infinite order series expansions.