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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.

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