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I am interested in an overview over the connection and correspondence between time series models in continuous vs. discrete time in finance. E.g. take ARMA(p,q) or GARCH(s,r) or ARMA(p,q)-GARCH(s,r) from discrete time, list their counterparts in continuous time and show how they are related (e.g. how a continuous time process is a limit for the discrete time process as observation frequency increases without bound, or something like that); and do the same with the popular continuous time models (whatever these are) giving their discrete-time counterparts. It could be a full answer with worked out examples or just a reference to a textbook or a paper.

Richard Hardy
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  • Related: [What are good references to jump from discrete to continuous time series models?](https://stats.stackexchange.com/questions/28714/what-are-good-references-to-jump-from-discrete-to-continuous-time-series-models). – Richard Hardy Apr 15 '19 at 12:16
  • A continuous ARMA is CARMA; see [this paper](https://arxiv.org/abs/1402.5978). – corey979 Apr 15 '19 at 13:36
  • @corey979, thanks for the reference! I knew the model existed but never had the time to explore it deeper, which is (a small) part of the reason why I am asking the question. Does the paper show the connection between the discrete and the continuous case, or does it treat the continuous case alone? – Richard Hardy Apr 15 '19 at 14:00
  • I asked about how to discretize stochastic volatility diffusions using the Euler method a while back: https://math.stackexchange.com/questions/2845825/discretizing-a-stochastic-volatility-sde/3070624#3070624 – Taylor May 10 '19 at 19:22

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