Which financial time series have a PDF and/or a CDF? Consider the following types of financial time series for a single publicly-listed stock:

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*Price data

*Log returns

*Cumulative returns

Each is computed from the item listed before it: log returns are based on differences of prices, and cumulative returns are cumulative products of log returns.

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*Which of the random variables listed above possess a probability distribution function (PDF),

*which have a cumulative distribution function (CDF), and

*which have both a PDF and CDF?

*for what financial applications do the CDF versus PDF, and vice versa, come in handy?

I ask because the following post says all random variables have a CDF, but not all of them have a PDF. So I wanted to see how this applies to commonly used financial data, which are prices and returns. Graphical depictions of the above datas' CDF and PDFs displayed side-by-side would help in the explanation.
I'm particularly curious about cumulative returns. Since they're cumulative, it automatically makes me think it corresponds and is represented best by a CDF, so in a way I'm wondering if cumulative returns are more useful than they're made out to be, despite being non-stationary.
 A: Real world phenomena do not imply probability models in a strict mathematical sense. An analyst can choose a probability model for the phenomenon of interest based on multiple considerations. In particular, declaring financial phenomena such as prices, log-returns and cumulative returns to be random variables is not unambiguous mathematically. Thus the question Which of the random variables listed above... cannot be answered without qualifications.
Stock prices as well as log-returns understood as random phenomena are discrete; see "Do financial return series have a probability mass function (pmf)?". Cumulative returns are cumulative sums of log-returns, and thus they are discrete, too. Discrete phenomena are naturally and most accurately modeled by discrete random variables, though it is not impossible to approximate them with continuous random variables. But if we go with the natural choice, the corresponding discrete random variables will have PMFs and CDFs but not PDFs; PDFs are reserved for continuous random variables.
Handiness of PDFs vs. CDFs in finance merits a separate question.
