Do discrete-time series always have a continuous-time underlying?

Question

Can one argue that discrete time-series coming from stocks or commodities (prices) are derived from a continuous-time process?

One can probably argue that stocks or commodities at any time have a value and therefore a continuous-time price/value curve exist.

The problem I have with this argumentation is that although one can think of the existence of a continuous-time process it doesn't seem possible to measure/record/observe the continuous-time process.

So what is really the answer to this question: Can one argue that discrete time-series coming from stocks or commodities (prices) are derived from a continuous-time process?

Edit: I know that some literature model prices as continuous-time processes. Is there any advantage compared to a discrete-time model? If one does not believe that stock-market prices is inherently a continuous-time process it seems weird to use a continuous-time model.

Answer 1

Can one argue that discrete time-series coming from stocks or commodities (prices) are derived from a continuous-time process?

I would say "no". The price of a stock is determined by a trade and each trade is a discrete event in time. I would argue that between trades the price is undefined since there is no experiment you could do to determine what the price between trades would be.

You could certainly construct an interpolation mechanism, but different interpolations will give you different answers and I don't think there is a way of telling which one would be "better" or "right".

The question here: what are you planning to do with the answer? What scenario can you imagine where the assumption of an underlying continuous process (or not) would make a difference ?

Answer 2

The official answer to your question is this : random processes (time-series) can be classified as being either continuous-time or discrete-time. Some of those discrete-time processes happen from sampling of continuous-time processes, but some of the discrete-time processes occur naturally, such as daily stock-market values, or Sun-Spot numbers.

This paragraph exists on most statistical digital signal processing text books. And It's clear that stock market data is considered as being a naturally discrete-time process.

However, from fundamental probability theory, it's known that if the continuous-time Dirac impulse function $\delta(x)$ is allowed to be used, then any probability distribution (or density) function can be modeled as a continuous function of its variable. This includes discrete distributions such as tossing a coin or die, the number of people on a queue etc... And from this point of view, every random process can be given a continuous-time model (including dirac impulses though).

On the other hand, the "quantum" guy would argue that, every continuous looking phenomenon is actually discrete underneath, (which also denies the existance of dirac impulses) and thus you can always model any practical thing using discrete quantities.

The practitioner's answer would be this: choose the model which yields the most useful and most tractable description of the phenomenon.

In the field of economics, most things naturally happen discretely: number of items produced and sold, number of customers, number of transactions, time of price changes etc... These are generally discrete.

But you can model the price as a continuous-time variable, which changes its value only at specific instants of time. Or you can even model it according to a continuously changing function of time, which is outlined in the answer of RBJ, resulting from a bandlimited interpolation of time series. However, assuming a bandlimited model for stock market prices may be inadequate at times...

Answer 3

1. For the world in the whole, with our current technical abilities we can't say for sure whether continuous time processes exist in the real world or not.

   The problem is degree of difference between time measurements. If your degree of difference is $\Delta t$ then you can't say something happened at the moment $t_0 + \Delta t/2$ or $t_0 + \Delta t$. From your point of view these events happen at the same time. So, your observations are always discrete.

   Physical (theoretical) degree of difference is Planck time, it's about $10^{-44}\ \text{s}$.

   And the best technical resolution is of $10^{-19}\ \text{s}$ order.

   So, there are two options:

   1. World is discrete with Planck time as $\Delta t$
   2. World is continuous, but we can't see it

   We can choose between these options if we could measure time close to Planck and compare predictions generated by these two models. But we can't yet.

2. For stocks and commodities an underlying process is (almost) definitely discrete.

   Degree of difference is the resolution of timer on the server running trading on exchange. I don't have numbers for servers, but for Windows on PC resolution of timer is about 60 ms and resolution of multimedia high-precision timer is 1 ms. But, regardless, minimal degree is one CPU or memory cycle. There is nothing in between.

   (Almost, if we don't consider physical implementation: how data are written to memory, storage or transmitted through networks, and don't consider soft errors from radiodecay)

3. One can probably argue that stocks or commodities at any time have a value and therefore a continuous-time price/value curve exist

   It's wrong. Consider oversold or just non-liquid market. Stock or commodity has different values for buyers and sellers. Sellers value this stock higher than buyers, so they can't reach a deal. But you as outside browser can't see these values and see just the price of the last deal.

   If you'd like to go in this direction, look for models of order book.

   And if we consider value of stock or commodity (or any possessed or desired thing) for just a single person, he can't assign this value constantly. Putting a price, valuation, is voluntary action requiring time. Even if a stock trader thinks about just one stock all his work time (and nothing else) he still can't reevaluate it immediatelly. So can't computers and algorighms. And even if they could we still couldn't measure it constantly. So, it is not continuous either.

4. Is there any advantage compared to a discrete-time model?

   Yes. math for continuos processes is much easies and more developed.

   Because of that it could be difficult to answer question formulated in #1 and to follow conclusion in #2. One model is advanced but probably slightly incorrect in presuppositions. Another one may be correct from the start but can deduce less useful insights from it.

   Which one will give you better answers? Most researchers go with continuous models.

   IMO, it makes sense to seriously consider discrete model if you analyze timeframe 1 second and less, especially HFT. For all longer timeframes difference between continuous and discrete model is insignificant, assuming the process is not something studied by chaos theory. But that's just an opinion, I didn't check it.

P.S. If you think more you can find that for things like prices there are no time processes and probabilities too. But I'm not sure you should.

Answer 4

Well I think we normally assume that to be the case. And that the continuous-time signal "underlying" is bandlimited to below the Nyquist frequency.

If the discrete-time sequence is $x[n]$ (and $n$ can only be an integer), then the "underlying" continuous-time signal is:

$$ x(t) = \sum\limits_{n=-\infty}^{\infty} x[n] \operatorname{sinc}(t-n)$$

where

$$ \operatorname{sinc}(u) \triangleq \begin{cases} 1 \qquad & \text{for } \ u = 0 \\ \frac{\sin(\pi u)}{\pi u} \qquad & \text{for } \ u \ne 0 \\ \end{cases}$$

You will notice that $x(t)$ is defined for all real $t \in \mathbb{R}$, even though $x[n]$ is defined only for $n \in \mathbb{Z}$. And $x(t)$ is smooth with continuous derivatives, as long as all of the $x[n]$ are finite.

And when $t=n$ then

$$ x(t)\Big|_{t=n} = x[n] $$

Answer 5

I would vote for "non-inherently continuous" for the following reasons.

First the notion of discrete time series may apply to a discretization either in time or in the amplitude domain (discrete sampling vs discrete valuation). For the latter, the question is really dependent on the generative or physical process.

On the valuation: If you do photon counting, or biological read, this part is really discrete (until we discover that photons are not elementary particles). For current intensities or weight of heavy objects, I would assume that such quantities evolve in a continuous manner (up to the macroscopic precision of the measurement).

For commodities or stocks, I would say that they are measured as (integer) units of objects in sub-quantities of some currency, within a certain precision: 7.1123 dollars, and that I cannot make sense of a continuous variations of the above.

On the sampling: There are many scheme that ensure perfect (or error-bounded) preservation of information from a continuous model to discrete times, either uniform or not. The Nyquist-Sahnnon one, some in compressive sensing, the concept called Finite-Rate-of-Innovation (FRI).

For commodities (like goods) or stocks, I would say that this time process could be continuous. With high-frequency trading, I expect that they could be sold or bought at any time (this is a thought experiment), even if you cannot observe this, and that in economics, experiments are REALLY not independent on the observation/capture/buy. There might by exceptions, like (who knows) products that can only be bought on Saturday, or between 10 pm and 11 pm everyday.

On both discrete aspects: All in all, I have not problem with thinking that stocks or commodities may be discrete. Then, what else? The classical saying "All models are wrong, some are useful" attributed to British statistician George Box fully applies here. For instance, some believe that fractal or other continuous stochastic processes are useful to describe stock variations. They are quite often very continuous in nature, think about differential equations, Itô calculus, the famous Black-Scholes model.

The underlying model you will use can be useful for several reasons: ensure a target precision, suggest regression methods, justify prediction, help you to understand it, encompass mathematical operations like time or value averaging that "change the discrete nature" of the original data, etc. And it does not have to fully fit a difficult-to-grasp underlying reality. This is quite a philosophical question.

[EDITED] In image processing for instance, the discretized pixel location and value can be processed by quite discrete tools, like discrete lines and contours, or a lot of mathematical morphology, with very good results. Note that one advantage of the continuous case is that one can use more easily variational formulation, derivatives, integrals. Check a related topic at: Examples of problems that are easier in the infinite case than in the finite case.

[Personal opinion: I kind of like the concept that time could be somehow discrete/quantic]

Answer 6

For a camera sensor, light arrive as discrete quanta (photons) according to Poisson statistics. One usually regard that as an «artifact», while the (more) continuous scene reflectance/illuminance is seen as the desired signal that photon quantization corrupts. If the scene allows, one could increase exposure time in order to get more photons in order to get closer to the «true» scene representation.

Could we do something similar with stock? Say that there is an underlaying continuous process describing «market sentiment» and that actual stock transactions is a discrete process partially obfuscating the true and smooth market perception due to people selling discrete quanta of stock at discrete points in time where their perception is continous while the choice to actually buy/sell is somewhat stochastic approximation to that perception.