“When are the paths of a stochastic process continuous?” is a question one might like to ask. But we need to ask more precise questions than that, because things are complicated in probability land. If we are concerned about whether the paths sampled from the process are almost-surely continuous functions then we probably mean something like:
“Does the process admit a modification such that is a.e. Hölder-continuous with probability 1?” or some other such mouthful. There are many notions of continuity of stochastic processes. Continuous with respect to what, with what probability, etc.? Feller-continuity, etc. This notebook is not an exhaustive taxonomy; this is just a list of notions I need to remember. Commonly useful notions for a stochastic process include the following.
- Continuity in probability:
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for each and each
- Continuity in mean square, or continuity:
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- Sample continuity:
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I have given these as continuity properties for all but they can also be considered pointwise for fixed . Since is continuous, this can lead to subtle problems with uncountable unions of events, etc.
Jump processes show the difference between these. A Poisson process has paths which are not continuous with probability 1, but which are continuous in mean square and in probability.
Kolmogorov continuity theorem
The Kolmogorov continuity theorem gives us sufficient conditions for admitting a modification possessing a version which is Hölder of the process based on how rapidly moments of the process increments grow. Question: What gives us sufficient conditions? Lowther is good on this.
Connection to strong solutions of SDEs
TBD.
Continuity of Gaussian processes
Todo: Read Kanagawa et al. (2018) section 4, for the startling revelations:
… it is easy to show that a GP sample path does not belong to the corresponding RKHS with probability 1 if is infinite dimensional… This implies that GP samples are “rougher”, or less regular, than RKHS functions … Note that this fact has been well known in the literature; see e.g., (Wahba 1990, 5) and (Lukić and Beder 2001 Corollary 7.1).
Let be a positive definite kernel on a set and be its RKHS, and consider with satisfying Then if is infinite dimensional, then with probability If is finite dimensional, then there is a version of such that with probability 1.
derivatives of random fields
Robert J. Adler, Taylor, and Worsley (2016) defines derivatives thus: Choose a point and a sequence of ‘directions’ in , and write these as From context I assume this means that these directions are supposed to have unit norm, We say that has a -th order partial derivative at , in the direction , if the limit exists in mean square, where . is usually axis aligned, e.g. . Here is the symmetrized difference and the limit is taken sequentially, i.e. first send then , etc.
That is a lot, so let us examine that for the special case of and We choose a point and a direction w.l.o.g. The symmetrised difference in this first order case becomes We say that has a first order partial derivative at , in the direction , if the limit exists in mean square. This should look like the usual first order (partial) derivative, just with the term mean-square thrown in front.
By choosing , where is the vector with -th element 1 and all others zero, we can talk of the mean square partial derivatives of various orders of Then we see that the covariance function of partial derivatives of a random field must, if it exists and is finite, be given by Note that we have not assumed stationarity here, or Gaussianity, and still this process covariance function encodes a lot of information.
In the case that is stationary, we can use the spectral representation to analyse these derivatives. In this case, the corresponding variances have an interpretation in terms of spectral moments. We define the spectral moments for all multi-indices with . Assuming that the underlying random field, and so the covariance function, are real valued, so that, as described above, stationarity implies that and , it follows that the odd ordered spectral moments, when they exist, are zero; specifically,
For example, if has mean square partial derivatives of orders and for $, , $, , then Note that although this equation seems to have some asymmetries in the powers, these disappear due to the fact that all odd ordered spectral moments, like all odd ordered derivatives of , are identically zero.
References
Adler, Robert J. 2010. The Geometry of Random Fields.
Adler, Robert J., and Taylor. 2007.
Random Fields and Geometry. Springer Monographs in Mathematics 115.
Chevyrev, and Kormilitzin. 2016.
“A Primer on the Signature Method in Machine Learning.” arXiv:1603.03788 [Cs, Stat].
Lukić, and Beder. 2001.
“Stochastic Processes with Sample Paths in Reproducing Kernel Hilbert Spaces.” Transactions of the American Mathematical Society.
Lyons, Terry J. 1998.
“Differential Equations Driven by Rough Signals.” Revista Matemática Iberoamericana.
Lyons, Terry J., Caruana, and Lévy. 2007.
Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics.
Lyons, Terry J., and Sidorova. 2005.
“Sound Compression: A Rough Path Approach.” In
Proceedings of the 4th International Symposium on Information and Communication Technologies. WISICT ’05.
Pugachev, and Sinit︠s︡yn. 2001. Stochastic systems: theory and applications.
Teye. 2010. “Stochastic Invariance via Wong-Zakai Theorem.”
Wahba. 1990. Spline Models for Observational Data.