Path continuity of stochastic processes



\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

“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 this is 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 \(\{\rv{x}(t)\}_t\) admit a modification such that \(t\mapsto \rv{x}(t)\) is a.e. Hölder-continuous with probability 1?”

There are many notions of continuity of stochastic processes.

I should catalogue somewhere the different notions of continuity that are in play here. Continuous wrt what etc? Feller-continuity etc.

Kolmogorov continuity theorem

The Kolmogorov continuity theorem gives us sufficient conditions for Hölder-path-continuity of the process based on how rapidly moments of the process increments grow. Question: What gives us sufficient conditions? Lowther is good on this.

SDEs with rough paths

See signatures and rough paths.

Connection to strong solutions of SDEs

TBD.

Continuity of Gaussian processes

Todo: Read Kanagawa et al. (2018) section 4, which has some startling revelations:

… it is easy to show that a GP sample path \(\mathrm{f} \sim \mathcal{G P}(0, k)\) does not belong to the corresponding RKHS \(\mathcal{H}_{k}\) with probability 1 if \(\mathcal{H}_{k}\) is infinite dimensional, as summarized in Corollary \(4.10\) below. This implies that GP samples are “rougher”, or less regular, than RKHS functions (see also Figure 2). 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 \(k\) be a positive definite kernel on a set \(\mathcal{X}\) and \(\mathcal{H}_{k}\) be its RKHS, and consider \(\mathrm{f} \sim \mathcal{G} \mathcal{P}(m, k)\) with \(m: \mathcal{X} \rightarrow \mathbb{R}\) satisfying \(m \in \mathcal{H}_{k} .\) Then if \(\mathcal{H}_{k}\) is infinite dimensional, then \(\mathrm{f} \in \mathcal{H}_{k}\) with probability \(0 .\) If \(\mathcal{H}_{k}\) is finite dimensional, then there is a version \(\tilde{\mathrm{f}}\) of \(\mathrm{f}\) such that \(\tilde{\mathrm{f}} \in \mathcal{H}_{k}\) with probability \(1 .\)

References

Adler, Robert J. 2010. The Geometry of Random Fields. SIAM ed. Philadelphia: Society for Industrial and Applied Mathematics.
Adler, Robert J., and Jonathan E. Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115. New York: Springer. https://doi.org/10.1007/978-0-387-48116-6.
Adler, Robert J, Jonathan E Taylor, and Keith J Worsley. 2016. Applications of Random Fields and Geometry Draft. https://robert.net.technion.ac.il/files/2016/08/hrf1.pdf.
Bongers, Stephan, and Joris M. Mooij. 2018. “From Random Differential Equations to Structural Causal Models: The Stochastic Case.” March 27, 2018. http://arxiv.org/abs/1803.08784.
Chevyrev, Ilya, and Andrey Kormilitzin. 2016. “A Primer on the Signature Method in Machine Learning.” March 11, 2016. http://arxiv.org/abs/1603.03788.
Kanagawa, Motonobu, Philipp Hennig, Dino Sejdinovic, and Bharath K. Sriperumbudur. 2018. “Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences.” July 6, 2018. http://arxiv.org/abs/1807.02582.
Lukić, Milan, and Jay Beder. 2001. “Stochastic Processes with Sample Paths in Reproducing Kernel Hilbert Spaces.” Transactions of the American Mathematical Society 353 (10): 3945–69. https://doi.org/10.1090/S0002-9947-01-02852-5.
Lyons, Terry. 2014. “Rough Paths, Signatures and the Modelling of Functions on Streams.” May 18, 2014. http://arxiv.org/abs/1405.4537.
Lyons, Terry J. 1998. “Differential Equations Driven by Rough Signals.” Revista Matemática Iberoamericana 14 (2): 215–310. https://doi.org/10.4171/RMI/240.
Lyons, Terry J., Michael Caruana, and Thierry Lévy. 2007. Differential Equations Driven by Rough Paths. Vol. 1908. Lecture Notes in Mathematics. Springer, Berlin. https://mathscinet.ams.org/mathscinet-getitem?mr=2314753.
Lyons, Terry J, and Nadia Sidorova. n.d. “Sound Compression – a Rough Path Approach,” 6.
Pugachev, V. S., and I. N. Sinitsyn. 2001. Stochastic Systems: Theory and Applications. River Edge, NJ: World Scientific. http://books.google.com?id=367TCNo0Mi4C.
Teye, Alfred Larm. 2010. “Stochastic Invariance via Wong-Zakai Theorem.” PhD Thesis, University of Amsterdam.
Wahba, Grace. 1990. Spline Models for Observational Data. SIAM.

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