Signal sampling is the study of approximating continuous signals with discrete ones and vice versa. What if the signal you are trying to recover is random, but you have a model for that randomness, and can thus assign likelihoods (posterior probabilities even) to some sample paths.? Now you are sampling a stochastic process.
This is a particular take on a classic inverse problem that arises in many areas, framed how electrical engineers frame it. In the presence of observation noise they might also frame it in terms of state filtering/smoothing. There is a brief summary of that framing in Draščić (2016).
I am especially interested in this in the context of non-Gaussian-process models, because everything more or less works already for stationary gaussian processes. If you consider Lévy noise driving a linear SDE there is some work done, under the heading of sparse stochastic processes.
Bostan, E., U. S. Kamilov, M. Nilchian, and M. Unser. 2013. “Sparse Stochastic Processes and Discretization of Linear Inverse Problems.” IEEE Transactions on Image Processing 22 (7): 2699–2710. https://doi.org/10.1109/TIP.2013.2255305.
Broersen, Piet M. T. 2005. “Time Series Analysis for Irregularly Sampled Data.” IFAC Proceedings Volumes, 16th IFAC World Congress, 38 (1): 154–59. https://doi.org/10.1016/S1474-6670(16)36038-4.
Broersen, P. M. T., and R. Bos. 2006. “Estimating Time-Series Models from Irregularly Spaced Data.” In IEEE Transactions on Instrumentation and Measurement, 55:1124–31. https://doi.org/10.1109/TIM.2006.876389.
Coulaud, Benjamin, and Frédéric JP Richard. 2018. “A Consistent Framework for a Statistical Analysis of Surfaces Based on Generalized Stochastic Processes.” https://hal.archives-ouvertes.fr/hal-01863312.
Draščić, Biserka. 2016. “Sampling Reconstruction of Stochastic Signals– the Roots in the Fifties.” Austrian Journal of Statistics 36 (1): 65. https://doi.org/10.17713/ajs.v36i1.321.
Glynn, Peter, and Karl Sigman. 1998. “Independent Sampling of a Stochastic Process.” Stochastic Processes and Their Applications 74 (2): 151–64. https://doi.org/10.1016/S0304-4149(97)00114-2.
Jones, Richard H. 1981. “Fitting a Continuous Time Autoregression to Discrete Data.” In Applied Time Series Analysis II, 651–82.
———. 1984. “Fitting Multivariate Models to Unequally Spaced Data.” In Time Series Analysis of Irregularly Observed Data, 158–88. Springer. http://link.springer.com/chapter/10.1007/978-1-4684-9403-7_8.
Lahalle, E., G. Fleury, and A. Rivoira. 2004. “Continuous ARMA Spectral Estimation from Irregularly Sampled Observations.” In Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference, 2004. IMTC 04, 2:923–27 Vol.2. https://doi.org/10.1109/IMTC.2004.1351213.
Larsson, Erik K., and Torsten Söderström. 2002. “Identification of Continuous-Time AR Processes from Unevenly Sampled Data.” Automatica 38 (4): 709–18. https://doi.org/10.1016/S0005-1098(01)00244-8.
Lii, Keh-Shin, and Elias Masry. 1992. “Model Fitting for Continuous-Time Stationary Processes from Discrete-Time Data.” Journal of Multivariate Analysis 41 (1): 56–79. https://doi.org/10.1016/0047-259X(92)90057-M.
Maravic, I., and M. Vetterli. 2005. “Sampling and Reconstruction of Signals with Finite Rate of Innovation in the Presence of Noise.” IEEE Transactions on Signal Processing 53 (8): 2788–2805. https://doi.org/10.1109/TSP.2005.850321.
Marziliano, P., M. Vetterli, and T. Blu. 2006. “Sampling and Exact Reconstruction of Bandlimited Signals with Additive Shot Noise.” IEEE Transactions on Information Theory 52 (5): 2230–3. https://doi.org/10.1109/TIT.2006.872844.
Matheron, G. 1973. “The Intrinsic Random Functions and Their Applications.” Advances in Applied Probability 5 (3): 439–68. https://doi.org/10.2307/1425829.
Murray-Smith, Roderick, and Barak A. Pearlmutter. 2005. “Transformations of Gaussian Process Priors.” In Deterministic and Statistical Methods in Machine Learning, edited by Joab Winkler, Mahesan Niranjan, and Neil Lawrence, 110–23. Lecture Notes in Computer Science. Springer Berlin Heidelberg. http://bcl.hamilton.ie/~barak/papers/MLW-Jul-2005.pdf.
O’Callaghan, Simon Timothy, and Fabio T. Ramos. 2011. “Continuous Occupancy Mapping with Integral Kernels.” In Twenty-Fifth AAAI Conference on Artificial Intelligence. https://www.aaai.org/ocs/index.php/AAAI/AAAI11/paper/view/3784.
Scargle, Jeffrey D. 1981. “Studies in Astronomical Time Series Analysis. I-Modeling Random Processes in the Time Domain.” The Astrophysical Journal Supplement Series 45: 1–71.
Söderström, T., and M. Mossberg. 2000. “Performance Evaluation of Methods for Identifying Continuous-Time Autoregressive Processes.” Automatica 1 (36): 53–59. https://doi.org/10.1016/S0005-1098(99)00104-1.
Sun, Qiyu, and Michael Unser. 2012. “Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes.” Advances in Computational Mathematics 36 (3): 399–441. https://doi.org/10.1007/s10444-011-9183-6.
Tan, V. Y. F., and V. K. Goyal. 2008. “Estimating Signals with Finite Rate of Innovation from Noisy Samples: A Stochastic Algorithm.” IEEE Transactions on Signal Processing 56 (10): 5135–46. https://doi.org/10.1109/TSP.2008.928510.
Unser, M. 2015. “Sampling and (Sparse) Stochastic Processes: A Tale of Splines and Innovation.” In 2015 International Conference on Sampling Theory and Applications (SampTA), 221–25. https://doi.org/10.1109/SAMPTA.2015.7148884.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press. http://www.sparseprocesses.org/sparseprocesses-123456.pdf.
Unser, M., P. D. Tafti, A. Amini, and H. Kirshner. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory 60 (5): 3036–51. https://doi.org/10.1109/TIT.2014.2311903.
Unser, M., P. D. Tafti, and Q. Sun. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory 60 (3): 1945–62. https://doi.org/10.1109/TIT.2014.2298453.
Wolfe, Stephen James. 1982. “On a Continuous Analogue of the Stochastic Difference Equation Xn=[Rho]Xn-1+Bn.” Stochastic Processes and Their Applications 12 (3): 301–12. https://doi.org/10.1016/0304-4149(82)90050-3.
Yadrenko, Mikhail Iosifovich. 1983. Spectral Theory of Random Fields. Translation Series in Mathematics and Engineering. New York, NY: Optimization Software.
Yaglom, A. M. 1987. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics. New York, NY: Springer Science & Business Media.