Stochastic differential equations driven by Lévy noise are not so tidy as Itô diffusions driven by Brownian noise, so they are frequently brushed aside in stochastic calculus texts. But I need ’em! There is a developed sampling theory for these creatures called sparse stochastic process theory.
Schoutens, Leuven, and Studer (2001) give a tractable example for Poisson processes.
Possibly also chaos expansions might be a useful tool, and/or Malliavin calculus, whatever that is.
But I will express no opinions, because this is a placeholder for a topic on which I do not know enough yet. 🏗
References
Applebaum, David. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.
Bolin, David. 2014. “Spatial Matérn Fields Driven by Non-Gaussian Noise.” Scandinavian Journal of Statistics 41 (3): 557–79.
Çinlar, E., and J. Jacod. 1981. “Representation of Semimartingale Markov Processes in Terms of Wiener Processes and Poisson Random Measures.” In Seminar on Stochastic Processes, 1981, edited by E. Çinlar, K. L. Chung, and R. K. Getoor, 159–242. Progress in Probability and Statistics. Boston, MA: Birkhäuser.
Di Nunno, Giulia, B. K. Øksendal, and Frank Proske. 2009. Malliavin Calculus for Lévy Processes with Applications to Finance. Universitext. Berlin ; New York: Springer.
Jacob, Niels, and René L. Schilling. 2001. “Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser.
Kloeden, Peter E., and Eckhard Platen. 1992. “Stochastic Taylor Expansions.” In Numerical Solution of Stochastic Differential Equations, edited by Peter E. Kloeden and Eckhard Platen, 161–226. Applications of Mathematics. Berlin, Heidelberg: Springer.
———. 2010. Numerical Solution of Stochastic Differential Equations. Berlin, Heidelberg: Springer Berlin Heidelberg.
Kurisu, Daisuke. 2020. “Nonparametric Regression for Locally Stationary Random Fields Under Stochastic Sampling Design.” arXiv:2005.06371 [Math, Stat], December.
Luo, Wuan. 2006. “Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations.” Phd, California Institute of Technology.
Osswald, Horst. 2012. Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion: An Introduction. Cambridge Tracts in Mathematics 191. Cambridge: Cambridge University Press.
Schoutens, Wim. 2000. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics. New York: Springer-Verlag.
Schoutens, Wim, K U Leuven, and Michael Studer. 2001. “Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management,” February, 24.
Şimşekli, Umut, Ozan Sener, George Deligiannidis, and Murat A. Erdogdu. 2020. “Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks.” CoRR abs/2006.09313.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press.
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.
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.
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