Lévy stochastic differential equations

May 23, 2020 — May 23, 2020

dynamical systems
Hilbert space
signal processing
sparser than thou
stochastic processes
time series
Figure 1

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. 🏗

Figure 2

1 References

Applebaum. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 116.
Bolin. 2014. Spatial Matérn Fields Driven by Non-Gaussian Noise.” Scandinavian Journal of Statistics.
Çinlar, and Jacod. 1981. Representation of Semimartingale Markov Processes in Terms of Wiener Processes and Poisson Random Measures.” In Seminar on Stochastic Processes, 1981. Progress in Probability and Statistics.
Di Nunno, Øksendal, and Proske. 2009. Malliavin Calculus for Lévy Processes with Applications to Finance. Universitext.
Jacob, and Schilling. 2001. Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications.
Kloeden, and Platen. 1992. Stochastic Taylor Expansions.” In Numerical Solution of Stochastic Differential Equations. Applications of Mathematics.
———. 2010. Numerical Solution of Stochastic Differential Equations.
Kurisu. 2020. Nonparametric Regression for Locally Stationary Random Fields Under Stochastic Sampling Design.” arXiv:2005.06371 [Math, Stat].
Luo. 2006. Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations.”
Osswald. 2012. Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion: An Introduction. Cambridge Tracts in Mathematics 191.
Schoutens. 2000. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics.
Schoutens, Leuven, and Studer. 2001. Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management.”
Şimşekli, Sener, Deligiannidis, et al. 2020. Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks.” CoRR.
Unser, Michael A., and Tafti. 2014. An Introduction to Sparse Stochastic Processes.
Unser, M., Tafti, Amini, et al. 2014. A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory.
Unser, M., Tafti, and Sun. 2014. A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory.