Lévy stochastic differential equations

2020-05-23 — 2020-05-23

Wherein Lévy-driven stochastic differential equations are examined, sparse stochastic process sampling theory is invoked and a Poisson-driven example is exhibited, and Malliavin calculus is mentioned as a potential tool.

dynamical systems

Hilbert space

SDEs

signal processing

sparser than thou

statistics

stochastic processes

time series

Stochastic differential equations driven by Lévy noise are not as 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) gives 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. 🏗

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.

Ikeda, and Watanabe. 2014. Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, v.v.

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.