There is a mathematically rich theory about particle filters, and the central tool to make that go seems to be Feynman-Kac formulae. The notoriously abstruse Del Moral (2004);Doucet, Freitas, and Gordon (2001) are universally commended for unifying and making consistent the diffusion processes and Feynman-Kac formulae and “propagation of chaos” and their delicate relationships. I will get around to them eventually, maybe?
Relate, apparently: Backward SDEs.
References
Beck, Christian, Weinan E, and Arnulf Jentzen. 2019. “Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations.” Journal of Nonlinear Science 29 (4): 1563–1619.
Cérou, F., P. Del Moral, T. Furon, and A. Guyader. 2011. “Sequential Monte Carlo for Rare Event Estimation.” Statistics and Computing 22 (3): 795–808.
Chan-Wai-Nam, Quentin, Joseph Mikael, and Xavier Warin. 2019. “Machine Learning for Semi Linear PDEs.” Journal of Scientific Computing 79 (3): 1667–1712.
Chopin, Nicolas, and Omiros Papaspiliopoulos. 2020. An Introduction to Sequential Monte Carlo. Springer Series in Statistics. Springer International Publishing.
Chuang, Howard Jen-Hao. 2010. “The Feynman-Kac Formula:Relationships Between Stochastic DifferentialEquations and Partial Differential Equations.” Honours, ANU.
Del Moral, Pierre. 2004. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. 2004 edition. Latheronwheel, Caithness: Springer.
Del Moral, Pierre, and Arnaud Doucet. 2010. “Interacting Markov Chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations.” The Annals of Applied Probability 20 (2): 593–639.
Del Moral, Pierre, Peng Hu, and Liming Wu. 2011. On the Concentration Properties of Interacting Particle Processes. Vol. 3. Now Publishers.
Del Moral, Pierre, and Laurent Miclo. 2000. “Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In Séminaire de Probabilités XXXIV, 1–145. Lecture Notes in Mathematics 1729. Springer.
Doucet, Arnaud, Nando Freitas, and Neil Gordon. 2001. Sequential Monte Carlo Methods in Practice. New York, NY: Springer New York.
Doucet, Arnaud, Simon Godsill, and Christophe Andrieu. 2000. “On Sequential Monte Carlo Sampling Methods for Bayesian Filtering.” Statistics and Computing 10 (3): 197–208.
E, Weinan, Jiequn Han, and Arnulf Jentzen. 2017. “Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations.” Communications in Mathematics and Statistics 5 (4): 349–80.
Han, Jiequn, Arnulf Jentzen, and Weinan E. 2018. “Solving High-Dimensional Partial Differential Equations Using Deep Learning.” Proceedings of the National Academy of Sciences 115 (34): 8505–10.
Hartmann, Carsten, Lorenz Richter, Christof Schütte, and Wei Zhang. 2017. “Variational Characterization of Free Energy: Theory and Algorithms.” Entropy 19 (11): 626.
Hutzenthaler, Martin, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen, and Philippe von Wurstemberger. 2020. “Overcoming the Curse of Dimensionality in the Numerical Approximation of Semilinear Parabolic Partial Differential Equations.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 (2244): 20190630.
Hutzenthaler, Martin, and Thomas Kruse. 2020. “Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities.” SIAM Journal on Numerical Analysis 58 (2): 929–61.
Kebiri, Omar, Lara Neureither, and Carsten Hartmann. 2019. “Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations.” In Stochastic Dynamics Out of Equilibrium, edited by Giambattista Giacomin, Stefano Olla, Ellen Saada, Herbert Spohn, and Gabriel Stoltz, 265–81. Springer Proceedings in Mathematics & Statistics. Cham: Springer International Publishing.
Nualart, David, and Wim Schoutens. 2001. “Backward Stochastic Differential Equations and Feynman-Kac Formula for Lévy Processes, with Applications in Finance.” Bernoulli 7 (5): 761–76.
Nüsken, Nikolas, and Lorenz Richter. 2021. “Solving High-Dimensional Hamilton–Jacobi–Bellman PDEs Using Neural Networks: Perspectives from the Theory of Controlled Diffusions and Measures on Path Space.” Partial Differential Equations and Applications 2 (4): 48.
Papanicolaou, Andrew. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.” arXiv:1504.05309 [Math, q-Fin], January.
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