A stochastic interpretation of the parametrix method

  • A. Kohatsu-Higa Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, Japan

Анотація

UDK 519.21

Стохастична інтерпретація параметричного методу

У дидактичній формі та з використанням стохастичного аналізу повторно розглянуто метод параметриксу та його застосування до незміщеного моделювання. Розглянуто, зокрема, випадок одновимірної дифузії без дрейфу.

Посилання

A. Ancona, K. Elworthy, M. Emery, H. Kunita, Stochastic differential geometry at saint-flour, Probab. St.-Flour, Springer etc. (2012); https://books.google.co.jp/books?id=nlCeMQEACAAJ.

P. Andersson, A. Kohatsu-Higa, Unbiased simulation of stochastic differential equations using parametrix expansions, Bernoulli, 23, № 3, 2028–2057 (2016). DOI: https://doi.org/10.3150/16-BEJ803

P. Andersson, A. Kohatsu-Higa, T. Yuasa, Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations, Stochast. Process. and Appl., 130, № 9, 5543–5574 (2020); DOI: 10.1016/ j.spa.2020.03.016; https://ideas. repec.org/a/eee/spapps/v130y2020i9p5543-5574.html. DOI: https://doi.org/10.1016/j.spa.2020.03.016

R. F. Bass, E. A. Perkins, On uniqueness in law for parabolic SPDEs and infinite-dimensional SDEs, Electron. J. Probab., 17, № 36, 1–54 (2012); DOI: 10.1214/EJP.v17-2049; https://doi.org/10.1214/EJP.v17-2049. DOI: https://doi.org/10.1214/EJP.v17-2049

S. Bodnarchuk, D. Ivanenko, A. Kohatsu-Higa, A. Kulik, Improved local approximation for multidimensional diffusions: the $g$-rates, Theory Probab. and Math. Statist., 101, № 1 (2021); DOI: 10.1090/tpms/1109. DOI: https://doi.org/10.1090/tpms/1109

P.-E. Chaudru de Raynal, N. Frikha, Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space, J. Math. Pures et Appl., 159, 1–167 (2022); DOI: https://doi.org/10.1016/j.matpur.2021.12.001; https://www.sciencedirect.com/science/article/pii/ S0021782421001884. DOI: https://doi.org/10.1016/j.matpur.2021.12.001

F. Delarue, Estimates of the solutions of a system of quasi-linear PDEs. A probabilistic scheme, Sémin. Probab. XXXVII, Lecture Notes in Math., 1832, 290–332 (2003); DOI: 10.1007/978-3-540-40004-2_12; https://doi.org/ 10.1007/978-3-540-40004-2_12. DOI: https://doi.org/10.1007/978-3-540-40004-2_12

A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, NJ (1964).

N. Frikha, L. Li, Weak uniqueness and density estimates for SDEs with coefficients depending on some path-functionals, Ann. Inst. Henri Poincaré, Probab. et Stat., 56, № 2, 1002–1040 (2020); DOI: 10.1214/19-AIHP992; https://doi.org/10.1214/19-AIHP992. DOI: https://doi.org/10.1214/19-AIHP992

I. Karatzas, S. Shreve, Brownian motion and stochastic calculus, Grad. Texts in Math., Springer-Verlag (1988); https://books.google.co.jp/books?id=uxnvAAAAMAAJ. DOI: https://doi.org/10.1007/978-1-4684-0302-2

I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Grad. Texts in Math., 113, Springer-Verlag, New York (1991); DOI: 10.1007/978-1-4612-0949-2; http://dx.doi.org/10.1007/ 978-1-4612-0949-2.

T. Kulczycki, A. M. Kulik, M. Ryznar, On weak solution of SDE driven by inhomogeneous singular Lévy noise, Trans. Amer. Math. Soc., 375, 4567–4618 (2022). DOI: https://doi.org/10.1090/tran/8612

E. E. Levi, Sulle equazioni lineari alle derivate parziali totalmente ellittiche, Rend. Real. Accad. Lincei, Cl. Sci., Fis., Mat., Natur. (5), 16, № 12, 932–938 (1907).

E. E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Mat. Palermo, 24, № 1, 275–317 (1907). DOI: https://doi.org/10.1007/BF03015067

P. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability, Springer, Berlin, Heidelberg (2003).

S. Watanabe, Stochastic Levi sums, Commun. Pure and Appl. Math., 47, № 5, 767–786 (1994); DOI: https://doi.org/ 10.1002/cpa.3160470509; https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160470509. DOI: https://doi.org/10.1002/cpa.3160470509

Опубліковано
30.11.2023
Як цитувати
Kohatsu-HigaA. «A Stochastic Interpretation of the Parametrix Method». Український математичний журнал, вип. 75, вип. 11, Листопад 2023, с. 1479 -6, doi:10.3842/umzh.v75i11.7382.
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