Improved local approximation for multidimensional diffusions: The G-rates
Authors:
S. Bodnarchuk, D. Ivanenko, A. Kohatsu-Higa and A. Kulik
Journal:
Theor. Probability and Math. Statist. 101 (2020), 13-38
MSC (2020):
Primary 60H35
DOI:
https://doi.org/10.1090/tpms/1109
Published electronically:
January 5, 2021
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Abstract: We consider the problem of improving the local approximations for multidimensional diffusions. In particular, our proposed explicit approximation improves the Milshtein approximation. We also provide a semi-explicit convergence rate estimate (we call it G-rate) for the proposed local approximation. The main error term in the difference of densities is bounded by a polynomial multiplied by a Gaussian density and the remainder is exponentially small as time goes to zero.
References
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References
- Y. Ait-Sahahia, Closed-form likelihood expansions for multivariate diffusions, Annals of Statistics 36 (2008), 906–937. MR 2396819
- A. Alfonsi, B. Jourdain, and A. Kohatsu-Higa, Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme, Electronic Journal of Probability 20 (2015), 1–31. MR 3361258
- R. Azencott, Densité des diffusions en temps petit: développements asymptotiques. I, Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, Springer, Berlin, 1984, pp. 402–498. MR 770974
- O. E. Barndorff-Nielsen and D. R. Cox, Asymptotic Techniques for Use in Statistics, Chapman and Hall, 1989. MR 1010226
- A. B. Cruzeiro, P. Malliavin, and A. Thalmaier, Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation, C. R. Math. Acad. Sci. Paris 338 (2004), 481–486. MR 2057730
- A. Friedman, Partial Differential Equations of Parabolic Type, Dover Publications, Inc., 1964. MR 0181836
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Series: Stochastic Modelling and Applied Probability, vol. 23, 1992. MR 1214374
- D. O. Ivanenko, A. Kohatsu-Higa, and A. M. Kulik, LA(M)N property for diffusion models with low regularity coefficients, Manuscript.
- S. Kusuoka, Approximation of expectation of diffusion process and mathematical finance, Taniguchi Conference on Mathematics Nara ’98, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo 131 (2001), 147–165. MR 1865091
- K. Oshima, J. Teichmann, and D. Velušček, A new extrapolation method for weak approximation schemes with applications, Ann. Appl. Probab. 22 (2012), 1008–1045. MR 2977984
- R. Ramer, On nonlinear transformations of Gaussian measures, J. Funct. Anal. 15 (1974), 166–187. MR 0349945
- A. Kulik, Integral representation for functionals on a space with a smooth measure, Theory of Stochastic Processes 3 (1997), 235–243.
- A. Üstünel and M. Zakai, The change of variables formula on Wiener space, Séminaire de probabilités de Strasbourg 31 (1997), 24–39. MR 1478713
- S. Watanabe, Short time asymptotic problems in Wiener functional integration theory, Applications to heat kernels and index theorems, Stochastic analysis and related topics, II (Silivri, 1988), Lecture Notes in Mathematics, vol. 1444, Springer, Berlin, 1990, pp. 1–62. MR 1078842
- S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 (1987), 1–39. MR 877589
- N. Yoshida, Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin–Watanabe, Probab. Theory Related Fields 92 (1992), 275–311. MR 1165514
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Additional Information
S. Bodnarchuk
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Prospect Peremogy 37, 03056 Kyiv, Ukraine
Email:
sem$_$bodn@ukr.net
D. Ivanenko
Affiliation:
Department of Radio Physics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv, Hlushkova Avenue 4g, 03127 Kyiv, Ukraine
Email:
divanenko1979@gmail.com
A. Kohatsu-Higa
Affiliation:
Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi Kusatsu, Shiga, 525-8577, Japan
Email:
khts00@fc.ritsumei.ac.jp
A. Kulik
Affiliation:
Wroclaw University of Science and Technology, Wybrzeźe Wyspiańskiego Str. 27, 50-370 Wroclaw, Poland
Email:
kulik.alex.m@gmail.com
Keywords:
Expansions,
stochastic differential equations,
total variation distance
Received by editor(s):
July 21, 2018
Published electronically:
January 5, 2021
Additional Notes:
The research of the first author and the second author was supported by Alexander von Humboldt Foundation within the Research Group Linkage Programme between the Institute of Mathematics at the University of Potsdam and the Institute of Mathematics of National Academy of Sciences of Ukraine.
The research of the third author was supported by KAKENHI grants 24340022 and 16H03642.
Article copyright:
© Copyright 2020
American Mathematical Society
