Tail Distribution, Smoothness and Gaussian Density Estimates of Stochastic Differential Delay Equations
Main Article Content
Abstract
In this paper, we study the tail distribution, smoothness and density estimates of the solution of a fundamental class stochastic differential delay equations. Base on the techniques of the Malliavin calculus we obtain an explicit estimate for tail distributions and upper and lower Gaussian estimates for density.
Keywords:
In this paper, we consider the basic stochastic differential delay equations of the form
References
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Vol. 125, No. 1-2, 2000, pp. 297-307, https://doi.org/10.1016/S0377-0427(00)00475-1.
[2] X. Mao, S. Sabanis, Numerical Solutions of Stochastic Differential Delay Equations Under Local Lipschitz Condition, J. Comput. Appl. Math, Vol. 151, No. 1, 2003, 215-227, https://doi.org/10.1016/S0377-0427(02)00750-1.
[3] B. Akhtari, Numerical Solution of Stochastic State-Dependent Delay Differential Equations: Convergence and Stability, Adv. Difference Equ., No. 396, 2019, 34 pp, https://doi.org/10.1186/s13662-019-2323-x.
[4] E. Buckwar, R. Kuske, S. E. Mohammed, T. Shardlow, Weak Convergence of the Euler Scheme for Stochastic Differential Delay Equations, LMS J. Comput. Math, Vol. 11, 2008, pp. 60-99, https://doi.org/10.1112/S146115700000053X.
[5] T. C. Son, N. T. Dung, N. V. Tan, T. M. Cuong, H. T. P. Thao, P. D. Tung, Weak Convergence of Delay SDEs with Applications to Carathéodory Approximation, Discrete Contin. Dyn. Syst. Ser. B., Vol. 27, No. 9, 2022,
pp. 4725-4747, https://doi.org/10.3934/dcdsb.2021249.
[6] N. T. Dung, Tail Probability Estimates for Additive Functionals, Statist. Probab. Lett., Vol. 119, 2016,
pp. 349-356, https://doi.org/10.1016/j.spl.2016.09.002.
[7] N. T. Dung, Tail Estimates for Exponential Functionals and Applications to SDEs, Stochastic Process, Appl., Vol. 128, No. 12, 2018, pp. 4154-4170, https://doi.org/10.1016/j.spa.2018.02.003.
[8] N. T. Dung, T. C. Son, Tail Distribution Estimates for One-dimensional Diffusion Processes, J. Math. Anal. Appl., Vol. 479, No. 2, 2019, pp. 2119-2138, https://doi.org/10.1016/j.spa.2018.02.003.
[9] N. T. Dung, The Density of Solutions to Multifractional Stochastic Volterra Integro-Differential Equations, Nonlinear Anal, Vol. 130, 2016, pp. 176-189, https://doi.org/10.1016/j.na.2015.10.003.
[10] N. T. Dung, Gaussian Lower Bounds for the Density Via Malliavin Calculus, C. R. Math. Acad. Sci.Paris,
Vol. 358, No. 1, 2020, pp. 79-88, https://doi.org/10.5802/crmath.13.
[11] N. T. Hang, P. T. P.Thuy, Tail Distribution Estimates of the Mixed-Fractional CEV Model, Open Journal of Mathematical Sciences,Vol. 5, Issue 1, 2021, pp. 371-379, https://doi.org/10.30538/oms2021.0172.
[12] N. T. Hang, N. V. Tan, Tail Distribution Estimates of Fractional CIR Model, VNU Journal of Science: Mathematics – Physics, Vol. 38, No. 3, 2022, pp. 70-78, http://dx.doi.org/10.25073/2588-1124/vnumap.4710.
[13] D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, Berlin, Second Edition, 2006.
[14] I. Nourdin, F. G. Viens, Density Formula and Concentration Inequalities with Malliavin Calculus, Electron J Probab, Vol. 14, 2009, pp. 2287-2309, https://doi.org/10.48550/arXiv.0808.2088.