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On density functions related to discrete time maximum of some one-dimensional diffusion processes

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  • Nakatsu, Tomonori

Abstract

This article focuses on the probability density functions related to the discrete time maximum of some one-dimensional diffusion processes. Firstly, we shall consider solutions of one-dimensional stochastic differential equations and prove an integration by parts formula on the discrete time maximum of the solutions. The smoothness, expressions and upper bounds of the density function will be obtained by the formula. Secondly, Gaussian processes will be dealt with. For some Gaussian processes, we shall obtain asymptotic behaviors of the density functions related to the discrete time maximum of the processes. The Malliavin calculus and Laplace’s method play important roles for the proofs.

Suggested Citation

  • Nakatsu, Tomonori, 2023. "On density functions related to discrete time maximum of some one-dimensional diffusion processes," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    • Handle: RePEc:eee:apmaco:v:441:y:2023:i:c:s0096300322007408
    DOI: 10.1016/j.amc.2022.127672
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    1. Guillaume Bernis & Emmanuel Gobet & Arturo Kohatsu‐Higa, 2003. "Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 99-113, January.
    2. Emmanuel Gobet, 2004. "Revisiting the Greeks for European and American Options," World Scientific Book Chapters, in: Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), Stochastic Processes And Applications To Mathematical Finance, chapter 3, pages 53-71, World Scientific Publishing Co. Pte. Ltd..
    3. Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
    4. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    5. Florit, Carme & Nualart, David, 1995. "A local criterion for smoothness of densities and application to the supremum of the Brownian sheet," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 25-31, January.
    6. Nakatsu, Tomonori, 2013. "Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2499-2506.
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