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Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations

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  • Tomonori Nakatsu

    (Shibaura Institute of Technology)

Abstract

This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.

Suggested Citation

  • Tomonori Nakatsu, 2019. "Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1746-1779, December.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:4:d:10.1007_s10959-019-00885-1
    DOI: 10.1007/s10959-019-00885-1
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    References listed on IDEAS

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    1. Kohatsu-Higa, Arturo & Makhlouf, Azmi, 2013. "Estimates for the density of functionals of SDEs with irregular drift," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1716-1728.
    2. 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.
    3. Masafumi Hayashi & Arturo Kohatsu-Higa & Gô Yûki, 2013. "Local Hölder Continuity Property of the Densities of Solutions of SDEs with Singular Coefficients," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1117-1134, December.
    4. 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.
    5. 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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