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Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum

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  • Nakatsu, Tomonori

Abstract

In this article, we consider an m-dimensional stochastic differential equation with coefficients which depend on the maximum of the solution. First, we prove the absolute continuity of the law of the solution. Then we prove that the joint law of the maximum of the ith component of the solution and the i′th component of the solution is absolutely continuous with respect to the Lebesgue measure in a particular case. The main tool to prove the absolute continuity of the laws is Malliavin calculus.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:11:p:2499-2506
    DOI: 10.1016/j.spl.2013.07.011
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.
    2. 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).

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