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Maximum and minimum of one-dimensional diffusions

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  • Davis, Richard A.

Abstract

Let Mt be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that P(Mt[less-than-or-equals, slant]x) - Ft(x)-->0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the maximum of independent and identically distributed random variables with distribution function F. A new proof of this fact is given which is based on a time change of the Ornstein-Uhlenbeck process. Using this technique, the asymptotic independence of the maximum and minimum is also established. Moreover, this method allows one to construct stationary processes in which the limiting behavior of Mt is essentially unaffected by the stationary distribution. That is, there may be no relationship between the distribution F above and the marginal distribution of the process.

Suggested Citation

  • Davis, Richard A., 1982. "Maximum and minimum of one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 13(1), pages 1-9, July.
    • Handle: RePEc:eee:spapps:v:13:y:1982:i:1:p:1-9
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    Cited by:

    1. Laurent Denis & Begoña Fernández & Ana Meda, 2009. "Estimation Of Value At Risk And Ruin Probability For Diffusion Processes With Jumps," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 281-302, April.
    2. Kim, Jihyun & Park, Joon Y., 2017. "Asymptotics for recurrent diffusions with application to high frequency regression," Journal of Econometrics, Elsevier, vol. 196(1), pages 37-54.
    3. Choi, Hwan-sik & Jeong, Minsoo & Park, Joon Y., 2014. "An asymptotic analysis of likelihood-based diffusion model selection using high frequency data," Journal of Econometrics, Elsevier, vol. 178(P3), pages 539-557.
    4. Choi, Hwan-sik, 2016. "Information theory for maximum likelihood estimation of diffusion models," Journal of Econometrics, Elsevier, vol. 191(1), pages 110-128.
    5. Grigelionis, Bronius, 2003. "On point measures of [var epsilon]-upcrossings for stationary diffusions," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 403-410, February.
    6. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility regressions with fat tails," Journal of Econometrics, Elsevier, vol. 218(2), pages 690-713.
    7. Ohad Perry & Ward Whitt, 2013. "A Fluid Limit for an Overloaded X Model via a Stochastic Averaging Principle," Mathematics of Operations Research, INFORMS, vol. 38(2), pages 294-349, May.
    8. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility Regressions with Fat Tails," TSE Working Papers 20-1097, Toulouse School of Economics (TSE).
    9. Jihyun Kim & Nour Meddahi, 2020. "Volatility Regressions with Fat Tails," Post-Print hal-03142647, HAL.

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