A Class of Copulas Associated with Brownian Motion Processes and Their Maxima
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Abstract
The copulas being introduced in this paper are derived from distributions associated with the Brownian motion and related processes. Useful background information is initially presented. Attention is focused on univariate Brownian motion processes having a drift parameter and their running maxima as well as correlated bivariate Brownian motion processes in conjunction with the cumulative maxima of one of their components. The copulas generated therefrom as well as the corresponding density functions are explicitly provided and graphically represented. The derivations are thorough, with the requisite preliminary results being stated beforehand. As well, various potential applications are suggested. A numerical example involving an actual data set consisting of daily stock prices is worked out in detail. The associated empirical copula density function is determined in terms of Bernstein's polynomials and compared to one of our proposed theoretical copula densities. The steps that are provided, including an initial transformation of the observations which likens them to a Wiener process, should enable one to efficiently apply the novel results introduced herein to data sets arising in other areas.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
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| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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