35 Pages Posted: 13 Jan 2020
Date Written: December 23, 2019
This paper focuses on the development of a new class of diffusion processes that allows for direct and dynamic modelling of quantile diffusions. We constructed quantile diffusion processes by transforming each marginal of a given univariate diffusion process under a composite map consisting of a distribution function and quantile function, which in turn produces the marginals of the resulting quantile process. The transformation allows for the moments of the underlying process to be directly interpreted with regard to parameters of the transformation. For instance, skewness or kurtosis may be introduced to enable more realistic modelling of data such as financial asset returns, as well as the recycling of samples of the underlying process to make simulation of the transformed quantile process easier. We derive the stochastic differential equation satisfied by the quantile diffusion, and characterise the conditions under which strong and weak solutions exist, both in the general case and for the more specific Tukey g-h, g-transform and h-transform families of quantile diffusions.
Keywords: Continuous-time diffusion processes, quantiles, rank transmutation map, Tukey g-h transforms, skew, kurtosis, SDE, weak/strong solutions
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