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 .
We propose a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: The first, which we largely focus on, features a random quantile level and allows for direct interpretation of model parameters such as skewness and kurtosis. The second type are function-valued quantile diffusions and are driven by stochastic parameter processes, which determine the entire quantile function at each point in time. The quantile processes are obtained by transforming the marginals of a diffusion process under a composite map consisting of a distribution and a quantile function. Such maps, analogous to rank transmutation maps, produce the marginals of the resulting quantile process. We discuss the relationship and differences between our approach and existing methods and characterisations of quantile processes in discrete and continuous time. As an example of an application of quantile diffusions, we show how measure distortions, often found in financial mathematics and actuarial science, may be induced .
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This paper focuses on the development of a new class of diffusion processes that
We propose a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: The first, which we largely focus on, features a random quantile level and
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and dynamic modelling of quantile diffusions. We constructed quantile diffusion processes by transforming each marginal of a given univariate
interpretation of model parameters such as skewness and kurtosis. The second type are function-valued quantile diffusions and are driven by stochastic parameter processes, which determine the entire quantile function at each point in time. The quantile processes are obtained by transforming the marginals of a
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function and quantile function, which in turn produces
and a quantile function. Such maps, analogous to rank transmutation maps, produce
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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
We discuss the relationship and differences between our approach and existing methods and characterisations of quantile processes in discrete and continuous time. As an example of an application of quantile diffusions, we show how measure distortions, often found in financial mathematics and actuarial science, may be induced