Abstract
The paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a ν-stable subordinator is used for the time-change. The submartingale property is assumed to guarantee the net-profit condition. The long-range dependence behavior is proven. The infinite-horizon ruin probability, a specialized version of the Gerber–Shiu function, is considered and investigated. In particular, we prove that the distribution function of the infinite-horizon ruin time satisfies an integral-differential equation. The case of the dividends paid according to a multi-layer dividend strategy is also considered.
Highlights
The motivation of such a contribution relies on the need to specialize the Cramer–Lundberg-type risk model with a random initial surplus.The choice of the time change by means of the inverse of a ν-stable subordinator ([4])applied to the classical risk model is related to the possibility to make the last one more flexible for applications: the fractional model evolves on a stochastic time scale, and this aspect is revealed to be optimal in the financial application context in which the changes in the capital value evolve on a time scale strictly linked with the occurrence of other stochastic events ([5,6,7,8])
Applied to the classical risk model is related to the possibility to make the last one more flexible for applications: the fractional model evolves on a stochastic time scale, and this aspect is revealed to be optimal in the financial application context in which the changes in the capital value evolve on a time scale strictly linked with the occurrence of other stochastic events ([5,6,7,8])
It can be useful to model a surplus process of an insurance company with an initial capital, with probability density function f, subject to the dividend payment in a random time, regulated by a ν-stable subordinator, and subject to further random variations due to the premiums and claims occurring in random times and with f random sizes, respectively
Summary
The motivation of such a contribution relies on the need to specialize (in the fractional context) the Cramer–Lundberg-type risk model with a random initial surplus (cf. [1,2,3]). The real life examples useful to understand why such kinds of time-changed models are advantageous include financial (as well as biological and other nature) dynamics subject to random changes in random times ([5,9]) This means that random variations are applied in correspondence of the occurrence of other phenomenological random events affecting the evolution of the focused process. A further feature of time-changed stochastic processes is to show a long-range dependence in the correlation function This is often used to construct models with the so-called long-memory properties (see, for instance, in the financial context, [10] and the references therein). By keeping in mind all these advantageous properties, here, we introduce a fractional time-changed risk model, and in order to provide a further generalization, we consider a random initial capital and a multi-layer dividend strategy.
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