Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

An insurance risk process with a generalized income process: A solvency analysis

Abstract: In ruin theory, an insurer’s income process is usually assumed to grow at a deterministic rate of c > 0 over time. For instance, both the well-known Cramer–Lundberg risk process and the Sparre Andersen risk model have this assumption built in the construction of their respective surplus processes. This assumption is mainly considered for purposes of mathematical tractability, but generally fails to accurately model an insurer’s income dynamics. To better characterize the variability and uncertainty of an insurer’s income process, several papers have studied insurance risk models with random incomes where the main emphasis is placed on carrying the related Gerber–Shiu analysis. However, a systematic and quantitative understanding of how the more volatile income processes impact an insurer’s solvency risk is still lacking. This paper aims to fill this gap in the literature by quantitatively assessing the impact of the choice of income process on some finite-time and infinite-time ruin quantities. To carry this analysis, we consider a generalized Sparre Andersen risk model with a random income process which renews at claim instants. For exponentially distributed claim sizes, we derive explicit expressions for some joint distributions involving the time to ruin and the number of claims until ruin. As special cases of the proposed insurance risk process, we consider income processes modelled by a subordinator or a particular varying premium rate model. Numerical examples are then carried to draw important risk management implications of a solvency nature for the insurer.

Local asymptotic properties for the growth rate of a jump-type CIR process

Abstract: In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is a unknown parameter. The L\'evy measure of the subordinator is finite or infinite. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Three cases are distinguished: subcritical, critical and supercritical. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To do so, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.

Canonical correlations for dependent gamma processes

Abstract: The present paper provides a characterisation of exchangeable pairs of random measures $(\widetilde\mu_1,\widetilde\mu_2)$ whose identical margins are fixed to coincide with the distribution of a gamma completely random measure, and whose dependence structure is given in terms of canonical correlations. It is first shown that canonical correlation sequences for the finite-dimensional distributions of $(\widetilde\mu_1,\widetilde\mu_2)$ are moments of means of a Dirichlet process having random base measure. Necessary and sufficient conditions are further given for canonically correlated gamma completely random measures to have independent joint increments. Finally, time-homogeneous Feller processes with gamma reversible measure and canonical autocorrelations are characterised as Dawson--Watanabe diffusions with independent homogeneous immigration, time-changed via an independent subordinator. It is thus shown that Dawson--Watanabe diffusions subordinated by pure drift are the only processes in this class whose time-finite-dimensional distributions have, jointly, independent increments.

Extinction time of logistic branching processes in a Brownian environment

Abstract: In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Levy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment converges to a specified distribution; otherwise, it becomes extinct a.s. In the latter scenario, and following a similar approach to Lambert (Lambert, Ann. Appl. Probab., 2005), we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Ricatti differential equation. In particular, the latter characterises the law of the process coming down from infinity.

Fractional Lévy stable motion time-changed by gamma subordinator

Abstract: In this paper a new stochastic process is introduced by subordinating fractional Levy stable motion (FLSM) with gamma process. This new process incorporates stochastic volatility in the parent proc...