On the time spent in the red by a refracted Lévy risk process

On occupation times in the red of Lévy risk models

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level $0$) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running ma…

Optimality of impulse control problem in refracted Lévy model with Parisian ruin and transaction costs

In this paper we investigate an optimal dividend problem with transaction costs, where the surplus process is modelled by a refracted Lévy process and the ruin time is considered with Parisian delay. Presence of the transaction costs implies that one need to consider the impulse control problem as a control strategy in such model. An impulse policy $(c_1,c_2)$, which is to reduce the reserves to some fixed level $c_1$ whenever they are above another level $c_2$ is an important strategy for the impulse control problem. Therefore, we give sufficient conditions under which the above described imp…

带指数巴黎破产的最优红利控制：在折射莱维风险模型下考虑交易成本

Optimal Dividend Control with Transaction Costs under Exponential Parisian Ruin for a Refracted Levy Risk Model

本文研究了在折射莱维风险模型下，考虑交易成本的最优红利控制问题，通过巴黎指数延迟和下限破产障碍确定破产时间，并利用HJB不等式验证最优策略的最优性。

On the distribution of cumulative Parisian ruin

We introduce the concept of cumulative Parisian ruin, which is based on the time spent in the red by the underlying surplus process. Our main result is an explicit representation for the distribution of the occupation time, over a finite-time horizon, for a compound Poisson process with drift and exponential claims. The Brownian ruin model is also studied in details. Finally, we analyze for a general framework the relationships between cumulative Parisian ruin and classical ruin, as well as with Parisian ruin based on exponential implementation delays.