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In this note, we introduce a spectrally positive Levy risk process with Parisian implementation delays in dividend payments, which means that the dividends can only be paid when the surplus of the Levy risk process has stayed continuously above the barrier b for a certain time r ( > 0 ) . Using the scale functions and the distribution of the risk process at time r , the Laplace transform of the ruin time is derived.

The literature has been witnessing an aroused interest in the study of the two-side exit problems for various models. Motivated by Kyprianou and Zhou (2009) and Li et al. (2018), the present paper concerns the two-side exit problems of the taxed spectrally negative Levy risk process involving the general draw-down time. Our two-side exit problem is separated into two sub-problems: one being the Laplace transform of the up-exiting time of a certain level on the event that the taxed risk process up-crosses that level before the general draw-down time; the other being the Laplace transform of the draw-down time on the event that draw-down of the taxed risk process occurs before it up-crosses a certain level. Using a modified approximating method of Li et al. (2018) together with the excursion theory, solutions for the aforementioned two-side exit problems are obtained.

We consider a Levy risk process and a Sparre-Andersen risk process with Parisian ruin in the presence of a constant dividend barrier. We demonstrate that with few exceptions, ruin occurs with probability one. Subsequently, generalizations to certain dependent risk processes are discussed. Despite the mathematical nature of this paper, its goal is to convey some simple conclusions to the actuarial community. The reader may focus solely on the introduction and conclusion sections (Sections 1 and 5, respectively) as well as the numerical illustrations.

In recent years there has been some focus on quasi-stationary behavior of an one-dimensional Levy process X , where we ask for the law P ( X t ∈ d y | τ 0 − > t ) for t → ∞ and τ 0 − = inf { t ≥ 0 : X t 0 } . In this paper we address the same question for so-called Parisian ruin time τ θ , that happens when process stays below zero longer than independent exponential random variable with intensity θ .

In risk theory, the problem of maximizing the expected cumulated discounted loss-carry-forward tax payments until ruin is a widely discussed topic since the taxation system was proposed by Albrecher and Hipp (2007). In the present paper, we discuss this maximization problem in the Cramer–Lundberg risk model including a constant force of interest. The optimal taxation return function is identified as the classical solution of the associated Hamilton–Jacobi–Bellman equation and the optimal taxation strategy in this risk model with interest is derived, which is of band type. Finally, an example is constructed for exponential claim sizes, in which closed-form expression for the optimal taxation return function is given.

Let B ( t ) , t ∈ R be a standard Brownian motion. Define a risk process (0.1) R u δ ( t ) = e δ t ( u + c ∫ 0 t e − δ s d s − σ ∫ 0 t e − δ s d B ( s ) ) , t ≥ 0 , where u ≥ 0 is the initial reserve, δ ≥ 0 is the force of interest, c > 0 is the rate of premium and σ > 0 is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability K S δ ( u , T u ) : = P { inf t ∈ [ 0 , S ] sup s ∈ [ t , t + T u ] R u δ ( s ) 0 } , S ≥ 0 , as u → ∞ where T u is a bounded function. Further, we show that the Parisian ruin time of this risk process can be approximated by an exponential random variable. Our results are new even for the classical ruin probability and ruin time which correspond to T u ≡ 0 in the Parisian setting.

In this paper, we present a result on the distribution of the maximum severity of ruin in a perturbed risk process with Markovian arrivals. We show that the distribution of the maximum severity of ruin is closely related to the distributions of one-sided and two-sided passage times.

In this paper, we present a new approach to the study of the Gerber–Shiu discounted function for the risk model with multi-layer dividend strategy. The formulae for the Gerber–Shiu discounted function and ruin probability were obtained and the special case where the claim size distribution is a combination of exponentials is considered in detail.

We provide a saddlepoint approximation which allows us to compute the probability of ruin of the risk process where the number of claims is Poisson distributed and where an additional Wiener term is considered. This approximation is very efficient and accurate. Some numerical illustrations and comparison with alternative methods are provided.