RATIO MONOTONICITY FOR TAIL PROBABILITIES IN THE RENEWAL RISK MODEL

A renewal model in risk theory is considered, where $\overline{H}(u,y)$ is the tail of the distribution of the deficit at ruin with initial surplus u and $\overline{F}(y)$ is the tail of the ladder height distribution. Conditions are derived under which the ratio $\overline{H}(u,y)/\overline{F}(u+y)$ is nondecreasing in u for any y≥0. In particular, it is proven that if the ladder height distribution is stable and DFR or phase type, then the above ratio is nondecreasing in u. As a byproduct of this monotonicity, an upper bound and an asymptotic result for $\overline{H}(u,y)$ are derived. Examples are given to illustrate the monotonicity results.

Asymptotics for the moments of the overshoot and undershoot of a random walk

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

Asymptotics for the moments of the overshoot and undershoot of a random walk

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

Let {(X n , S n ), n ≥ 0} be a Markov random walk in which X n takes values in a general state space and S n takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

On ladder height densities and Laguerre series in the study of stochastic functionals. I. Basic methods and results

In this paper we develop methods for reducing the study, the computation, and the construction of stochastic functionals to those of standard concepts such as the moments of the pertinent random variables. Principally, our methods are based on the notion of ladder height densities and their Laguerre expansions, and our results provide a unifying framework for the distinct approaches of Dufresne (2000) and Schröder (2005).

Asymptotics of the density of the supremum of a random walk with heavy-tailed increments

Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a random walk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.