arXiv subjects

Martijn R. Pistorius

Publications and source records attributed to Martijn R. Pistorius.

3 recordsLinked to original sources

arXiv2012

On the drawdown of completely asymmetric Levy processes

The {\em drawdown} process $Y$ of a completely asymmetric Lévy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the Lévy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time $\bar{G}_{τ_a}$ of the last supremum of $X$ prior to $τ_a$, the infimum $\unl X_{τ_a}$ and supremum $\ovl X_{τ_a}$ of $X$ at $τ_a$ and the undershoot $a - Y_{τ_a-}$ and overshoot $Y_{τ_a}-a$ of $Y$ at $τ_a$. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential Lévy model.

math.PR↗

arXiv2009

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramér light-tail assumption on the claim size distribution.

math.PR↗

arXiv2007

On the optimal dividend problem for a spectrally negative Lévy process

In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.

math.PR↗