Martijn R. Pistorius

On the optimal dividend problem for a spectrally negative Levy process

In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative L{e}vy 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 L{e}vy 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.

On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum

Consider a spectrally one-sided Lévy process X and reflect it at its past infimum I. Call this process Y. For spectrally positive X, Avram et al.(2) found an explicit expression for the law of the first time that Y=X−I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y, if X is spectrally negative. Subsequently, we find an expression for the resolvent measure for Y killed upon leaving [0,a]. We determine the exponential decay parameter ϱ for the transition probabilities of Y killed upon leaving [0,a], prove that this killed process is ϱ-positive and specify the ϱ-invariant function and measure. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined in [0,a] and prove some properties of this process.

On perpetual American put valuation and first-passage in a regime-switching model with jumps

In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching Lévy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first-passage problem under a state-dependent level rests on a path transformation and a new matrix Wiener–Hopf factorization result for this class of processes.

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 Cram{e}r light-tail assumption on the claim size distribution.

A Potential-theoretical Review of some Exit Problems of Spectrally Negative Lévy Processes

In this note we consider first exit problems of completely asymmetric (reflected) Levy processes and present an alternative derivation of their Laplace transforms essentially based on potential theory of Markov processes.

An Excursion-Theoretical Approach to Some Boundary Crossing Problems and the Skorokhod Embedding for Reflected Lévy Processes

Let X be a spectrally negative Levy process, reflect X at its supremum X and call this process Y . Let τa denote the first time Y crosses the level a. Using excursion theory we solve the problem of Lehoczky for a spectrally negative Levy process, that is, we express the joint law of (τa, Xτa , Yτa−, ∆Xτa) in terms of socalled scale functions that also turn up in the solution of the two-sided exit problem, thereby extending results of Avram et al. [2], who solved for the joint law of (τa, Yτa). Next we obtain an explicit and non-randomised solution to the Skorokhod embedding problem of Y : we find a stopping time T such that YT ∼ ν for a measure ν on (0,∞) without atoms.

On doubly reflected completely asymmetric Lévy processes

Consider a completely asymmetric Levy process X and let Z be X reflected at 0 and at a>0. In applied probability (e.g. The Single Server Queue, 2nd Edition, North-Holland, Amsterdam, 1982) the process Z turns up in the study of the virtual waiting time in an M/G/1-queue with finite buffer a or the water level in a finite dam of size a. We find an expression for the resolvent density of Z. We show Z is positive recurrent and determine the invariant measure. Using the regenerative property of Z, we determine the asymptotic law of for an appropriate class of functions f. Finally, the long time average of the local time of Z in x[set membership, variant][0,a] is studied.

On an optimal consumption problem for p-integrable consumption plans

Abstract. A generalization is presented of the existence results for an optimal consumption problem of Aumann and Perles [4] and Cox and Huang [10]. In addition, we present a very general optimality principle.

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ± such that the stopping time T=infu2009{t>0:Xt∈{−φ−(Lt),φ+(Lt)}} induces XT∼μ, where (Lt) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup.Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.