Refracted Lévy processes

Kyprianou, A. E.; Loeffen, R. L.

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 24-44.

Motivé par des considérations classiques de la théorie du risque, nous étudions des problèmes de croisement de frontière par des processus de Lévy réfractés. Un processus de Lévy réfracté est un processus de Lévy dont la dynamique possède une dérive linéaire fixe. Plus formellement, un processus de Lévy réfracté est décrit par l'unique solution forte, si elle existe, de l'équation différentielle stochastique

$$d{U}_t=-\delta {\mathbf{1}}_{\{U_t>b\}}dt+d{X}_t$$

, où $X$ est un processus de Lévy et $b$, $\delta \in ℝ$, de telle manière que le processus $U$ peut visiter l'intervalle $(b,\infty )$ avec probabilité positive. En particulier, nous considérons le cas où le processus de Lévy n'a pas de saut positif et nous fournissons des calculs explicites pour le problème de sortie d'un intervalle. Toutes les identités peuvent être écrites en terme de la fonction d'échelle associée au processus de Lévy sous-jacent et sa version perturbée. Finalement, nous appliquons les identités obtenues aux processus de risque.

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation

$$d{U}_t=-\delta {\mathbf{1}}_{\{U_t>b\}}dt+d{X}_t$$

, where $X=\{X_t:t\ge 0\}$ is a Lévy process with law $\mathbb{P}$ and $b$, $\delta \in ℝ$ such that the resulting process $U$ may visit the half line $(b,\infty )$ with positive probability. We consider in particular the case that $X$ is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the $q$-scale function of the driving Lévy process and its perturbed version describing motion above the level $b$. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

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