Optimal control with absolutely continuous strategies for spectrally negative lévy processes

Lévy processes

Optimality of a refraction strategy in the optimal dividends problem with absolutely continuous controls subject to Parisian ruin

We consider de Finetti's optimal dividends problem with absolutely continuous strategies in a spectrally negative L\'evy model with Parisian ruin as the termination time. The problem considered is essentially a generalization of both the control problems considered by Kyprianou, Loeffen & P\'erez (2012) and by Renaud (2019). Using the language of scale functions for Parisian fluctuation theory, and under the assumption that the density of the L\'evy measure is completely monotone, we prove that a refraction dividend strategy is optimal and we characterize the optimal threshold. In particular, we study the effect of the rate of Parisian implementation delays on this optimal threshold

On the time spent in the red by a refracted L\'evy risk process

In this paper, we introduce an insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model, the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the process recovers. The analysis is mainly focused on the time a refracted L\'evy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from Kyprianou and Loeffen (2010) and Loeffen et al. (2012), we identify the distribution of various functionals related to occupation times of refracted spectrally negative L\'evy processes. For example, these results are used to compute the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring

Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps)

On the Dual Risk Models

Abstract This thesis focuses on developing and computing ruin-related quantities that are potentially measurements for the dual risk models which was proposed to describe the annuity-type businesses from the perspective of the collective risk theory in 1950’s. In recent years, the dual risk models are revisited by many researchers to quantify the risk of the similar businesses as the annuity-type businesses. The major extensions included in this thesis consist of two aspects: the first is to search for new ruin-related quantities that are potentially indices of the risk for well-established dual models; the other aspect is to generalize the settings of the dual models instead of the ruin quantities. There are four separate articles in this thesis, in which the first (Chapter 2) and the last (Chapter 5) belong to the first type of extensions while the others (Chapter 3 and Chapter 4) belong to the generalizations of the dual models. The first article (Chapter 2) studies the discounted moments of the surplus at the time of the last jump before ruin for the compound Poisson dual risk model. The idea comes from that the ruin of the compound Poisson dual models is caused by absence of positive jumps within a period with length being propotional to the surplus at the time of the last jump. As a quantity related to a non-stopping time, the explicit expression of the target quantity is obtained through integro-differential equations. The second article (Chapter 3) investigate the Sparre-Andersen dual risk models in which the epochs are independently, identically distributed generalized Erlang-n random variables. An important difference between this model and some other models such as the Erlang-n dual risk models is that the roots to the generalized Lundberg’s equation are not necessarily distinct. By taking the multiple roots into account, the explicit expressions of the Laplace transform of the time to ruin and expected discounted aggregate dividends under the threshold strategy and exponential distributed revenues are derived. The third article (Chapter 4) revisits the the dual Lévy risk model. The target ruin quantity is the expected discounted aggregate dividends paid up to ruin under the threshold dividend strategy. The explicit expression is obtained in terms of the q-scale functions through constructing a new dividend strategy having the target ruin quantity converging to that under the threshold strategy. Also, the optimality of the threshold strategy among all the absolutely continuous stategies when evaluating the target quantity as a value function is discussed. The fourth article (Chapter 5) initiate the study of the Parisian ruin problem for the general dual Lévy risk models. Unlike the regular ruin for the dual models, the deficit at Parisian ruin is not necessarily equal to zero. Hence we introduce the Gerber-Shiu expected discounted penalty function (EPDF) at the Parisian ruin and obtain an explicit expression for this function. Keywords: Sparre-Andersen dual models, expected discounted aggregate dividends, dual Levy risk models, Parisian ruin, Gerber-Shiu function ii

Optimal dividend policies with random profitability

We study an optimal dividend problem under a bankruptcy constraint. Firms face a trade-off between potential bankruptcy and extraction of profits. In contrast to previous works, general cash flow drifts, including Ornstein--Uhlenbeck and CIR processes, are considered. We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as comparison between the sub- and supersolutions of the Hamilton--Jacobi--Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution. The value function is given by a nonlinear PDE with a gradient constraint from below in one dimension. We find that the optimal strategy is both a barrier and a band strategy and that it includes voluntary liquidation in parts of the state space. Finally, we present and numerically study extensions of the model, including equity issuance and credit lines

Beiträge zur Theorie des optimalen Stoppens

This thesis deals with the explicit solution of optimal stopping problems with infinite time horizon. To solve Markovian problems in continuous time we introduce an approach that gives rise to explicit results in various situations. The main idea is to characterize the optimal stopping set as the union of the maximum points of explicitly given functions involving the harmonic functions for the underlying stochastic process. This provides elementary solutions for a variety of optimal stopping problems and answers questions concerning the geometric shape of the optimal stopping set. The approach is shown to work well for one- and multidimensional diffusion processes, spectrally negative Lévy processes and problems containing the running maxima process. Furthermore we introduce a new class of problems, which we call problems with guarantee. For continuous one-dimensional driving processes and certain Lévy processes we prove that the optimal strategies are of two-sided type and establish first-order ODEs that characterize the solution. In the second part we consider optimal stopping problems for autoregressive processes in discrete time. This class of processes is intensively studied in statistics and other fields of applied probability. We establish elementary conditions to ensure that the optimal stopping time is of threshold type and find the joint distribution of the threshold-time and the overshoot for a wide class of innovations. Using the principle of continuous fit this leads to explicit solutions.Gegenstand dieser Arbeit ist die explizite Lösung von Problemen des optimalen Stoppens mit unendlichem Zeithorizont. Im ersten Teil führen wir zur Lösung Markovscher Probleme in stetiger Zeit einen Ansatz ein, der in einer Vielzahl von Situationen zu expliziten Ergebnissen führt. Unter Benutzung der harmonischen Funktionen des zugrunde liegenden Prozesses charakterisieren wir dazu zunächst das Stoppgebiet als Menge von Maximalstellen konkret gegebener Funktionen. Dies führt in vielen Fällen zu elementaren Lösungen und ermöglicht Aussagen zur geometrischen Form des Stoppgebiets. Der Ansatz ist anwendbar auf ein- und mehrdimensionale Diffusionen, spektral-negative Lévyprozesse und Probleme, die den Supremumsprozess enthalten. Des Weiteren führen wir eine neue Klasse von Problemen ein, die wir Stoppprobleme mit Garantien nennen. Für stetige eindimensionale Prozesse zeigen wir mithilfe des obigen Ansatzes, dass die optimalen Strategien zweiseitig sind und charakterisieren die optimalen Grenzen mittels gewöhnlicher Differentialgleichungen erster Ordnung. Diese Ergebnisse übertragen wir anschließend auf Lévyprozesse. Im zweiten Teil beschäftigen wir uns mit Problemen des optimalen Stoppens autoregressiver Folgen, welche zur Beantwortung von Fragen in der Statistik und in anderen Feldern der angewandten Mathematik untersucht werden. Wir geben elementare Bedingungen an, die sicherstellen, dass die optimalen Stoppzeiten Erstübertrittszeiten sind und bestimmen die gemeinsame Verteilung von Erstübertrittszeit und Overshoot für eine große Klasse von Innovationen. Mithilfe des Prinzips des stetigen Übergangs erhält man explizite Lösungen