72 research outputs found

    Exit Problems for LĂ©vy and Markov Processes with One-Sided Jumps and Related Topics

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    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)

    A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems

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    International audienceThe Segerdahl-Tichy Process, characterized by exponential claims and state dependent drift, has drawn a considerable amount of interest, due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest non-LĂ©vy, non-diffusion example of a spectrally negative Markov risk model. Note that for both spectrally negative LĂ©vy and diffusion processes, first passage theories which are based on identifying two "basic" monotone harmonic functions/martingales have been developed. This means that for these processes many control problems involving dividends, capital injections, etc., may be solved explicitly once the two basic functions have been obtained. Furthermore, extensions to general spectrally negative Markov processes are possible; unfortunately, methods for computing the basic functions are still lacking outside the LĂ©vy and diffusion classes. This divergence between theoretical and numerical is strikingly illustrated by the Segerdahl process, for which there exist today six theoretical approaches, but for which almost nothing has been computed, with the exception of the ruin probability. Below, we review four of these methods, with the purpose of drawing attention to connections between them, to underline open problems, and to stimulate further work

    On Central Branch/Reinsurance Risk Networks: Exact Results and Heuristics

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    Modeling the interactions between a reinsurer and several insurers, or between a central management branch (CB) and several subsidiary business branches, or between a coalition and its members, are fascinating problems, which suggest many interesting questions. Beyond two dimensions, one cannot expect exact answers. Occasionally, reductions to one dimension or heuristic simplifications yield explicit approximations, which may be useful for getting qualitative insights. In this paper, we study two such problems: the ruin problem for a two-dimensional CB network under a new mathematical model, and the problem of valuation of two-dimensional CB networks by optimal dividends. A common thread between these two problems is that the one dimensional reduction exploits the concept of invariant cones. Perhaps the most important contribution of the paper is the questions it raises; for that reason, we have found it useful to complement the particular examples solved by providing one possible formalization of the concept of a multi-dimensional risk network, which seems to us an appropriate umbrella for the kind of questions raised her

    Optimal stopping problems for maxima and minima in models with asymmetric information

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    We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normal-reflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations

    On Some Stochastic Optimal Control Problems in Actuarial Mathematics

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    The event of ruin (bankruptcy) has long been a core concept of risk management interest in the literature of actuarial science. There are two major research lines. The first one focuses on distributional studies of some crucial ruin-related variables such as the deficit at ruin or the time to ruin. The second one focuses on dynamically controlling the probability that ruin occurs by imposing controls such as investment, reinsurance, or dividend payouts. The content of the thesis will be in line with the second research direction, but under a relaxed definition of ruin, for the reason that ruin is often too harsh a criteria to be implemented in practice. Relaxation of the concept of ruin through the consideration of "exotic ruin" features, including for instance, ruin under discrete observations, Parisian ruin setup, two-sided exit framework, and drawdown setup, received considerable attention in recent years. While there has been a rich literature on the distributional studies of those new features in insurance surplus processes, comparably less contributions have been made to dynamically controlling the corresponding risk. The thesis proposes to analytically study stochastic control problems related to some "exotic ruin" features in the broad area of insurance and finance. In particular, in Chapter 3, we study an optimal investment problem by minimizing the probability that a significant drawdown occurs. In Chapter 4, we take this analysis one step further by proposing a general drawdown-based penalty structure, which include for example, the probability of drawdown considered in Chapter 3 as a special case. Subsequently, we apply it in an optimal investment problem of maximizing a fund manager's expected cumulative income. Moreover, in Chapter 5 we study an optimal investment-reinsurance problem in a two-sided exit framework. All problems mentioned above are considered in a random time horizon. Although the random time horizon is mainly determined by the nature of the problem, we point out that under suitable assumptions, a random time horizon is analytically more tractable in comparison to its finite deterministic counterpart. For each problem considered in Chapters 3--5, we will adopt the dynamic programming principle (DPP) to derive a partial differential equation (PDE), commonly referred to as a Hamilton-Jacobi-Bellman (HJB) equation in the literature, and subsequently show that the value function of each problem is equivalent to a strong solution to the associated HJB equation via a verification argument. The remaining problem is then to solve the HJB equations explicitly. We will develop a new decomposition method in Chapter 3, which decomposes a nonlinear second-order ordinary differential equation (ODE) into two solvable nonlinear first-order ODEs. In Chapters 4 and 5, we use the Legendre transform to build respectively one-to-one correspondence between the original problem and its dual problem, with the latter being a linear free boundary problem that can be solved in explicit forms. It is worth mentioning that additional difficulties arise in the drawdown related problems of Chapters 3 and 4 for the reason that the underlying problems involve the maximum process as an additional dimension. We overcome this difficulty by utilizing a dimension reduction technique. Chapter 6 will be devoted to the study of an optimal investment-reinsurance problem of maximizing the expected mean-variance utility function, which is a typical time-inconsistent problem in the sense that DPP fails. The problem is then formulated as a non-cooperative game, and a subgame perfect Nash equilibrium is subsequently solved. The thesis is finally ended with some concluding remarks and some future research directions in Chapter 7

    Poissonian potential measures for LĂ©vy risk models

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    The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.insmatheco.2018.07.004 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/This paper studies the potential (or resolvent) measures of spectrally negative Lévy processes killed on exiting (bounded or unbounded) intervals, when the underlying process is observed at the arrival epochs of an independent Poisson process. Explicit representations of these so-called Poissonian potential measures are established in terms of newly defined Poissonian scale functions. Moreover, Poissonian exit measures are explicitly solved by finding a direct relation with Poissonian potential measures. Our results generalize Albrecher et al. (2016) in which Poissonian exit identities are solved. As an application of Poissonian potential measures, we extend the Gerber–Shiu analysis in Baurdoux et al. (2016) to a (more general) Parisian risk model subject to Poissonian observations.Natural Sciences and Engineering Research Council of Canada (341316; 05828)Canada Research Chair ProgramJames C. Hickman Scholar program of the Society of Actuaries, USAEducational Institution Grant of the Society of Actuarie

    On the design of customized risk measures in insurance, the problem of capital allocation and the theory of fluctuations for LĂ©vy processes

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    Dans cette thèse, nous étudions quelques problèmes fondamentaux en mathématiques financières et actuarielles, ainsi que leurs applications. Cette thèse est constituée de trois contributions portant principalement sur la théorie de la mesure de risques, le problème de l’allocation du capital et la théorie des fluctuations. Dans le chapitre 2, nous construisons de nouvelles mesures de risque cohérentes et étudions l’allocation de capital dans le cadre de la théorie des risques collectifs. Pour ce faire, nous introduisons la famille des "mesures de risque entropique cumulatifs" (Cumulative Entropic Risk Measures). Le chapitre 3 étudie le problème du portefeuille optimal pour le Entropic Value at Risk dans le cas où les rendements sont modélisés par un processus de diffusion à sauts (Jump-Diffusion). Dans le chapitre 4, nous généralisons la notion de "statistiques naturelles de risque" (natural risk statistics) au cadre multivarié. Cette extension non-triviale produit des mesures de risque multivariées construites à partir des données financiéres et de données d’assurance. Le chapitre 5 introduit les concepts de "drawdown" et de la "vitesse d’épuisement" (speed of depletion) dans la théorie de la ruine. Nous étudions ces concepts pour des modeles de risque décrits par une famille de processus de Lévy spectrallement négatifs.The aim of this thesis is to study fundamental problems in financial and insurance mathematics particularly the problem of measuring risk and its application within financial and insurance frameworks. The main contributions of this thesis can be classified in three main axes: the theory of risk measures, the problem of capital allocation and the theory of fluctuation. In Chapter 2, we design new coherent risk measures and study the associated capital allocation in the context of collective risk theory. We introduce the family of Cumulative Entropic Risk Measures. In Chapter 3, we study the optimal portfolio problem for the Entropic Value at Risk coherent risk measure for particular return models which are based on relevant cases of Jump-Diffusion models. In Chapter 4, we extending the notion of natural risk statistics to the multivariate setting. This non-trivial extension will endow us with multivariate data-based risk measures that are bound to have applications in finance and insurance. In Chapter 5, we introduce the concepts of drawdown and speed of depletion to the ruin theory literature and study them for the class of spectrally negative Lévy risk processes

    Stable Processes

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