David Landriault

• PhD, Laval University
• University of Waterloo

In this paper, we explore a non-cooperative optimal reinsurance problem incorporating likelihood ratio uncertainty, aiming to minimize the worst-case risk of the total retained loss for the insurer. We establish a general relation between the optimal reinsurance strategy under the reference probability measure and the strategy in the worst-case sce...

In the literature, insurance and reinsurance pricing is typically determined by a premium principle, characterized by a risk measure that reflects the policy seller's risk attitude. Building on the work of Meyers (1980) and Chen et al. (2016), we propose a new performance-based variable premium scheme for reinsurance policies, where the premium dep...

This paper introduces an economic framework to assess optimal longevity risk transfers between institutions, focusing on the interactions between a buyer exposed to long-term longevity risk and a seller offering longevity protection. While most longevity risk transfers have occurred in the reinsurance sector, where global reinsurers provide long-te...

In this paper, we consider an optimal reinsurance contract under a mean-variance criterion in a Stackelberg game theoretical framework. The rein-surer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal per-loss reinsurance to purchase. The objective...

This paper considers an optimal stopping problem with weighted discounting, and the state process is modelled by a general exponential Lévy process. Due to the time inconsistency, we provide a new martingale method based verification theorem for the equilibrium stopping strategies. As an application, we generalize an investment problem with non-exp...

The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a predetermined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend t...

Research in classical ruin theory has largely focused on the rst passage time analysis of a surplus process below level 0. Recently, inspired by numerous applications in nance, physics, and optimization, there has been an accrued interest in the analysis of the last passage time (below level 0). In this paper, we aim to bridge the rst and the last...

p>In this paper, we study the magnitude and the duration of deep drawdowns for the Lévy insurance risk model through the characterization of the Laplace transform of a related stopping time. Relying on a temporal approximation approach (e.g., Li et al. (2018)), the proposed methodology allows for a unified treatment of processes with bounded and un...

This paper proposes a novel high-water mark fee structure and investigates its impact on the marketability of variable annuities. To evaluate the welfare effects of holding a variable annuity, we set up a mean-variance preference model. By also examining the welfare effects of holding two alternative investments, a risk-free bond and a pure fund, w...

In this paper, we study the magnitude and the duration of deep drawdowns for the Lévy insurance risk model through the characterization of the Laplace transform of a related stopping time. Relying on a temporal approximation approach (e.g., Li et al. (2018)), the proposed methodology allows for a unified treatment of processes with bounded and unbo...

In ruin theory, an insurer’s income process is usually assumed to grow at a deterministic rate of c>0 over time. For instance, both the well-known Cramér-Lundberg risk process and the Sparre Andersen risk model have this assumption built in the construction of their respective surplus processes. This assumption is mainly considered for purposes of...

In this paper, we consider a dynamic Pareto optimal risk‐sharing problem under the time‐consistent mean‐variance criterion. A group of n insurers is assumed to share an exogenous risk whose dynamics is modeled by a Lévy process. By solving the extended Hamilton–Jacobi–Bellman equation using the Lagrange multiplier method, an explicit form of the ti...

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level 0) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only the Laplace transforms are know...

In this note, we consider a nonstandard analytic approach to the examination of scale functions in some special cases of spectrally negative Lévy processes. In particular, we consider the compound Poisson risk process with or without perturbation from an independent Brownian motion. New explicit expressions for the first and second scale functions...

The time to ruin has been the primary focus of many ruin-related analyses, mainly due to its significance in the assessment of an insurer’s solvency risk. The finite-time ruin probability and more recently, the t-year deferred ruin probability have drawn considerable attention over the years. Embedded into the expected discounted penalty function (...

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level $0$) up to an (independent) exponential horizon for spectrally negative L\'{e}vy risk processes and refracted spectrally negative L\'{e}vy risk processes. This result improves the existing literature in which only the Laplace transform...

In this paper, we complement the existing literature on the occupation time in the red (below level $0$) of a spectrally negative L\'evy process, and later extend the analysis to the refracted spectrally negative L\'evy process. For both classes of processes, we derive an explicit expression for the distribution of such occupation time up to an ind...

We study equilibrium feedback strategies for a dynamic mean-variance problem of investing in a risky financial market. We assume the time horizon is random, and we consider both discrete-time and continuous-time frameworks. The random time horizon is assumed to have a distribution that is independent of the underlying asset processes. By applying s...

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 newl...

In this paper, we model an entity's surplus process X using the drawdown-based regime-switching (DBRS) dynamics proposed in [9]. We introduce the state-dependent termination time to the model, and provide rationale for its introduction in insurance contexts. By examining some related potential measures, we first derive an explicit expression for th...

Drawdown (resp. drawup) of a stochastic process, also referred as the reflected process at its supremum (resp. infimum), has wide applications in many areas including financial risk management, actuarial mathematics and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the...

Drawdown (resp. drawup) of a stochastic process, also referred as the reflected process at its supremum (resp. infimum), has wide applications in many areas including financial risk management, actuarial mathematics and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the...

In recent years, multivariate insurance risk processes have received increasing attention in risk theory. First-passage-time problems in the context of these insurance risk processes are of primary interest for risk management purposes. In this article we study joint-ruin problems of two risk undertakers in a proportionally shared Markovian claim a...

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization prob...

In this paper, we derive and study a pair of optimal reinsurance-investment strategies under the two-sided exit framework which aims to (1) maximize the probability that the surplus reaches the target before ruin occurs over the time horizon (where is an independent exponentially distributed random time); (2) minimize the probability that ruin occu...

Incurred but not reported (IBNR) claims, which arise naturally in insurance contexts, are of central importance to insurers for risk management and financial reporting purposes. In this paper, we first examine the moments of the total discounted IBNR claim amount at a given time when claim events occur according to a compound renewal process. Under...

In this paper, we study some drawdown-related quantities in the context of the renewal insurance risk process with general interarrival times and phase-type distributed jump sizes. We make use of some recent results on the two-sided exit problem for the spectrally negative Markov additive process and a fluid flow analogy between certain queues and...

The order statistics of an independent sample of mixed Erlang random variables are shown to also be of mixed Erlang form. The mixing weights involve easily evaluated sums of multinomial type terms. A life contingency application is also considered.

Drawdown measures the decline of portfolio value from its historic high-water mark. In this paper, we study a lifetime investment problem aiming at minimizing the risk of drawdown occurrences. Under the Black-Scholes framework, we examine two …nancial market models: a market with two risky assets, and a market with a risk-free asset and a risky ass...

In this paper, we propose a new drawdown-based regime-switching (DBRS) Lévy insurance model in which the underlying drawdown process is used to model the level of financial distress over time, and trigger regime-switching transitions. By some analytical arguments, we derive explicit formulas for a generalized two-sided exit problem. We specifically...

In this paper, we consider a risk model which allows the insurer to partially reflect the recent claim experience in the determination of the next period’s premium rate. In a ruin context, similar mechanisms to the one proposed in this paper have been studied by, e.g., Tsai and Parker (2004), Afonso et al. (2009) and Loisel and Trufin (2013). In ou...

Occupation times have so far been primarily analyzed in the class of Levy processes, most notably some of its special cases, by capitalizing on the stationary and independence property of the process increments. In this paper, we relax this assumption and provide a closed-form expression for the Laplace transform of occupation times for surplus pro...

Inspired by Parisian barrier options in finance (see e.g. Chesney et al. (1997)), a new definition of the event "ruin" for an insurance risk model is considered. As in Dassios and Wu (2009), the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized. In this paper, we capitalize on the idea of Erlangi...

In this paper, we propose to revisit Kendall’s identity (see, e.g. Kendall (1957)) related to the distribution of the first passage time for spectrally negative Lévy processes. We provide an alternative proof to Kendall’s identity for a given class of spectrally negative Lévy processes, namely compound Poisson processes with diffusion, through the...

In this paper, we consider a fairly large class of dependent Sparre Andersen risk models where the claim sizes belong to the class of Coxian distributions. We analyze the Gerber-Shiu discounted penalty function when the penalty function depends on the deficit at ruin. We show that the system of equations needed to solve for this quantity is surpris...

A model for the number or amount of aggregate claim values on a portfolio of insurance business is analysed. The number of claims process is assumed to be a (possibly time transformed) mixed Poisson process, and the value of a claim is allowed to depend on the time of incurral as well as the end point of the observation period. The mixed Erlang ass...

The finite-time ruin problem, which implicitly involves the inversion of the Laplace transform of the time to ruin, has been a long-standing research problem in risk theory. Existing results in the Sparre Andersen risk models are mainly based on an exponential assumption either on the interclaim times or on the claim sizes. In this paper, we utiliz...

Risk theory Ruin theory Aggregate loss process Maximal aggregate loss process Ascending ladder heights Discounted sum of ascending ladder heights a b s t r a c t Within the Sparre Andersen risk model, the ruin probability corresponds to the survival function of the maximal aggregate loss. It is well known that the maximum aggregate loss follows a c...

In this paper, we consider an extension of the classical risk model in which the premium rate policy is adaptive to claim experience. We assume that the premium rate is reviewed each time the surplus reaches a new descending ladder height. A choice between a finite number mm of rates is then made depending on the time elapsed between successive lad...

In this paper, the compound Poisson risk model with surplus-dependent premium rate is analyzed in the taxation system proposed by Albrecher and Hipp (Blätter der DGVFM 28(1):13–28, 2007). In the compound Poisson risk model, Albrecher and Hipp (Blätter der DGVFM 28(1):13–28, 2007) showed that a simple relationship between the ruin probabilities in t...

In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative Lévy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative Lévy process and...

Recent research into the nature of the distribution of the time of ruin in some Sparre Andersen risk models has resulted in series expansions for the associated density function. Examples include Dickson and Willmot (2005) in the classical Poisson model with exponential interclaim times, and Borovkov and Dickson (2008), who used a duality argument...

The class of risk models with Markovian arrival process (MAP) (see e.g., Neuts[ 15 15. Neuts , M.F. A versatile Markovian point process . Journal of Applied Probability 1979 , 16 ( 4 ), 764 – 779 . [CrossRef], [Web of Science ®]View all references ]) is generalized by allowing the waiting times between two successive events (which can be a change i...

A generalization of the Gerber–Shiu function proposed by (Cheung et al., Scand. Actuarial J., in press, 2010) is used to derive some ordering properties for certain ruin-related quantities in a Sparre Andersen type risk model. Additional bounds and/or refinements can be obtained by further assuming that the claim size and the interclaim time distri...

A generalization of the usual penalty function is proposed, and a defective renewal equation is derived for the Gerber–Shiu discounted penalty function in the classical risk model. This is used to derive the trivariate distribution of the deficit at ruin, the surplus prior to ruin, and the surplus immediately following the second last claim before...

The structure of various Gerber–Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the...

In this paper, a risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber-Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber-Shiu functi...

In this paper an extension of the semi-Markovian risk model studied by Albrecher and Boxma (2005) is considered by allowing for general interclaim times. In such a model, we follow the ideas of Cheung et al. (2010b) and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty functi...

In the context of a dividend barrier strategy (see, e.g.~Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identi...

In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identi...

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that ut...

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf (1984) and Kulkarni (1...

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulk...

In this paper, we propose a generalization of the expected discounted penalty function and analyze the proposed analytic tool in the framework of the compound binomial model with a general premium rate c (c ∈ ℕ+) received per period. We derive an explicit expression for this generalized analytic tool in terms of the zeros of a matrix determinant. W...

In this paper, we study the dual risk process in ruin theory (see e.g. Cramér, H. 1955. Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes. Ab Nordiska Bokhandeln, Stockholm, Takacs, L. 1967. Combinatorial methods in the Theory of Stochastic Processes. Wiley, New York and Avanzi, B., Gerber,...

In this paper, we consider the Sparre Andersen risk model with an arbitrary interclaim time distribution and a fairly general class of distributions for the claim sizes. Via a two-step procedure which involves a combination of a probabilitic and an analytic argument, an explicit expression is derived for the Gerber–Shiu discounted penalty function,...

The risk model with interclaim-dependent claim sizes proposed by Boudreault et al. [Boudreault, M., Cossette, H., Landriault, D., Marceau, E., 2006. On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actur. J., 265-285] is studied in the presence of a constant dividend barrier. An integro-differential equation...

In this article, we consider the class of risk models with Markovian claim arrivals studied by Badescu et al. (2005) and Ramaswami (2006), among others. Under a multi-threshold dividend structure, we develop a recursive algorithm for the calculation of the moments of the discounted dividend payments before ruin. Capitalizing on the connection betwe...

We consider a class of Markovian risk models in which the insurer collects premiums at rate c1(c2) whenever the surplus level is below (above) a constant threshold level b. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin,...

We consider a class of Markovian risk models perturbed by a multiple threshold dividend strategy in which the insurer collects premiums at rate c i whenever the surplus level resides in the i-th surplus layer, i=1, 2, …,n+1 where n<∞. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the discounted j...

In this paper, we derive an explicit expression for the n-th moment of the discounted dividend payments prior to ruin, generalizing the results on the first moment in Badescu et al. (2007). Based on the connection between an insurer's surplus process and its corresponding fluid flow process, we propose a recursive algorithm to compute the higher mo...

We consider an extension to the classical compound Poisson risk model for which the increments of the aggregate claim amount process are independent. In Albrecher and Teugels (2006), an arbitrary dependence structure among the interclaim time and the subsequent claim size expressed through a copula is considered and they derived asymptotic results...

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ru...

In this paper, we propose a compound binomial model defined in a markovian environment which is an extension to the compound binomial model presented by Gerber (1988) [Mathematical fun with the compound bionomial process. ASTIN Bull. 18, 109–123; Mathematical fun with ruin theory. Ins. Math. Econ. 7, 15–23]. An algorithm is presented for the comput...

The compound Markov binomial model was first proposed by Cossette et al. [Scandinavian Actuarial Journal (2003) 301] to introduce time-dependence in the aggregate claim amount increments. As pointed out in [Scandinavian Actuarial Journal (2003) 301], this model can be seen as an extension to Gerber’s compound binomial model. In this paper, we pursu...

In this paper, we present a compound Markov binomial model which is an extension of the compound binomial model proposed by Gerber (1988a, b) and further examined by Shiu (1989) and Willmot (1993). The compound Markov binomial model is based on the Markov Bernoulli process which introduces dependency between claim occurrences. Recursive formulas ar...

We consider a class of Markovian risk models in which the insurer collects premiums at rate c 1 (c 2) whenever the surplus level is below (above) a constant barrier level b. We derive the Laplace­Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin,...