Ruin Problem Research Articles

Ruin problem of a two-dimensional fractional Brownian motion risk process

ABSTRACTThis paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.

On the analysis of a random walk-jump chain with tree-based transitions and its applications to faulty dichotomous search

ABSTRACTRandom walks (RWs) have been extensively studied for more than a century (Feller, 1968). These walks have traditionally been on a line, and the generalizations for two and three dimensions have been by extending the random steps to the corresponding neighboring positions in one or many of the dimensions. Among the most popular RWs on a line are the various models for birth and death processes, renewal processes, and the gambler’s ruin problem. All of these RWs operate on a discretized line, and the walk is achieved by performing small steps to the current state’s neighbor states. Indeed, it is this neighbor-step motion that renders their analyses tractable. When some of the transitions are to non-neighbour states, a formal analysis is typically impossible because the difference equations of the steady-state probabilities are not solvable. One endeavor on such an analysis is found in Yazidi et al. (2011). The problem is far more complex when the transitions of the walk follow an underlying tree-like structure. The analysis of RWs on a treehave received little attention, even though it is an important topic since a tree is a counterpart space representation of a line whenever there is some ordering on the nodes on the line. Nevertheless, RWs on a tree entail moving to non-neighborstates in the space, which makes the analysis involved and, in many cases, impossible. In this article, we consider the analysis of one such fascinating RW. We demonstrate that an analysis of the chain is feasible because we can invoke the phenomenon time reversibility. Apart from the analysis being interesting in itself from an analytical perspective, the RW on the tree that this article models is a type of generalization of dichotomous search with faulty feedback about the direction of the search, rendering the real-life application of the model to be pertinent. To resolve this, we advocate the concept of “backtracking” transitions in order to efficiently explore the search space. Interestingly, it is precisely these backtracking transitions that naturally render the chain to be time reversible. By doing this, we are able to bridge the gap between deterministic dichotomous search and its faulty version. The article contains the analysis of the chain, reports some fascinating limiting properties, and also includes simulations that justify the analytic steady-state results.

On the Parisian ruin of the dual Lévy risk model

AbstractIn this paper we investigate the Parisian ruin problem of the general dual Lévy risk model. Unlike the usual concept of ultimate ruin, allowing the surplus level to be negative within a prespecified period indicates that the deficit at Parisian ruin is not necessarily equal to zero. Hence, we consider a Gerber–Shiu type expected discounted penalty function at the Parisian ruin and obtain an explicit expression for this function under the dual Lévy risk model. As particular cases, we calculate the Parisian ruin probability and the expected discountedkth moments of the deficit at the Parisian ruin for the compound Poisson dual risk model and a drift-diffusion model. Numerical examples are given to illustrate the behavior of Parisian ruin and the expected discounted deficit at Parisian ruin.

Dependent risk models with Archimedean copulas: A computational strategy based on common mixtures and applications

In this paper, we investigate dependent risk models in which the dependence structure is defined by an Archimedean copula. Using such a structure with specific marginals, we derive explicit expressions for the pdf of the aggregated risk and other related quantities. The common mixture representation of Archimedean copulas is at the basis of a computational strategy proposed to find exact or approximated values of the distribution of the sum of risks in a general setup. Such results are then used to investigate risk models in regard to aggregation, capital allocation and ruin problems. An extension to nested Archimedean copulas is also discussed.

Duality in ruin problems for ordered risk models

On one hand, an ordered dual risk model is considered where the profit arrivals are governed by an order statistic point process (OSPP). First, the ruin time distribution is obtained in terms of Abel–Gontcharov polynomials. Then, by duality, the ruin time distribution is deduced for an insurance model where the claim amounts correspond to the inter-arrival times in an OSPP. On the other hand, an ordered insurance model is considered with an OSPP as claim arrival process. Lefèvre and Picard (2011) determined the finite-time ruin probability in terms of Appell polynomials. Duality is used to derive the ruin probability in a dual model where the profit sizes correspond to the inter-arrival times of an OSPP.

Discounted penalty function at Parisian ruin for Lévy insurance risk process

In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b), which generalizes known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q-scale function and the distribution of the Lévy process.

Archimedean Copulas: Aggregation, Capital Allocation and Other Applications

In this paper, we investigate dependent risk models in which the dependence structure is defined by an Archimedean copula. Using such a structure with specific marginals, we derive explicit expressions for the pdf of the aggregated risk and other related quantities. The common mixture representation of Archimedean copulas is at the basis of a computational strategy proposed to find exact or approximated values of the distribution of the sum of risks in a general setup. Such results are then used to investigate risk models in regard to aggregation, capital allocation and ruin problems. An extension to nested Archimedean copulas is also discussed.

SIEGMUND DUALITY FOR MARKOV CHAINS ON PARTIALLY ORDERED STATE SPACES

For a Markov chain on a finite partially ordered state space, we show that its Siegmund dual exists if and only if the chain is Möbius monotone. This is an extension of Siegmund's result for totally ordered state spaces, in which case the existence of the dual is equivalent to the usual stochastic monotonicity. Exploiting the relation between the stationary distribution of an ergodic chain and the absorption probabilities of its Siegmund dual, we present three applications: calculating the absorption probabilities of a chain with two absorbing states knowing the stationary distribution of the other chain; calculating the stationary distribution of an ergodic chain knowing the absorption probabilities of the other chain; and providing a stable simulation scheme for the stationary distribution of a chain provided we can simulate its Siegmund dual. These are accompanied by concrete examples: the gambler's ruin problem with arbitrary winning/losing probabilities; a non-symmetric game; an extension of a birth and death chain; a chain corresponding to the Fisher–Wright model; a non-standard tandem network of two servers, and the Ising model on a circle. We also show that one can construct a strong stationary dual chain by taking the appropriate Doob transform of the Siegmund dual of the time-reversed chain.

Variance of the asymmetric n-player gambler’s ruin problem with ties allowed

ABSTRACTIn this article, the variance of the duration of play in the asymmetric n-player gambler’s ruin problem is considered, when the players use equal initial fortunes of d dollars, 1 ⩽ d ⩽ n + 1, and ties allowed in each round. Some special games are simulated and the simulation results verify the validity of the proposed formulas. It is shown that when we do not have the possibility of tie in the game, the increase in the number of players will change the ruin time from a random variable to a degenerate random variable. Finally, the three-tower problem with one of its different definitions are introduced and their expected times as well as their variances of the duration are considered.

Ruin problems in Markov-modulated risk models

AbstractChen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

Dividend barrier and ruin problems for a risk model with delayed claims

ABSTRACTIn this paper, a compound Poisson risk model in the presence of a constant dividend barrier is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and and the time of delay for the claim is assumed to be random. A system of integro-differential equations with certain boundary conditions for the expected discounted penalty function is derived. We show that its solution can be expressed as the solution to the expected discounted penalty function in the same risk model with the absence of a barrier plus a linear combination of two linearly independent solutions to the associated homogeneous integro-differential equation. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the sum of discounted dividend payments until ruin, a matrix version of the dividends–penalty type relationship is derived. We also prove that ruin is certain under constant dividend barrier strategy. The closed form expressions are given when the claim amounts from both classes are exponentially distributed. Finally, a numerical example is presented to illustrate the solution procedure.

A note on Lévy risk model with two-sided phase-type jumps

In this paper we establish some results on a phase-type Lévy risk model with two-sided jumps and a barrier dividend strategy. Following [4] we describe the connection between the ruin problem of the risk model with barrier dividend strategy and the first passage problem of the risk model reflected at its running supremum. Then we give some results for the joint Laplace transform of the upward entrance time and the overshoot for the phase-type jump diffusion reflected at its running supremum. Finally we find some expressions for the Laplace transform of the time of ruin and the expected discounted dividends up to ruin. All our results on the ruin problem are expressed in terms of the solutions to the Cramér-Lundberg equation corresponding to the underlying phase-type jump diffusion.

Closed-Form Absolute Ruin Problems of the Risk Models with State-Dependent Switched Claims

This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the differential-integral equations for the Gerber-Shiu discounted penalty function, and applying the hypergeometric functions, the closed-form absolute ruin probability is derived.

Gambler’s ruin problem on Erdős–Rényi graphs

Abstract A multiagent ruin-game is studied on Erdős–Renyi type graphs. Initially the players have the same wealth. At each time step a monopolist game is played on all active links (links that connect nodes with nonzero wealth). In such a game each player puts a unit wealth in the pot and the pot is won with equal probability by one of the players. The game ends when there are no connected players such that both of them have non-zero wealth. In order to characterize the final state for dense graphs a compact formula is given for the expected number of the remaining players with non-zero wealth and the wealth distribution among these players. Theoretical predictions are given for the expected duration of the ruin game. The dynamics of the number of active players is also investigated. Validity of the theoretical predictions is investigated by Monte Carlo experiments.

A note on ruin problems in perturbed classical risk models

In this short note, we derive explicit formulas for the joint densities of the time to ruin and the number of claims until ruin in perturbed classical risk models, by constructing several auxiliary random processes.

Generalized Gambler\u2019s Ruin Problem: Explicit Formulas via Siegmund Duality

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler’s ruin problem. The generalization is best interpreted as a game of one player against d other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of Möbius monotonicity.

Linear algebra-based solution of the gambler’s ruin problem

ABSTRACTIn this paper, we derive the result of the classical gambler’s ruin problem using elementary linear algebra. Moreover, the pedagogical advantage of the derivation is briefly discussed.

A multidimensional ruin problem and an associated notion of duality

ABSTRACTWe consider a dimensional general Cramer–Lundberg type network, with initial capital operating under a risk diversifying treaty; this is described in terms of the appropriate Skorokhod problem (SP) in determined by a reflection matrix R. Ruin of the insurance network is defined as the marginal deficit of each company being positive (and hence surplus of each company being 0) at the same time. The corresponding ruin problem is studied using the associated regulated random walk {(Y(a)n, Z(a)n)} with {Z(a)n} denoting the reflected part, and {Y(a)n} the pushing part. A dual process is introduced through time reversal at sample path level over finite time horizon; the stochastic analogue is again a regulated random walk in starting at 0; let {Wn} denote the reflected part of the dual regulated random walk. It is shown that ruin for the insurance network corresponds to {Wn} hitting the open upper orthant with vertex at R− 1a before hitting the boundary of even at the sample path level. Under natural hypotheses, we show that the ruin probability is for some n ⩾ 1) boundary hitting time of storage process) A notion of dimensional ladder height distribution is defined, and a Pollaczek–Khinchine formula is derived. An expression for the ladder height distribution is given. (One-dimensional ladder height distributions are known for about 50 years.) Some examples are presented.

On joint ruin probability for a bidimensional Lévy-driven risk model with stochastic returns and heavy-tailed claims

In this paper we study the joint ruin problem for two insurance companies that divide between them the losses in positive proportions δ1 and δ2 (modeling an insurance and a re-insurance company). Assume that the surplus process of i-th (i=1,2) company Ui has the form of dUi(t)=Ui(t−)dRi(t)−δidP(t), t>0, with Ui(0)=xi>0, and P and Ri two Lévy processes representing, respectively, a loss process and a stochastic return process. Supposing that the loss process P has a Lévy measure of consistent variation and the Laplace exponent of Ri(i=1,2) satisfies some conditions, an asymptotic estimate for this joint ruin problem is established.

Ruin problems in the generalized Erlang(n) risk model

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (Insur Math Econ 37:324–334, 2005), by considering a general claim amount distribution.