Ruin probabilities in the discrete time renewal risk model

Cossette, Hélène; Landriault, David; Marceau, Étienne

doi:10.1016/j.insmatheco.2005.09.005

Cited by 17 publications

(19 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Results follow from random walk theory and renewal theory and as such seem to be classical ones. In the ruin theory for discrete risk process such results are presented e.g., in Landriault et al [12] (see also Willmot and Lin [51] and Rolski et al [45, p. 255-259]). However, there exist different representation of the constant C used in this results.…”

Section: Cramér's Estimate Of the Ultimate Parisian Ruin Probabilitymentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…Results for the discrete-time risk models were also used as approximations or bounds for the corresponding results in continuous time, see Cossette et al [11] and Dickson et al [20] for the approximating procedures. Other references on the related topics are: Cossette et al [10,12] [54,55].The name for the problem considered in this paper is borrowed from Parisian option, where prices are activated or canceled depending on a type of option when underlying asset stays above or below barrier long enough (see Albrecher et al [1], Chesney et al [9], Dassios and Wu [14]). We believe that Parisian ruin probability could be a better measure of risk in many situations giving possibility for insurance company to get solvency.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Discrete time ruin probability with Parisian delay

Czarna

Palmowski

Świa̧tek³

2016

Scandinavian Actuarial Journal

View full text Add to dashboard Cite

Abstract. In this paper we evaluate the probability of the discrete time Parisian ruin that occurs when surplus process stays below or at zero at least for some fixed duration of time d > 0. We identify expressions for the ruin probabilities within finite and infinite-time horizon. We also find their light and heavy-tailed asymptotics when initial reserves approach infinity. Finally, we calculate these probabilities for a few explicit examples.Keywords. Discrete time risk process ruin probability asymptotic Parisian ruin. IntroductionIn the present paper we consider the following process:where u > 0 denotes the initial reserve andY i , n = 1, 2, 3, . . . .We assume that Y i (i = 1, 2, . . . ) are i.i.d. claims and we also assume that premium rate equals to 1. We denote P(Y 1 = k) = p k for k = 0, 1, 2, . . . and we assume that µ = E(Y 1 ) < 1, hence R n → +∞ a.s. The risk process R starts from R 0 = u and later we use convention P(·|R 0 = u) = P u (·) and P 0 = P. The discrete-time model (1) is very important for actuarial practice, since many crucial quantities related to this model have a recursive nature and are readily programmable in practice; see e.g., [39,49] and references therein. One of the most important characteristics in risk theory is finite-time ruin probability defined by P u (τ 0 < t) for the ruin moment τ 0 = inf{n ∈ N : R n ≤ 0} and fixed time horizon t. Let us note here that our definition is compatible with many papers, see e.g., Gerber [25] and Dickson [17]. Other authors define the ruin moment when the reserve takes strictly negative value (see e.g., Willmot [49]). In this paper we extend this notion to so-called Parisian ruin probability, which occurs if the process R stays below or at zero at least for a fixed time period d ∈ {1, 2, . . .}. Formally, we define Parisian ruin time by: τ d = inf{n ∈ N : n − sup{s < n : R s > 0} > d, R n ≤ 0}and we consider Parisian ruin probabilities P u (τ d < t) and P u (τ d < ∞). The case d = 0 corresponds to the classical ruin problem. There are already a number of relevant results analyzing this case, e.g., Dickson and Hipp [22], Gerber and Shiu [26,27], Li and Garrido [35,36], Lin and Willmot [37,38], Shiu [46], Willmot [50]. Moreover, Li et al. in [39] presented a comprehensive review. The discrete model was first proposed by seminal paper of Gerber [25]. In this paper the ruin probability was expressed in terms of total amount S t of claims cumulated up to time t. Explicit formulas for the ruin probability were also derived by Willmot [49] (see also Cheng et al. [8]), where the author used analytical techniques, such as Lagrange's expansions of moment generating functions.Other related results concern the expected discounted penalty function (so-called Gerber-Shiu function), corresponding to the joint distribution of the surplus immediately before and at ruin moment. For example Cheng et al. in [8] considered the discounted probability of ruin:where the surplus just before ruin is x, the deficit at ruin equals y and υ is a discount factor (0 < ...

show abstract

Section: Cramér's Estimate Of the Ultimate Parisian Ruin Probabilitymentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Discrete time ruin probability with Parisian delay

Czarna

Palmowski

Świa̧tek³

2016

Scandinavian Actuarial Journal

View full text Add to dashboard Cite

show abstract

“…With these preliminaries in place, we adopt the principle of conditioning on the first claim time as in Cossette et al [10] or Drekic and Mera [4]. Measured from our initial time point (which we label as time 0), the lower limit of the time until the first claim occurs is 1, but its pmf is now conditional on the value of m. Morever, in evaluating σ(u, f , n, m), we condition on first claim times ranging from 1 up to c n,m,u, f , and on the event that the time until the first claim occurs is greater than c n,m,u, f .…”

Section: Calculation Of Finite-time Ruin Probabilitiesmentioning

confidence: 99%

Ruin Analysis of a Discrete-Time Dependent Sparre Andersen Model with External Financial Activities and Randomized Dividends

Kim

Drekic

2016

Risks

View full text Add to dashboard Cite

Abstract:We consider a discrete-time dependent Sparre Andersen risk model which incorporates multiple threshold levels characterizing an insurer's minimal capital requirement, dividend paying situations, and external financial activities. We focus on the development of a recursive computational procedure to calculate the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin associated with this model. We investigate several numerical examples and make some observations concerning the impact our threshold levels have on the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin.

show abstract

“…For the compound binomial model, however, Dickson et al (1995) developed recursive numerical procedures for calculating the joint and marginal probability distributions of the surplus immediately prior to ruin and the deficit at ruin. While recent papers by Pavlova and Willmot (2004), Li (2005aLi ( , 2005b, and Cossette et al (2006) have analyzed variants of the discrete-time Sparre Andersen model described above, the emphasis in these papers has been primarily theoretical, focussing on bounds on ruin probabilities as well as distributional properties and mathematical connections to other related renewal risk models of interest via the methodology of the Gerber-Shiu discounted penalty function (e.g., see Gerber and Shiu (1998)). …”

Section: Introductionmentioning

confidence: 99%

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

Alfa¹,

Drekic²

2007

ASTIN Bull.

View full text Add to dashboard Cite

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples. KEYWORDSSparre Andersen model, discrete time, matrix analytic methods, time of ruin, surplus immediately prior to ruin, deficit at ruin.

show abstract

Ruin probabilities in the discrete time renewal risk model

Cited by 17 publications

References 10 publications

Discrete time ruin probability with Parisian delay

Discrete time ruin probability with Parisian delay

Ruin Analysis of a Discrete-Time Dependent Sparre Andersen Model with External Financial Activities and Randomized Dividends

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

Contact Info

Product

Resources

About