Discrete time ruin probability with Parisian delay

Abstract

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 < ...

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