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 < ...

In this paper we consider a general Lévy process X reflected at a downward periodic barrier A t and a constant upper barrier K, giving a processWe find the expression for a loss rate defined by l K = E L K 1 and identify its asymptotics as K → ∞ when X has light-tailed jumps and E X 1 < 0.

In this paper we consider a general Lévy process X reflected at a downward periodic barrier A t and a constant upper barrier K, giving a processWe find the expression for a loss rate defined by l K = E L K 1 and identify its asymptotics as K → ∞ when X has light-tailed jumps and E X 1 < 0.

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A NOTE ON FIRST PASSAGE PROBABILITIES OF A LÉVY PROCESS REFLECTED AT A GENERAL BARRIERIn this paper we analyze a Lévy process reflected at a general possibly random barrier. For this process we prove the Central Limit Theorem for the first passage time. We also give the finite-time first passage probability asymptotics.