Parisian Excursion Below a Fixed Level from the Last Record Maximum of Lévy Insurance Risk Process

Abstract

This paper presents some new results on Parisian ruin under Lévy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Lévy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Lévy process. In contrast to the Parisian ruin of Lévy process below a fixed level, ruin under drawdown occurs in finite time with probability one.AMS 2000 subject classifications. 60G40, 35R35, 60J65, 60G25, 45G10.

On the analysis of deep drawdowns for the Lévy insurance risk model

“…Relying on a temporal approximation approach (e.g., Li et al[17]), the proposed methodology allows for a unied treatment of processes with bounded and unbounded variation paths whereas these two cases used to be treated separately (e.g., Yin and Yuen [30]). In particular, we extend the results of Landriault et al [14] and Surya [28]. We later analyze certain limiting cases of our main results where consistency with some known drawdown results in the literature will be shown.…”

Draw-down Parisian ruin for spectrally negative Lévy processes

“…which recovers Theorem 2.1 of [26]. It can then be checked via a change of measure that Proposition 2.3 of [26] can also be recovered from our Theorem 4.2.…”

“…Notice that to compute the draw-down Parisian ruin probability (see Remark 4.1), we only need the expression for r := (0) r . To find the Laplace transform associated to the two-sided exit solution (26), the kth moment of dividends (see Theorem 4.4), or the potential measure of X (see Theorem 4.3) involving the draw-down Parisian ruin time, we also need the expression for (q) r . If we want to compute the probability density of the draw-down Parisian ruin time, we must first use the expression for (q) r to compute the right-hand side of (30), then numerically invert it using the algorithms in [7] or the method of Fourier series expansion proposed by [23], which has been proven to be an efficient method for the numerical inversion of Laplace transforms.…”

“…We recently noticed that the two-sided exit problem involving the draw-down Parisian ruin time was studied in [26] for the classical draw-down process using fluctuation theory; the author investigated only the classical draw-down rather than the general draw-down. We remark that the solution to the two-sided exit problem in [26] can be recovered from one of our results; see Remark 4.2 concerning the general draw-down.…”

“…For k = 1 we present an alternative, more transparent argument. Given a > b, by(26) we haveV 1 (b) = E b ∞ {L(t)<L(τ + a ∧κ ξ r )} e −qt dL(t) = ∞ b e −qL −1 (y) 1 {y<L(τ + a )=a−b, y<L(κ ξ r )} dy = a−b 0 E b e −qL −1 (y) 1 {L −1 (y)<κ ξ r }dy Application of draw-down Parisian ruin results to a spectrally negative Lévy process reflected at its past supremumRecall the dividend process D defined in(5). Let the corresponding risk process with dividends deducted according to the barrier strategy with barrier level b be defined asY(t) := X(t) − D(t), t ≥ 0.For fixed b ∈ (0, ∞), if we choose the general draw-down function ξ such that ξ (z) := ξ b (z) = (z − b) ∨ 0, z ∈ (− ∞, ∞), then we have κ ξ r := inf{t > r:t − g ξ t > r}, where g ξ t := sup{0 ≤ s ≤ t : Y(s) ≥ 0}; i.e., κ ξ r = κ ξ b r degenerates to the classical Parisian ruin time for the risk process Y.…”