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On the depletion problem for an insurance risk process: new non-ruin quantities in collective risk theory. (English) Zbl 1396.91292

The authors study the depletion problem of the reserve for Lévy insurance risk processes, i.e., how fast and how frequent drawdowns of a certain size occur, and derive expressions for the distribution of several depletion-related random variables.
Let \(X = (X_{t})_{t \geq 0}\) be a spectrally negative Lévy process starting at an initial surplus \(x \geq 0\). Consider the following notions:
\(\bullet\)
The running infinum and supremum of \(X\): \[ \underline X_{t}=\inf_{0 \leq s \leq t} X_{s},~\overline X_{t}=\sup_{0 \leq s \leq t} X_{s}. \]
\(\bullet\)
The drawdown process \(Y\): \(Y_{t} = \overline X_{t} - X_{t}\) \((t \geq 0)\).
\(\bullet\)
The first-passage time over a level \(a > 0\) of the drawdown process \(Y\): \[ \tau_{a} = \inf\{t \geq 0 : Y_{t} > a\}. \]
\(\bullet\)
The last time before \(t\) that \(X\) reaches its running supremum: \[ \overline G_{t}= \sup\{s \leq t : X_{s} \text{ or } X_{s-} = \overline X_{s}\}. \]
\(\bullet\)
The last time the reserve was at its maximum level prior to critical drawdown: \(\overline G_{\tau_{a}}\).
\(\bullet\)
The speed of depletion: \(\tau_{a} - \overline G_{\tau_{a}}\).
\(\bullet\)
The maximum reserve level attained before critical drawdown is observed: \(\overline X_{\tau_{a}}\).
\(\bullet\)
The minimum reserve level prior to critical drawdown: \(\underline X_{\tau_{a}}\).
\(\bullet\)
The largest drawdown observed before critical drawdown of size \(a\): \(Y_{\tau_{a}-}.\)
\(\bullet\)
The overshoot of the critical drawdown over level \(a\): \(Y_{\tau_{a}} - a.\)
The authors give general expressions for the probability density functions of \(Y_{\tau_{a}-}\), \(Y_{\tau_{a}} - a\), and \(\overline X_{\tau_{a}}\); establish that \(\overline G_{\tau_{a}}\) and \(\tau_{a} - \overline G_{\tau_{a}}\) are independent; and compute the bivariate Laplace transform of \(\tau_{a}\) and \(\overline G_{\tau_{a}}\). The distributions of \(Y_{\tau_{a}-}\), \(Y_{\tau_{a}} - a\), and \(\overline X_{\tau_{a}}\) are also computed conditional on \(\{\underline X_{\tau_{a}} \geq 0\}\), i.e.that ruin does not occur before the critical drawdown time. These quantities are explicitly computed in three special models: the Cramer-Lundberg model with exponential claims, the gamma risk process, and the spectrally negative stable risk process.

MSC:

91B30 Risk theory, insurance (MSC2010)
60J75 Jump processes (MSC2010)
60G51 Processes with independent increments; Lévy processes
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