Lévy systems and the time value of ruin for Markov additive processes
Let \(\{J_t\}\) be a Markov chain on \(S = \{1,2,\dotsc, N\}\). We denote by \(\{T_k\}\) the jump times of \(J\), and \(T_0 = 0\). For each \(i \in S\), let \(Y^i\) be a spectrally positive Lévy process. The processes are assumed to be independent. We construct a Markov additive process \(Y\) by \[ Y_t = \sum_{k=1}^\infty \sum_{i=1}^S \bigl[ (Y^i_{T_k\wedge t} - Y^i_{T_{k-1}\wedge t}) \operatorname{1}_{\{J_{T_{k-1}} = i\}} + \sum_{j=1}^S U_k^{i j} \operatorname{1}_{\{T_{k-1} = i, T_k = j\}} \operatorname{1}_{\{T_k \leq t\}}\bigr]. \] The variables \(\{U_k^{i j}\}\) are all independent of each other and independent of \(J\) and of the Lévy processes, and, for a fixed pair \((i,j)\), they have the same distribution. The time to ruin is defined as \(\tau_x = \inf\{t: Y_t > x\}\). The quantity of interest is the matrix valued Gerber-Shiu function \[ \phi(x;w,q) = \operatorname{E}[e^{-q \tau_x} w(x-Y_{\tau_x-}, Y_{\tau_x}-x) \operatorname{1}_{\{\tau_x < \infty\}} \operatorname{1}_{\{J_{\tau_x} = j\}}| J_0 = i], \] where \(w: [0,\infty)^2 \to \mathbb R\) is a bounded measurable function and \(q \geq 0\). Since the authors are not interested in creeping, it is assumed that \(w(\cdot, 0) = 0\). The function \(\phi\) is calculated in terms of a scale and potential measure. There are no explicit expressions for these measures, except in some special cases. A discussion of these measures is given.
Markov modulated spectrally positive Lévy process
scientific article; zbMATH DE number 6134560
scientific article