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A quintuple law for Markov additive processes with phase-type jumps. (English) Zbl 1205.60095
Let \(\{J_t\}\) be an irreducible Markov process on a finite state space. On the set \(\{J_t = i\}\), the process \(\{X_t\}\) behaves like a Lévy process with drift \(\mu_i\), diffusion parameter \(\sigma_i^2\) and a jump part modelled as a compound Poisson process with two-sided phase-type distributed jumps, with characteristics dependent on \(i\). At the times of a jump of \(\{J_t\}\), a phase type distributed jump may occur, with characteristics dependent on \(J_{t-}\) and \(J_t\).
The process starts in \(0\). Let \(\tau(u) = \inf\{t: X_t > u\}\) denote the time of the first passage of the level \(u\), \(M(u)\) be the maximal value of \(\{X_t\}\) before \(\tau(u)\), \(G(u)\) be the time at which the maximum is attained, \(U(u) = u-X_{\tau(u)-}\) the difference to the level prior to the passage over \(u\) and \(O(u) = X_{\tau(u)} -u\) the overshoot. The goal is to find the quantity \[ E[\exp\{ -\gamma G(u) - \gamma^* (\tau(u) - G(u))\}; M(u) \in d z, U(u) \in d x, O(u) \in d y, J_{\tau(u)} = j \mid J_0 = i]\;. \] The approach is based on a formula for the first passage time \[ E[\exp\{ - \gamma \tau(x)\}; J_{\tau(x)} = j \mid J_0 = i]\;. \] In order to obtain an explicit expression, some matrix equation has to be solved. The path of interest is split into three parts: the path to the maximum, the path from the maximum to \(u - U(u)\), and the jump over the threshold. If the maximum before \(\tau\) is \(z\), the level \(z\) has to be reached (for the first time) at time \(G(u)\). By time reversal, the path from \(M(u)\) to \(u - x\) can be seen as a first passage over the level \(z - u + x\). Finally, a jump of size \(x + y\) is necessary.
For the proof, some additional states are added for the jumps. At the jump times, the paths increase or decrease at rate \(1\). This yields in addition the state of the generator of the phase-type distribution when the level is crossed. By the Markov property, the overshoot distribution is obtained. This gives then \[ E[\exp\{ - \gamma \tau(x)\}; J_{\tau(x)} = j, O(u) \in d y \mid J_0 = i]\;. \] Putting the expressions for the three parts of the process together, gives the desired formula.

MSC:
60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
91B30 Risk theory, insurance (MSC2010)
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