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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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