Occupation densities in solving exit problems for Markov additive processes and their reflections (Q444361)

A process \((X,J)\) is called a Markov additive process if \(J\) is a finite-state continuous-time Markov chain, \(X\) evolves as a Lévy process \(X_i\) while \(J=i\), and a transition of \(J\) from state \(i\) to state \(j \neq i\) triggers a jump of \(X\) distributed as \(U_{i,j}\). Moreover, these components are independent. One can say that \(J\) is a background process representing a random Markovian environment for the additive component \(X\). The above process is called spectrally negative if \(X\) has no positive jumps and each \(U_{i,j} \leq 0\), but the paths of \(X\) are not a.s. non-increasing.NEWLINENEWLINEThe first passage-time of \(X\) over the level \(\pm x\) \((x \geq 0)\) is NEWLINE\[NEWLINE \tau_x^{\pm} := \inf\{t \geq 0 : \pm X(t) > x\} . NEWLINE\]NEWLINE A main result of the paper is that NEWLINE\[NEWLINE \operatorname{P}[\tau_a^+ < \tau_b^-, J(\tau_a^+) ] =W(b) W(a+b)^{-1} \qquad (a,b \geq 0,\;a+b > 0), NEWLINE\]NEWLINE where \(W(x)\) is the scale matrix of the process, a generalization of the scale function of a spectrally negative Lévy process. (Here, \(\operatorname{P}[A, J(\tau)]\) denotes a matrix with \((i,j)\)-th element \(\operatorname{P}_{J(0)=i}[A, J(\tau)=j]\).) In addition, a new explicit formula is given for the scale matrix, i.e., NEWLINE\[NEWLINE W(x) = e^{-\Lambda x} \mathbf{L}(x), NEWLINE\]NEWLINE where \(\Lambda\) is the transition rate matrix of the Markov chain \(J(\tau_x^+)\): \(e^{\Lambda x} = \mathrm{L\'evy}{P}[J(\tau_x^+)]\) and \(\mathbf{L}(x)\) is the matrix of expected local times at \(0\) up to the first passage-time \(\tau_x^+\).NEWLINENEWLINERelated results are given for reflected versions of the process as well.