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Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps).

Research & information: general --- Mathematics & science --- Lévy processes --- non-random overshoots --- skip-free random walks --- fluctuation theory --- scale functions --- capital surplus process --- dividend payment --- optimal control --- capital injection constraint --- spectrally negative Lévy processes --- reflected Lévy processes --- first passage --- drawdown process --- spectrally negative process --- dividends --- de Finetti valuation objective --- variational problem --- stochastic control --- optimal dividends --- Parisian ruin --- log-convexity --- barrier strategies --- adjustment coefficient --- logarithmic asymptotics --- quadratic programming problem --- ruin probability --- two-dimensional Brownian motion --- spectrally negative Lévy process --- general tax structure --- first crossing time --- joint Laplace transform --- potential measure --- Laplace transform --- first hitting time --- diffusion-type process --- running maximum and minimum processes --- boundary-value problem --- normal reflection --- Sparre Andersen model --- heavy tails --- completely monotone distributions --- error bounds --- hyperexponential distribution --- reflected Brownian motion --- linear diffusions --- drawdown --- Segerdahl process --- affine coefficients --- spectrally negative Markov process --- hypergeometric functions --- capital injections --- bankruptcy --- reflection and absorption --- Pollaczek–Khinchine formula --- scale function --- Padé approximations --- Laguerre series --- Tricomi–Weeks Laplace inversion