Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Abstract

Preface to Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics It has long been known that exit problems for one-dimensional Lévy processes are easier when there are jumps in one direction only. In the last few years, this intuition became more precise.We know now that a great variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two "q-harmonic functions" W and Z (or scale functions, or q-martingales). See paper 1, https://www.mdpi.com/2227-9091/7/4/121 for a brief introduction to W and two numerical methods to compute it.The reader may then get an idea of some important applications in risk theory by looking at the next six papers:1. The paper of J. F. Renaud considers the Finetti's stochastic control problem when the controlled process is allowed to spend time under the critical level (the so-called Parisian ruin). It is shown that if the tail of the Lévy measure is log-convex, the optimal strategy is of barrier type. An interesting implied question is whether this continues to be true when this assumption is not satisfied. https: //www.mdpi.com/2227-9091/7/3/73; 2. M. Junca, H.A. Moreno-Franco, and J.L. Pérez consider the optimal bail-out dividend problem with fixed transaction cost for a Lévy risk model with a constraint on the expected present value of injected capital, and establish the optimality of reflected (c1, c2)-policies. https://www.mdpi.com/ 2227-9091/7/1/13; 3. F. Avram, D. Goreac, and J.F. Renaud prove a so-called Løkka-Zervos alternative, for Cramér-Lundberg risk processes with exponential claims. This means that if the proportional cost of capital injections is low, then it is optimal to pay dividends and inject capital according to a double-barrier strategy, meaning that ruin never occurs; and if the cost of capital injections is high, then it is optimal to pay dividends according to a single-barrier strategy and never inject capital.Note, however, that this paper only addresses de Finetti and Shreve -Lehoczky-Gaver policies. The non-restricted stochastic control problem has been solved only recently, and, again, only with exponential claims. https://www.mdpi.com/2227-9091/7/4/120; 4. WenyuanWang and Xiaowen Zhou provide an in-depth study of spectrally negative Lévy risk process with general tax structure https://www.mdpi.com/2227-9091/7/3/85; 5. Eberhard Mayerhofer's paper https://www.mdpi.com/2227-9091/7/4/105 provides self-contained proofs concerning processes stopped at draw-down times; 6. P.V. Gapeev, N. Rodosthenous, and V.L. Chinthalapati obtain in https://www.mdpi.com/ 2227-9091/7/3/87 closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion process stopped at the first time at which the associated drawdown or drawup process hits a constant level. This paper studies this problem for Lévy processes with state-dependent coefficients. The next three papers concern similar stochastic models. Note that since the essence of "W,Z" pr...