**Preface to "Exit Problems for L´evy and Markov Processes with One-Sided Jumps and Related Topics"**

Preface to Exit Problems for Levy and Markov Processes with One-Sided Jumps and Related ´ Topics It has long been known that exit problems for one-dimensional Levy 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 grea<sup>t</sup> variety of identities for exit problems of spectrally-negative Levy 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 ge<sup>t</sup> 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 Levy 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. Perez consider the optimal bail-out dividend problem ´ with fixed transaction cost for a Levy 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 Cramer–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 Levy 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 Levy processes with state-dependent coefficients. The next three papers concern similar stochastic ´ models. Note that since the essence of "W,Z" proofs is the strong Markov property applied at smooth-crossing times and variations, the results, in principle, are expected to hold for the wider class of spectrally-negative strong Markov processes.

This is established in the particular cases of certain random walks by M. Vidmar—see https://www.mdpi.com/2227-9091/6/3/102—and seems to be true for general strong Markov processes, subject to technical conditions—see the paper https://www.mdpi.com/2227-9091/7/1/18 of F. Avram, D. Grahovac, and C. Vardar-Acar.

Note, however, that computing the functions W, Z is still essentially an open problem outside the Levy and diffusion classes. One exception is the simplest Segerdahl risk model with affine drift ´ and exponential jumps—see https://www.mdpi.com/2227-9091/7/4/117 for this case, and also for a review of certain generalizations of the Segerdahl process.

The final two papers deal with problems concerning more general models:

1. H Albrecher, E Vatamidou https://www.mdpi.com/2227-9091/7/4/104 construct error bounds for the ruin probability of the Sparre Andersen risk process with interclaim times that belong to the class of distributions with rational Laplace transform. An exciting extension would be to Levy ´ perturbed Sparre Andersen risk processes.

2. Finally, K. Debicki, L. Ji and T. Rolski go multidimensional and obtain in https:// www.mdpi.com/2227-9091/7/3/83 logarithmic asymptotics (large deviations) for probability of a component-wise ruin in a two-dimensional Brownian model.

> **Florin Avram** *Editor*

*Article*
