*1.1. The Model*

In what follows, we will use the following notation: the law of a Markov process *X* when starting from *X*0 = *x* will be denoted by P*<sup>x</sup>*, and the corresponding expectation by E*<sup>x</sup>*. We write P and E when *x* = 0.

To fix ideas, let us start with the Cramér–Lundberg risk model for *t* ≥ 0 (see, for example, Dufresne and Gerber 1991; Albrecher and Asmussen 2010):

$$X\_t = x + ct - S\_{t\_f} \quad \text{where } S\_t = \sum\_{i=1}^{N\_l} C\_i. \tag{1}$$

Here, *x* ≥ 0 is the initial surplus, *c* ≥ 0 is the linear premium rate, and {*Ci*, *i* = 1, 2, . . . } are independent and identically distributed random variables, with distribution function *F* and mean *m*1 = ∞0 *zF*(*dz*) representing non-negative jumps/claims. The inter-arrival times between these jumps are independent and exponentially distributed with mean 1/*λ*, and *Nt* denotes the time-*t* value of the associated Poisson process counting the arrivals of claims on the interval [0, *t*]. We will assume the positive profit condition *p* := *c* − *λ<sup>m</sup>*1 > 0.

The process given in (1) is a particular case of a spectrally negative Lévy process (SNLP), that is, a Lévy process without positive jumps, where in this case there is also a finite mean. More precisely, such a process is defined by adding a Brownian perturbation to (1), and by assuming that *St* is a subordinator with a *σ*-finite Lévy measure <sup>Π</sup>(*dx*), having possibly infinite activity near the origin, that is, Π(0, ∞) = ∞. For a SNLP, the positive profit condition becomes *p* = *c* − ∞0 *x*<sup>Π</sup>(d*x*) > 0. Note that for the SNLP given in (1), we have Π(*dx*) = *λF*(*dx*) so Π(0, ∞) = *λ*. See, for example, (Bertoin 1998) for more details.

The main result of our paper assumes that the claim sizes/jumps are exponentially distributed with mean 1/*μ*, that is, that *<sup>F</sup>*(*z*) = 1 − e<sup>−</sup>*μ<sup>z</sup>* when *z* > 0. However, as most of our intermediate results hold for a general SNLP, they will be stated in this more general context. Unfortunately, one key fact below holds only for a Cramér–Lundberg process with exponential jumps. Consequently, in the general SNLP case, the Løkka–Zervos alternative is still an open problem.

Recall that a SNLP *X* is characterized by its Laplace exponent defined by *ψ*(*θ*) = lnE e*θX*<sup>1</sup> . For the Cramér-Lundberg process *X* given in (1), we have

$$\psi(\theta) = c\theta + \int\_0^\infty \left(\mathbf{e}^{-\theta z} - 1\right) \lambda F(\mathbf{d}z)$$

and, in the case of exponential jumps, we further have

$$
\psi(\theta) = \theta \left( c - \frac{\lambda}{\mu + \theta} \right) . \tag{2}
$$
