In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévymeasure ν. We consider the first passage time T x of(Xt, t ≥ 0) to level x > 0, and Kx := XTx - x theovershoot and Lx := x- XTx- the undershoot. We first provethat the Laplace transform of the random triple (Tx,Kx,Lx )satisfies some kind of integral equation. Second, assuming thatν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution asx → ∞, where $\widetilde{T_x}$ denotes a suitablerenormalization of T x .


This short note shows that the Lundberg exponential upper bound in the ruin problem of non-life insurance with compound Poisson claims is also valid for the Poisson shot noise delayed-claims model, and that the optimal exponent depends only on the distribution of the total claim per accident, not on the time it takes to honour the claim. This result holds under Cramer's condition.

Let {Yn | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {PM | M > 0} by PM(B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Yn}, the family {PM} satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.

Let {Yn | n = 1, 2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n = 1, 2,…). Using the techniques of large deviations theory we obtain rough exponential estimates for ruin probabilities for a general class of processes. Special attention is given to the probability that ruin occurs up to a certain time point. We also generalize the concept of the safety loading and consider its importance to ruin probabilities.