We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005) and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent work in the actuarial literature concerning calculations for the n-th moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than existing literature in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér-Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and for the case of the n-th moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

Abstract We consider a self-similar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. We show that under certain conditions the typical size in the ensemble is of the order t −1/α and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law. 1

Abstract: We consider a stochastic process (Xt)t≥0 that grows linearly in time and experiences collapses at times governed by a Poisson process with rate λ. The collapses are modeled by multiplying the process level by a random variable supported on [0, 1). For the hitting time defined as τy = inf{t> 0|Xt = y} we derive power series for the Laplace transform and all moments. We further discuss the asymptotic behavior of the mean of τy as y tends to infinity. 1.

We consider a diffusion process X in a random Lévy potential V which is a solution of the informal stochastic differential equation dXt = dβt −

Abstract. We find a simple expression for the probability density of R exp(Bs − s/2)ds in terms of its distribution function and the distribution function for the time integral of exp(Bs + s/2). The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, E [ exp ` θ / R exp(Bs)ds ´ ] is finite iff θ < 2. We present a new formula for the price of an Asian call option.

Abstract. The representation of continuous-state branching processes (CSBPs) as time-changed Lévy processes with no negative jumps was discovered by John Lamperti in 1967 but was never proved. The goal of this paper is to provide a proof, and we actually provide two. The first one relies on studying the timechange, using martingales and the Lévy-Itô representation of Lévy processes. It gives insight into a stochastic differential equation satisfied by CSBPs and on its relevance to the branching property. The other method studies the time-change in a discrete model, where an analogous Lamperti representation is evident, and provides functional approximations to Lamperti transforms by introducing a new topology on Skorohod space. Some classical arguments used to study CSBPs are reconsidered and simplified. 1.

We give necessary and su#cient conditions for the law of a positive self-similar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive self-similar process to a unique Levy process. Then we show that the convergence mentioned above holds if and only if the process of the overshoots of the underlying Levy process # in the Lamperti's representation converges weakly at infinity and E T1 where T 1 = inf{t : # t 1}. Under these conditions, we give a pathwise construction of the limit law. Key words: Self-similar process, Levy process, Lamperti's representation, overshoot, weak convergence, first passage time.

Compounding processes, also known as perpetuities, play an important role in many applications; in particular, in time series analysis and mathematical finance. Apart from some special cases, the distribution of a perpetuity is hard to compute, and large deviations estimates sometimes involve complicated constants which depend on the complete distribution. Motivated by this, we propose provably efficient importance sampling algorithms which apply to qualitatively different cases, leading to light and heavy tails. Both algorithms have the non-standard feature of being statedependent. In addition, in order to verify the efficiency, we apply recently developed techniques based on Lyapunov inequalities. 1

We consider a self-similar fragmentation process in which the generic particle of mass x is replaced by the offspring particles at probability rate x , with positive parameter #. The total of offspring masses may be both larger or smaller than x with positive probability. We show that under certain conditions the typical mass in the ensemble is of the order t -1/# and that the empirical distribution of masses converges to a random limit which we characterise in terms of the reproduction law.

Abstract: This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti’s 1967 representation of CSBPs in terms of spectrally positive Lévy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.