Ronnie Loeffen

On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes

We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433-443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156-180] studied the case when the risk process is modeled by a general spectrally negative Levy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.

Parisian ruin probability for spectrally negative Lévy processes

In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally negative Levy process and the distribution of the process at time r.

Refracted Lévy processes

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Levy processes. The latter is a Levy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Levy process is described by the unique strong solution to the stochastic differential equation dU(t) = -delta 1({Ut > b})dt + dX(t), where X = {X-t: t >= 0) is a Levy process with law P and b, delta is an element of R such that the resulting process U may visit the half line (b, infinity) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Levy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

Semi-closed form cubature and applications to financial diffusion models

Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ., 2004, 6, 69–83] and Lyons–Victoir [Proc. R. Soc. Lond. Ser. A, 2004, 460, 169–198], involve the solution to numerous auxiliary ordinary differential equations (ODEs). With focus on the Ninomiya–Victoir algorithm [Appl. Math. Finance, 2008, 15, 107–121], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and show the structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies by introducing a (model-dependent) variation of the Ninomiya–Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.

Option Pricing in a One-Dimensional Affine Term Structure Model via Spectral Representations

Under a mild condition on the branching mechanism, we provide an eigenvalue expansion for the pricing semigroup in a one-dimensional positive affine term structure model. This representation, which...