Erik J. Baurdoux

The Shepp-Shiryaev stochastic game driven by a spectrally negative Levy process.

In [A. E. Kyprianou, Finance Stoch., 8 (2004), pp. 73–86], the stochastic-game-analogue of Shepp and Shiryaevs optimal stopping problem (cf. [L. A. Shepp and A. N. Shiryaev, Ann. Appl. Probab., 3 (1993), pp. 631–640] and [L. A. Shepp and A. N. Shiryaev, Theory Probab. Appl., 39 (1994), pp. 103–119]) was considered when driven by an exponential Brownian motion. We consider the same stochastic game, which we call the Shepp–Shiryaev stochastic game, but driven by a spectrally negative Levy process and for a wider parameter range. Unlike [A. E. Kyprianou, Finance Stoch., 8 (2004), pp. 73–86], we do not appeal predominantly to stochastic analytic methods. Principally, this is due to difficulties in writing down variational inequalities of candidate solutions on account of then having to work with nonlocal integro-differential operators. We appeal instead to a mixture of techniques including fluctuation theory, stochastic analytic methods associated with martingale characterizations, and reduction of the stochastic game to an optimal stopping problem.

Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

Inspired by works of Landriault et al. [11, 12], we study the Gerber{Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Levy insurance risk process. To be more specific, we study the so-called Gerber{Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Levy processes and relies on the theory of so-called scale functions. In particular, we extend recent results of Landriault et al. [11, 12].

Examples of optimal stopping via measure transformation for processes with one-sided jumps

In this short note, we show that the method introduced by Beibel and Lerche (1997) for solving certain optimal stopping problems for Brownian motion can be applied as well to some optimal stopping problems involving processes with one-sided jumps.

Optimal double stopping of a Brownian bridge

We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of a threshold type.

Optimal prediction for positive self-similar Markov processes

This paper addresses the question of predicting when a positive self-similar Markov process XX attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in [9] under the assumption that XX is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to [3], where the same question is studied for a Levy process drifting to −∞−∞. The connection to [3] relies on the so-called Lamperti transformation [15] which links the class of positive self-similar Markov processes with that of Levy processes. Our approach shows that the results in [9] for Bessel processes can also be seen as a consequence of self-similarity.