Potentials of stable processes

For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.Comment: 10 page

The Gapeev-Kuhn stochastic game driven by a spectrally positive Levy process

In Gapeev and Kuhn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Levy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurrence of continuous and smooth fit. In Gapeev and Kuhn (2005) [8], the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game

The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process

In Gapeev and Kühn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurrence of continuous and smooth fit. In Gapeev and Kühn (2005) [8], the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game.Stochastic games Optimal stopping Pasting principles Fluctuation theory Levy processes

Predicting the time at which a Lévy process attains its ultimate supremum

We consider the problem of finding a stopping time that minimises the L 1-distance to θ, the time at which a Lévy process attains its ultimate supremum. This problem was studied in Du Toit and Peskir (Proc. Math. Control Theory Finance, pp. 95–112, 2008) for a Brownian motion with drift and a finite time horizon. We consider a general Lévy process and an infinite time horizon (only compound Poisson processes are excluded. Furthermore due to the infinite horizon the problem is interesting only when the Lévy process drifts to −∞). Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We show the following. If θ has infinite mean there exists no stopping time with a finite L 1-distance to θ, whereas if θ has finite mean it is either optimal to stop immediately or to stop when the process reflected in its supremum exceeds a positive level, depending on whether the median of the law of the ultimate supremum equals zero or is positive. Furthermore, pasting properties are derived. Finally, the result is made more explicit in terms of scale functions in the case when the Lévy process has no positive jumps