18 research outputs found
Optimality of doubly reflected Lévy processes in singular control
We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Lévy process so as to minimize the total costs comprising of the running and controlling costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Lévy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results
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 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 Lévy 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 Lévy processes. Our approach shows that the results in [9] for Bessel processes can also be seen as a consequence of self-similarity
optimal prediction of the last zero of a spectrally negative L\'evy process
Given a spectrally negative L\'evy process drifting to infinity, we are
interested in finding a stopping time which minimises the distance
() with , the last time is negative. The solution is substantially
more difficult compared to the case for which it was shown in
\cite{baurdoux2018predicting} that it is optimal to stop as soon as exceeds
a constant barrier. In the case of treated here, we prove that solving
this optimal prediction problem is equivalent to solving an optimal stopping
problem in terms of a two-dimensional strong Markov process which incorporates
the length of the current excursion away from . We show that an optimal
stopping time is now given by the first time that exceeds a non-increasing
and non-negative curve depending on the length of the current excursion away
from . We also show that smooth fit holds at the boundary
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]