arXiv subjects

Ceren Vardar-Acar

Publications and source records attributed to Ceren Vardar-Acar.

4 recordsLinked to original sources

arXiv2021

An optimal stopping problem for spectrally negative Markov additive processes

Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$, as well as of a one-dimensional diffusion. Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle. In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated Lévy processes). Similarly to previous works in the Lévy setting, the optimal stopping boundary is characterised by a system of ordinary first-order differential equations, one for each state of the modulating component of the Markov additive process. Moreover, whereas scale functions played an important role in the previously mentioned work, we work instead with scale matrices for Markov additive processes here. We exemplify our calculations in the setting of the Shepp-Shiryaev optimal stopping problem, as well as a family of capped maximum optimal stopping problems.

math.PR↗

arXiv2019

The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends

First passage problems for spectrally negative Lévy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" $W$ (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function $Z$ (Avram, Kyprianou and Pistorius 2004). Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "$W,Z$ alphabet" for a great variety of first passage problems. We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem. One interesting use of the kit is for recognizing relationships between apparently unrelated problems -- see Lemma 3. Last but not least, it turned out recently that once the classic $W,Z$ are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) Lévy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative Lévy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with Omega killing (Li and Palmowski 2017).

math.PR↗

arXiv2019

A Generalized Correlated Random Walk Converging to Fractional Brownian Motion

We propose a new algorithm to generate a fractional Brownian motion, with a given Hurst parameter, 1/2<H<1 using the correlated Bernoulli random variables with parameter p; having a certain density. This density is constructed using the link between the correlation of multivariate Gaussian random variables and the correlation of their dichotomized binary variables and the relation between the correlation coefficient and the persistence parameter. We prove that the normalized sum of trajectories of this proposed random walk yields a Gaussian process whose scaling limit is the desired fractional Brownian motion with the given Hurst parameter, 1/2<H<1

stat.CO↗

arXiv2019

Path Decomposition of Spectrally Negative Levy Processes

Path decomposition is performed to analyze the pre-supremum, post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T as motivated by the aim of finding the joint distribution of the maximum loss and maximum gain. In addition, the joint distribution of the supremum and the infimum before an exponential time is displayed. As an application of path decomposition, the distributions of supremum of the post-infimum process and the maximum loss of the post-supremum process are obtained.

math.PR↗