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 L\'evy 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 L\'evy processes). Similarly to previous works in the L\'evy 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.Comment: 31 page

The W,ZW,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 L\'evy 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) L\'evy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative L\'evy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with Omega killing (Li and Palmowski 2017)

Hunting for molecules in schizophrenia through omics technologies

Schizophrenia is a complex mental disorder that affects 1% of the population worldwide with ~80% heritability rate. This mental disorder has dramatic impacts not only on the patients but also on the society as well. Unfortunately our knowledge about the molecular mechanisms underlying the disease is limited. To understand the pathological mechanisms that lead to disease phenotype we need to use genomics, transcriptomics, epigenomics, proteomics, metabolomics like approaches with newly developed technologies. These approaches will also help scientist to find out new diagnostic tools that can be used as biomarkers in a complex disease like schizophrenia or personalized therapy strategies. It is possible to map the molecular changings in disease and healthy state with the help of the OMICS based technologies. This review sheds light on these OMICS based approaches to hunt the biomarkers that can be used as diagnostic tools for schizophrenia and other mental disorders or to figure out the candidate molecules for new treatment options. [Med-Science 2020; 9(1.000): 270-3

The Supremum the Infimum Maximum Gain and Maximum Loss of Brownian Motion with Drift

Related to risk and to hedging investors would be interested insupremum, infimum, maximum gain and maximum loss. Price ofstock can be modeled using Brownian motion. Here, we presentthe marginal and joint distributions of supremum, infimum, maxi-mum gain and maximum loss of Brownian motion with drift 0. Asan extension, we provide calculations of correlations for Brownianmotion with drift. We give number of results related to distributionsover various time. We present numerical studies of Brownian mo-tion with drift and collect conjectures on relation between maximumgain and maximum los