Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is the (absolute) drawdown Δ of the surplus X, defined as the distance to its running maximum X¯. Large, long-lasting drawdowns are unfavourable for the insurance company. We consider the stochastic optimisation problem of minimising the expected time that the drawdown is larger than a positive critical value (weighted by a discounting factor) under investment. A fixed-point argument is used to show that the value function is the unique solution to the Hamilton–Jacobi–Bellman equation related to the problem. It turns out that the optimal investment strategy is given by a piecewise monotone and continuously differentiable function of the current drawdown. Several numerical examples illustrate our findings.

Comparative Study on the Effect of Diverse Substrates on the Cultivation of Pleurotus ostreatus (Jacq.) P. Kumm. (Oyster Mushroom)

Introduction: Pleurotus ostreatus (Oyster mushroom) is an eatable mushroom with an exceptional aroma and savour. It is suited to be cultivated in temperate as well as tropical climate. Aim: This research was aimed at comparing the effects of different substrates on the cultivation of P. ostreatus. Method and Materials: The substrates used in this study were cornhusk, sawdust with wood shavings, banana leaves and a combination of all the substrates. All substrates bags were inoculated with 52.5 mL teaspoon of spawn, autoclaved at 1210C and 1.1 kg/cm2 pressure and incubated under appropriate conditions both at the cropping room and fruiting chambers. The linear mycelia growth and biological efficiency were determined. Result: The time for spawn running varied between 20-60 days and time for harvesting took between 60-85 days. The maximum linear mycelia growth after spawn running, were observed on cornhusk and the longest mycelia growth time was observed on sawdust which took 60 days after spawn running. Maximum yield of P. ostreatus was obtained in cornhusk treatments with an average weight value of 92.1 g. The minimum yield observed on sawdust was 22 g and combination of all was 23 g while the banana leaves treatment gave no yield at all. The biological efficiency obtained on cornhusk was 12.43% and the lowest 2.2% was obtained on sawdust. Conclusion: Among all the substrates, cornhusk was established as the most effective substrate for the production of P. ostreatus as it best supported both the spawn running phase and the yield phase.

On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

We obtain closed-form expressions for the value of the joint Laplace transform of therunning maximum and minimum of a diffusion-type process stopped at the first time at which theassociated drawdown or drawup process hits a constant level before an independent exponentialrandom time. It is assumed that the coefficients of the diffusion-type process are regular functionsof the current values of its running maximum and minimum. The proof is based on the solution tothe equivalent inhomogeneous ordinary differential boundary-value problem and the applicationof the normal-reflection conditions for the value function at the edges of the state space of theresulting three-dimensional Markov process. The result is related to the computation of probabilitycharacteristics of the take-profit and stop-loss values of a market trader during a given time period.

Exit problems for general draw-down times of spectrally negative Lévy processes

For spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

Augmented Interval List: a novel data structure for efficient genomic interval search

Motivation Genomic data is frequently stored as segments or intervals. Because this data type is so common, interval-based comparisons are fundamental to genomic analysis. As the volume of available genomic data grows, developing efficient and scalable methods for searching interval data is necessary. Results We present a new data structure, the Augmented Interval List (AIList), to enumerate intersections between a query interval q and an interval set R. An AIList is constructed by first sorting R as a list by the interval start coordinate, then decomposing it into a few approximately flattened components (sublists), and then augmenting each sublist with the running maximum interval end. The query time for AIList is O(log2N+n+m), where n is the number of overlaps between R and q, N is the number of intervals in the set R and m is the average number of extra comparisons required to find the n overlaps. Tested on real genomic interval datasets, AIList code runs 5–18 times faster than standard high-performance code based on augmented interval-trees, nested containment lists or R-trees (BEDTools). For large datasets, the memory-usage for AIList is 4–60% of other methods. The AIList data structure, therefore, provides a significantly improved fundamental operation for highly scalable genomic data analysis. Availability and implementation An implementation of the AIList data structure with both construction and search algorithms is available at http://ailist.databio.org. Supplementary information Supplementary data are available at Bioinformatics online.

Augmented Interval List: a novel data structure for efficient genomic interval search

MotivationGenomic data is frequently stored as segments or intervals. Because this data type is so common, interval-based comparisons are fundamental to genomic analysis. As the volume of available genomic data grows, developing efficient and scalable methods for searching interval data is necessary.ResultsWe present a new data structure, the augmented interval list (AIList), to enumerate intersections between a query interval q and an interval set R. An AIList is constructed by first sorting R as a list by the interval start coordinate, then decomposing it into a few approximately flattened components (sublists), and then augmenting each sublist with the running maximum interval end. The query time for AIList is O(log2N + n + m), where n is the number of overlaps between R and q, N is the number of intervals in the set R, and m is the average number of extra comparisons required to find the n overlaps. Tested on real genomic interval datasets, AIList code runs 5 - 18 times faster than standard high-performance code based on augmented interval-trees (AITree), nested containment lists (NCList), or R-trees (BEDTools). For large datasets, the memory-usage for AIList is 4% - 60% of other methods. The AIList data structure, therefore, provides a significantly improved fundamental operation for highly scalable genomic data analysis.AvailabilityAn implementation of the AIList data structure with both construction and search algorithms is available at code.databio.org/AIList.