On an Effective Solution of the Optimal Stopping Problem for Random Walks

We find a solution of the optimal stopping problem for the case when a reward function is an integer function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, ? , T} converges with an exponential rate as T approaches infinity to the limit under the assumption that jumps of the random walk are exponentially bounded.optimal stopping; random walk; rate of convergence; Appell polynomials

Translational tilings by a polytope, with multiplicity

We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-tiling polytopes. Conversely, in the case that P is a rational polytope, we also prove that if P is centrally symmetric and has centrally symmetric facets, then P must k-tile R^d for some positive integer k

On a Solution of the Optimal Stopping Problem for Processes with Independent Increments

We discuss a solution of the optimal stopping problem for the case when a reward function is a power function of a process with independent stationary increments (random walks or Levy processes) on an infinite time interval. It is shown that an optimal stopping time is the first crossing time through a level defined as the largest root of the Appell function associated with the maximum of the underlying process.

Small-Angle X-ray and neutron scattering from diamond single crystals

Results of Small-Angle Scattering study of diamonds with various types of point and extended defects and different degrees of annealing are presented. It is shown that thermal annealing and/or mechanical deformation cause formation of nanosized planar and threedimensional defects giving rise to Small-Angle Scattering. The defects are often facetted by crystallographic planes 111, 100, 110, 311, 211 common for diamond. The scattering defects likely consist of clusters of intrinsic and impurity-related defects; boundaries of mechanical twins also contribute to the SAS signal. There is no clear correlation between concentration of nitrogen impurity and intensity of the scattering.Comment: 6 pages, 5 figures; presented at SANS-YuMO User Meeting 2011, Dubna, Russi

A quickest detection problem with an observation cost

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift "changes" into a Brownian motion with positive drift. The change occurs at an unknown "disorder" time with exponential distribution. There is a penalty for declaring too early that the change has occurred, and a cost for late detection proportional to the time between occurrence of the change and the time when the change is declared. Here, we consider the case where there is also a cost for observing the process. This stochastic control problem can be formulated using either the notion of strong solution or of weak solution of the s.d.e. that defines the observation process. We show that the value function is the same in both cases, even though no optimal strategy exists in the strong formulation. We determine the optimal strategy in the weak formulation and show, using a form of the "principle of smooth fit" and under natural hypotheses on the parameters of the problem, that the optimal strategy takes the form of a two-threshold policy: observe only when the posterior probability that the change has already occurred, given the observations, is larger than a threshold $A\geq0$, and declare that the disorder time has occurred when this posterior probability exceeds a threshold $B\geq A$. The constants $A$ and $B$ are determined explicitly from the parameters of the problem.Comment: Published at http://dx.doi.org/10.1214/14-AAP1028 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Esscher transform and the duality principle for multidimensional semimartingales

The duality principle in option pricing aims at simplifying valuation problems that depend on several variables by associating them to the corresponding dual option pricing problem. Here, we analyze the duality principle for options that depend on several assets. The asset price processes are driven by general semimartingales, and the dual measures are constructed via an Esscher transformation. As an application, we can relate swap and quanto options to standard call and put options. Explicit calculations for jump models are also provided.Comment: Published in at http://dx.doi.org/10.1214/09-AAP600 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Square summability of variations and convergence of the transfer operator

In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [13], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in [7]. We also prove uniqueness of so-called G-measures, introduced by Brown and Dooley [2], under square summability of variations.Comment: 8 page

Unspecified distribution in single disorder problem

We register a stochastic sequence affected by one disorder. Monitoring of the sequence is made in the circumstances when not full information about distributions before and after the change is available. The initial problem of disorder detection is transformed to optimal stopping of observed sequence. Formula for optimal decision functions is derived.Comment: 23 page