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

Microphase separation in thin block copolymer films: a weak segregation mean-field approach

In this paper we consider thin films of AB block copolymer melts confined between two parallel plates. The plates are identical and may have a preference for one of the monomer types over the other. The system is characterized by four parameters: the Flory-Huggins chi-parameter, the fraction f of A-monomers in the block copolymer molecules, the film thickness d, and a parameter h quantifying the preference of the plates for the monomers of type A. In certain regions of parameter space, the film will be microphase separated. Various structures have been observed experimentally, each of them characterized by a certain symmetry, orientation, and periodicity. We study the system theoretically using the weak segregation approximation to mean field theory. We restrict our analysis to the region of the parameter space where the film thickness d is close to a small multiple of the natural periodicity. We will present our results in the form of phase diagrams in which the absolute value of the deviation of the film thickness from a multiple of the bulk periodicity is placed along the horizontal axis, and the chi-parameter is placed along the vertical axis; both axes are rescaled with a factor which depends on the A-monomer fraction f. We present a series of such phase diagrams for increasing values of the surface affinity for the A-monomers. We find that if the film thickness is almost commensurate with the bulk periodicity, parallel orientations of the structures are favoured over perpendicular orientations. We also predict that on increasing the surface affinity, the region of stability of the bcc phase shrinks.Comment: 35 pages, 20 figure

An experimental study of the effects of autocratic, student-centered and laissez-faire group leadership on college orientation classes

Thesis (Ed.M.)--Boston UniversityThe purpose of this study is to attempt to determine the effect of different types of leadership for freshman orientation groups at Boston University School of Educatio