Optimal detection of homogeneous segment of observations in stochastic sequence

A Markov process is registered. At random moment $\theta$ the distribution of observed sequence changes. Using probability maximizing approach the optimal stopping rule for detecting the change is identified. Some explicit solution is obtained.Comment: 13 page

Probing the role of Nd3+ ions in the weak multiferroic character of NdMn2O5 by optical spectroscopies

Raman and infrared spectroscopies are used as local probes to study the dynamics of the Nd-O bonds in the weakly multiferroic NdMn2O5 system. The temperature dependence of selected Raman excitations reveals the splitting of the Nd-O bonds in NdMn2O5. The Nd3+ ion crystal field (CF) excitations in NdMn2O5 single crystals are studied by infrared transmission as a function of temperature, in the 1800-8000 cm-1 range, and under an applied magnetic field up to 11 T. The frequencies of all 4Ij crystal-field levels of Nd3+ are determined. We find that the degeneracy of the ground-state Kramers doublet is lifted ({\Delta}0 ~7.5 cm-1) due to the Nd3+-Mn3+ interaction in the ferroelectric phase, below TC ~ 28 K. The Nd3+ magnetic moment mNd(T) and its contribution to the magnetic susceptibility and the specific heat are evaluated from {\Delta}0(T) indicating that the Nd3+ ions are involved in the magnetic and the ferroelectric ordering observed below ~ 28 K. The Zeeman splitting of the excited crystal field levels of the Nd3+ ions at low temperature is also analyzed.Comment: This paper is accepted for publication as a Regular Article in Physical Review

When to sell Apple and the NASDAQ? Trading bubbles with a stochastic disorder model

In this paper, the authors apply a continuous time stochastic process model developed by Shiryaev and Zhutlukhin for optimal stopping of random price processes that appear to be bubbles. By a bubble we mean the rising price is largely based on the expectation of higher and higher future prices. Futures traders such as George Soros attempt to trade such markets. The idea is to exit near the peak from a starting long position. The model applies equally well on the short side, that is when to enter and exit a short position. In this paper we test the model in two technology markets. These include the price of Apple computer stock AAPL from various times in 2009-2012 after the local low of March 6, 2009; plus a market where it is known that the generally very successful bubble trader George Soros lost money by shorting the NASDAQ-100 stock index too soon in 2000. The Shiryaev-Zhitlukhin model provides good exit points in both situations that would have been profitable to speculators following the model

An Exact Formula for the Average Run Length to False Alarm of the Generalized Shiryaev-Roberts Procedure for Change-Point Detection under Exponential Observations

We derive analytically an exact closed-form formula for the standard minimax Average Run Length (ARL) to false alarm delivered by the Generalized Shiryaev-Roberts (GSR) change-point detection procedure devised to detect a shift in the baseline mean of a sequence of independent exponentially distributed observations. Specifically, the formula is found through direct solution of the respective integral (renewal) equation, and is a general result in that the GSR procedure's headstart is not restricted to a bounded range, nor is there a "ceiling" value for the detection threshold. Apart from the theoretical significance (in change-point detection, exact closed-form performance formulae are typically either difficult or impossible to get, especially for the GSR procedure), the obtained formula is also useful to a practitioner: in cases of practical interest, the formula is a function linear in both the detection threshold and the headstart, and, therefore, the ARL to false alarm of the GSR procedure can be easily computed.Comment: 9 pages; Accepted for publication in Proceedings of the 12-th German-Polish Workshop on Stochastic Models, Statistics and Their Application

Solving the dual Russian option problem by using change-of-measure arguments

We apply the change-of-measure arguments of Shepp and Shiryaev [38]to study the dual Russian option pricing problem proposed by Shepp and Shiryaev [39] as an optimal stopping problem for a one-dimensional diffusion process with reflection. We recall the solution to the associated free-boundary problem and give a solution to the resulting onedimensional optimal stopping problem by using the martingale approach of Beibel and Lerche [6] and [7]

The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature

We study a spin system on a large box with both Ising interaction and Sherrington-Kirpatrick couplings, in the presence of an external field. Our results are: (i) existence of the pressure in the limit of an infinite box. When both Ising and Sherrington-Kirpatrick temperatures are high enough, we prove that: (ii) the value of the pressure is given by a suitable replica symmetric solution, and (iii) the fluctuations of the pressure are of order of the inverse of the square of the volume with a normal distribution in the limit. In this regime, the pressure can be expressed in terms of random field Ising models

Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists

We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community.Comment: 10 pages, 2 figure