Hitting-Time Densities of a Two-Dimensional Markov Process

In this paper, we discuss almost sure asymptotic stability for deterministic systems, which are randomized by adding one-dimensional standard Wiener processes. First, we clarify the difference between the deterministic Lyapunov stability theory and the stochastic stability theory proposed by Hasminskii, and show that local asymptotic stability in probability causes trouble in randomization problems. Second, we describe the Lyapunov stability theory based on almost sure stability, as proposed by Bardi and Cesaroni, and show that the stability is “almost the same” as the deterministic Lyapunov stability. Third, we survey randomization problems briefly, and show that Stratonovich integrals are valid for such problems based on one-dimensional Wiener processes. Finally, we derive the necessary conditions for stochastic Lyapunov functions to ensure almost sure stabilities and discuss the difference between Hasminskii's stabilities and those of Bardi and Cesaroni via linear stochastic systems with numerical simulations.

The study of the First Passage Time (FPT) problem (also known as first passage problem, FPP) started more than a century ago, but its diverse applications in science and engineering mostly emerged in the last two to three decades. Assuming that X(t) is a one-dimensional stochastic process, the First Passage Time is defined as the time (T) when X(t) first crosses a threshold. Engineering reliability is obviously a suitable application domain, and indeed applications such as optimal dam design in hydrology and analysis of structural failure in civil and mechanical engineering are typical examples. Although we envision that the FPT problem has great potential in network and software reliability, it should be more useful for network security and survivability because the approaches developed for the FPT problem are mostly analytical. The assumption for this inference is that in reliability analysis, experimental or historical data are often more readily available, which makes statistical approaches such as survival analysis more convenient and likely more realistic. In contrast, data is generally more difficult to obtain in security and survivability analyses, and analytical approaches can be leveraged to play more important roles. Furthermore, security and survivability often have to deal with malicious actions that may be driven by sophisticated cognition and behavioral processes, which are highly variable over time and very difficult to detect with short term data. If the behavior of an intruder can be characterized with some stochastic process such as Brownian motion, then the FPT approach may be applied to find the closed-form solution of the probability density function (PDF) of the first passage time, which can be the time when the system breaks down or when the hacker is successful in compromising a network. In addition, the solutions to FPT depend on boundary and initial conditions of the corresponding partial differential equations, and they also describe the evolution of PDF over time. This may suggest that it is possible to model the behavior changes of an intruder over time and circumstances. Another advantage of FPT analysis is that it may help solve some non-Markov stochastic process problems in reliability analysis and survival analysis. In this article, we first briefly introduce the FPT problem with Brownian motion as an example, and then suggest its potential applications in software reliability and network security.

Let dx(t) = y(t) dt, where y(t) is a one-dimensional Wiener process. In this note, we obtain a formula for the moment-generating function of y(T), where T is the 1/2-winding time about the origin of the integrated Wiener process x(t).

Motivated by the dynamics of resonant neurons we discuss the properties of the first passage time (FPT) densities for non-Markovian differentiable random processes. We start from an exact expression for the FPT density in terms of an infinite series of integrals over joint densities of level crossings, and consider different approximations based on truncation or on approximate summation of this series. Thus the first few terms of the series give good approximations for the FPT density on short times. For rapidly decaying correlations the decoupling approximations perform well in the whole time domain. As an example we consider resonate-and-fire neurons representing stochastic underdamped or moderately damped harmonic oscillators driven by white Gaussian or by Ornstein-Uhlenbeck noise. We show that approximations reproduce all qualitatively different structures of the FPT densities: from monomodal to multimodal densities with decaying peaks. The approximations work for the systems of whatever dimension and are especially effective for the processes with narrow spectral density, exactly when Markovian approximations fail.

Theoretical considerations and analysis of survey data suggest that live loads on floors of buildings can be represented by a two-dimensional stochastic process with independent increments, the prototype of which is the compound poisson process. An essential assumption, which is vindicated by analysis of the data, is the statistical independence of the loads on nonoverlapping areas, provided the areas are not too small. This is closely analysed. Rational probabilistic design is based on the upper tail of the load probability distribution, and it is shown that the tail can be well approximated by the Pareto formula Cx -α . The fit to survey data obtained is compared with that of the alternatives of a normal and an exponential tail. From the representation of the tail distribution, rational load-reduction formulas are derived. A further consideration which has been so far neglected is the necessity to assure that none of the panels into which the floor area is divided collapse, and this leads to an increase of the design load with increasing floor area.

The value of gas-fired power plants in markets with high shares of renewable energy: A real options application

In various astronomical applications, the impracticability of describing the phenomena under study with a relatively simple model leads to the failure of any data fitting adjustment. Our inability to relate the observable quantities to the unknown physical parameters means that the residuals appear larger than expected and that they are strongly correlated. A powerful tool to attack this problem is the method of least-squares collocation, borrowed from geodesy, combined with a technique for the estimation of the covariance function of the residuals

This paper considers algorithms that compute fixed points (or more generally solve equations) which are based on complementary pivoting. These algorithms, starting with a map f0 and its fixed point x0, deform ft to f∞ = f as t goes from 0 to ∞, and follow the path xt of fixed points of ft. In this paper we study these paths. In particular, we treat some special applications of these algorithms from a global point of view, and thus look at the behavior and properties of these paths sufficiently far from the solution. Our methods are motivated by methods of global analysis. We show that in many implementations of these algorithms the path can be specified. These include the cases when the mapping is linear and when it is smooth and monotone. The results of the linear analysis are used to study the local behavior of these paths (i.e., properties and behavior sufficiently close to the solution) for smooth mappings. An important implication of this study is that the paths can be modified and thus the work done by these algorithms can be controlled. We also show, for smooth mappings, how our results can be implemented to increase the efficiency of some standard algorithms.

This special issue on yeast proteomics provides an opportunity to acknowledge Hélian Boucherie, who retired at the beginning of this year, for his contributions to 2-D gel electrophoresis and yeast proteomics. Hélian received his Ph.D. in 1979. This was followed by a post-doctoral fellowship in 1980 in the lab of Calvin McLaughlin in California, where he had his first experience with 2-D gel electrophoresis. In 1990, he became Research Director at the CNRS (Centre National de la Recherche Scientifique). He has been the head of a research team at the Institute of Biochemistry and Cellular Genetics in Bordeaux, France, where he spent his entire career dedicated to the development and applications of 2-D gels. Hélian was also President of the SFEAP (French Society of Electrophoresis and Proteomics) from 2002 to 2005. After the brilliant demonstration of 2-D gel electrophoresis technique by O'Farrell in 1975 1, the prospects seemed enormous. However, a decade later, the results were somewhat disappointing. Though the method proved successful for a variety of applications, two main difficulties hindered the wide-scale development of 2-D gels: (1) the methodology was technically demanding and confined to a few expert laboratories and (2) the approach remained limited as long as the identity of the protein spots on the maps was unknown. Much work remained to make 2-D gel technology easier and more popular. In this context, while some colleagues (Angelika Görg and collaborators) developed immobilised pH gradients, Hélian worked intensively to improve the classical pH gradient technique in gel tubes. He defined protein extraction protocols of cell extracts that avoided any trace of proteolysis. He developed his home-made electrophoresis system and after years of effort, he succeeded in establishing 2-D maps of yeast proteins of outstanding quality, resolution and reproducibility (2, see this issue). This was the situation in 1994, the year I first met Hélian. Thanks to the contribution of Hélian and the other pioneers of 2-D gels, the technique had reached the status of the only global analysis method available for functional genomics studies, since transcriptome analysis methods did not yet exist. At this time, the yeast genome sequence was not completed and the notion of functional genomics studies was still in its infancy. The concept of untargeted global approaches was not always well understood or welcome in the scientific community. Fortunately, some influential and visionary researchers, such as Piotr Slonimski and André Goffeau, actively encouraged these approaches and supported Hélian's projects. However, a major problem still hindered the wide utilisation of 2-D gels: the question of the identifications. Tirelessly, Hélian has tackled this task. All methods were welcome to identify or provide functional information about the unknown spots. He tested nearly all the possible identification methods: immunoprecipitation, co-migration, the use of deletion or overexpression mutants, as well as continuously evolving analytical techniques such as microsequencing, amino acid composition and of course mass spectrometry. Finally, he tried to characterize more and more faint spots, including invisible spots (2, see this issue). Throughout these years, the yeast protein map was continuously enriched by new identifications and was made accessible to the scientific community on a web server (Yeast Protein Map server). The aim of all this intense technical investment was to study biological questions. Of course, Hélian never forgot the biological objective. In particular, the 2-D gel approach allowed him to make remarkable contributions in the analysis of yeast adaptation to glucose exhaustion, a physiological transition called the diauxic phase. All throughout his career, Hélian devoted himself to these two areas: 2-D gels and yeast physiology. It is symbolic that Helian's final two articles, published in this issue, deal with new spot identifications for the first area 2 and half lives of yeast proteins for the second one 3. It is thus with great pleasure that I take the opportunity of this special issue on yeast proteomics, on behalf of researchers of both the yeast and of the proteomic communities, to thank Hélian for his intense participation in the yeast proteomic adventure and to wish him a happy retirement near Bordeaux. Of course Hélian was not alone in this endeavor and the strong involvement of his colleagues should also be acknowledged. More widely speaking, this tribute to Hélian's career can also be the occasion to thank all the researchers who dedicated themselves to the technical improvements that have allowed major scientific breakthroughs. As the yeast strains present on Bordeaux vineyards contain a very abundant protein (identified in his lab as alcohol dehydrogenase), one can expect that, although retired, Hélian will continue to joyfully taste some functional aspects of yeast proteomics for many years to come. Jean Labarre

A Bayesian 2D functional linear model for gray-level co-occurrence matrices in texture analysis of lower grade gliomas

The stochastic finite element method designed to deal with indeterminate elements in actual magnetic materials was studied. Permeability is considered as a variable in a homogeneous two-dimensional spatial stochastic process, and the analysis results are compared with those of the usual determinate finite-element method. Both methods produced identical results over a wide range. But the advantage of the probabilistic approach is that the variance can be evaluated directly without performing recursion.

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

Motivated by a range of biological applications related to the transport of molecules in cells, we present a modular framework to treat first-passage problems for diffusion in partitioned spaces. The spatial domains can differ with respect to their diffusivity, geometry, and dimensionality, but can also refer to transport modes alternating between diffusive, driven, or anomalous motion. The approach relies on a coarse-graining of the motion by dissecting the trajectories on domain boundaries or when the mode of transport changes, yielding a small set of states. The time evolution of the reduced model follows a generalized master equation (GME) for non-Markovian jump processes; the GME takes the form of a set of linear integro-differential equations in the occupation probabilities of the states and the corresponding probability fluxes. Further building blocks of the model are partial first-passage time (FPT) densities, which encode the transport behavior in each domain or state. After an outline of the general framework for multiple domains, the approach is exemplified and validated for a target search problem with two domains in one- and three-dimensional space, first by exactly reproducing known results for an artificially divided, homogeneous space, and second by considering the situation of domains with distinct diffusivities. Analytical solutions for the FPT densities are given in Laplace domain and are complemented by numerical backtransforms yielding FPT densities over many decades in time, confirming that the geometry and heterogeneity of the space can introduce additional characteristic time scales.

Let Y(t) be a one-dimensional jump-diffusion process and X(t) be defined by dX(t)=ρ[X(t),Y(t)]dt, where ρ(·,·) is either a strictly positive or negative function. First-passage-time problems for the degenerate two-dimensional process (X(t),Y(t)) are considered in the case when the process leaves the continuation region at the latest at the moment of the first jump, and the problem of optimally controlling the process is treated as well. A particular problem, in which ρ[X(t),Y(t)]=Y(t)−X(t) and Y(t) is a standard Brownian motion with jumps, is solved explicitly.

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation. The theoretical details of the solution procedure are first presented. The method is then used to solve the Fokker–Planck equations for both one- and two-dimensional processes that possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared with analytic (one-dimensional process) and Monte Carlo (two-dimensional process) solutions. Trends in the standard deviation and coefficient of kurtosis with respecttotheadditivenoiselevelandbifurcationparameterarereportedforwhatappearstobethe firsttimeforthe saddle-node/subcritical Hopf bifucation problem. The deterministic form of this problem exhibits hysteresis, which is often a phenomenon present in more-complex nonlinear aeroelastic systems. Finally, statistical results are shown for a typical section airfoil with nonlinear stiffness subjected to random aerodynamic loading. The constants for the nonlinear stiffness are chosen such that the deterministic behavior exhibits both a saddle-node and subcritcal bifurcation. Standard deviation results computed using the direct quadrature method of moments for a reduced frequency below the deterministic saddle-node bifurcation point compare well with Monte Carlo results.