Markov Methods Research Articles

Mathematics of Derivative Securities

Foreword R. C. Merton Part I. Introduction: 1. Editors' introduction 2. Stochastic calculus and Markov methods L. C. G. Rogers 3. The risk premium in trading equilibria which support Black-Scholes option pricing S. D. Hodges and M. J. P. Selby 4. On the numeraire portfolio P. Artzner Part II. Option Pricing and Hedging: 5. Convergence of Snell envelopes and critical prices in the American Put N. J. Cutland, P. E. Kopp, W. Willinger and M. C. Wyman 6. Some combination of Asian, Parisian and Barrier options M. Yor, M. Chesnay, H. Geman and M. Jeanblanc-Pique 7. Co-movement term structure and the valuation of crack energy spread options A. Mbanefo 8. Pricing and hedging with Smiles B. Dupire 9. Filtering derivative security valuations from market prices R. J. Elliott, C. H. Lahaie and D. B. Madan 10. Option pricing in the presence of extreme fluctuations J.-P. Bouchard, D. Sornette and M. Potters 11. Hedging long maturity commodity commitments with short-dated futures contracts M. J. Brennan and N. I. Crew 12. Nonlinear financial markets: hedging and portfolio optimization J. Cvitanic 13. Semimartingales and asset pricing under constraints M. Frittelli 14. Option pricing in incomplete markets M. H. A. Davis 15. Option pricing and hedging in discrete time with transaction costs F. Mercurio and T. C. F. Vorst Part III. Term Structure and Interest Rate Derivatives: 16. Bond and bond option pricing based on the current term structure P. H. Dybvig 17. Dynamic models for yield curve evolution B. Flesaker and L. P. Hughston 18. General interest rate models and the universality of HJM M. W. Baxter 19. Swap derivatives in a Gaussian HJM framework A. Brace and M. Musiela 20. Modelling bonds and derivatives with default risk D. Lando 21. Term structure modelling under alternative official regimes S. H. Babbs and N. J. Webber 22. Interest rate distributions, yield curve modelling and monetary policy L. El-Jahel, H. Lindberg and W. Perraudin Part IV. Numerical Methods: 23. Numerical option pricing using conditioned diffusions S. K. Gandhi and P. J. Hunt 24. Numerical valuation of cross-currency swaps and swaptions M. A. H. Dempster and J. P. Hutton 25. Numerical methods for stochastic control problems in finance H. J. Kushner 26. Simulation methods for option pricing J. P. Lehoczky 27. New methodologies for valuing derivatives S. H. Paskov.

Microbial gene identification using interpolated Markov models.

This paper describes a new system, GLIMMER, for finding genes in microbial genomes. In a series of tests on Haemophilus influenzae , Helicobacter pylori and other complete microbial genomes, this system has proven to be very accurate at locating virtually all the genes in these sequences, outperforming previous methods. A conservative estimate based on experiments on H.pylori and H. influenzae is that the system finds >97% of all genes. GLIMMER uses interpolated Markov models (IMMs) as a framework for capturing dependencies between nearby nucleotides in a DNA sequence. An IMM-based method makes predictions based on a variable context; i.e., a variable-length oligomer in a DNA sequence. The context used by GLIMMER changes depending on the local composition of the sequence. As a result, GLIMMER is more flexible and more powerful than fixed-order Markov methods, which have previously been the primary content-based technique for finding genes in microbial DNA.

Pseudopodium extension and amoeboid locomotion in Dictyostelium discoideum: Possible autowave behaviour of F-actin

Supramolecular patterns of filamentous (F-)actin up to several micrometres across were visualized within projections of locomotory amoebae after cell fixation and staining with phalloidin-rhodamine. The patterns included rings, single and double spirals, some apparently colliding and disintegrating. Cell stimulation with a pulse of the chemoattractant cyclic AMP induced damping oscillations in F-actin ring frequency with a period of 6–7 s. Ring front propagation after stimulation was modelled by Markov and Fourier methods at 3.1–17.5 μm/min, similar to actual cell speed. We argue that the dynamics and detailed morphological correspondence of these F-actin structures to wave patterns in chemical reaction-diffusion systems strongly supports the interpretation that Dictyostelium cytoplasm behaves as an unstable, excitable medium enabling the propagation of self-organized, physico-chemical relaxation oscillations, i.e. autowaves, of reversible F-actin assembly or aggregation — a new state of actin - fundamental to pseudopodium extension, cell locomotion, chemotaxis and other cell functions.

Stochastic analysis of an active-stand by redundant network with two types of common-cause failures

This paper presents probabilistic analysis of a reliability system composed of two active and one standby units. The system can fail either due to normal failures or two types of common-cause failures. The failed system is repairable. The supplementary variable and Markov methods were used to perform probabilistic analysis. Expressions for system availability and state probabilities are presented along with various plots

State Increment Method of Life-Cycle Cost Analysis for Highway Management

A state increment method of life-cycle cost analysis (SILCCA) is presented for project-level evaluation of highway infrastructure maintenance options. The method involves: (1) Characterization of facility's physical and functional characteristics into discrete states; (2) specification of treatment alternatives for each state; (3) prediction of facility life-cycle performance as a series of state transitions; and (4) application of dynamic programming for life-cycle cost analysis. Four important advantages of SILCCA are: (1) A systematic procedure is used to identify treatment alternatives based on assessed condition; (2) engineering expertise is used to capture the effect of treatments on pavement condition and in turn improve the accuracy of prediction; (3) analytical and computational efficiency over traditional, Markov, and semi-Markov methods; and (4) a convenient framework for integration with network-level optimization methods. Based on the findings of a case study, it is concluded that SILCCA is systematic, accurate, and efficient.

Recent Results in Random Vibrations of Nonlinear Mechanical Systems

The influence of random vibration on the design of mechanical components has been considered within the framework of the linear theory of small oscillations. However, in some important cases this theory is inadequate and fails to predict some complex response characteristics that have been observed experimentally and which can only be predicted by nonlinear analyses. This paper describes some recent developments in the theory of nonlinear random vibration based on Markov methods and related problems in the design of dynamical systems. Research efforts have been focused on stability/bifurcation conditions, response statistics and reliability problems. Significant progress has been made in developing new analytical methods and conducting experimental testing. These developments have helped to resolve some controversies, and to enhance our understanding of difficult issues. Experimental and numerical simulations have revealed new phenomena that were not predicted analytically. These include on-off intermittency, snap-through phenomena, and the dependence of the response bandwidth on the excitation level. The main results of studying the responses of nonlinear single-and two-degree-of-freedom systems to random excitations obtained by the author and others are discussed in this paper.

Recent Results in Random Vibrations of Nonlinear Mechanical Systems

The influence of random vibration on the design of mechanical components has been considered within the framework of the linear theory of small oscillations. However, in some important cases this theory is inadequate and fails to predict some complex response characteristics that have been observed experimentally and which can only be predicted by nonlinear analyses. This paper describes some recent developments in the theory of nonlinear random vibration based on Markov methods and related problems in the design of dynamical systems. Research efforts have been focused on stability/bifurcation conditions, response statistics and reliability problems. Significant progress has been made in developing new analytical methods and conducting experimental testing. These developments have helped to resolve some controversies, and to enhance our understanding of difficult issues. Experimental and numerical simulations have revealed new phenomena that were not predicted analytically. These include on-off intermittency, snap-through phenomena, and the dependence of the response bandwidth on the excitation level. The main results of studying the responses of nonlinear single-and two-degree-of-freedom systems to random excitations obtained by the author and others are discussed in this paper.

Covering properties of convolutional codes and associated lattices

The paper describes Markov methods for analyzing the expected and worst case performance of sequence-based methods of quantization. We suppose that the quantization algorithm is dynamic programming, where the current step depends on a vector of path metrics, which we call a metric function. Our principal objective is a concise representation of these metric functions and the possible trajectories of the dynamic programming algorithm. We shall consider quantization of equiprobable binary data using a convolutional code. Here the additive group of the code splits the set of metric functions into a finite collection of subsets. The subsets form the vertices of a directed graph, where edges are labeled by aggregate incremental increases in mean squared error (MSE). Paths in this graph correspond both to trajectories of the Viterbi algorithm and to cosets of the code. For the rate 1/2 convolutional code [1+D/sup 2/, 1+D+D/sup 2/], this graph has only nine vertices. In this case it is particularly simple to calculate per dimension expected and worst case MSE, and performance is slightly better than the binary [24, 12] Golay code. Our methods also apply to quantization of arbitrary symmetric probability distributions on [0, 1] using convolutional codes. For the uniform distribution on [0, 1], the expected MSE is the second moment of the Voronoi region of an infinite-dimensional lattice determined by the convolutional code. It may also be interpreted as an increase in the reliability of a transmission scheme obtained by nonequiprobable signaling. For certain convolutional codes we obtain a formula for expected MSE that depends only on the distribution of differences for a single pair of path metrics. >

Stochastic modelling ofk-out-of-nunits family of systems

Two newly developed stochastic models to perform reliability and availability analyses of (n − 1)-out-of-n units and (n − 2)-out-of-n units systems with common-cause failures and critical human errors are presented. The supplementary variables and Markov methods are used to perform analyses. For both models, generalized expressions for Laplace transforms of state probabilities are developed along with generalized formulae for system steady-state availability and mean time to failure. Some plots are shown to demonstrate the effect of common-cause failures and critical human errors on system steady-state availability and mean time to failure.

Parametric and External Excitation of Marine Risers

The response of a nonlinear marine riser model to both parametric and external excitation is examined using Markov methods. Moment-closure techniques are used to predict the response of the riser in a random seaway. An approximate nonlinear polynomial representation of the hydrodynamic damping term is developed in order to implement Gaussian and non-Gaussian moment-closure schemes. These methods are applied to solve the case of pure parametric excitation and the results are compared with those obtained using stochastic averaging. It is shown that as the parametric excitation level is varied, the response predictions from different techniques exhibit the same linear trends. However, the non-Gaussian closure scheme provides improved response estimates that are closer to stochastic averaging results. Moment-closure techniques are then extended to consider a riser under combined parametric and external excitation. For a fixed level of parametric excitation, the response statistics are found to vary nonlinearly with external excitation levels. The Gaussian closure scheme is found to overpredict structural response for the case of pure parametric excitation and to underpredict structural response for the case of pure external excitation.

Filter approach to ocean structure response prediction

The design and application of shaping filters for random response analysis of offshore structures is presented. The approach is based upon Markov methods and its suitability for use in offshore engineering is demonstrated through a series of numerical examples. Shaping filters are designed for generating the wave elevation, wave kinematics and wave forces from white noise. The filter equations are combined with state-space equation models of two different types of offshore structures. Moment equations are generated using Ito's rule and solved for the structural response in terms of statistical moments. This filter methodology is applied to both linear and non-linear structural models of offshore platforms. The strengths and limitations of the filter approach are examined through direct comparison with time domain simulations.

Possible correlation between uranium content in host rocks and radon and daughters concentration in air for two Italian ZnS-PbS mines

Research was carried out to evaluate the possible correlation between the uranium content in host rocks and the Rn and daughters concentration in air for two Italian ZnS-PbS mines. As a great variability had been ascertained for222Rn and daughters concentration in the different areas, it appeared of some interest to verify whether this fact was due to a different uranium concentration in the host dolomitic rocks or simply to a ventilation problem. Some rock samples were collected in several places where also radon and daughters were drawn; uranium was determined by fluorimetry and by alpha spectrometry after a chemical separation with a Microthene-TOPO column.222Rn was measured by the scintillation cell method, while the decay products were determined by the Markov and Tsivoglou methods. No correlation could be found between uranium content and radon concentration, but a good linearity was detected between the uranium content and the concentration of Rn decay products (alpha potential energy).

FORECAST UNCERTAINTY IN WATER SUPPLY RESERVOIR OPERATION1

ABSTRACT: This research investigates the benefits of forecasting in water supply systems. Questions relating operational losses to forecast period and accuracy are addressed. Some simple available forecasting techniques are assessed for their accuracy and applicability. These issues are addressed through the use of a simulation model of the Cedar and South Fork Tolt Rivers, where the system is modeled as a single purpose reservoir supplying municipal and industrial water to the Seattle metropolitan area. The following conclusions were made for this system: (1) reservoir operation deteriorates markedly with the loss of forecast accuracy; (2) the optimal length of forecasting period is five months; (3) reservoir operation may be improved by as much as 88 percent if perfect predictive abilities are available; (4) the mean of the historic data is not recommended to predict future flows because Markov methods are always superior; and (5) lag‐one autoregressive Markov schemes exhibit about a 9 percent improvement in operation over no forecasting.

A Markov Chain Model of Population Growth with an Application in Epidemiology

AbstractThis paper is concerned with a class of population growth processes in discrete time; the simple epidemic process is considered as a specific example. A Markov chain model is constructed and standard Markov methods are used to study the main biological concepts. A simple and explicit formula is obtained for the transient distribution of the population size. Then, the cost of the process is defined and the joint probability generating function of its components is derived. Finally, the results are extended to the case where the inter‐transition periods are bounded i.i.d. random variables.

Joint determination of the concentrations of the222Rn and220Rn decay products in air

The authors describe a modification of the Kuznets and Markov methods normally employed for the determination of radon 220 and 222 daughter alpha product concentration in air in which an air sample is taken for 10 minutes on a filter at a flow rate of 10-40 liters per minute. After the conclusion of sampling the filter activity is measured for another 10 minutes. In order to then determine the latent energy of the radon 222 daughter products and to bring into account the radon 220 daughter products in the total activity measurements of the filter are taken for a second time for 30 minutes five hours after initial sampling. The level of latent energy of the combined daughter products are calculated by an equation which incorporates alpha particle detection efficiency, aerosol retention efficiency in the filter, and the Kuznets coefficients, and analyzes the separate and combined contributions of both daughter products from both sampling periods. A statistical analysis employing the Markov method is also depicted in modified form and is recommended when a more rapid analysis of air radioactivity is mandated.

Radon (thoron) daughter measurements with an automated, programmable, radiation monitor

A technical evaluation of an automated, programmable, grab-sampler manufactured by Pylon Electronic Development under the commercial name WL-1000C has been conducted. Six different methods are implemented in the instrument for analyzing radiation data. Any one method can be used on command and easily selected by means of a keyboard. Available radiation data that can be retrieved on command include radon (thoron) daughter concentrations and radon (thoron) Working Levels. Measurements were carried out under laboratory-controlled conditions in a large (26 m(3)) radon/thoron test facility, designed for calibration purposes, and at an underground uranium mine. Data obtained with the WL-1000C have been compared with conventional grab-sampling (e.g., Kusnetz, Thomas-Tsivoglou and Markov methods) and with other automated radiation instrumentation previously tested at our laboratories. Tests were done under constant radiation conditions and also under rapidly fluctuating conditions in order to determine the response of the instruments and methods in these two cases of practial interest. The Working Level used in these experiments was in the approximate range of 0.01 to 10 WL. Tests were conducted under a variety of environmental conditions. Good agreement with grab-sampling data was found for radon daughters. Some discrepancies with grab-sampling data were found for radon daughter/thoron daughter mixtures. Disagreement in the latter case is to be expected because of α-energy overlap between RaA and ThC.

Reliability & Availability of Duplex Systems: Some Simple Models

The reliability and availability of a system can be enhanced by redundancy and repair. The reliability and availability are shown for some simple models of duplex systems, by using Markov methods. All transition rates between states are constant.

Continuous-time Markov methods in the solution of practical reliability problems

The use of continuous-time Markov matrix methods in the prediction of reliability for parallel active and redundant subgroups of identical components, subject to negative exponential failure and repair distributions, is described. Decomposition of parallel subgroups in the assessment of reliability for systems of greater complexity, using solutions obtained by these methods, is considered in terms of the single-element equivalent of such subgroups. Also, the difficulty associated with non-exponential failure of the single-element equivalent is addressed. Finally, the technique is generalised by varying initial conditions to enable phased mission reliability problems to be considered.

A random wave process

The parabolic or forward scattering approximation to the equation describing wave propagation in a random medium leads to a stochastic partial differential equation which has the form of a random Schrodinger equation. Existence, uniqueness and continuity of solutions to this equation are established. The resulting process is a Markov diffusion process on the unit sphere in complex Hilbert space. Using Markov methods a limiting Markov process is identified in the case of a narrow beam limit; this limiting process corresponds to a simple random translation of the beam known as “spot-dancing.”

Markov methods of analysis of stochastic quantum dynamics of quasiclassical open systems

Markov methods of analysis of stochastic quantum dynamics of quasiclassical open systems