Robust Dynamics and Control of a Partially Observed Markov Chain

In a seminal paper, Martin Clark (Communications Systems and Random Process Theory, Darlington, 1977, pp. 721–734, 1978) showed how the filtered dynamics giving the optimal estimate of a Markov chain observed in Gaussian noise can be expressed using an ordinary differential equation. These results offer substantial benefits in filtering and in control, often simplifying the analysis and an in some settings providing numerical benefits, see, for example Malcolm et al. (J. Appl. Math. Stoch. Anal., 2007, to appear). Clark’s method uses a gauge transformation and, in effect, solves the Wonham-Zakai equation using variation of constants. In this article, we consider the optimal control of a partially observed Markov chain. This problem is discussed in Elliott et al. (Hidden Markov Models Estimation and Control, Applications of Mathematics Series, vol. 29, 1995). The innovation in our results is that the robust dynamics of Clark are used to compute forward in time dynamics for a simplified adjoint process. A stochastic minimum principle is established.     

Using equity options to imply credit information

The evolution of credit derivatives has inspired many researchers to investigate the behaviour of credit spreads. Today the growing consensus is that the equity option market provides sufficient information to estimate latent credit parameters. Hull et al. (J. Credit Risk 1(1):3–28, 2005) propose a clever approach to estimate credit spreads from the equity option market. In this paper we first perform a time series analysis to test the conjecture of an existing relationship between credit spreads and implied equity volatility and find strong evidence of a positive relationship. We also propose an extension to Hull et al.’s paper that significantly improves credit spread estimation.     

Returns to scale in multiplicative models in data envelopment analysis

One class of models introduced in DEA is called multiplicative models, in which, as shown by Banker and Maindiratta (Manag. Sci. 32:126–135, 1986), the piecewise linear frontiers usually employed in DEA are replaced by a frontier that is piecewise Cobb-Douglas(=log  linear). Banker and Maindiratta (Manag. Sci. 32:126–135, 1986) introduced a model to identify the most productive scale size pattern, and Banker et al. (Eur. J. Oper. Res. 154:345–362, 2004) presented a two-stage method for the identification of returns to scale (RTS) in multiplicative models. In this paper it is shown that both the RTS situation and the MPSS pattern could be determined by a single model in one step. The new method is important in the computational point of view.     

Gaussian Stationary Processes: Adaptive Wavelet Decompositions,Discrete Approximations,and Their Convergence

We establish particular wavelet-based decompositions of Gaussian stationary processes in continuous time. These decompositions have a multiscale structure, independent Gaussian random variables in high-frequency terms, and the random coefficients of low-frequency terms approximating the Gaussian stationary process itself. They can also be viewed as extensions of the earlier wavelet-based decompositions of Zhang and Walter (IEEE Trans. Signal Process. 42(7):1737–1745, [1994]) for stationary processes, and Meyer et al. (J. Fourier Anal. Appl. 5(5):465–494, [1999]) for fractional Brownian motion. Several examples of Gaussian random processes are considered such as the processes with rational spectral densities. An application to simulation is presented where an associated Fast Wavelet Transform-like algorithm plays a key role. The second author was supported in part by the NSF grant DMS-0505628.     

Aggregation in age and space structured population models: an asymptotic analysis approach

In this paper we describe how techniques of asymptotic analysis can be used in a systematic way to perform ‘aggregation’ of variables, based on a separation of different time scales, in a population model with age and space structure. The main result of the paper is proving the convergence of the formal asymptotic expansion to the solution of the original equation. This result improves and clarifies earlier results of Arino et al. (SIAM J Appl Math 60(2):408–436, 1999), Auger et al. (Structured population models in biology and epidemiology. Springer Verlag, Berlin, 2008), Lisi and Totaro (Math Biosci 196(2):153–186, 2005).     

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney, Tremaine in Galactic Dynamics, Princeton University Press, Princeton, 1987) is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov (Sov. Astron. 4:859–867, 1961). In the previous work (Lemou et al. in A new variational approach to the stability of gravitational systems, submitted, 2011), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature (Lynden-Bell in Mon. Not. R. Astron. Soc. 144:189–217, 1969; Gardner in Phys. Fluids 6:839–840, 1963; Wiechen et al. in Mon. Not. R. Astron. Soc. 223:623–646, 1988; Aly in Mon. Not. R. Astron. Soc. 241:15, 1989). In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.     

Diffusion processes with delay at the endpoints of a segment

A continuous semi-Markov process with a segment as the range of values is considered. This process coincides with a diffusion process inside the segment, i.e., up to the first hitting time of the boundary of the segment and at any time when the process leaves the boundary. The class of such processes consists of Markov processes with reflection at the boundaries (instantaneously or with a delay) and semi-Markov processes with intervals of constancy on some boundary. We derive conditions of existence of such a process in terms of a semi-Markov transition generating function on the boundary. The method of imbedded alternating renewal processes is applied to find a stationary distribution of the process. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 284–297.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

Economic value of weather forecasting: the role of risk aversion

Extreme meteorological events have increased over the last decades and it is widely accepted that it is due to climate change (IPCC, Climate Change 2007, Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, 2007; Beniston et al., Clim. Change 81:71–95, 2007). Some of these extremes, like drought or frost episodes, largely affect agricultural outputs, and risk management becomes crucial. The goal of this paper it is to analyze farmers’ decisions about risk management, taking into account climatological and meteorological information. We consider a situation in which the farmer, as part of crop management, has available technology to protect the harvest from weather effects. This approach has been used by Murphy et al. (Mon. Weather Rev. 113:801–813, 1985), Katz and Murphy (J. Forecast. 9:75–86, 1990 and Economic Value of Weather and Climate Forecasts, pp. 183–217, Cambridge University Press, Cambridge, 1997) and others in the case when the farmer maximizes the expected returns. In our model, we introduce the attitude towards risk. Thus we can evaluate how the optimal decision is affected by the absolute risk aversion coefficient of Arrow and Pratt, and compute the economic value of the information in this context, while proposing a measure to estimate the amount of money that the farmer is willing to pay for this information in terms of the certainty equivalent.     

RETRACTED ARTICLE: Convergence of Weighted Sums for Arrays of Negatively Dependent Random Variables and Its Applications

We discuss the complete convergence of weighted sums for arrays of rowwise negatively dependent random variables (ND r.v.’s) to linear processes. As an application, we obtain the complete convergence of linear processes based on ND r.v.’s which extends the result of Li et al. (Stat. Probab. Lett. 14:111–114, 1992), including the results of Baum and Katz (Trans. Am. Math. Soc. 120:108–123, 1965), from the i.i.d. case to a negatively dependent (ND) setting. We complement the results of Ahmed et al. (Stat. Probab. Lett. 58:185–194, 2002) and confirm their conjecture on linear processes in the ND case.     

Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom. Différentielle Catég., 44: 281–316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81–108, 2004). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517, Springer, Berlin Heidelberg New York, 2005), corresponding to partially ordered subspaces of R ⁿ minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.      

Downward migration credit risk problem: a non-homogeneous backward semi-Markov reliability approach

International organizations evaluate credit risk and rank firms according to risk by assigning them a ‘rating’. The time evolution of a rating can be studied by means of Markov models. Some papers have outlined the problem pertaining to the unsuitable fitting of Markov processes in a credit risk environment. This paper presents a model that overcomes the problems given by the Markov rating models. It includes non-homogeneity, the downward problem and the randomness of time in the transitions of states, thus making it possible to consider the duration inside a state in a complete way. In this paper, both, the transient and asymptotic analyses are presented. The asymptotic analysis is performed by using a mono-unireducible topological structure. Moreover, a real data application is conducted using the historical database of Standard & Poor’s as the source.     

Bayesian Hierarchical Space–time Modeling of Earthquake Data

Stochastic earthquake models are often based on a marked point process approach as for instance presented in Vere-Jones (Int. J. Forecast., 11:503–538, 1995). This gives a fine resolution both in space and time making it possible to represent each earthquake. However, it is not obvious that this approach is advantageous when aiming at earthquake predictions. In the present paper we take a coarse point of view considering grid cells of 0.5 × 0.5°, or about 50 × 50 km, and time periods of 4 months, which seems suitable for predictions. More specifically, we will discuss different alternatives of a Bayesian hierarchical space–time model in the spirit of Wikle et al. (Environ. Ecol. Stat., 5:117–154, 1998). For each time period the observations are the magnitudes of the largest observed earthquake within each grid cell. As data we apply parts of an earthquake catalogue provided by The Northern California Earthquake Data Center where we limit ourselves to the area 32–37° N, 115–120° W for the time period January 1981 through December 1999 containing the Landers and Hector Mine earthquakes of magnitudes, respectively, 7.3 and 7.1 on the Richter scale. Based on space-time model alternatives one step earthquake predictions for the time periods containing these two events for all grid cells are arrived at. The model alternatives are implemented within an MCMC framework in Matlab. The model alternative that gives the overall best predictions based on a standard loss is claimed to give new knowledge on the spatial and time related dependencies between earthquakes. Also considering a specially designed loss using spatially averages of the 90th percentiles of the predicted values distribution of each cell it is clear that the best model predicts the high risk areas rather well. By using these percentiles we believe that one has a valuable tool for defining high and low risk areas in a region in short term predictions.      

Corrigendum to “The distribution of the number of arrivals in a subinterval of a busy period of a single server queue”

The purpose of this corrigendum is two-fold. First, we acknowledge that two results in our paper (Novak et al. in Queueing Syst. 53:105–114, 2006) can be obtained from earlier results of Prabhu and Bhat. Second, we make corrections to Theorem 2.2, Corollary 2.1 and Theorem 4.2 of Novak et al. (Queueing Syst. 53:105–114, 2006).      

On the Surrogate Gradient Algorithm for Lagrangian Relaxation

When applied to large-scale separable optimization problems, the recently developed surrogate subgradient method for Lagrangian relaxation (Zhao et al.: J. Optim. Theory Appl. 100, 699–712, 1999) does not need to solve optimally all the subproblems to update the multipliers, as the traditional subgradient method requires. Based on it, the penalty surrogate subgradient algorithm was further developed to address the homogenous solution issue (Guan et al.: J. Optim. Theory Appl. 113, 65–82, 2002; Zhai et al.: IEEE Trans. Power Syst. 17, 1250–1257, 2002). There were flaws in the proofs of Zhao et al., Guan et al., and Zhai et al.: for problems with inequality constraints, projection is necessary to keep the multipliers nonnegative; however, the effects of projection were not properly considered. This note corrects the flaw, completes the proofs, and asserts the correctness of the methods. This work is supported by the NSFC Grant Nos. 60274011, 60574067, the NCET program (No. NCET-04-0094) of China. The third author was supported in part by US National Science Foundation under Grants ECS-0323685 and DMI-0423607.     

Convergence of the Law of the Environment Seen by the Particle for Directed Polymers in Random Media in the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Region

We consider the model of directed polymers in an i.i.d. Gaussian or bounded environment (Imbrie and Spencer in J. Stat. Phys. 52(3/4), 609–626, 1988; Carmona and Hu in Probab. Theory Relat. Fields 124(3), 431–457, 2002; Comets et al. in Adv. Stud. Pure Math. 39, 115–142, 2004) in the L ² region. We prove the convergence of the law of the environment seen by the particle.     

Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

In compressed sensing, we seek to gain information about a vector x∈ℝ ^N from d ≪ N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, 2006; Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006; Donoho, IEEE Trans. Inf. Theory 52:1289–1306, 2006) proposed to seek a good approximation to x via ℓ ₁ minimization. In this paper, we show that in the case of Gaussian measurements, ℓ ₁ minimization recovers the signal well from inaccurate measurements, thus improving the result from Candes et al. (Commun. Pure Appl. Math. 59:1207–1223, 2006). We also show that this numerically friendly algorithm (see Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006) with overwhelming probability recovers the signal with accuracy, comparable to the accuracy of the best k-term approximation in the Euclidean norm when k∼d/ln N.     

Semi-Markov processes and α-invariant distributions

A semi-Markov process is easily made Markov by adding some auxiliary random variables. This paper discusses the I-type quasi-stationary distributions of such “extended” processes, and the α-invariant distributions for the corresponding Markov transition probabilities; and we show that there is an intimate relation between the two. The results have relevance in the study of the time to “absorption” or “death” of semi-Markov processes. The particular case of a terminating renewal process is studied as an example.     

Organizational learning: effects of (network) structure and (individual) strategy

Earlier theoretical accounts of collective learning relied on rules and operating procedures as the organizational memory (March in Organ. Sci. 2(1):71–87, 1991; Rodan in Scand. J. Manag. 21:407–428, 2005). This paper builds on this tradition drawing on ideas from social network theory. Learning is modeled as a social-psychological process (Darr and Kurtzberg in Organ. Behav. Hum. Decis. Process. 82(1):28–44, 2000; Rulke et al. in Organ. Behav. Hum. Decis. Process. 82(1):134–149, 2000), in which organizations learn by exchanging information internally between their members (Argote et al. in Organ. Behav. Hum. Decis. Process. 82(1):1–8, 2000; Carley in Am. Soc. Rev. 56(3):331–354, 1991; Carley in Soc. Perspect. 48(4):547–571, 1995). Learning is also characterized as stochastic and creative (Gruenfeld et al. in Organ. Behav. Hum. Decis. Process. 82(1):45–59, 2000). This model is used to explore predictions about the effect social networks have on idea generation and learning and alternative strategies for choosing from whom to seek information.

On a Risk Model with Surplus-dependent Premium and Tax Rates

In this paper, the compound Poisson risk model with surplus-dependent premium rate is analyzed in the taxation system proposed by Albrecher and Hipp (Bl?tter der DGVFM 28(1):13–28, 2007). In the compound Poisson risk model, Albrecher and Hipp (Bl?tter der DGVFM 28(1):13–28, 2007) showed that a simple relationship between the ruin probabilities in the risk model with and without tax exists. This so-called tax identity was later generalized to a surplus-dependent tax rate by Albrecher et al. (Insur Math Econ 44(2):304–306, 2009). The present paper further generalizes these results to the Gerber–Shiu function with a generalized penalty function involving the maximum surplus prior to ruin. We show that this generalized Gerber–Shiu function in the risk model with tax is closely related to the ‘original’ Gerber–Shiu function in the risk model without tax defined in a dividend barrier framework. The moments of the discounted tax payments before ruin and the optimal threshold level for the tax authority to start collecting tax payments are also examined.