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Gaussian concentration graph models and covariance graph models are two classes of graphical models that are useful for uncovering latent dependence structures among multivariate variables. In the Bayesian literature, graphs are often determined through the use of priors over the space of positive definite matrices with fixed zeros, but these methods present daunting computational burdens in large problems. Motivated by the superior computational efficiency of continuous shrinkage priors for regression analysis, we propose a new framework for structure learning that is based on continuous spike and slab priors and uses latent variables to identify graphs. We discuss model specification, computation, and inference for both concentration and covariance graph models. The new approach produces reliable estimates of graphs and efficiently handles problems with hundreds of variables.

Heavy-tailed distributions are found throughout many naturally occurring phenomena. We have reviewed the models of stochastic dynamics that lead to heavy-tailed distributions (and power law distributions, in particular) including the multiplicative noise models, the models subjected to the Degree-Mass-Action principle (the generalized preferential attachment principle), the intermittent behavior occurring in complex physical systems near a bifurcation point, queuing systems, and the models of Self-organized criticality. Heavy-tailed distributions appear in them as the emergent phenomena sensitive for coupling rules essential for the entire dynamics.

Heavy-tailed distributions are found throughout many naturally occurring phenomena. We have reviewed the models of stochastic dynamics that lead to heavy-tailed distributions (and power law distributions, in particular) including the multiplicative noise models, the models subjected to the Degree-Mass-Action principle (the generalized preferential attachment principle), the intermittent behavior occurring in complex physical systems near a bifurcation point, queuing systems, and the models of Self-organized criticality. Heavy-tailed distributions appear in them as the emergent phenomena sensitive for coupling rules essential for the entire dynamics.

We present a simulation workflow for efficient investigations of the interplay between 3D lithium-ion electrode microstructures and electrochemical performance, with emphasis on lithium plating. Our approach addresses several challenges. First, the 3D microstructures of porous electrodes are generated by a parametric stochastic model, in order to significantly reduce the necessity of tomographic imaging. Secondly, we integrate a consistent microscopic, 3D spatially-resolved physical model for the electrochemical behaviour of the lithium-ion cells taking lithium plating and stripping into account. This highly non-linear mathematical model is solved numerically on the complex 3D microstructures to compute the transient cell behaviour. Due to the complexity of the model and the considerable size of realistic microstructures even a single charging cycle of the battery requires several hours computing time. This renders large scale parameter studies extremely time consuming. Hence, we develop a mathematical model order reduction scheme. We demonstrate how these aspects are integrated into one unified workflow, which is a step towards computer aided engineering for the development of more efficient lithium-ion cells.

This article is from Frontiers in Neuroscience , volume 7 . Abstract The number and variety of connectivity estimation methods is likely to continue to grow over the coming decade. Comparisons between methods are necessary to prune this growth to only the most accurate and robust methods. However, the nature of connectivity is elusive with different methods potentially attempting to identify different aspects of connectivity. Commonalities of connectivity definitions across methods upon which base direct comparisons can be difficult to derive. Here, we explicitly define “effective connectivity” using a common set of observation and state equations that are appropriate for three connectivity methods: dynamic causal modeling (DCM), multivariate autoregressive modeling (MAR), and switching linear dynamic systems for fMRI (sLDSf). In addition while deriving this set, we show how many other popular functional and effective connectivity methods are actually simplifications of these equations. We discuss implications of these connections for the practice of using one method to simulate data for another method. After mathematically connecting the three effective connectivity methods, simulated fMRI data with varying numbers of regions and task conditions is generated from the common equation. This simulated data explicitly contains the type of the connectivity that the three models were intended to identify. Each method is applied to the simulated data sets and the accuracy of parameter identification is analyzed. All methods perform above chance levels at identifying correct connectivity parameters. The sLDSf method was superior in parameter estimation accuracy to both DCM and MAR for all types of comparisons.

We study the joint effect of thermal bath fluctuations and an external noise tuning activity of cytotoxic cells on the triggered immune response in a growing cancerous tissue. The immune response is assumed to be primarily mediated by effector cells that develop a cytotoxic activity against the abnormal tissue. The kinetics of such a reaction is represented by an enzymatic-like Michaelis-Menten two step process. Effective free-energy surface for the process is further parametrized by the fluctuating energy barrier between the states of high and low concentration of cancerous cells. By analyzing the far from equilibrium escape problem across the fluctuating potential barrier, we determine condtions of the most efficient decay kinetics of the cancer cell-population in the presence of dichotomously fluctuating concentration of cytotoxic cells.

We consider data fusion for the purpose of smoothing and interpolation based on observation records with missing data. Stochastic processes are generated by linear stochastic models. The paper begins by drawing a connection between time reversal in stochastic systems and all-pass extensions. A particular normalization (choice of basis) between the two time-directions allows the two to share the same orthonormalized state process and simplifies the mathematics of data fusion. In this framework we derive symmetric and balanced Mayne-Fraser-like formulas that apply simultaneously to smoothing and interpolation.

We consider the problem of selecting deterministic or stochastic models for a biological, ecological, or environmental dynamical process. In most cases, one prefers either deterministic or stochastic models as candidate models based on experience or subjective judgment. Due to the complex or intractable likelihood in most dynamical models, likelihood-based approaches for model selection are not suitable. We use approximate Bayesian computation for parameter estimation and model selection to gain further understanding of the dynamics of two epidemics of chronic wasting disease in mule deer. The main novel contribution of this work is that under a hierarchical model framework we compare three types of dynamical models: ordinary differential equation, continuous time Markov chain, and stochastic differential equation models. To our knowledge model selection between these types of models has not appeared previously. Since the practice of incorporating dynamical models into data models is becoming more common, the proposed approach may be very useful in a variety of applications.

Using an artificial system of self-replicating strings, we show a correlation between the age of a genotype and its abundance that reflects a punctuated rather than gradual picture of evolution, as suggested long ago by Willis. In support of this correlation, we measure genotype abundance distributions and find universal coefficients. Finally, we propose a simple stochastic model which describes the dynamics of equilibrium periods and which correctly predicts most of the observed distributions.

In this paper, we establish a set of criteria which are applied to discuss various formulations under which Lagrangian stochastic models can be found. These models are used for the simulation of fluid particles in single-phase turbulence as well as for the fluid seen by discrete particles in dispersed turbulent two-phase flows. A central issue is to put forward reliable requirements which must be met by Lagrangian stochastic models and a new element brought by the present analysis is to address the single- and two-phase flow situations from a unified point of view.This analysis does not address the question of the relative predictive capacities of different models but concentrates on their formulation since advantages and disadvantages of different formulations are not always clear. Indeed, hidden in the changes from one structure to another are some possible pitfalls which can lead to flaws in the construction of practical models and to physically-unsound numerical calculations. A first interest of the present approach is illustrated by considering some models proposed in the literature and by showing that these criteria help to assess whether these Lagrangian stochastic models can be regarded as acceptable descriptions. A second interest is to indicate how future developments can be safely built, which is also relevant for stochastic subgrid models for particle-laden flows in the context of Large Eddy Simulations.

We model the growth dynamics for repeated measurements of tumor volumes in mice. We consider a two compartments representation corresponding to the fractions of tumor cells killed by and survived to a treatment, respectively. Dynamics are modelled with stochastic differential equations, resulting in a new state-space stochastic differential equation mixed effects model (SDEMEM) for response to treatment and regrowth. Inference for SDEMEMs is challenging due to the intractable likelihood function. We were able to estimate the model parameters, using both exact Bayesian methodology and an approximate method using the synthetic likelihoods approach. As a case study we consider data from two treatment groups and one control from a tumor xenography study, each consisting of 7-8 mice. Results from the case study and from a further simulation study shows that our models are able to reproduce the observed patterns and that Bayesian synthetic likelihoods is a reliable inference tool for SDEMEMs.

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Applied Stochastic Models and Data Analysis 1993: Volume 9 , Issue CONTENTS. Digitized from IA1534624-02 . Previous issue: sim_applied-stochastic-models-in-business-and-industry_1992-12_8_4 . Next issue: sim_applied-stochastic-models-in-business-and-industry_1993-03_9_1 .

A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with $A_n$ symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these $A_n$ symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains whose spectra of the transition matrices (resp. intensity matrices) are the same as the ones of the corresponding integrable models.

We discuss the space-time determinantal random field which arises for the PNG model in one dimension and resembles the one for Dyson's Brownian motion. The information of interest for growth processes is carried by the edge statistics of the random field and therefore their universal scaling is related to the edge properties of Gaussian multi-matrix models.

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Combat attrition can be modelled as either a deterministic or a stochastic process. In this thesis the forecasts of combat outcomes generated by deterministic and stochastic Lanchaster-type models for combat between two homogeneous forces are compared. Using stochastic formulations, the probability of winning is studied and contrasted with a deterministic win criterion for some idealized combat situations. Force ratios required to assure victory in an attack are studied from this standpoint.

Combat attrition can be modelled as either a deterministic or a stochastic process. In this thesis the forecasts of combat outcomes generated by deterministic and stochastic Lanchaster-type models for combat between two homogeneous forces are compared. Using stochastic formulations, the probability of winning is studied and contrasted with a deterministic win criterion for some idealized combat situations. Force ratios required to assure victory in an attack are studied from this standpoint.

Combat attrition can be modelled as either a deterministic or a stochastic process. In this thesis the forecasts of combat outcomes generated by deterministic and stochastic Lanchaster-type models for combat between two homogeneous forces are compared. Using stochastic formulations, the probability of winning is studied and contrasted with a deterministic win criterion for some idealized combat situations. Force ratios required to assure victory in an attack are studied from this standpoint.

The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are continuous time Markov chains. Simulation of such models is often relatively straightforward as there are easily implementable methods for the generation of exact sample paths. However, when using ensemble averages to approximate expected values, the computational complexity can become prohibitive as the number of computations per path scales linearly with the number of jumps of the process. When such methods become computationally intractable, approximate methods, which introduce a bias, can become advantageous. In this paper, we provide a general framework for understanding the weak error, or bias, induced by different numerical approximation techniques in the current setting. The analysis takes into account both the natural scalings within a given system and the step-size of the numerical method. Examples are provided to demonstrate the main analytical results as well as the reduction in computational complexity achieved by the approximate methods.

We describe an approach to model biochemical reaction networks at the level of promotion-inhibition circuitry through a class of stochastic spin models that depart from the usual chemical kinetics setup and includes spatial and temporal density fluctuations in a most natural way. A particular but otherwise generally applicable choice for the microscopic transition rates of the models also makes them of independent interest. To illustrate the formalism, we investigate some stationary state properties of the repressilator, a synthetic three-gene network of transcriptional regulators that possesses a rich dynamical behaviour.

The analysis of massive graphs is now becoming a very important part of science and industrial research. This has led to the construction of a large variety of graph models, each with their own advantages. The Stochastic Kronecker Graph (SKG) model has been chosen by the Graph500 steering committee to create supercomputer benchmarks for graph algorithms. The major reasons for this are its easy parallelization and ability to mirror real data. Although SKG is easy to implement, there is little understanding of the properties and behavior of this model. We show that the parallel variant of the edge-configuration model given by Chung and Lu (referred to as CL) is notably similar to the SKG model. The graph properties of an SKG are extremely close to those of a CL graph generated with the appropriate parameters. Indeed, the final probability matrix used by SKG is almost identical to that of a CL model. This implies that the graph distribution represented by SKG is almost the same as that given by a CL model. We also show that when it comes to fitting real data, CL performs as well as SKG based on empirical studies of graph properties. CL has the added benefit of a trivially simple fitting procedure and exactly matching the degree distribution. Our results suggest that users of the SKG model should consider the CL model because of its similar properties, simpler structure, and ability to fit a wider range of degree distributions. At the very least, CL is a good control model to compare against.

The analysis of massive graphs is now becoming a very important part of science and industrial research. This has led to the construction of a large variety of graph models, each with their own advantages. The Stochastic Kronecker Graph (SKG) model has been chosen by the Graph500 steering committee to create supercomputer benchmarks for graph algorithms. The major reasons for this are its easy parallelization and ability to mirror real data. Although SKG is easy to implement, there is little understanding of the properties and behavior of this model. We show that the parallel variant of the edge-configuration model given by Chung and Lu (referred to as CL) is notably similar to the SKG model. The graph properties of an SKG are extremely close to those of a CL graph generated with the appropriate parameters. Indeed, the final probability matrix used by SKG is almost identical to that of a CL model. This implies that the graph distribution represented by SKG is almost the same as that given by a CL model. We also show that when it comes to fitting real data, CL performs as well as SKG based on empirical studies of graph properties. CL has the added benefit of a trivially simple fitting procedure and exactly matching the degree distribution. Our results suggest that users of the SKG model should consider the CL model because of its similar properties, simpler structure, and ability to fit a wider range of degree distributions. At the very least, CL is a good control model to compare against.

The analysis of massive graphs is now becoming a very important part of science and industrial research. This has led to the construction of a large variety of graph models, each with their own advantages. The Stochastic Kronecker Graph (SKG) model has been chosen by the Graph500 steering committee to create supercomputer benchmarks for graph algorithms. The major reasons for this are its easy parallelization and ability to mirror real data. Although SKG is easy to implement, there is little understanding of the properties and behavior of this model. We show that the parallel variant of the edge-configuration model given by Chung and Lu (referred to as CL) is notably similar to the SKG model. The graph properties of an SKG are extremely close to those of a CL graph generated with the appropriate parameters. Indeed, the final probability matrix used by SKG is almost identical to that of a CL model. This implies that the graph distribution represented by SKG is almost the same as that given by a CL model. We also show that when it comes to fitting real data, CL performs as well as SKG based on empirical studies of graph properties. CL has the added benefit of a trivially simple fitting procedure and exactly matching the degree distribution. Our results suggest that users of the SKG model should consider the CL model because of its similar properties, simpler structure, and ability to fit a wider range of degree distributions. At the very least, CL is a good control model to compare against.

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment generating function computation in the particular case of the Heston model.

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment generating function computation in the particular case of the Heston model.

We study two models with n equivalent absorbing states that generalize the Domany-Kinzel cellular automaton and the contact process. Numerical investigations show that for n=2 both models belong to the same universality class as branching annihilating walks with an even number of offspring. Unlike previously known models, these models have no explicit parity conservation law.

The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued second-order differential operator). The matrix-valued spherical functions associated to the compact symmetric pair $(\mathrm{SU}(2)\times \mathrm{SU}(2), \mathrm{diag} \, \mathrm{SU}(2))$ will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright-Fisher models involving only mutation effects.

In this article we discuss several aspects of the stochastic dynamics of spin models. The paper has two independent parts. Firstly, we explore a few properties of the multi-point correlations and responses of generic systems evolving in equilibrium with a thermal bath. We propose a fluctuation principle that allows us to derive fluctuation-dissipation relations for many-time correlations and linear responses. We also speculate on how these features will be modified in systems evolving slowly out of equilibrium, as finite-dimensional or dilute spin-glasses. Secondly, we present a formalism that allows one to derive a series of approximated equations that determine the dynamics of disordered spin models on random (hyper) graphs.

The Gibbs sampler is a particularly popular Markov chain used for learning and inference problems in Graphical Models (GMs). These tasks are computationally intractable in general, and the Gibbs sampler often suffers from slow mixing. In this paper, we study the Swendsen-Wang dynamics which is a more sophisticated Markov chain designed to overcome bottlenecks that impede the Gibbs sampler. We prove O(\log n) mixing time for attractive binary pairwise GMs (i.e., ferromagnetic Ising models) on stochastic partitioned graphs having n vertices, under some mild conditions, including low temperature regions where the Gibbs sampler provably mixes exponentially slow. Our experiments also confirm that the Swendsen-Wang sampler significantly outperforms the Gibbs sampler when they are used for learning parameters of attractive GMs.

The intrinsic stochasticity of gene expression can lead to large variations in protein levels across a population of cells. To explain this variability, different sources of mRNA fluctuations ('Poisson' and 'Telegraph' processes) have been proposed in stochastic models of gene expression. Both Poisson and Telegraph scenario models explain experimental observations of noise in protein levels in terms of 'bursts' of protein expression. Correspondingly, there is considerable interest in establishing relations between burst and steady-state protein distributions for general stochastic models of gene expression. In this work, we address this issue by considering a mapping between stochastic models of gene expression and problems of interest in queueing theory. By applying a general theorem from queueing theory, Little's Law, we derive exact relations which connect burst and steady-state distribution means for models with arbitrary waiting-time distributions for arrival and degradation of mRNAs and proteins. The derived relations have implications for approaches to quantify the degree of transcriptional bursting and hence to discriminate between different sources of intrinsic noise in gene expression. To illustrate this, we consider a model for regulation of protein expression bursts by small RNAs. For a broad range of parameters, we derive analytical expressions (validated by stochastic simulations) for the mean protein levels as the levels of regulatory small RNAs are varied. The results obtained show that the degree of transcriptional bursting can, in principle, be determined from changes in mean steady-state protein levels for general stochastic models of gene expression.

There is discussion if traffic displays multiple phases (e.g. laminar, jammed, synchronized) or not. This paper presents evidence that a stochastic car following model, by changing one of its parameters, can be moved from showing two phases (laminar and jammed) to showing only one phase. Models with two phases show three states: two being homogeneous states corresponding to each phase, and a third state which consists of a mix between the two phases (phase coexistence). Although the gas-liquid analogy to traffic models has been widely discussed, no traffic-related model ever displayed a completely understood stochastic version of that transition. Having a stochastic model is however important to understand the potentially probabilistic nature of the transition. Most importantly, if indeed 2-phase models describe certain aspects correctly, then this leads to predictions for breakdown probabilities. Alternatively, if 1-phase models describe these aspects better, then there is no breakdown.Interestingly, such 1-phase models can still allow for jam formation on small scales, which may give the impression of having a 2-phase dynamics.

Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GMWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GMWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal, optimal surrender and optimal withdrawal strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.

Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GMWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GMWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal, optimal surrender and optimal withdrawal strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.

Block graphons (also called stochastic block models) are an important and widely-studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error at least on the order of $\min(\rho, \sqrt{\rho k^2/n^2})$ in the $\delta_2$ metric with constant probability, in the worst case over graphons. In particular, our bound rules out any nontrivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the minimax accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp, Tsybakov and Verzelen (2016).

Stochastic models are necessary for the realistic description of an increasing number of applications. The ability to identify influential parameters and variables is critical to a thorough analysis and understanding of the underlying phenomena. We present a new global sensitivity analysis approach for stochastic models, i.e., models with both uncertain parameters and intrinsic stochasticity. Our method relies on an analysis of variance through a generalization of Sobol' indices and on the use of surrogate models. We show how to efficiently compute the statistical properties of the resulting indices and illustrate the effectiveness of our approach by computing first order Sobol' indices for two stochastic models.

Stochastic models are necessary for the realistic description of an increasing number of applications. The ability to identify influential parameters and variables is critical to a thorough analysis and understanding of the underlying phenomena. We present a new global sensitivity analysis approach for stochastic models, i.e., models with both uncertain parameters and intrinsic stochasticity. Our method relies on an analysis of variance through a generalization of Sobol' indices and on the use of surrogate models. We show how to efficiently compute the statistical properties of the resulting indices and illustrate the effectiveness of our approach by computing first order Sobol' indices for two stochastic models.

There is vast empirical evidence that given a set of assumptions on the real-world dynamics of an asset, the European options on this asset are not efficiently priced in options markets, giving rise to arbitrage opportunities. We study these opportunities in a generic stochastic volatility model and exhibit the strategies which maximize the arbitrage profit. In the case when the misspecified dynamics is a classical Black-Scholes one, we give a new interpretation of the classical butterfly and risk reversal contracts in terms of their (near) optimality for arbitrage strategies. Our results are illustrated by a numerical example including transaction costs.

The dynamic properties and independence structure of stochastic kinetic models (SKMs) are analyzed. An SKM is a highly multivariate jump process used to model chemical reaction networks, particularly those in biochemical and cellular systems. We identify SKM subprocesses with the corresponding counting processes and propose a directed, cyclic graph (the kinetic independence graph or KIG) that encodes the local independence structure of their conditional intensities. Given a partition $[A,D,B]$ of the vertices, the graphical separation $A\perp B|D$ in the undirected KIG has an intuitive chemical interpretation and implies that $A$ is locally independent of $B$ given $A\cup D$. It is proved that this separation also results in global independence of the internal histories of $A$ and $B$ conditional on a history of the jumps in $D$ which, under conditions we derive, corresponds to the internal history of $D$. The results enable mathematical definition of a modularization of an SKM using its implied dynamics. Graphical decomposition methods are developed for the identification and efficient computation of nested modularizations. Application to an SKM of the red blood cell advances understanding of this biochemical system.

In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup $P_t$. A fundamental and still largely open problem is the understanding of the long time behavior of $\d_\h P_t$ when the initial configuration $\h$ is sampled from a highly disordered state $\nu$ (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular $b$-ary tree $\Tree^b$, we tackle the above problem for the Ising and hard core gas (independent sets) models on $\Tree^b$. If $\nu$ is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove $\nu$-almost sure weak convergence of $\d_\h P_t$ to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time $t$. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L\'evy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors -- one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of \citet*{fpss} to models of the exponential L\'evy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility. To illustrate the flexibility of our modeling framework we extend five exponential L\'evy processes to include stochastic volatility and jump-intensity. For each of the extended models, using a single fast-varying factor of volatility and jump-intensity, we perform a calibration to the S&P500 implied volatility surface. Our results show decisively that the extended framework provides a significantly better fit to implied volatility than both the traditional exponential L\'evy models and the fast mean-reverting stochastic volatility models of \citet{fpss}.

We consider community detection in Degree-Corrected Stochastic Block Models (DC-SBM). We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$(n)$ or higher. Recovery succeeds even for very heterogeneous degree-distributions. The used algorithm does not rely on parameters as input. In particular, it does not need to know the number of communities.

It is known that the implied volatility skew of FX options demonstrates a stochastic behavior which is called stochastic skew. In this paper we create stochastic skew by assuming the spot/instantaneous variance correlation to be stochastic. Accordingly, we consider a class of SLV models with stochastic correlation where all drivers - the spot, instantaneous variance and their correlation are modeled by Levy processes. We assume all diffusion components to be fully correlated as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate capacity of the presented approach in modeling stochastic skew.

In the last decade, stochastic models have shown to be very useful for quantitative modelling of social processes. Here, a configurational master equation for the description of behavioral changes by pair interactions of individuals is developed. Three kinds of social pair interactions are distinguished: Avoidance processes, compromising processes, and imitative processes. Computational results are presented for a special case of imitative processes: the competition of two equivalent strategies. They show a phase transition that describes the selforganization of a behavioral convention. This phase transition is further analyzed by examining the equations for the most probable behavioral distribution, which are Boltzmann-like equations. Special cases of Boltzmann-like equations do not obey the H-theorem and have oscillatory or even chaotic solutions. A suitable Taylor approximation leads to the socalled game dynamical equations (also known as selection-mutation equations in the theory of evolution).

This volume was digitized and made accessible online due to deterioration of the original print copy.

This volume was digitized and made accessible online due to deterioration of the original print copy.

Stochastic Mixed Integer Programs (SMIP) are recognized as one of the most formidable classes of mathematical programming problems. Not only are there significant challenges due to potentially large number of scenarios, but, SMIP with integers in the second stage give rise to a non-convex and discontinuous recourse function that may be difficult to optimize. As a result of this project, there have been significant advances in the design of algorithms for solving SMIP problems. Thanks to this project, we are able to report on solution to SMIP problems with over a million binary variables in less than 3 hours of computing on a desk-top machine! These problems arise in a variety of Air Force applications, such as adaptive command and control (AC2) under uncertainty. For instance, consider a situation in which assignments of pilots/aircrafts to targets may be contingent of sensor data revealed during the mission. In such situations, a preliminary set of assignments are made, recognizing that these will be revised once more reliable observations (regarding targets) are available. While such adaptive methods can enhance the effectiveness of C2, uncertainties can vastly enlarge the set of choices, and new algorithmic tools are necessary. Our algorithms solve such problems.

Stochastic Mixed Integer Programs (SMIP) are recognized as one of the most formidable classes of mathematical programming problems. Not only are there significant challenges due to potentially large number of scenarios, but, SMIP with integers in the second stage give rise to a non-convex and discontinuous recourse function that may be difficult to optimize. As a result of this project, there have been significant advances in the design of algorithms for solving SMIP problems. Thanks to this project, we are able to report on solution to SMIP problems with over a million binary variables in less than 3 hours of computing on a desk-top machine! These problems arise in a variety of Air Force applications, such as adaptive command and control (AC2) under uncertainty. For instance, consider a situation in which assignments of pilots/aircrafts to targets may be contingent of sensor data revealed during the mission. In such situations, a preliminary set of assignments are made, recognizing that these will be revised once more reliable observations (regarding targets) are available. While such adaptive methods can enhance the effectiveness of C2, uncertainties can vastly enlarge the set of choices, and new algorithmic tools are necessary. Our algorithms solve such problems.