Gaussian Diffusions Research Articles

Fluctuations for the outbreak prevalence of the SIR epidemics in complex networks

The contacting heterogeneity is crucial for the propagation of directed transmitted infectious diseases. Here we construct a stochastic model on the level of subgroups which are classified according to the activity degrees of the hosts. Using the central limit theorem, we present the analytical results concerning with the fluctuations around the outbreak prevalence. In particular, we get the forms of the covariance matrix for Gaussian random diffusions as well as the laws of outbreak prevalence for SIR epidemics. So that the analytical distributions for the outbreak prevalence of the SIR epidemics in complex networks are presented, which can be used in outbreak prevalence predicting and transmission intensity inferring.

A Bayesian race model for response times under cyclic stimulus discriminability

Response time (RT) data from psychology experiments are often used to validate theories of how the brain processes information and how long it takes a person to make a decision. When an RT results from a task involving two or more possible responses, the cognitive process that determines the RT may be modeled as the first-passage time of underlying competing (racing) processes with each process describing accumulation of information in favor of one of the responses. In one popular model the racers are assumed to be Gaussian diffusions. Their first-passage times are inverse Gaussian random variables and the resulting RT has a min-inverse Gaussian distribution. The RT data analyzed in this paper were collected in an experiment requiring people to perform a two-choice task in response to a regularly repeating sequence of stimuli. Starting from a min-inverse Gaussian likelihood for the RTs we build a Bayesian hierarchy for the rates and thresholds of the racing diffusions. The analysis allows us to characterize patterns in a person’s sequence of responses on the basis of features of the person’s diffusion rates (the “footprint” of the stimuli) and a person’s gradual changes in speed as trends in the diffusion thresholds. Last, we propose that a small fraction of RTs arise from distinct, noncognitive processes that are included as components of a mixture model. In the absence of sharp prior information, the inclusion of these mixture components is accomplished via a two-stage, empirical Bayes approach. The resulting framework may be generalized readily to RTs collected under a variety of experimental designs.

Large Deviations for Gaussian Diffusions with Delay

Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes $X_t$ driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for $X_t$ and of the most likely paths transiting from $X_0 = p$ to $X_T=q$. Via linear noise local approximations, we can then compute most likely routes of escape from metastable states for nonlinear delay SDEs. We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle switch, which may be described by a nonlinear chemical Langevin SDE with delay.

Conic martingales from stochastic integrals

AbstractIn this paper, we introduce the concept of conic martingales. This class refers to stochastic processes that have the martingale property but that evolve within given (possibly time‐dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in [0, 1]. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion, and geometric Brownian motion) having a separable diffusion coefficient and that can be obtained via a time‐homogeneous mapping of Gaussian diffusions. The approach is exemplified by modeling stochastic conditional survival probabilities in the univariate and bivariate cases.

On the sequential testing and quickest change-point detection problems for Gaussian processes

We study the sequential hypothesis testing and quickest change-point (disorder) detection problems with linear delay penalty costs for certain observable time-inhomogeneous Gaussian diffusions and fractional Brownian motions. The method of proof consists of the reduction of the initial problems into the associated optimal stopping problems for one-dimensional time-inhomogeneous diffusion processes and the analysis of the associated free boundary problems for partial differential operators. We derive explicit estimates for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratios and obtain their exact asymptotic growth rates under large time values.

Sequential testing of hypotheses about drift for Gaussian diffusions

In statistical inference on the drift parameter θ in the process Xt=θa(t)+∫0tb(s)dWs, where a(t) and b(t) are known, deterministic functions, there is known a large number of options how to do it. We may, for example, base this inference on the differences between the observed values of the process at discrete times and their normality. Although such methods are very simple, it turns out that it is more appropriate to use sequential methods. For the hypotheses testing about the drift parameter θ, it is more proper to standardize the observed process and to use sequential methods based on the first exit time of the observed process of a pre-specified interval until some given time. These methods can be generalized to the case of random part being a symmetric Itô integral or continuous symmetric martingale.

Conic Martingales from Stochastic Integrals

In this paper we introduce the concept of conic martingales. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in [0,1]. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion and Geometric Brownian motion) having a separable coefficient (t; y) = g(t)h(y) and that can be obtained via a time-homogeneous mapping of Gaussian diffusions. The approach is exemplified to the modeling of stochastic conditional survival probabilities in the univariate (both conditional and unconditional to survival) and bivariate cases.

Large-time and small-ball asymptotics for quadratic functionals of Gaussian diffusions

Using asymptotic analysis of the Laplace transform, we establish almost sure divergence of certain integrals and derive logarithmic asymptotic of small ball probabilities for quadratic forms of Gaussian diffusion processes. The large time behavior of the quadratic forms exhibits little dependence on the drift and diffusion matrices or the initial conditions, and, if the noise driving the equation is not degenerate, then similar universality also holds for small ball probabilities. On the other hand, degenerate noise leads to a variety of different asymptotics of small ball probabilities, including unexpected influence of the initial conditions.

Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates

In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims.

Identification and Inference for Multivariate Cointegrated and Ergodic Gaussian Diffusions

Inference is considered in the multivariate continuous time Gaussian Ornstein-Uhlenbeck (OU) model on the basis of observations in discrete time. Under the hypothesis of ergodicity as well as cointegration, the classical identification or ‘aliasing’ problem is re-addressed and new results given. Exact conditions are given for (i) identification of individual parameters, as well as results for, (ii) identification of rank and cointegration parameters, and, furthermore (iii) for the existence of a continuous time OU process which embeds a discrete time vector autoregression. Estimation and cointegration rank inference are discussed. An empirical illustration is given in which the ‘cost-of-carry’ hypothesis is investigated.

Asymptotic Likelihood Based Inference for Co‐integrated Homogenous Gaussian Diffusions

In this paper we consider inference for a multivariate Gaussian homogenous diffusion which is co‐integrated, i.e. admits a representation in terms of stable relations (ergodic diffusions) plus Brownian motions. We show that inference on co‐integration rank (the number of stable relations) in continuous time can be based on existing asymptotic distributions from discrete time co‐integration analysis. Likewise the asymptotic distributions of the co‐integration parameters are shown to be mixed Gaussian. Special attention is given to the parametrization of the drift terms. It is shown that the asymptotic distribution of the co‐integration rank test statistic does not depend on the level of the process as a result of the chosen parametrization.

Gussian diffusions and continuous state branching processes with killing

This work considers a diffusion process , subject to killing at a rate which depends on time and current value of a diffusion. A conditioned process is defined, on , by restricting the sample space of , to the sample paths which survived up to time t. It is shown that conditioned process is a diffusion whose infinitesimal parameters are functions of the infinitesimal parameters of , and of the survival function of . This result is applied to the problem of identification of a parametric form of a killing-rate function such that an unconditioned and an associated conditioned diffusion belong to the same family of diffusions. The specific results are obtained for Gaussian diffusions and for continuous state branching processes. For these diffusions the required killing-rate functions are identified and the properties of associated diffusions are discussed. In particular formulas for marginal survival functions of these diffusions are obtained. A general formula for the survival function of a diffusion when the values of a diffusion are known at discrete times is also discussed. The applications of this work to survival analysis and possible extensions are indicated

Phase Space Transformations of Gaussian Diffusions

It is shown that for Gaussian diffusions, the transformation back to Brownian motion, usually accomplished via the Girsanov (or Feynman–Kac) formula and time-shift, can be obtained by a classical canonical, i.e. symplectic, transformation in phase space. The method is based on constants of motion, in this case the Wronskian. Similar transformations for general diffusions are briefly discussed.

Symplectic structure for Gaussian diffusions

Using information exclusively about classical Hamiltonian boundary value problems (with quadratic time-dependent potentials) we construct probability measures providing a symplectic structure associated with some inhomogeneous Gaussian diffusion processes introduced earlier [T. Kolsrod and J. C. Zambrini, J. Math. Phys. 33, 1301 (1992)].

Characterization of Multivariate Stationary Gaussian Reciprocal Diffusions

Jamison's classification of scalar stationary Gaussian reciprocal processes is extended to multivariate Gaussian reciprocal diffusions (GRDs). The second-order self-adjoint differential equation satisfied by the covariance of a GRD specifies a Hamiltonian matrix whose eigenstructure is employed to parametrize the covariance of stationary GRDs. Characterizations of the conservation matrix and of the covariance matrix of the end point values of a stationary GRD are also provided.

Gaussian reciprocal diffusions and positive definite sturm-liouville operators

This paper examines the connection between Gaussian reciprocal diffusions (GRDs) on a finite interval [0, T] and positive definite Sturm-Liouville boundary values problems (BVPs) on the same interval. We show that there exists a bijection between the set of GRDs on [0, T] and the set of positive definite Sturm-Liouville BVPs on [0, T]. The bijection occurs through the identification of GRD covariances with Sturm-Liouville Green's functions. Furthermore, every GRD x(t) can be formulated as a weak solution of its corresponding Sturm-Liouville BVP, where the interior forcing terms and the boundary forcing terms form a stochastic object whose covariance structure is determined by that of x(t). This formulation differs from the one presented in [20] in that it represents x(t) as the solution of its matching positive definite Sturm-Liouville BVP, as opposed to a Dirichlet BVP. Finally, for GRDs with a negative stress tensor [11], it is shown that the Sturm-Liouville BVP they satisfy can be reformulated as a fir...

Gaussian reciprocal diffusions and their associated conservation laws

Krener and collaborators have shown that Gaussian Reciprocal Diffusions (GRDs) x(t) must satisfy on [0,T] a 2nd order stochastic differential equation of the form , where ξ(t) is a generalized noise process. Krener has also shown that such GRDs have conditional moments satisfying a system of conservation laws similar to those of continuum mechanics In this paper we further investigate the relationship between GRDs. their conditional moments and their associated conservation laws. We show that a GRD is determined by its lifetime, its reciprocal invariants and the initial values of its conditional moments. A characterization is then given of those functions that are allowable as the initial values for the conditional moments of a GRD having a given lifetime and given reciprocal invariants. We show further that the conservation laws associated with a GRD form a hyperbolic system for either all or none of the process lifetime, and we give criteria for determining which of these alternatives holds

Second-Order Markov and Reciprocal Stationary Gaussian Processes

Previous article Next article Second-Order Markov and Reciprocal Stationary Gaussian ProcessesR. N. MiroshinR. N. Miroshinhttps://doi.org/10.1137/1124099PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernshtei˘n, On the dependencies among random variablesCollected Works, Vol. IV, Nauka, Moscow, 1964, 235–254, (In Russian.) Google Scholar[2] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[3] Benton Jamison, Reciprocal processes: The stationary Gaussian case, Ann. Math. Statist., 41 (1970), 1624–1630 MR0267637 0248.60030 CrossrefGoogle Scholar[4] I. I. Gikhman and , A. V. Skorokhod, The Theory of Stochastic Processes, Vol. I, Springer-Verlag, New York, 1974 CrossrefGoogle Scholar[5] E. Kamke, Differentialgleichungen. Lösungsmethoden and Lösungen, Vol. 1, Chelsea, New York, 1974 Google Scholar[6] E. Wong, Some results concerning the zero-crossings of Gaussian noise, SIAM J. Appl. Math., 14 (1966), 1246–1254 10.1137/0114099 MR0207059 0147.15905 LinkGoogle Scholar[7] E. Wong, The distribution of intervals between zeros for a stationary Gaussian process, SIAM J. Appl. Math., 18 (1970), 67–73 10.1137/0118008 MR0256456 0221.60024 LinkGoogle Scholar[8] R. N. Miroshin, Convergence of Rice and Longuet-Higgins series for a Wong process, Theory Prob. Applications, 21 (1976), 863–866 10.1137/1121103 0374.60049 LinkGoogle Scholar[9] R. N. Miroshin, Asymptotics of the second factorial moment of the number of upward crossings of the line $kt+a$ by a Gaussian stationary process, Vestnik Leningrad. Univ., 23 (1968), 109–120 MR0235612 0165.52602 Google Scholar[10] M. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 17–22 MR0004333 0022.35302 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Characterization of Multivariate Stationary Gaussian Reciprocal DiffusionsJournal of Multivariate Analysis, Vol. 62, No. 1 Cross Ref Gaussian reciprocal diffusions and positive definite sturm-liouville operators4 April 2007 | Stochastics and Stochastic Reports, Vol. 55, No. 3-4 Cross Ref Gaussian reciprocal processes revisitedStatistics & Probability Letters, Vol. 12, No. 4 Cross Ref Reciprocal covariance solutions of some matrix differential equationsStochastic Processes and their Applications, Vol. 37, No. 1 Cross Ref Characterization and Classification of Gaussian Second Order Reciprocal Processes Cross Ref Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order2 May 2007 | Stochastics and Stochastic Reports, Vol. 34, No. 1-2 Cross Ref Reciprocal diffusions and stochastic differential equations of second order ∗Stochastics, Vol. 24, No. 4 Cross Ref Multivariate reciprocal stationary Gaussian processesJournal of Multivariate Analysis, Vol. 23, No. 1 Cross Ref On Miroshin’s Second-Order Reciprocal ProcessesJulia Abrahams3 August 2006 | SIAM Journal on Applied Mathematics, Vol. 44, No. 1AbstractPDF (316 KB)The Use of Rice SeriesR. N. Miroshin28 July 2006 | Theory of Probability & Its Applications, Vol. 28, No. 4AbstractPDF (1011 KB)The zero-crossing problem for some nonstationary Gaussian processes (Corresp.)IEEE Transactions on Information Theory, Vol. 28, No. 4 Cross Ref Convergence of Longuet-Higgins Series for Stationary Gaussian Markov Processes of First OrderR. N. Miroshin17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 1AbstractPDF (1546 KB) Volume 24, Issue 4| 1980Theory of Probability & Its Applications History Submitted:11 July 1977Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1124099Article page range:pp. 845-852ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

An alternative to angle lap and stain analysis of Gaussian diffusions

The junction depth and surface concentration for a Gaussian diffusion into a known uniform background can be determined by measuring the depletion layer capacitance and the sheet resistance. A family of sheet resistance versus depletion layer capacitance curves is computed. They facilitate the direct conversion of experimental data to diffusion data. The technique offers a fast, accurate, nondestructive technique for the analysis of Gaussian diffusions.

Approximation of Stochastic Processes by Gaussian Diffusions, and Applications to Wright-Fisher Genetic Models

For each $N\geqq 1$, let $\{ {X_n^N ,n\geq 0} \}$ be a discrete-time stochastic process, and let $\Delta X_n^N = X_{n + 1}^N - X_n^N $. Suppose that $E( {\Delta X_n^N | {X_n^N } } ) = O( {\varepsilon ^N } )$ and $\operatorname{var} ( {\Delta X_n^N | {X_n^N } } ) = O( {\tau ^N } )$, where $\varepsilon ^N \to 0$ and ${\tau ^N / \varepsilon ^N \to 0}$ as $N \to \infty $. Conditions are given under which there are constants $\gamma _n^N $ such that $Z_n^N ( {X_n^N - \gamma _n^N } )( \varepsilon ^N / \tau ^N )^{1 / 2} $ can be approximated by a Gaussian diffusion when N is large. It is shown that these conditions are satisfied by the Wright–Fisher models for fluctuations in gene frequency under theinfluence of mutation, selection and random drift. For these models, N is the population size and the constants $\gamma _n^N $ are the gene frequencies specified by Haldane’s deterministic theory of evolution.