Space Of Stochastic Processes Research Articles

ВОССТАНОВЛЕНИЕ ДИНАМИЧЕСКИ ИСКАЖЕННЫХ СИГНАЛОВ НА ОСНОВЕ ТЕОРИИ ОПТИМАЛЬНОГО УПРАВЛЕНИЯ РЕШЕНИЯМИ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА В ПРОСТРАНСТВАХ СЛУЧАЙНЫХ ПРОЦЕССОВ

This paper investigates the solvability of the optimal control problem for solutions of stochastic Sobolev type equations. It is shown that the optimal dynamic measurement problem can be considered as an optimal control problem. To do this, the mathematical model of dynamic measurements is reduced to a stochastic Sobolev type equation of the first order in the spaces of stochastic processes. The article presents theorems on the existence of a unique classical and strong solutions of the Sobolev type equation with initial condition of Showalter–Sidorov in the spaces of stochastic processes. The theorem of the unique solvability of the optimal control problem for such equation is proved. The abstract results obtained for Sobolev type equation are applied to the problem of restoring a dynamically distorted signal as an optimal dynamic measurement.

A Stochastic View on the Effect of Random Rail Irregularities on Railway Bridge Vibrations

This contribution studies dynamic amplification factors that consider the impact of random rail irregularities on the deflection and acceleration response of simple railway bridges. Response statistics for a set of plane fully coupled bridge–vehicle interaction models with random irregularity profiles are derived by Monte Carlo simulations. The underlying samples of random irregularity profiles are generated from a power spectral density function, assuming that the uneven rail surface is governed by a Gaussian stochastic process in space. A modified bootstrap method is used to identify the required sample size of profiles for sufficiently accurate description of the random response. On the basis of this pilot study, a parametric response examination of short span railway bridges with variable length, fundamental frequency, structural mass and damping, and track quality in a range commonly observed in real structures is conducted. A statistical representation of the uncertain bridge response is employed for analytical modelling of the response amplification due to random rail irregularities considering limit state-based exceedance probabilities. The results are compared to outcomes of existing simplified approaches in design codes used to take into account the effect of rail irregularities on the dynamic bridge response, showing the improvement of the developed framework.

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.

Operator differential-algebraic equations with noise arising in fluid dynamics

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

Input-to-State Stability of Linear Stochastic Functional Differential Equations

The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “theW-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.

On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces

This paper is concerned with solution in weighted Holder spaces for backward stochastic partial differential equations (BSPDEs) in a half space. Considering the solution as functional with value in Banach spaces of stochastic processes, and using the methods of partial differential equations (PDEs), we establish the existence and uniqueness of classical solution for BSPDE in functional weighted Holder spaces.

The Measurable Space of Stochastic Processes

We introduce a stochastic extension of CCS endowed with structural operational semantics expressed in terms of measure theory. The set of processes is organised as a measurable space by the sigma-algebra generated by structural congruence. The structural operational semantics associates to each process a set of measures over the space of processes. The measures encode the rates of the transitions from a process state of a system to a measurable set of processes. We prove that the stochastic bisimilarity is a congruence, which extends the structural congruence. In addition to an elegant operational semantics, our calculus provides a canonic way to define metrics on processes that measure how similar two processes are in terms of behaviour.

Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer

We perform global sensitivity analysis (GSA) through polynomial chaos expansion (PCE) on a contaminant transport model for the assessment of radionuclide concentration at a given control location in a heterogeneous aquifer, following a release from a near surface repository of radioactive waste. The aquifer hydraulic conductivity is modeled as a stationary stochastic process in space. We examine the uncertainty in the first two (ensemble) moments of the peak concentration, as a consequence of incomplete knowledge of (a) the parameters characterizing the variogram of hydraulic conductivity, (b) the partition coefficient associated with the migrating radionuclide, and (c) dispersivity parameters at the scale of interest. These quantities are treated as random variables and a variance-based GSA is performed in a numerical Monte Carlo framework. This entails solving groundwater flow and transport processes within an ensemble of hydraulic conductivity realizations generated upon sampling the space of the considered random variables. The Sobol indices are adopted as sensitivity measures to provide an estimate of the role of uncertain parameters on the (ensemble) target moments. Calculation of the indices is performed by employing PCE as a surrogate model of the migration process to reduce the computational burden. We show that the proposed methodology (a) allows identifying the influence of uncertain parameters on key statistical moments of the peak concentration (b) enables extending the number of Monte Carlo iterations to attain convergence of the (ensemble) target moments, and (c) leads to considerable saving of computational time while keeping acceptable accuracy.

On the existence of continuous selections of solution and reachable sets of quantum stochastic differential inclusions

We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection continuous from the locally convex space of stochastic processes to the adapted and weakly absolutely continuous space of solutions. As a corollary, we show that the reachable sets admit some continuous selections. In the framework of the Hudson - Parthasarathy formulations of quantum stochastic calculus, our results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion. The results here compliment similar results in our previous work in [3] where continuous selections defined on the set of the matrix elements of initial values were established. JONAMP Vol. 11 2007: pp. 71-82

Risk Neutrality and Ordered Vector Spaces

The following result clarifies when preferences over time and under risk correspond to discounting and are not risk neutral. If a binary relation on a real vector space V satisfies four axioms, then there is a utility function U=fu from V to R where u from V to R is linear as a map of vector spaces and f from R to R is continuous and weakly monotone. Three axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is X-Y is weakly preferred to 0 (where 0 is the zero element in V) implies X is weakly preferred to Y (X and Y in V). The function f from R to R is linear if and only if the binary relation also satisfies the converse of the fourth axiom. When V is a real vector space of stochastic processes and 0 is the zero process, it is known that the four axioms imply the existence of discount factors and the linearity of an intra-period utility function. So preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satisfied.

Covariance kernel representations of multidimensional second-order stochastic processes

The dynamics of stationary stochastic processes in space is not exactly analogous to that of stationary stochastic processes in the time domain. This is due to the unilateral nature of the time series that is only influenced by past values as opposed to the dependence in all directions of the spatial process. In this work, we unfold the connection that exits between the covariance kernel of a multi-dimensional second-order autoregressive random process and its underlying discrete random dynamical system. Starting from a discrete random dynamical system, we show that the random process satisfying that system is governed by the modified Helmholtz equation in the continuous limit. We establish the dependence of the correlation constant on the grid size of the discretization. We also show that the random forcing term in the continuous case turns out to be a white noise process. A number of covariance functions are worked out for simple and more complex geometrical domains with various boundary conditions in multi-dimensions. We use both the discrete and the continuous systems in our computations.

Weighted BMO and discrete time hedging within the Black-Scholes model

The paper combines two objects rather different at first glance: spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) and the approximation of certain stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands. The consideration of the approximation error with respect to weighted BMO implies Lp and uniform distributional estimates for the approximation error by a John-Nirenberg type theorem. The general results about weighted BMO are given in the first part of the paper and applied to our approximation problem in the second one.

Mathematical models of vaccination.

Mathematical models of epidemics have a long history of contributing to the understanding of the impact of vaccination programmes. Simple, one-line models can predict target vaccination coverage that will eradicate an infectious agent, whilst other questions require complex simulations of stochastic processes in space and time. This review introduces some simple ordinary differential equation models of mass vaccination that can be used to address important questions about the predicted impact of vaccination programmes. We show how to calculate the threshold vaccination coverage rate that will eradicate an infection, explore the impact of vaccine-induced immunity that wanes through time, and study the competitive interactions between vaccine susceptible and vaccine resistant strains of infectious agent.

Head and flux variability in heterogeneous unsaturated soils under transient flow conditions

A numerical model for the analysis of uncertainty propagation in flow through unsaturated soils is developed. This model is based on a first‐order Taylor series expansion of the discretized Richards equation. Soil hydrologic properties (the saturated hydraulic conductivity and the pore size distribution parameter) are assumed to be stochastic processes in space. The surface boundary conditions can be considered as deterministic variables in time or stochastic time series. Spectral analysis and Monte Carlo simulations were used to verify this numerical model for flow under both steady and transient conditions. The model is then used to examine the effect of uncertainty in boundary conditions and the effect of heterogeneity on the pressure head and flux variance profiles at various times for one‐dimensional vertical flow cases. Dependence of pressure head variance on the flow conditions (drying or wetting) is examined. On the basis of the analysis it is found that the propagation of the head variance is similar to that of the concentration variance for solute transport in saturated aquifers. The head variance is proportional to the mean pressure gradient, and thus large head variances are associated with the wetting and the drying front of a moisture pulse. The peak head variance is smaller at the wetting front than it is at the drying front. This difference is attributed to the difference in the magnitude of mean hydraulic gradient and should not necessarily be interpreted as a hysteresis effect. In addition, it is shown how the variance of the flux of a moisture pulse increases with depth.

Nonlinear random wave equations

Nonlinear wave equations describing forced vibrations of a string subject to random perturbations of intensity ε are studied herein. Two kinds of random perturbations are considered; one is Gaussian in the form of initial white noise and the other is a non-Gaussian random forcing which involves the formal derivative of a Brownian sheet. The solutions uε(t,x) of the complete forced equation are a stochastic process in space and time which has a unique stochastic equilibrium. The time variable t varies in a finite interval I:=[0, T], where T is a fixed positive time, and the space variable x varies in an interval J. Assuming that the initial conditions are of sufficient regularity, the fluctuations uε(t,x) from u0(t,x), the unperturbed solution, are analyzed as the intensity of perturbation ε↓0.

Korovkin inequalities for stochastic processes

The rate of convergence of a sequence of positive linear stochastic operators { T j } j ∈ n to the unit operator I on spaces of stochastic processes X is studied. This is mainly done for stochastic processes that are smooth over a compact and convex index set Q ⊂ R k, k ⩾ 1. Nearly best upper bounds are given for ¦E(T jX)(x 0) − (EX)(x 0)¦, x 0 ϵ Q , where E is the expectation operator and T j are E-commutative. These lead to strong and elegant inequalities many times sharp, which involve the first modulus of continuity of EX α, where X α is a (partial) derivative of X. The case of Q being a compact convex subset of a real normed vector space is also met and there the upper bound is the best possible.

A discrete Stochastic Korovkin theorem

In this article we give a sufficient condition for the pointwise −− in the first mean Korovkin property on B0(P), the space of stochastic processes with real state space and countable index set Γ and bounded first moments.

Vector valued stochastic processes. IV. Integral representation of linear operations on spaces of stochastic processes

On etudie les operations lineaires continues J sur l'espace des processus a valeurs vectorielles. On obtient une representation integrale de J

Large fluctuations for a nonlinear heat equation with noise

Studies a nonlinear heat equation in a finite interval of space subject to a white noise forcing term. The equation without the forcing term exhibits several equilibrium configurations, two of which are stable. The solution of the complete forced equation is a stochastic process in space and time that has a unique stochastic equilibrium. The authors study this process in the limit of small noise, and obtain lower and upper bounds for the probability of large fluctuations. They then apply these estimates to calculate the transition probability between the stable configurations (tunnelling). This model problem can be interpreted as a rigorous version of some recent attempts to describe Euclidean quantum systems in terms of stochastic equilibrium states of a nonlinear stochastic differential equation in infinite dimensions. However, its significance goes beyond this situation and the authors' methods may be applicable to models in other areas of natural science.