Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics

We consider a d-dimensional random field that solves a possibly non-linear system of stochastic partial differential equations, such as stochastic heat or wave equations. We present results, obtained in joint works with Davar Khoshnevisan and Eulalia Nualart, and with Marta Sanz-Sole, on upper and lower bounds on the probabilities that the random field visits a deterministic subset of \(\mathbb {R}^d\), in terms, respectively, of Hausdorff measure and Newtonian capacity of the subset. These bounds determine the critical dimension above which points are polar, but do not, in general, determine whether points are polar in the critical dimension. For linear SPDEs, we discuss, based on joint work with Carl Mueller and Yimin Xiao, how the issue of polarity of points can be resolved in the critical dimension.

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

Comparison of systems of stochastic partial differential equations

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

In some problems, partial differential equations are reduced to ordinary differential equations. In special cases, when incorporating randomness, equations can be reduced to systems of stochastic differential Equations (SDEs). Stochastic averaging for a class of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise is considered. Stability criteria for systems of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise do not currently exist. Usually, studies on determining the sensitivity of solutions to the accuracy of setting the initial conditions are being conducted to explain the phenomenon of deterministic chaos. These studies show both convergence in mean square and convergence in probability to averaged systems of stochastic differential equations driven by fractional Brownian motion and Lévy process. The solutions to systems can be approximated by solutions to averaged stochastic differential equations by using the stochastic averaging.

2. Statistical Simulation of the Cauchy Problem Solution for Systems of Stochastic Differential Equations. 2.8. Mean Square Consistency of Methods was published in Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations on page 94.

2. Statistical Simulation of the Cauchy Problem Solution for Systems of Stochastic Differential Equations. 2.7. Families of Numerical Methods for Solving SDE Systems. Theorem of Convergence was published in Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations on page 90.

This article presents an analysis of deterministic models of computer virus epidemic propagation (SIR models) and their classification. The main areas of research into these models are highlighted. An analysis of existing stochastic models based on the SIR model and their diversity are presented. A method for constructing a stochastic SIR model based on the classical SIR model, represented by a system of Ito stochastic differential equations with a Wiener process, is proposed. The possibility of constructing a stochastic and deterministic model of an information system protected against computer virus infection with probability 1 is demonstrated: a stochastic model, in which infection by viruses occurs continuously, and a deterministic model, in which the virus is present in the information system. A mathematical stochastic model of an information system protected from computer virus outbreaks is constructed as a system of stochastic differential equations whose first integrals are invariants preserved with probability 1. A certain functional relationship between model variables, maintaining a constant value, is considered as the system's security indicator. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Similarly, based on the proposed algorithm, a deterministic model of an information system protected from computer virus infection is constructed. A control similar to programmed control with probability 1 (PCP1) is introduced into the constructed model, which allows the invariants to be maintained. A distinctive feature of the proposed models is that they preserve invariants associated with the properties that ensure the security of the information system. The behavior of the constructed models is studied using numerical simulation in the MathCad environment. Based on the research results, conclusions were drawn on the possibility of using the proposed method in constructing stochastic models based on other models of epidemic spread, as well as for models of protecting an information system from the spread of a computer virus epidemic.

Fractional Brownian Motions, Fractional Noises and Applications

In this paper, we study a two-species model in the form of a coupled system of nonlinear stochastic differential equations (SDEs) that arises from a variety of applications such as aggregation of biological cells and pedestrian movements. The evolution of each process is influenced by four different forces, namely an external force, a self-interacting force, a cross-interacting force and a stochastic noise where the two interactions depend on the laws of the two processes. We also consider a many-particle system and a (nonlinear) partial differential equation (PDE) system that associate to the model. We prove the wellposedness of the SDEs, the propagation of chaos of the particle system, and the existence and (non)-uniqueness of invariant measures of the PDE system.

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by cylindrical $ \alpha $-stable process with $ \alpha\in (1, 2) $. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.

The main goal of this paper is to prove that there are solutions to stochastic Burgers type equations that have an oblique reflection direction. For penalized systems of stochastic partial differential equations (SPDEs), we use approximations. The method makes use of approximations of SPDE penalized systems. We establish the existence of solutions for stochastic Burgers type heat equations with oblique reflection by using a priori estimates for the solutions of the penalized systems of SPDEs.

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time tau and provide a novel asymptotic estimate of the cumulative distribution function of tau . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

In this article we test the ability of the stochastic differential model proposed by Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329–362] of forecasting the returns of a long-short equity hedge fund index and of a market index, that is, of the Hedge Fund Research performance Index (HFRI)-Equity index and of the S&P 500 (Standard & Poor 500 New York Stock Exchange) index, respectively. The model is based on the assumptions that the value of the variation of the log-return of the hedge fund index (HFRI-Equity) is proportional up to an additive stochastic error to the value of the variation of the log-return of a market index (S&P 500) and that the log-return of the market index can be satisfactorily modelled using the Heston stochastic volatility model. The model consists of a system of three stochastic differential equations, two of them are the Heston stochastic volatility model and the third one is the equation that models the behaviour of the hedge fund index and its relation with the market index. The model is calibrated on observed data using a method based on filtering and maximum likelihood proposed by Mariani et al. [Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: An application of nonlinear filtering theory, Opt. Lett., 2 (2008), pp. 177–222] and further developed in Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329–362; The calibration of the Heston stochastic volatility model using filtering and maximum likelihood methodsin Proceedings of Dynamic Systems and Applications, Vol. 5, G.S. Ladde, N.G. Medhin, C. Peng, and M. Sambandham, eds., Dynamic Publishers, Atlanta, USA, 2008, pp. 170–181]. That is, an inverse problem for the stochastic dynamical system representing the model is solved using the calibration procedure. The data analysed is from January 1990 to June 2007, and are monthly data. For each observation time, they consist of the value at the observation time of the log-returns of the HFRI-Equity and of the S&P 500 indices. The calibration procedure uses appropriate subsets of data, that is the data observed in a 6 months time period. The 6 months data time period used in the calibration is rolled through the time series generating a sequence of calibration problems. The values of the HFRI-Equity and S&P 500 indices forecasted using the calibrated models are compared to the values of the indices observed. The result of the comparison is very satisfactory. The website http://www.econ.univpm.it/recchioni/finance/w8 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of some of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).