Random Differential Equations Research Articles

Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P′(t,ω)=A(t,ω)(1−P(t,ω))P(t,ω), t∈[t0,T], P(t0,ω)=P0(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P0 to approximate the density function f1(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P0, and the density of P0, , is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L2([t0,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P0, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For , only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence of density functions in terms of expectations that tends to f1(p,t) pointwise. Numerical examples illustrate our theoretical results.

Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

In this article, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic differential equations. The second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties are stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss, in particular, previous results obtained by Étoré et al. [J. Differ. Equations 257, 2115–2135 (2014)] and we finally propose a new family of stochastic Landau-Lifshitz equations.

Using the stochastic Galerkin method as a predictive tool during an epidemic

The ability to accurately predict the course of an epidemic is extremely important. This article looks at an influenza outbreak that spread through a small boarding school. Predictions are made on multiple days throughout the epidemic using the stochastic Galerkin method to consider a range of plausible values for the parameters. These predictions are then compared to known data points. Predictions made before the peak of the epidemic had much larger variances compared to predictions made after the peak of the epidemic.
 
 References B. M. Chen-Charpentier, J. C. Cortes, J. V. Romero, and M. D. Rosello. Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation, 219(9):4208 4218, 2013. doi:10.1016/j.amc.2012.11.007 B. M. Chen-Charpentier and D. Stanescu. Epidemic models with random coefficients. Mathematical and Computer Modelling, 52:1004 1010, 2010. doi:10.1016/j.mcm.2010.01.014 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with individualised parameter distributions. In Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C160C176, August 2016. doi:10.21914/anziamj.v57i0.10394 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with uncertainty in the parameters. Mathematical Biosciences, 277:25 37, 2016. doi:10.1016/j.mbs.2016.03.012 D. B. Harman and P. R. Johnston. Boarding house: find border. 2019. doi:10.6084/m9.figshare.7699844.v1 D. B. Harman and P. R. Johnston. SIR uniform equations. 2 2019. doi:10.6084/m9.figshare.7692392.v1 H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4):599653, 2000. doi:10.1137/S0036144500371907 R.I. Hickson and M.G. Roberts. How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. Journal of Theoretical Biology, 350(0):70 80, 2014. doi:10.1016/j.jtbi.2014.01.014 W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115(772):700721, August 1927. doi:10.1098/rspa.1927.0118 M. G. Roberts. A two-strain epidemic model with uncertainty in the interaction. The ANZIAM Journal, 54:108115, 10 2012. doi:10.1017/S1446181112000326 M. G. Roberts. Epidemic models with uncertainty in the reproduction number. Journal of Mathematical Biology, 66(7):14631474, 2013. doi:10.1007/s00285-012-0540-y F. Santonja and B. Chen-Charpentier. Uncertainty quantification in simulations of epidemics using polynomial chaos. Computational and Mathematical Methods in Medicine, 2012:742086, 2012. doi:10.1155/2012/742086 Communicable Disease Surveillance Centre (Public Health Laboratory Service) and Communicable Diseases (Scotland) Unit. Influenza in a boarding school. BMJ, 1(6112):587, 1978. doi:10.1136/bmj.1.6112.586 G. Strang. Linear Algebra and Its Applications. Thomson, Brooks/Cole, 2006. D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, 2010.

Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem

ABSTRACT A computational approach to approximate the probability density function of random differential equations is based on transformation of random variables and finite difference schemes. The theoretical analysis of this computational method has not been performed in the extant literature. In this paper, we deal with a particular random differential equation: a random diffusion-reaction Poisson-type problem of the form , , with boundary conditions , . Here, α, A and B are random variables and is a stochastic process. The term is a stochastic process that solves the random problem in the sample path sense. Via a finite difference scheme, we approximate with a sequence of stochastic processes in both the almost sure and senses. This allows us to find mild conditions under which the probability density function of can be approximated. Illustrative examples are included.

Study of a boundary value problem for fractional order psi -Hilfer fractional derivative

This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in psi -Hilfer sense. By utilizing classical fixed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justification of our analysis we provide two examples.

A fractional mathematical model of breast cancer competition model

In this paper, a mathematical model which considers population dynamics among cancer stem cells, tumor cells, healthy cells, the effects of excess estrogen and the body’s natural immune response on the cell populations was considered. Fractional derivatives with power law and exponential decay law in Liouville–Caputo sense were considered. Special solutions using an iterative scheme via Laplace transform were obtained. Furthermore, numerical simulations of the model considering both derivatives were obtained using the Atangana–Toufik numerical method. Also, random model described by a system of random differential equations was presented. The use of fractional derivatives provides more useful information about the complexity of the dynamics of the breast cancer competition model.

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay $$\tau >0$$ : $$x'(t)=ax(t)+bx(t-\tau )$$ , $$t\ge 0$$ , with initial condition $$x(t)=g(t)$$ , $$-\tau \le t\le 0$$ . The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using $$\mathrm {L}^p$$ -calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an $$\mathrm {L}^p$$ -solution too. An analysis of $$\mathrm {L}^p$$ -convergence when the delay $$\tau $$ tends to 0 is also performed in detail.

Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.

Random fractional differential equations with Riemann-Liouville-type fuzzy differentiability concept

Random fractional differential equations with Riemann-Liouville-type fuzzy differentiability concept

On the initial value problem for random fuzzy differential equations with Riemann-Liouville fractional derivative: Existence theory and analytical solution

On the initial value problem for random fuzzy differential equations with Riemann-Liouville fractional derivative: Existence theory and analytical solution

On a stochastic arc furnace model

One of the approaches in modeling of electric arc furnace is based on the power balance equation and results in a nonlinear ordinary differential equation. In reality it can be observed that the graph of the arc voltage varies randomly in time, in fact it oscillate with a random time-varying amplitude and a slight shiver. To get a more realistic model, at least one of the model parameters should be modeled as a stochastic process, which leads to a random differential equation. We propose a stochastic model using the stationary Ornstein-Uhlenbeck process for modeling stochastic influences. Results, gained by applying Monte Carlo method and polynomial chaos expansion, are given here.

A systematic path to non-Markovian dynamics: new response probability density function evolution equations under Gaussian coloured noise excitation

Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic dynamics, advanced statistical physics and uncertainty quantification of macroscopic systems. In the present work, such equations are derived for a scalar, nonlinear RDE under additive coloured Gaussian noise excitation, through the stochastic Liouville equation. The latter is an exact, yet non-closed equation, involving averages over the time history of the non-Markovian response. This non-locality is treated by applying an extension of the Novikov–Furutsu theorem and a novel approximation, employing a stochastic Volterra–Taylor functional expansion around instantaneous response moments, leading to efficient, closed, approximate equations for the response pdf. These equations retain a tractable amount of non-locality and nonlinearity, and they are valid in both the transient and long-time regimes for any correlation function of the excitation. Also, they include as special cases various existing relevant models, and generalize Hänggi's ansatz in a rational way. Numerical results for a bistable nonlinear RDE confirm the accuracy and the efficiency of the new equations. Extension to the multidimensional case (systems of RDEs) is feasible, yet laborious.

Hilfer and Hadamard random fractional differential equations in Fréchet spaces

Hilfer and Hadamard random fractional differential equations in Fréchet spaces

Pricing and Hedging Equity-Linked Life Insurance Contracts Beyond the Classical Paradigm: The Principle of Equivalent Forward Preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.

A deep learning solution approach for high-dimensional random differential equations

Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution approach for these problems based on deep learning. This approach is intrusive, entirely unsupervised, and mesh-free. Specifically, the random PDE is approximated by a feed-forward fully-connected deep residual network, with either strong or weak enforcement of initial and boundary constraints. Parameters of the approximating deep neural network are determined iteratively using variants of the Stochastic Gradient Descent (SGD) algorithm. The satisfactory accuracy of the proposed approach is numerically demonstrated on diffusion and heat conduction problems, in comparison with the converged Monte Carlo-based finite element results.

Z-Differential Equations

This paper is devoted to make a framework for studying a class of uncertain differential equations called Z -differential equations. In order to achieve the purpose, we first introduce four basic operations on $Z^{+}$ -numbers based on semigranular function. Then, the limit and continuity concepts of a Z -number-valued function are given, under a definition of a metric on the space of $Z^{+}$ -numbers. Moreover, the concepts of Z -differentiability, Z -integral, and Z -Laplace transform of a Z -number-valued function are introduced. In addition, by giving some theories proved in this paper, a basis for calculus— Z -calculus—is established. We further give theories based on which existence and uniqueness of Z -differential equations are investigated. A conceptual unity between Z -differential equations and $Z^{+}$ -numbers is also shown. The conceptual unity demonstrates that a Z -differential equation may be expressed as a bimodal differential equation combining a fuzzy differential equation (FDE) and a random differential equation. Moreover, the concept of a bimodal cut called $ (s, \mu)$ -cut is introduced and its relation to other new concepts such as acceptable time and acceptable information area is explained. Using an example, the application of Z -differential equations in medicine is clarified. It is demonstrated that Z -differential equations outperform FDEs in making a decision under uncertainty.

Random differential equations with discrete delay

In this article, we study random differential equations with discrete delay with initial condition The uncertainty in the problem is reflected via the outcome ω. The initial condition g(t) is a stochastic process. The term x(t) is a stochastic process that solves the random differential equation with delay in a probabilistic sense. In our case, we use the random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.

Practical trajectory tracking of random Lagrange systems

The problem of trajectory tracking is considered in this paper for Lagrange systems disturbed by second moment processes. For random differential equations, the concept of noise-to-state practical stability and its criterion are proposed. A state-feedback tracking control is designed by using vectorial backstepping method, which covers Slotine–Li controller and “PD+” controller as special cases. As natural extension, adaptive control is further researched, and a practical equivalence principle is presented. For the above two cases, results of global noise-to-state stability of closed-loop systems are obtained, and practical trajectory tracking can be achieved under a practical parameters-tuning principle. Simulations are conducted for a nonlinear benchmark system to illustrate the effectiveness and advantages of the proposed new control strategies.

Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay

We randomize the following class of linear differential equations with delay, xτ′(t)=axτ(t)+bxτ(t−τ),t > 0, and initial condition, xτ(t)=g(t),−τ≤t≤0, by assuming that coefficients a and b are random variables and the initial condition g(t) is a stochastic process. We consider two cases, depending on the functional form of the stochastic process g(t), and then we solve, from a probabilistic point of view, both random initial value problems by determining explicit expressions to the first probability density function, f(x, t; τ), of the corresponding solution stochastic processes. Afterwards, we establish sufficient conditions on the involved random input parameters in order to guarantee that f(x, t; τ) converges, as τ→0+, to the first probability density function, say f(x, t), of the corresponding associated random linear problem without delay (τ=0). The paper concludes with several numerical experiments illustrating our theoretical findings.

Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative

In this paper, the existence, uniqueness and stability of random implicit fractional differential equations (RIFDs) with nonlocal condition and impulsive effect involving a generalized Hilfer fractional derivative (HFD) are discussed. The arguments are discussed via Krasnoselskii's fixed point theorems, Schaefer's fixed point theorems, Banach contraction principle and Ulam type stability. Some examples are included to ensure the abstract results.