Ito Differential Equations Research Articles

Analyzing reciprocity dynamics in supply chains of public goods: a stochastic evolutionary game approach.

To start with an infinitely repeated game of supply chains of public goods, a stout reciprocity mechanism is introduced into income games to build a matric dynamic equation. The conventional evolutionary game method is employed to propose a model called the evolutionary game for the cooperative strategy of both the manufacturer and the seller groups in the supply chain of public goods. Also, white Gaussian noise (WGN) is added to reflect random interference in the evolution process. Then, a stochastic dynamic system is established, and Ito's differential equation is used to analyze both sides' strategy evolution in a game, interpret changes in system stability when random disturbance is added, and finally test the influence of different situations on the system stability by running a numerical simulation. The research shows that the stronger the reciprocity coefficient is, and the system is subjected to random interference, the greater the strategy choice change in players' decision-making procedures when the repeated game of public goods is conducted.

Resolvents of the Ito Differential Equations Multiplicative with Respect to the State Vector

Resolvents of the Ito Differential Equations Multiplicative with Respect to the State Vector

Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations

Abstract This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations

The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS

In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed.
 Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole.
 It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

On construction of a field of forces along given trajectories in the presence of random perturbations

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

Programmed Control with Probability 1 for Stochastic Dynamical Systems

In this paper, we suggest a new type of tasks for control theory for stochastic dynamical systems — programmed control with Probability 1 (PCP1). PCP1 is an application of an invariant theory. We use the PCP1 concept for dynamical processes described by a system of Ito differential equations with jump-diffusion (GSDES). The considered equations include the drift, the diffusion, and the jumps, together or not. Features of our approach are both a wide set of dynamical systems and investigation of such systems for their unique trajectories. Our method is based on the concept of a stochastic first integral (SFI) for GSDES and its equations which author studied before. The purpose of the present paper is to construct a differential equation system (both stochastic and deterministic) using a known set of FIs for the investigating process. Several examples are given.

Robust incentive Stackelberg strategy for Markov jump linear stochastic systems via static output feedback

In this study, a robust static output feedback (SOF) incentive Stackelberg game for a Markov jump linear stochastic system governed by Ito differential equations with multiple leaders and multiple followers is investigated. The existence conditions for the SOF incentive Stackelberg strategies are derived in terms of the solvability of a set of higher-order cross-coupled stochastic algebraic Lyapunov type equations (CCSALTEs). A classical Lagrange multiplier technique is employed to solve the CCSALTEs; therefore, the solution of the bilinear matrix inequality, which is a common NP-hard problem when designing SOF strategies, is not required. A heuristic algorithm is developed based on the CCSALTEs. In particular, it is shown that a robust convergence is guaranteed by combining the Krasnoselskii–Mann iterative algorithm with a new convergence condition. The performance of the proposed algorithm is discussed and a simple practical example is provided to demonstrate the effectiveness of the proposed algorithm and the SOF incentive Stackelberg strategies.

Stationary probability densities of generalized Maxwell-type viscoelastic systems under combined harmonic and Gaussian white noise excitations

Viscoelasticity has been widely studied over the past few decades as a combination of the viscous effects of energy dissipation and the elastic effects of energy storage. As an application, viscoelastic dampers are commonly used to suppress the dynamic behaviour of structures. However, in an actual physical environment, it is difficult to explore the stationary probability densities of random responses for viscoelastic systems. For this purpose, we present a numerical method to investigate the dynamic behaviour of a generalized Maxwell-type viscoelastic system under harmonic and Gaussian white noise excitations. Using approximate equivalent and stochastic averaging, we establish an averaged Ito differential equation for the amplitude of the system. For primary external resonance, we solve the reduced Fokker–Planck–Kolmogorov equation by using the successive over-relaxation technique combined with a finite difference method. Through Monte Carlo simulations, we verify the applicability, accuracy and efficiency of the proposed methodology and demonstrate that viscoelasticity has a significant impact on the dynamic behaviour of the viscoelastic systems. This work reveals the remarkable influences of viscoelasticity, excitation intensities, linear and nonlinear damping and linear and nonlinear stiffness on the probability density functions of system responses, which can guide the selection of materials, stiffnesses and structures in viscoelastic damper design.

An approach dealing with inertia nonlinearity of a cantilever model subject to lateral basal Gaussian white noise excitation

Abstract An uniform inextensible slender cantilever model with longitudinal inertia nonlinearity under lateral basal Gaussian white noise excitation was studied. The effect of inertia nonlinearity was especially taken into account, which is the main novelty of this study. A modified stochastic averaging method for strong nonlinearity was applied to transform the system into an Ito differential equation about the transient equivalent amplitude. After that, a prediction-correction method was presented to improve the predicting accuracy. The stationary probability density function (PDF) of transient equivalent amplitude, as well as the joint PDF of the displacement and velocity was studied. The reliability function and the probability density of first passage failure time were also investigated by the theoretical analysis and the Monte Carlo simulation. The Monte Carlo results vindicated these approaches.

Stochastic Modelling for Dynamics of Interacting Populations

Abstract The paper deals with a mathematical model for two interacting populations. Under the assumption of fast stochastic contacting of populations, we derive stochastic Poisson-type differential equations with a small parameter and propose an approximative algorithm for quantitative analysis of population dynamics that consists of two steps. First, we derive an ordinary differential equation for mean value of each population growth and analyse the average asymptotic population behaviour. Then, applying diffusion approximation procedure, we derive a stochastic Ito differential equation for small random deviations on the average motion in a form of a linear non-homogeneous Ito stochastic differential equation and analyse the probabilistic characteristics of the Gaussian process given by this equation.

The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator

We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

On Long Term Investment Optimality

We study the problem of optimal long term investment with a view to beat a benchmark for a diffusion model of asset prices. Two kinds of objectives are considered. One criterion concerns the probability of outperforming the benchmark and seeks either to minimise the decay rate of the probability that a portfolio exceeds the benchmark or to maximise the decay rate that the portfolio falls short. The other criterion concerns the growth rate of an expected risk-sensitised utility of wealth which has to be either minimised, for a risk-averse investor, or maximised, for a risk-seeking investor. It is assumed that the mean returns and volatilities of the securities are affected by an economic factor, possibly, in a nonlinear fashion. The economic factor and the benchmark are modelled with general Ito differential equations. The results identify asymptotically optimal portfolios and produce the decay, or growth, rates. The proportions of wealth invested in the individual securities are time-homogeneous functions of the economic factor. Furthermore, a uniform treatment is given to the out—and under—performance probability optimisation as well as to the risk-averse and risk-seeking portfolio optimisation. It is shown that there exists a portfolio that optimises the decay rates of both the outperformance probability and the underperformance probability. While earlier research on the subject has relied, for the most part, on the techniques of stochastic optimal control and dynamic programming, in this contribution the quantities of interest are studied directly by employing the methods of the large deviation theory. The key to the analysis is to recognise the setup in question as a case of coupled diffusions with time scale separation, with the economic factor representing “the fast motion”.

Dynamics of a Population Subject to Impulse Type Random Loss

Abstract We present a qualitative population growth analysis approach using the Pearl logistic population growth differential equation for a population with the intensity of birth a, stationary size K to instances of fast small random population size extractions proportional to εξ n at random time moments τ n , where ε is the small positive parameter. Assuming that the intervals τ n − τ n−1 are independent identically exponentially distributed random variables with the parameter λ / ε, and that ξ n are the independent identically distributed positive random variables with the mean h and the variance b 2, we analyse the population dynamics and the population asymptotic behaviour. We propose a probabilistic limit theorem based stochastic approximation algorithm for the qualitative analysis of the above model on any finite time interval. At first we derive the linear differential equation for mathematical expectation 𝔼{x(t)} of the population growth and the stochastic Ito differential equation for the normalised deviations (𝔼{x(t)} − x(t))ε−1/2. Assuming that the difference a − λh = εc is sufficiently small we derive the stochastic differential equation for the scaled population growth in accelerated time ε−1 Kx(t / ε)and prove that under condition 2c < λ(h 2+b 2) the population disappears with probability one, otherwise the distribution of the scaled population size with increasing time tends to the Gamma-distribution Γ(k,q) with the shape k = 2c/λ(h 2 + b 2) and the scale θ = λ(h 2 + b 2)/2c.

Estimation of the evolution of a random set

An estimation problem for a random set that is a reachable domain of the Ito differential equation with respect to its initial data is considered. The Markov property of the reachable set in the space of closed sets is proved. For the purposes of numerical solution, a random initial set of the differential equation is approximated by a finite set on an integer multidimensional grid, and the differential equation is replaced by a multistep Markov chain. Examples are considered.

Solution of the Problem of Stochastic Stability of an Integral Manifold by the Second Lyapunov Method

By the method of Lyapunov functions, we establish sufficient conditions of stability and asymptotic stability in probability for the integral manifold of the Ito differential equations in the presence of random perturbations from the class of processes with independent increments. Theorems on stochastic stability of the analytically given integral manifold of differential equations are proved in the first approximation and under the permanent action of small (in the mean) random perturbations.

Stochastic Response of Dynamical Systems with Fractional Derivative Term under Wide-Band Excitation

Transient solution of a fractional stochastic dynamical system under wide-band noise excitation is investigated. Generalized Harmonic Balance technique is firstly used to approximate restoring force of the given system as an amplitude-dependent form. In this way, stochastic averaging method then can be applied to transform the system into an Ito differential equation. Furthermore, the fractional derivative in the integral-differential form can be equivalent to a combination of periodic functions after the averaging procedure. As the following, Galerkin method therein is utilized to obtain the transient probability density functions by solving associated Fokker-Planck-Kolmogorov (FPK) equation. As an example, the Rayleigh oscillator is studied to illustrate the efficiency and accuracy of the proposed approaches. Numerical results show that exact stationary solution and transient solution derived from Galerkin method are in good agreement with those from Monte Carlo Simulation.

Moment Lyapunov exponents of the Parametrical Hill's equation under the excitation of two correlated wideband noises

The Lyapunov exponent and moment Lyapunov exponents of Hill\' s equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged Ito differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.

The Nonlinear Random Vibration of a Clamped Rectangular Thin Plate in Magnetic Field

In this paper, the magneto-elastic nonlinear random vibration of a clamped rectangular thin plate in magnetic field is studied. According to the magneto-elastic theory of plates and shells and the theory of structural random vibration, the magneto-elastic nonlinear random vibration equation of a clamped rectangular thin plate in a magnetic field is derived. Then the nonlinear random vibration equation is transferred into the Ito differential equation, and the Ito differential equation is solved using FPK equation method. Thus the numerical characteristics of displacement response and velocity response of the rectangular thin plate are obtained. Finally, through a numerical example, the influences of magnetic field parameters on the numerical characteristics are discussed, and some methods which can be used to effectively control the random vibration responses of the plate are given.

Effects of stochastic noise on dynamical decoupling procedures

Dynamical decoupling is an important tool to counter decoherence and dissipation effects in quantum systems originating from environmental interactions. It has been used successfully in many experiments; however, there is still a gap between fidelity improvements achieved in practice compared to theoretical predictions. We propose a model for imperfect dynamical decoupling based on a stochastic Ito differential equation which could explain the observed gap. We discuss the impact of our model on the time evolution of various quantum systems in finite- and infinite-dimensional Hilbert spaces. Analytical results are given for the limit of continuous control, whereas we present numerical simulations and upper bounds for the case of finite control.