Research on Carbon Emission for Preventive Maintenance of Wind Turbine Gearbox Based on Stochastic Differential Equation

Time based maintenance(TBM)and condition based maintenance(CBM)are widely applied in many large wind farms to optimize the maintenance issues of wind turbine gearboxes,however,these maintenance strategies do not take into account environmental benefits during full life cycle such as carbon emissions issues.Hence,this article proposes a carbon emissions computing model for preventive maintenance activities of wind turbine gearboxes to solve the issue.Based on the change of the gearbox state during operation and the influence of external random factors on the gearbox state,a stochastic differential equation model(SDE)and corresponding carbon emission model are established,wherein SDE is applied to model the evolution of the device state,whereas carbon emission is used to implement carbon emissions computing.The simulation results indicate that the proposed preventive maintenance cannot ensure reliable operation of wind turbine gearboxes but reduce carbon emissions during their lifespan.Compared with TBM,CBM minimizes unit carbon emissions without influencing reliable operation,making it an effective maintenance method.

GENERAL COUPLED MEAN-FIELD REFLECTED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations(FBSDEs),whose coefficients not only depend on the solution but also on the law of the solution.The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations(BSDEs)under Lipschitz conditions,and for the one-dimensional case a comparison theorem is studied.With the help of this comparison result,we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions.It should be mentioned that,under appropriate assumptions,we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner.

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to stochastic delay differential equations. Based on the Caratheodory approximate procedure, it was proved that stochastic delay differential equations have unique solution and established that the Caratheodory approximate solution converges to the unique solution of stochastic delay differential equations under the Cauchy sequence and initial condition. This Caratheodory approximate procedure and Euler method both converge at the same rate. This is achieved by replacing the present state with past state. The existence and uniqueness of an approximate solution of the stochastic delay differential equation were shown and the approximate solution to the unique solution was also shown. .

RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.

Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM

In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.

Asymptotic Behavior of the Drift Coefficient Estimator of Stochastic Differential Equations Driven by Small Noises

The parametric estimation problem for diffusion processes with small white noise based on continuous time observations is well developed. However,in parametric inference,it is more realistic and interesting to consider asymptotic estimation for diffusion processes based on discrete observations. The least squares method is used to obtain the estimator of the drift parameter for stochastic differential equations( SDEs) driven by general Lévy noises when the process is observed discretely. Its strong consistency and the rate of convergence of the squares estimator are studied under some regularity conditions.

FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH STOPPING TIME

The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also proved.

EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.

RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.

ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS, WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS, AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO- DIFFERENTIAL EQUATIO

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.

Solutions to general forward-backward doubly stochastic differential equations

A general type of forward-backward doubly stochastic differential equations （FBDSDEs） is studied. It extends many important equations that have been well studied, including stochastic Hamiltonian systems. Under some much weaker monotonicity assumptions, the existence and uniqueness of measurable solutions are established with a incthod of continuation. Furthermore, the continuity and differentiability of the solutions to FBDSDEs depending on parameters is discussed.

FULLY COUPLED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERAL MARTINGALE

The article first studies the fully coupled Forward-Backward Stochastic Differential Equations （FBSDEs） with the continuous local martingale. The article is mainly divided into two parts. In the first part, it considers Backward Stochastic Differential Equations （BSDEs） with the continuous local martingale. Then, on the basis of it, in the second part it considers the fully coupled FBSDEs with the continuous local martingale. It is proved that their solutions exist and are unique under the monotonicity conditions.

Forward-backward Stochastic Differential Equations and Backward Linear Quadratic Stochastic Optimal Control Problem

In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedback regulator for the optimal control problem by using the solutions of a group of Riccati equations.

Numerical Analysis of Balanced Methods for the Impulsive Stochastic Differential Equations

Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.

Stochastic Approximate Solutions of Stochastic Differential Equations with Random Jump Magnitudes and Non-Lipschitz Coefficients

A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.

Variational Approach for the Adapted Solution of Backw ard Stochastic Differential Equations with Locally Lipschitz Diffusion Coefficients

One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".

AN AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS DRIVEN BY TIME-CHANGED LéVY NOISE

In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability,respectively.The convergence order is also estimated in terms of noise intensity.Finally,an example with numerical simulation is given to illustrate the theoretical result.

Data-Driven Discovery of Stochastic Differential Equations

Stochastic differential equations(SDEs)are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources.The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system’s dynamics.The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources.This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning(SBL)technique to search for a parsimonious,yet physically necessary representation from the space of candidate basis functions.More importantly,we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data.The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices,bearing variation,and wind speed,as well as simulated data on well-known stochastic dynamical systems,including the generalized Wiener process and Langevin equation.This framework aims to assist specialists in extracting stochastic mathematical models from random phenomena in the natural sciences,economics,and engineering fields for analysis,prediction,and decision making.

Two-stage Milstein Methods for Stochastic Differential Equations

In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations （SDEs）. Six fully explicit methods （TSM 1 -- TSM 6） are given in this paper. Their order of strong convergence is proved. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of Ito SDEs.