### 统计代写|随机控制代写Stochastic Control代考|MATH4406

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机控制代写Stochastic Control代考|Numerical simulations

Approximate evolution equations, equations (33) and (34), are intractable theoretically, since the global properties are replaced with the local. Numerical experiments under a variety of conditions allow examining the effectiveness of the approximate estimation procedure. The following set of initial conditions and system parameters can be chosen for the numerical testing:
\begin{aligned} &\alpha=-1, a=0.001, \beta=-0.2, b=0.8, \sigma_{B}=0.028, \sigma_{u}=0.07, \bar{x}{1}(0)=0.1, \bar{x}{2}(0)=0.5 \ &P_{11}(0)=1, P_{12}(0)=0, P_{22}(0)=2, n=3 . \end{aligned}
Here the initial variances are chosen ‘non-zero’ and covariances take zero values, which illustrate uncertainties in initial conditions and the uncertainties are initially uncorrelated respectively. The order $n$ of the state-dependent perturbation $\sigma_{B} x_{t}^{n} d B_{t}$ is three, since this choice of the order contributes to higher-order partials of the diffusion coefficient $\left(G G^{T}\right)\left(x_{t}, t\right)$ and allows to examine the efficacy of higher-order estimation algorithms.

## 统计代写|随机控制代写Stochastic Control代考|The Itô calculus for a noisy dynamical system

The deterministic versions of dynamical systems have been studied extensively in literature. The notion of noisy dynamical systems is attributed to random initial conditions and small perturbations felt by dynamical systems. The stochastic differential equation formalism is utilized to describe noisy dynamical systems. The Itô calculus, a pioneering contribution of Kiyoshi Itô, is regarded as a path-breaking discovery in the branch of mathematical science in which the term ‘ $d B_{t}^{\prime}=\dot{B}{t} d t$, where the Brownian motion $B=\left{B{t}, t_{0} \leq t<\infty\right}$. The Itô theory deals with multi-dimensional Itô differential rule, Itô stochastic integral and subsequently, can be exploited to analyse non-linear stochastic differential systems.

This chapter discusses the usefulness of Itô theory to analysing a noisy dynamical system. In this chapter, we consider a system of two coupled second-order fluctuation equations, which has central importance in noisy dynamical systems. Consider the system of the coupled fluctuation equations of the form
\begin{aligned} \ddot{x}{1} &=F{1}\left(t, x_{1}, \dot{x}{1}, x{2}, \dot{x}{2}, \dot{B}{1}\right), \ \ddot{x}{2} &=F{2}\left(t, x_{1}, \dot{x}{1}, x{2}, \dot{x}{2}, \dot{B}{2}\right) \end{aligned}
where the state vector $x_{t}=\left(x_{1}, x_{2}, \dot{x}{1}, \dot{x}{2}\right)^{T}$ and the vector Brownian motion $B_{t}=\left(B_{1}, B_{2}\right)^{T}$. Interestingly, a suitable choice of the right-hand side terms $F_{1}, F_{2}$ of the above formalism describes the motion of an orbiting satellite in noisy environment, which $w^{\prime} \mathrm{d}$ be the subject of discussion. After accomplishing the phase space formulation, the structure of the dynamical system of concern here becomes a multidimensional stochastic differential equation. Remarkably, in this chapter, the resulting SDE is analysed using the Itô differential rule in contrast to the Fokker-Planck approach. This chapter aims to open the topic to a broader audience as well as provides guidance for understanding the estimation-theoretic scenarios of stochastic differential systems.
Key words: Brownian motion, Itô differential rule, Fokker-Planck approach, second-order fluctuation equations, multi-dimensional stochastic differential equation.

## 统计代写|随机控制代写Stochastic Control代考|Numerical simulations

$$\alpha=-1, a=0.001, \beta=-0.2, b=0.8, \sigma_{B}=0.028, \sigma_{u}=0.07, \bar{x} 1(0)=0.1, \bar{x} 2(0)=0.5 \quad P_{11}(0)$$

## 统计代写|随机控制代写Stochastic Control代考|The Itô calculus for a noisy dynamical system

$$\ddot{x} 1=F 1\left(t, x_{1}, \dot{x} 1, x 2, \dot{x} 2, \dot{B} 1\right), \ddot{x} 2=F 2\left(t, x_{1}, \dot{x} 1, x 2, \dot{x} 2, \dot{B} 2\right)$$

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