### 统计代写|随机控制代写Stochastic Control代考|MAST90059

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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|Evolution of conditional probability density

The Fokker-Planck equation describes the evolution of conditional probability density for given initial states for the Itô stochastic differential system. The equation is also known as the prediction density evolution equation, since it can be utilized to develop prediction algorithms, especially where observations are not available at every time instant. One of the potential applications of the Fokker-Planck equation is to develop estimation algorithms for the satellite trajectory estimation. This chapter summarizes four different proofs to arrive at the Fokker-Planck equation. The first two proofs can be regarded as elementary proofs and the last two utilize the Itô differential rule. Moreover, the Fokker-Planck equation for the OU process-driven stochastic differential equation is discussed here, where the input process has non-zero, finite, relatively smaller correlation time.

The first proof of this chapter begins with the Chapman-Kolmogorov equation. The Chapman-Kolmogorov equation is a consequence of the theory of the Markov process. This plays a key role in proving the Kolmogorov backward equation (Feller 2000). Here, we describe briefly the Chapman-Kolmogorov equation and subsequently, the concept of the conditional probability density as well as transition probability density are introduced to derive the evolution of conditional probability density for the non-Markov process. The Fokker-Planck equation becomes a special case of the resulting equation. The conditional probability density
$$p\left(x_{1}, x_{2} \mid x_{3}\right)=p\left(x_{1} \mid x_{2}, x_{3}\right) \ldots p\left(x_{2} \mid x_{3}\right)$$
Consider the random variables $x_{t_{1}}, x_{t_{2}}, x_{t_{3}}$ at the time instants $t_{1}, t_{2}, t_{3}$, where $t_{1}>t_{2}>t_{3}$ and take values $x_{1}, x_{2}, x_{3}$. In the theory of the Markov process, the above can be re-stated as
$$p\left(x_{1}, x_{2} \mid x_{3}\right)=p\left(x_{1} \mid x_{2}\right) p\left(x_{2} \mid x_{3}\right)$$

## 统计代写|随机控制代写Stochastic Control代考|A stochastic Duffing-van der Pol system

The second-order fluctuation equation describes a dynamical system in noisy environment. The second-order fluctuation equation can be regarded as
$$\ddot{x}{t}=F\left(t, x{t}, \dot{x}{t}, \dot{B}{t}\right) .$$
The phase space formulation allows transforming a single equation of order $n$ into a system of $n$ first-order differential equations. Choose $x_{t}=x_{1}$
$$\begin{gathered} \dot{x}{1}=x{2}, \ \dot{x}{2}=F\left(t, x{1}, x_{2}, \dot{B}{t}\right) \end{gathered}$$ by considering a special case of the above system of equations, we have $$\begin{gathered} d x{1}=x_{2} d t \ d x_{2}=f_{2}\left(t, x_{1}, x_{2}\right) d t+g_{2}\left(t, x_{1}, x_{2}\right) d B_{t} \end{gathered}$$
in matrix-vector format
$$d \xi_{t}=f\left(t, x_{1}, x_{2}\right) d t+G\left(t, x_{1}, x_{2}\right) d B_{t^{\prime}}$$
where
$$\xi_{t}=\left(\xi_{1}, \xi_{2}\right)^{T}=\left(x_{1}, x_{2}\right)^{T}, f\left(t, \xi_{t}\right)=\left(x_{2}, f_{2}\right)^{T}, G\left(t, \xi_{t}\right)=\left(0_{2}, g_{2}\right)^{T}$$

## 统计代写|随机控制代写Stochastic Control代考|Evolution of conditional probability density

Fokker-Planck 方程描述了 Itô 随机微分系统给定初始状态的条件概率密度的演变。该方程也称为预测密度演化方 程，因为它可用于开发预测算法，尤其是在每个时刻都无法获得观测值的情况下。Fokker-Planck方程的潜在应 用之一是开发用于卫星轨迹估计的估计算法。本章总结了得出 Fokker-Planck方程的四种不同证明。前两个证明 可以看作基本证明，后两个利用伊藤微分规则。此外，这里讨论了 OU 过程驱动的随机微分方程的 FokkerPlanck 方程，其中输入过程具有非零、有限、

Kolmogorov方程，随后引入了条件概率密度和转移概率密度的概念，以推导非马尔可夫过程的条件概率密度的 演变。Fokker-Planck 方程成为所得方程的特例。条件概率密度
$$p\left(x_{1}, x_{2} \mid x_{3}\right)=p\left(x_{1} \mid x_{2}, x_{3}\right) \ldots p\left(x_{2} \mid x_{3}\right)$$

$$p\left(x_{1}, x_{2} \mid x_{3}\right)=p\left(x_{1} \mid x_{2}\right) p\left(x_{2} \mid x_{3}\right)$$

## 统计代写|随机控制代写Stochastic Control代考|A stochastic Duffing-van der Pol system

$$\ddot{x} t=F(t, x t, \dot{x} t, \dot{B} t) .$$

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

$$d x 1=x_{2} d t d x_{2}=f_{2}\left(t, x_{1}, x_{2}\right) d t+g_{2}\left(t, x_{1}, x_{2}\right) d B_{t}$$

$$d \xi_{t}=f\left(t, x_{1}, x_{2}\right) d t+G\left(t, x_{1}, x_{2}\right) d B_{t^{\prime}}$$

$$\xi_{t}=\left(\xi_{1}, \xi_{2}\right)^{T}=\left(x_{1}, x_{2}\right)^{T}, f\left(t, \xi_{t}\right)=\left(x_{2}, f_{2}\right)^{T}, G\left(t, \xi_{t}\right)=\left(0_{2}, g_{2}\right)^{T}$$

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