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金融代写|随机控制理论代写STOCHASTIC CONTROL代考|MATH691 Independent model

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金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Independent model

We shall consider an independent sequences as a special model. Let $\left(W_{n}\right){n \in N}$ be a sequence of independent distributed random variables with $P\left(\sup {n}\left|W_{n}\right|<\infty\right)=1$, and let $Z$ be a random variable which is independent of $\left(W_{n}\right){n \in N}$. For each $n \in N$ let $\mathcal{F}{n}$ be the $\sigma$-field generated by $\left{W_{k} ; k \leq n\right}$. Also, for each $n \in N$, let $X_{n}=W_{n}-c$ and $Y_{n}=W_{n}+d$, where $c$ and $d$ are positive constants.

Since $\mathcal{F}{n}$ is independent of $\left{W{k} ; k>n\right}$, the relation in Proposition $3.1$ (b) is represented as follows:
\begin{aligned} V_{n}(r) &=\operatorname{mid}\left(\tilde{X}{n}(r), \tilde{Y}{n}(r), E\left[V_{n+1}(r)\right]\right) \ &=\operatorname{mid}\left(I_{\left(W_{n} \leq r+c\right)}, I_{\left(W_{n} \leq r-d\right)}, E\left[V_{n+1}(r)\right]\right) . \end{aligned}

金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Conditional expectation and coherent conditional prevision

Partial knowledge is a natural interpretation of conditional probability. This interpretation can be formalized in a different way in the axiomatic approach and in the subjective approach where conditional probability is respectively defined by the Radon-Nikodym derivative or by the axioms of coherence. In both cases conditional probability is obtained as the restriction of conditional expectation or conditional prevision to the class of indicator functions of events. Some critical situations, which highlight as the axiomatic definition of conditional probability is not always a useful tool to represent partial knoweledge, are proposed in literature and analyzed in this section. In particular the role of the Radon-Nikodym derivative in the assessment of a coherent conditional prevision is investigated.

It is proven that, every time that the $\sigma$-field of the conditioning events is properly contained in the $\sigma$-field of the probability space and it contains all singletons, the Radon-Nikodym derivative cannot be used as a tool to define coherent conditional previsions. This is due to the fact that one of the defining properties of the Radon-Nikodym derivative, that is to be measurable with respect to the $\sigma$-field of the conditioning events, contradicts a necessary condition for the coherence.

Analysis done points out the necessity to introduce a different tool to define coherent conditional previsions.

金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Independent model

Yleft 的分隔符缺失或无法识别

$$V_{n}(r)=\operatorname{mid}\left(\tilde{X} n(r), \tilde{Y} n(r), E\left[V_{n+1}(r)\right]\right) \quad=\operatorname{mid}\left(I_{\left(W_{n} \leqslant r+c\right)}, I_{\left(W_{n} \leqslant r-d\right)}, E\left[V_{n+1}(r)\right]\right) .$$

金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Conditional expectation and coherent conditional prevision

Nikodym 导数或相干公理定义。在这两种情况下，条件概率都是作为对事件指标函数类别的条件期望或条件预知的限制而获得

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。