ManifoldCaffe 流形与咖啡

Optimal Stopping“System” evolves in time according to differential equation. “Controls” modify some parameters or stop the process. “Cost criterion” depends on the choice of control and the corresponding state of the the system. The goal is to discover an optimal choice of controls, to minimize the cost criterion. { \left\{ \begin{align} & \mathrm{d} X = b(X) \mathrm{d} t + B(X) \mathrm{d} W, \\ & X_{0} = x. \end{align} \right. }{ J_{x}(\theta) := \mathbb{E}\left( \underbrace{ \int_{0}...

LimitDefinition: The Limit of a Function over a Base.A topology space . A function on to . A neighborhood of point is a subset of containing an open set containing . A base is a family of sets s.t. . ${B{1}, B{2} \in \mathcal{B}}\implies{\exists\,B \in \mathcal{B}\,(B \in B{1} \cap B{2}))}$.{ (\lim_{\mathcal{B}} f = a) := \forall\,V(a)\,(\exists\,B \in \mathcal{B}\,(f(B) \subset V(a))). } DifferentialDefinition: Differential.Let be an open region on and be a map on to and a...

Mathematical LogicCommon symbols of mathematical logic: , , , , . Set Theory We take a set to be an assemblage of definite, perfectly distinguishable objects of our intuition or our thought into a coherent whole.— Georg Cantor Cantor, the great creator of set theory, put forward basic assumptions of set: A set may consist of any distinguishable objects. A set is unambiguously determined by the collection of objects that comprise it. Any property defines the set of objects having that...

量化之路酝酿良久，终于迈步。 导言这个项目的主要目的是 熟悉搭建量化交易系统的主要流程； 验证一个简单的套利策略的可行性。 交易的标的是BTC现货和合约。 架构设计1234graph LR A[数据模块] --> B[策略模块] B --> C[回测模块] --> E[可视化模块] B --> D[交易模块] --> E 模块搭建数据模块 数据源： Binance...

工作流 obsidian书写博文; 将博文放入网站文件夹; 在终端运行git commit -m <comments>和git push origin main. 1Obsidian安装latex-suite插件搭建博客Hexo安装创建博客配置数学支持butterfly主题安装个性化GitHub托管注册创建仓库连接本地仓库上传阿里云购买实名制备案Netlify注册连接GitHub配置域名cloudflare配置域名

在读, 随缘写.

Proposition: Kolmogorov ineq. : Let be an i.i.d r.v. seq., , then , .Supremum inequality, is a martingale Theorem.: Let be a iid r.v. and , then converges a.e. . Proof: W. L. O. G. , suppose . Let . converges a.s. as , Definition: A measurable space is called separable, if every atom of it is singleton. Two measurable spaces are called isomorphic if there exists a bijection that is measurable both sides (which is called measurable a isomorphism)....

§1 Metric Space§1.1 Contraction Mapping PrincipleDefinition:(Metric space) Let be a non-empty set. is called metric space if there is a function on into s.t. ; ; . (). Here is call metric on . A metric space with as metric is denoted as . {\sqrt{ }}§1.2 CompletionDefinition: A sequence is called a fundamental sequence iff s.t. . Definition: A sequence in converges to iff ().We denote it as . Definition: A subset of is closed iff . Definition: A subset of is called...

§1 Introduction§§1.2 Kolmogorov Thm. and Separability1. Kolmogorov TheoremProposition: Suppose is a Polish space with a compatible probability measure family , then there is a probability on s.t. coincides with on finite rectangle parabolic set on . 2. SeparabilityDefinition: Separability of stochastic process: open interval , closed set , , , a.eIf is separable, then , a.e. -measurable. (), when 3. Shift OperatorDefinition: (Shift operator) Let , is closed under addition. Define...