在接下来这个系列中，我将介绍线性规划中的对偶单纯形法的基本原理，并示范如何实现一个简单的对偶单纯形法求解器。

趁着年末整理一下最近的一些新的见解，留个记录，以便后续反思和打脸。

对于一组独立同分布的随机变量$\{X_n\}$及相应的和$S_n=\sum_{k=1}^nX_k$，中心极限定理指出，在一定的条件下，$\frac{1}{\sqrt{n}}S_n$的分布将随$n\to\infty$趋于一个正态分布。在这篇文章中，我将介绍随机过程理论中一个相似的结论：如果我们将序列$\{S_n\}$视为随机游走，并通过线性插值的方式将其延拓为连续时间过程$\{S_t\}$，那么在一定的条件下，该连续时间过程的缩放形式$\{\frac{1}{\sqrt{n}}S_{nt}\}$将随$n\to\infty$趋于一个布朗运动$B$。该结论被称为Donsker定理。Donsker定理一定程度上说明了布朗运动的重要性：对于很多关于某个具体随机游走$X$整体轨迹的性质，例如轨迹极值，Donsker定理指出从长远来看其分布可以通过布朗运动$B$的相应性质的分布进行刻画。
For a sequence of i.i.d. random variables $\{X_n\}$ and their sums $S_n=\sum_{k=1}^nX_k$, the central limit theorem states that, under certain conditions, the distribution of $\frac{1}{\sqrt{n}}S_n$ converges to a normal distribution as $n\to\infty$. In this post, I will introduce a similar result for stochastic processes: if we regard the sequence $\{S_n\}$ as a random walk and extend it to a continuous-time process $\{S_t\}$ via linear interpolation, then under certain conditions, the scaled process $\{\frac{1}{\sqrt{n}}S_{nt}\}$ converges to a Brownian motion $B$ as $n\to\infty$. This result is called the Donsker’s theorem. Donsker’s theorem to some extent illustrates the importance of Brownian motions in the theory of stochastic processes: for many path-properties of some specific random walks $X$ such as the maximum or minimum of the trajectories, Donsker’s theorem implies that their distributions in the long run can be characterized by the distribution of the corresponding properties of a Brownian motion $B$.

在本系列的之前几篇文章中，我介绍了离散时间场景下的滤子、停时、鞅、马尔科夫链等概念。在这篇文章中，我将介绍这些概念在连续时间场景中的定义，并指出这些定义与离散时间下的定义的主要区别。
In the previous posts in this series, I introduced the concepts of filters, stopping times, martingales, and Markov chains, in the discrete-time setting. In this post, I will introduce their counterparts in the continuous-time setting, with a focus on their differences.

在上一篇文章中，我介绍了马尔科夫链（在一定条件下）的时间平均与空间平均的一致性。在这一篇文章中，我将首先介绍一种可以解释这种时空一致性的动力系统视角。然后，我将基于这种理解介绍两组使得这种时空一致性成立、但较马尔科夫条件更普适的条件。最后，我将给出一系列例子来解释这些更普适条件的应用方式。
In the last post, I introduced the consistency between the time-average and the space-average of Markov chains (under certain regularity conditions). In this post, I will first introduce a dynamical system perspective that can explain this consistency. Then, based on this understanding, I will introduce two sets of conditions that lead to the consistency between the time-average and the space-average but are more general than the Markov condition. Finally, I will provide several examples to explain how to apply these more general conditions in practice.

在上一篇文章中，我介绍了如何根据关于条件期望的结构性信息（具体来说，条件期望的单调性）来推导一些理论性质和计算一些随机变量的期望值。在这一篇文章中，我将更进一步，讨论如何利用关于条件概率测度$P$的结构性信息。具体来说，我将关注具有以下形式的$P$

In the previous post, I discussed how to use structural information about conditional expectations (specifically, monotonicities of conditional expectations) to derive several theoretical properties and calculate the expected values of some random variables. In this post, I will discuss how to use structural information about the conditional probability measure $P$. Specifically, I will focus on $P$ with the following form

$$P(X_n|\mathcal{F}_{n-1}) = p(X_{n-1},X_n),$$

其中概率转移分布$p$是一个不随时间$n$变化的函数。

in which the transition probability $p$ is a function that is invariant across time $n$.

在上一篇文章中引入条件期望这一概念时，我曾提到：在某些情况下，仅通过期望值信息，我们就能够给出随机过程的一些重要结构性细节。在这篇文章中，我将通过鞅论对这一论断进行进一步阐述。

When introducing the concept of conditional expectations in the previous post, I mentioned that in some cases, “knowing the expected values alone is enough to provide many important structural details of the stochastic process”. In this post, I will further illustrate this idea with the theory of martingales.

In the past year, I have been in the ride-hailing industry and have encountered many interesting problems. In this post, I would like to summarize my experience in formulating and tackling these problems and share my personal ideas on future directions.

随机性在我们的日常生活中随处可见：股票价格的涨跌，天气的变化以及交通拥堵的产生等现象都与其有关。随机过程理论则提供了一系列简洁有力的工具，不仅能够对这些现象进行建模解释，还为大量工程领域的奠定了分析基础，例如排队系统、最优控制、衍生品定价等。

Stochasticity is pervasive in our daily lives and is related to phenomena like the rise and fall of stock prices, the changes in weather, and the occurrence of traffic congestion. The theory of stochastic processes provides a series of elegant and powerful tools that not only can model these phenomena, but also lay the analytical foundation for a large number of fields and applications, including queueing systems, optimal control problems, and derivative pricing.

在本文和接下来的几篇文章中，我将简要介绍一些随机过程理论中的基本概念及其应用。我主要有两个目标：一是对最近对Durrett的Probability: Theory and Examples一书的学习进行总结，便于未来参考；二则是想要分享和讨论我个人对这些工具的理解与看法。

In this and the next few posts, I will briefly introduce some of the basic concepts in the theory of stochastic processes and their applications. I have two objectives: first, I would like to summarize the recent study of Durrett’s Probability: Theory and Examples for future reference; secondly, I want to share and discuss my understanding and views on these tools.

在这篇短文中，我将基于过去一年中的研究经验，简单地分享一下我目前对行为决策建模的见解。

In this post, I will briefly discuss some of my current thoughts on decision behavior modeling based on my research experience over the past year.

抽象地来看，决策行为即在给定决策信息$s$（下称状态）的情况下，通过某种策略$\pi$采取动作$a$

From an abstract point of view, decision making is to take an action $a$ according to a certain strategy $\pi$ given decision information $s$ (hereinafter referred to as the state)

$$a \sim \pi(s).$$

而行为决策建模，则希望根据观察数据$\mathcal{D} = \{(s,a)\}_n$，获取决策机制$\pi$的相关信息。

While decision behavior modeling attempts to infer the strategy $\pi$ from observations $\mathcal{D} = \{(s,a)\}_n$.

由于数据集$\mathcal{D}$是有限的，能够很好地拟合该数据集的$\pi$可能有无穷多个。因此，研究者通常需要对$\pi$的形式做出额外假设。而一个最基础且核心的假设是$\pi$服从效用最大化原理。

Because the dataset $\mathcal{D}$ has finite datapoints, there can be an infinite number of $\pi$ that fits well into the dataset. Therefore, researchers often need to make additional assumptions about the form of $\pi$. And one of the most fundamental assumptions is that $\pi$ follows the utility-maximization principle.