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.
在下面的讨论中，我将多次使用一维布朗运动作为例子来解释概念和展示方法应用；因此，在这里我将先给出布朗运动的定义。一个一维实值连续时间随机过程$B:\mathbb{R}^+ \times \Omega \to \mathbb{R}$被称为布朗运动，若其满足以下三个条件：
(1) （独立增量）对任意$t_0 < t_1 < \cdots < t_n$，随机变量$B(t_0,\cdot), B(t_1,\cdot) - B(t_0,\cdot), \cdots, B(t_n,\cdot) - B(t_{n-1},\cdot)$独立。
(2) （正态分布）$B({s+t},\cdot) - B(s,\cdot) \sim N(0,t)$。
(3) （轨迹连续性）$P(\{\omega: t \to B(t,\omega) \text{ 连续}\}) = 1$。
In the following discussion, I will introduce several examples of the one-dimensional Brownian motion in order to explain concepts and illustrate applications of methods. Therefore, I will give the definition of Brownian motions first. A one-dimensional real-valued continuous-time stochastic process $B:\mathbb{R}^+ \times \Omega \to \mathbb{R}$ is called a Brownian motion, if it satisfies the following three conditions:
(1) (Independent increment) For any $t_0 < t_1 < \cdots < t_n$, random variables $B(t_0,\cdot)$,$ B(t_1,\cdot) - B(t_0,\cdot)$,$ \cdots$, $B(t_n,\cdot) - B(t_{n-1},\cdot)$ are independent.
(2) (Normally distributed) $B({s+t},\cdot) - B(s,\cdot) \sim N(0,t)$.
(3) (Sample-continuity) $P(\{\omega: t \to B(t,\omega) \text{ is continuous}\}) = 1$.
为简便起见，下面我将交替使用$X_t$和$X(t,\cdot)$表示随机过程$X$在$t$时刻截面对应的随机变量。
For simplicity, in the following discussion, I will use $X_t$ and $X(t,\cdot)$ interchangeably to represent the value of the stochastic process $X$ at the time $t$, which is a random variable.
样本轨迹连续性 Sample-continuity
在本系列的第一篇文章中，我曾提到理解随机过程的两种视角：由不同时间下相应随机变量组成的序列（以下简称随机变量序列）、或是取值在实值序列空间上的随机变量（以下简称序列随机变量），并指出在离散时间场景下，可以通过Kolmogorov延拓定理来说明这两种视角的一致性。但对于连续时间场景，这种一致性并不是必然的。具体来说，对一条满足一系列关于有限维分布的相容性条件的随机变量序列$\{X_t\}$，Kolmogorov延拓定理能够保证与该序列服从同样有限维分布的序列随机变量$X$的存在性；但是，该定理无法保证构造所得$X$在样本轨迹维度$X(\cdot,\omega)$上具有所期望的性质。例如，让我们考虑布朗运动的构造问题。在上述布朗运动的定义中，条件(1-2)是对有限维分布的要求，因此可以通过Kolmogorov延拓定理保证。但条件(3)要求样本轨迹是连续的；已有工具并不能实现该要求。因此，在连续时间场景下，我们需要引入新的工具和思路来解决与样本轨迹性质相关的问题；此即本节的主题。特别地，在这篇文章中，我们将主要关注样本轨迹的连续性。
In the first post of this series, I introduced two perspectives to model stochastic processes: the “random-variable-sequences” view (sequences of random variables sampled at different times), or the “sequence-random-variables” view (random variables taking values in the sequence space). In that post, I claimed that, under the discrete-time setting, the Kolmogorov extension theorem implies the consistency between these two perspectives. However, under the continuous-time setting, this consistency is no longer universal. Specifically, for a random-variable-sequence $\{X_t\}$ with consistent finite-dimensional distributions, the Kolmogorov extension theorem guarantees the existence of a sequence-random-variable $X$ with the same finite-dimensional distributions. However, the theorem does not guarantee that $X$ has the desired properties of sample trajectories $X(\cdot, \omega)$. For example, let us consider the problem of constructing Brownian motions. In the above definition of Brownian motions, conditions (1-2) are requirements on the finite-dimensional distributions and, therefore, can be guaranteed by the Kolmogorov extension theorem. However, condition (3) requires the sample trajectories to be continuous, which cannot be derived from any existing tools. Therefore, in the continuous-time setting, we need to introduce new tools and methods to resolve problems related to the properties of sample trajectories. In particular, in this post, I will focus on the continuity of sample trajectories.
对上述问题，一种自然的解决思路是：基于Kolmogorov延拓定理构造所得的$X$，我们尝试在序列空间$\Omega$里找一个与$X$足够接近且满足样本轨迹连续性的$Y$。从这种思路出发，首先要解决的问题是对“足够接近”的定义。
A natural solution to the above problem is to search in the sequence space $\Omega$ for a $Y$ that satisfies the sample-continuity condition and is sufficiently close to the $X$ constructed by Kolmogorov’s extension theorem. The first problem to be resolved in this solution is to define “sufficiently close”.
定义1.1：随机过程$X,Y$被称为不可区分，若
Definition 1.1: Stochastic processes $X,Y$ are called indistinguishable, if
$$P(\omega: X(t,\omega) = Y(t,\omega), \forall t \in \mathbb{R}) = 1.$$
定义1.2：随机过程$Y$被称为$X$的修正，如果
Definition 1.2: Stochastic processes $X,Y$ are called modifications of each other, if
$$P(\omega: X(t,\omega) = Y(t,\omega)) = 1, \forall t\in \mathbb{R}.$$
换句话说，不可区分性要求两个随机过程几乎处处相同，而修正性要求在每个具体的时间点$t$，相应的截面随机变量几乎处处相同。可以证明，在离散时间场景下，两者是等价的；但在连续时间场景下，不可区分性要强于修正性。（注：读者可以尝试构造两个互相是修正但可区分的随机过程$X,Y$。）然而，当所讨论的随机过程满足一定的样本轨迹连续性质时，我们可以证明以上两种定义之间的等价性。
In other words, the indistinguishability requires that the two processes are almost surely identical, while being modifications requires that for any specific time $t$ the corresponding random variables are almost surely identical. It can be proved that in the discrete-time setting, the two definitions are equivalent; while in the continuous-time setting, the indistinguishability is stronger. (Note: Readers can try to construct two processes $X,Y$ that are modifications of each other but distinguishable.) However, when the stochastic processes in question satisfy some sample-continuity conditions, we can show that the above two conditions are equivalent.
定理1.1：对两个互相是修正的随机过程$X,Y$，若它们的几乎任意轨迹都是（左）右连续，则它们不可区分。
Theorem 1.1: For two stochastic processes $X,Y$ that are modifications of each other, if almost every trajectory of them is (left) right-continuous, then they are indistinguishable.
简证：对任意样本$\omega$，若轨迹$X(\cdot,\omega)$和$Y(\cdot,\omega)$都是（左）右连续的，则条件$X(t,\omega) = Y(t,\omega), \forall t \in \mathbb{R}$与条件$X(t,\omega) = Y(t,\omega), \forall t \in \mathbb{Q}$等价，其中$\mathbb{Q}$是有理数集合。又由于$\mathbb{Q}$是可列的，因此单点零测事件$\{X(t,\omega) \neq Y(t,\omega)\}$的并集$\{X(t,\omega) \neq Y(t,\omega), \ \forall t \in \mathbb{Q}\}$也是零测的。
** Proof in words:** For any sample $\omega$, if the trajectories $X(\cdot,\omega)$ and $Y(\cdot,\omega)$ are both (left) right-continuous, then the condition $ X(t,\omega) = Y(t,\omega), \forall t \in \mathbb{R}$ and the condition $X(t,\omega) = Y(t,\omega), \forall t \in \mathbb{Q}$ are equivalent, where $\mathbb{Q}$ is the set of all rational numbers. Since $\mathbb{Q}$ is countable, the union of zero-measured events $\{X(t,\omega) \neq Y(t,\omega)\}$, $\{X(t,\omega) \neq Y(t,\omega), \ \forall t \in \mathbb{Q}\}$, also has measure zero.
通常来说，我们希望构造所得的$Y$与$X$不可区分，但修正$Y$要比满足不可区分性的$Y$更容易构造：要构造一个随机过程$X$的修正，只需要分别对每个时间点$t$上的随机变量$X_t$进行操作即可。定理1.1的意义在于，在一定的连续性条件下，修正性是不可区分性的充分条件，因此此时我们只需关注修正的构造。
Usually, we want the new process $Y$ to be indistinguishable from $X$; however, it is easier to construct a modification than an indistinguishable process, because for modifications we can for each time $t$ modify the corresponding random variable $X_t$ independently. The significance of theorem 1.1 is that when certain continuity conditions are provided, being a modification implies indistinguishability, and we are safe to look at modifications only.
在解决了“接近”的定义问题后，我们的问题转化为：对一个随机过程$X$，构造出一个样本轨迹连续的修正$Y$。Kolmogorov连续性定理给出了该问题的一个解决方案。
After clarifying the meaning of “sufficiently close”, we can now reformulate the construction problem as a problem to search for a sample-continuous modification $Y$ of $X$. One solution to this problem is provided by the Kolmogorov continuity theorem.
定理1.2（Kolmogorov连续性定理）：对于随机过程$X$，若$\exists \alpha,\beta,K > 0$使得$E|X_s - X_t|^{\beta} \leq K |s - t|^{\alpha + 1}$，则存在一个$X$的修正$Y$，使得$Y$是几乎处处样本轨迹连续的，且对于任意$\gamma < \frac{\alpha}{\beta}$，$Y$的几乎任意轨迹$Y(\cdot,\omega)$都是局部$\gamma-$Holder连续的。
**Theorem 1.2 (Kolmogorov’s Continuity Theorem):** For a stochastic process $X$, if there exists $\alpha,\beta, K > 0$ such that $E|X_s - X_t|^{\beta} \leq K |s - t|^{\alpha + 1}$, then there exists a modification $Y$ of $X$ such that $Y$ is almost surely sample-continuous. Moreover, for any $\gamma < \frac{\alpha}{\beta}$, almost every sample trajectory $Y(\cdot,\omega)$ of $Y$ is locally $\gamma-$Holder continuous.
定理1.2的显著性在于，样本轨迹连续性作为一个轨迹性质，可以通过有限维分布的性质来获得！这大大降低了样本轨迹连续性的构造难度。下面，我们回到构造布朗运动的例子中。假设我们通过Kolmogorov延拓定理已经得到了一个满足条件(1-2)的随机过程$B$。根据条件(1-2)，我们知对任意$s,t \in \mathbb{R}^+$有$E|B_s - B_t|^{2m} = |s-t|^m \cdot E|B_1 - B_0|^{2m}$，因此定理1.2对任意$\beta = 2m$, $\alpha = m-1, m\in \mathbb{N}$成立。现在，令$m\to \infty$，可知对任意$\gamma < 1/2$，布朗运动存在轨迹局部$\gamma$-Holder连续的连续修正。特别地，满足条件(1-3)的随机过程存在。
The significance of theorem 1.2 is that, as a property on sample trajectories, the sample-continuity condition can be derived from the property of finite time distributions! This greatly reduces the difficulty in the construction of sample-continuous processes. Now, let us return to the problem of constructing Brownian motions. Suppose we have obtained a stochastic process $B$ by the Kolmogorov extension theorem such that $B$ satisfies the condition (1-2). According to the condition (1-2), we know that for any $s,t \in \mathbb{R}^+$ we have $E|B_s - B_t|^{2m} = |s-t|^m \cdot E|B_1 - B_0|^{2m}$, so theorem 1.2 holds for any $\beta = 2m$, $\alpha = m-1, m\in \mathbb{N}$. Now, let $m\to \infty$, and we can show that for any $\gamma < 1/2$, the Brownian motion $B$ has a sample-continuous modification of which almost every sample trajectory is locally $\gamma$-Holder continuous. In particular, a stochastic process that satisfies the condition (1-3) exists.
注：在上面的讨论中，我跳过了对可测性的讨论。在连续时间场景中，随机过程的可测性会发生怎样的变化？特别地，令$C = \{\omega: t\to \omega(t) \text{ 连续}\}$。对由Kolmogorov延拓定理所得的概率空间$(\Omega,\mathcal{F},P)$，$C$是否可测？
Note: In the above discussion, I ignored the discussion of measurability. What happens to the measurability of stochastic processes in the continuous-time setting? In particular, let $C = \{\omega: t\to \omega(t) \text{ is continuous}\}$. Is $C$ measurable with respect to the probability space $(\Omega,\mathcal{F},P)$ generated by the Kolmogorov extension theorem?
滤子与停时 Filtrations and Stopping Times
在这一节中，我将介绍连续时间场景下的滤子和停时。首先让我们重温这两者在离散时间场景下的定义。对一个给定的随机变量序列$\{X_n\}$，相应的滤子$\mathcal{F}_n$代表由历史观测值形成的$\sigma$-代数：$\mathcal{F}_n = \sigma(X_{n}, X_{n-1}, \cdots, X_0)$。而对于一个给定的滤子$\{\mathcal{F}_n\}$，一个随机变量序列$\{X_n\}$被称为适应的，若$X_n\in \mathcal{F_n},\ \forall n\in\mathbb{N}$。最后，给定滤子$\{\mathcal{F}_n\}$，一个取值为非负整数的随机变量$T: \Omega \to \mathbb{N} \cup \{+\infty \}$被称为停时，若对于任意$n \in \mathbb{N}$，有$\{ T = n\} \in \mathcal{F}_n$。
In this section, I will introduce filtrations and stopping times under the continuous-time setting. First, let us review the definitions of these two concepts in the discrete-time setting. For a given random-variable-sequence $\{X_n\}$, the corresponding information set $\mathcal{F}_n$ in the filtration $\{\mathcal{F}_n\}$ represents the $\sigma$-algebra generated from historical observations: $\mathcal{F}_n = \sigma(X_{n}, X_{n-1}, \cdots, X_0)$. For a given filtration $\{\mathcal{F}_n\}$, a random-variable-sequence $\{X_n\}$ is called adapted, if $X_n\in \mathcal{F_n},\ \forall n\in\mathbb{N}$. Finally, given a filtration $\{\mathcal{F}_n\}$, a random variable $T: \Omega \to \mathbb{N} \cup \{+ \infty \}$ is called a stopping time, if for any $n \in \mathbb{N}$, $\{ T = n\} \in \mathcal{F}_n$.
根据上述讨论，容易设计出以下对连续时间场景下滤子和停时的定义：
According to the above discussion, it is straightforward to come up with the following definitions of filtrations and stopping times under the continuous-time setting:
给定随机过程$X$，集合$\mathcal{F}_t^o = \sigma(\{X_s: s\leq t\})$所形成的的集合族$\{\mathcal{F}_t^o\}$被称为$X$对应的自然滤子；
给定滤子$\{\mathcal{F}_t\}$，随机变量$T:\Omega \to \mathbb{R}^+ \cup \{+\infty\}$被称为停时，若对于任意$t \in \mathbb{R}^+$，有$\{ T \leq t\} \in \mathcal{F}_t$。
对于滤子$\{\mathcal{F}_t\}$上的停时$T$，定义$\mathcal{F}_T = \{A: A\cap \{T\leq t\} \in \mathcal{F}_t, \ \forall t \in \mathbb{R}^+\}$。
Given a stochastic process $X$, the family $\{\mathcal{F}_t^o\}$ formed by sets $\mathcal{F}_t^o = \sigma(\{X_s: s\leq t\})$ is called the natural filtration of $X$;
Given a filtration $\{\mathcal{F}_t\}$, a random variable $T:\Omega \to \mathbb{R}^+ \cup \{+\infty\}$ is called a stopping time, if for any $t \in \mathbb{R}^+$, $\{ T \leq t\} \in \mathcal{F}_t$.
For a stopping time $T$ defined on the filtration $\{\mathcal{F}_t\}$, we can define the corresponding information set $\mathcal{F}_T$ as $\mathcal{F}_T = \{A: A\cap \{T \leq t\} \in \mathcal{F}_t, \ \forall t \in \mathbb{R}^+\}$.
然而，在连续时间场景下，时间轴上的取极限操作变得复杂，此时简单采用上述定义可能会产生问题。首先，让我们考虑以下这种停时的定义：随机变量$T:\Omega \to \mathbb{R}^+ \cup \{+\infty\}$被称为宽停时，若对于任意$t \in \mathbb{R}^+$，有$\{ T < t\} \in \mathcal{F}_t$。容易看出，
However, in the continuous-time setting, taking limits in the time dimension is not trivial, and directly employing the above definitions may be problematic. For example, let us consider the following alternative definition of stopping times: a random variable $T:\Omega \to \mathbb{R}^+ \cup \{+\infty\}$ is called an optional time, if for any $t \in \mathbb{R}^+$, $\{ T < t\} \in \mathcal{F}_t$. Now, we can show that
$$\{T \leq t\} \in \mathcal{F}_t \Rightarrow \{T < t\} = \bigcup_{s < t} \{T \leq s\} \in \bigcup_{s < t} \mathcal{F}_s = \mathcal{F}_{t-} \subseteq \mathcal{F}_t;$$
$$\{T < t\} \in \mathcal{F}_t \Rightarrow \{T \leq t\} = \bigcap_{s > t} \{T < s\} \in \bigcap_{s > t} \mathcal{F}_s = \mathcal{F}_{t+} \not\subseteq \mathcal{F}_t;$$
因此，宽停时与停时的等价性取决于相应滤子$\{\mathcal{F}_t\}$的右连续性。然而，上述定义的自然滤子$\{\mathcal{F}_t^o\}$并不一定满足右连续性。例如，考虑$\Omega = \{-1,1\}, X(t,\omega) = \omega t$；容易看出$\mathcal{F}_0^o$是平凡的，而对任意$s > 0$，$\mathcal{F}_s^o = 2^{\Omega}$。因此，$F_{0+}^o \neq F_0^o$。一种简单的获取滤子右连续性、从而解决上述问题的方法是考虑$\mathcal{F}_t = \mathcal{F}_{t+}^o$。具体地说，对给定的随机过程$X$，我们称集合$\mathcal{F}_t = \bigcap_{t’ > t}\sigma(\{X_s: s\leq t’\})$所组成的集合族为$X$对应的常规滤子。这种定义的一种理解方式是：如果一些信息是可在无穷短的未来内可以获得的，那么这些信息应被认为是当前已知的。
therefore, the equivalence between optional times and stopping times depends on the right-continuity of the underlying filtration $\{\mathcal{F}_t\}$. However, the natural filtration $\{\mathcal{F}_t^o\}$ defined above does not always satisfy the right-continuous condition. For example, consider $\Omega = \{-1,1\}, X(t,\omega) = \omega t$; it is not hard to show that $\mathcal{F}_0^o$ is trivial, but for any $s > 0$, $\mathcal{F}_s^o = 2^{\Omega}$. Therefore, $F_{0+}^o \neq F_0^o$. A simple way to resolve this problem and to make the filtration right-continuous is to consider $\mathcal{F}_t = \mathcal{F}_{t+}^o$. Specifically, for a given stochastic process $X$, we call the family consisting of sets $\mathcal{F}_t = \bigcap_{t’ > t}\sigma(\{X_s: s\leq t’\})$ the usual filtration of $X$. This definition can be interpreted as follows: if some information is available in the near future, then it should be regarded as already known.
基于上述结论，在后续讨论中，在没有特殊注明的情况下我们都将假定所讨论的滤子是右连续滤子（特别地，若给定基本参照随机过程$X$，那么我们将考虑相应的完备化的常规滤子），并交替使用宽停时和停时这两个概念。容易看出，此时对$\mathcal{F}_T$的定义也不受到$\{T < t\}$和$\{T \leq t\}$之间差异的影响。
According to the above conclusions, in the following discussion, we will assume the filtration under discussion is right-continuous, if there is no further specification. In particular, if a stochastic process $X$ is given for reference, then we will consider the corresponding usual filtration. Under these assumptions, we can and will use the concept of stopping times and optional times interchangeably. It is also clear that in this case the definition of $\mathcal{F}_T$ is not affected by the difference between $\{T < t\}$ and $\{T \leq t\}$.
在本节的最后，我将介绍一些停时的相关性质。
At the end of this section, I will introduce some properties of stopping times.
性质2.1：若$S,T$为停时，则$\min(S,T),\max(S,T),S+T$也是停时。
性质2.2：对停时序列$\{T_n\}$，$\sup_n T_n,\inf_n T_n,\limsup_n T_n,\liminf_n T_n$都是停时。进一步地，若$T_n\downarrow T$或$\uparrow T$，则$T$也是停时。
性质2.3：若停时$S\leq T$，则$\mathcal{F}_S \subseteq \mathcal{F}_T$。进一步地，若$T_n \downarrow T$，则$\mathcal{F}_T = \bigcap_n \mathcal{F}_{T_n}$。
Property 2.1: If $S, T$ are stopping times, then $\min(S,T),\max(S,T), S+T$ are also stopping times.
Property 2.2: Given a stopping time sequence $\{T_n\}$, $\sup_n T_n,\inf_n T_n,\limsup_n T_n,\liminf _n T_n$ are stopping times. Furthermore, if $T_n\downarrow T$ or $\uparrow T$, then $T$ is also a stopping time.
Property 2.3: If stopping times $S\leq T$, then $\mathcal{F}_S \subseteq \mathcal{F}_T$. Furthermore, if $T_n \downarrow T$, then $\mathcal{F}_T = \bigcap_n \mathcal{F}_{T_n}$.
鞅 Martingales
在这一节中，我将介绍连续时间场景下鞅的定义及一些基本结论。给定滤子$\{\mathcal{F}_t\}$，随机过程$X$被称为（适应于$\{\mathcal{F}_t\}$的）下鞅，若其满足以下条件
In this section, I will introduce the definition of martingales under the continuous-time setting and some basic results. Given a filtration $\{\mathcal{F}_t\}$, a stochastic process $X$ is called a submartingale (adapted to $\{\mathcal{F}_t\}$), if it satisfies the following conditions
$$(1) \ E|X_t| < \infty; (2) \ X_t \in \mathcal{F}_t, \ \forall t \in \mathbb{R}^+; (3) \ E(X_s|\mathcal{F}_t) \geq X_t, \text{a.s.}, \ \forall s \geq t \geq 0. \tag{3.1}$$
特别地，$X$被称为鞅或上鞅，若上述条件(3)中的不小于关系“$\geq$”调整为等于“$=$”或不大于“$\leq$”。容易看出，该定义与离散时间场景下的定义是基本一致的。
In particular, $X$ is called a martingale or a supermartingale, if the relationship “$\geq$” in condition (3) is replaced by “$=$” or “$\leq$”. It is not hard to see that such a definition is consistent with the definition in the discrete-time setting.
下面，我将讨论连续时间场景下鞅的性质。假设我们给定了一个连续时间鞅$X$。如果我们将时间$t$的取值限制在一个可列集合$T_c$上，那么由相应截面随机变量$X_t$组成的随机过程将成为离散时间鞅，故可应用已有的关于离散时间鞅的结论。进一步地，如果$X$还满足一定的轨迹（右）连续性质，那么通过令$T_c$变得更稠密并取极限，就能将很多离散时间鞅的性质推广到连续时间鞅上。根据这种思路，我们可以证明以下定理：
Next, I will discuss the properties of martingales under the continuous-time setting. Suppose a continuous-time martingale $X$ is given. If we restrict the time $t$ on a countable set $T_c$, then the stochastic process consisting of the corresponding random variable $X_t$ is a discrete-time martingale, and existing tools can be applied. Furthermore, if $X$ also has some (right) continuous properties of sample trajectories, then by letting $T_c$ be finer and taking limits, many properties of discrete-time martingales can be generalized to continuous-time martingales. According to this idea, we can prove the following theorems:
定理3.1：假设$X$是一个右连续下鞅，且$T$是一个有界停时：$T(\omega) \leq t, \forall \omega\in\Omega$。则$EX_0 \leq EX_T \leq EX_t$，且$X_T \leq E(X_t|\mathcal{F}_T)$。
Theorem 3.1: Assume that $X$ is a right-continuous submartingale, and $T$ is a bounded stopping time: $T(\omega) \leq t, \forall \omega\in\Omega$. Then, $EX_0 \leq EX_T \leq EX_t$, and $X_T \leq E(X_t|\mathcal{F}_T)$.
证明：这里只给出第一式的证明。考虑$T_m = \lceil 2^mT \rceil/2^m$；容易看出$T_m \geq T$，且当$m\to\infty$时，$T_m \downarrow T$。又容易验证$X_k^m = X_{k2^{-m}}$构成了离散时间下鞅，因此对$X^m$和$T_m$应用离散时间下的可选停时定理可得
Proof: We only provide the proof of the first statement here. Consider $T_m = \lceil 2^mT \rceil/2^m$; we can see that $T_m \geq T$, and when $m\to\infty$, $T_m \downarrow T$. We can also verify that $X_k^m = X_{k2^{-m}}$ constitutes a discrete-time submartingale, so we can apply the optional stopping theorem for discrete-time submartingales to $X^m$ and $T_m$ and obtain the following equation
$$EX_0 \leq EX_{T_m} = EX_{\lceil 2^mT \rceil}^m \leq EX_{\lceil 2^mt \rceil}^m = EX_ {\lceil 2^mt \rceil/2^m}. \tag{3.2}$$
在式(3.2)中取极限$m\to\infty$即得证。
The result follows as we take the limit $m\to\infty$ in equation (3.2).
定理3.2：若$X$是一个一致可积的右连续下鞅，则存在随机变量$X_{\infty}$，使得$X_t$按$L^1$和a.s.-收敛到$X_{\infty}$。进一步地，对于任意停时$T$，有$EX_0 \leq EX_T \leq EX_{\infty}$，且$X_T\leq E(X_{\infty}|\mathcal{F}_T)$。
Theorem 3.2: If $X$ is an uniformly integrable right-continuous submartingale, then there exists a random variable $X_{\infty}$, such that $X_t$ converges to $X_{\infty}$ a.s. and in $L^1$. Furthermore, for any stopping time $T$, $EX_0 \leq EX_T \leq EX_{\infty}$, and $X_T\leq E(X_{\infty}|\mathcal{F}_T)$.
根据上面的讨论，样本轨迹右连续性能够帮助我们将一系列离散时间鞅的性质推广到连续时间场景中。下面的定理则指出，这并不是一个很苛刻的要求。
According to the discussion above, the right-continuity of sample trajectories can help us generalize several properties of discrete-time martingales to the continuous-time setting. The following theorem states that this is not a very strong or restrictive condition.
定理3.3：假设滤子$\{\mathcal{F}_t\}$是右连续的，而且是完备的（即对于任意满足$A\in\mathcal{F},P(A)=0$的集合$A$，有$A\in\mathcal{F}_t$）。则对于任意适应于该滤子的下鞅$X$，其存在一个右连续修正，当且仅当映射$t \to EX_t$是右连续的。特别地，若$X$是鞅，则其必然存在右连续修正。
Theorem 3.3: Suppose the filtration $\{\mathcal{F}_t\}$ is right-continuous and complete, i.e., for any set $A$ satisfying $A\in\mathcal{F},P(A)=0$, we have $A\in\mathcal{F}_t$. Then, for any submartingale $X$ adapted to this filtration, there exists a right-continuous modification if and only if the mapping $t \to EX_t$ is right-continuous. In particular, if $X$ is a martingale, then such a right-continuous modification always exists.
由于完备化的常规滤子满足定理3.3中关于滤子的条件，因此在一般情况下我们只需要考虑映射$t \to EX_t$的右连续性，而该条件是很容易构造和检验的。
Since the completed usual filtration satisfies the conditions in theorem 3.3, in general, we only need to consider the right-continuity of the mapping $t \to EX_t$, which is easy to construct and verify.
在本节的最后，我将结合布朗运动$B$给出定理3.1在计算上的一个应用。首先，我们可以证明以下结论：
At the end of this section, I will introduce an example that illustrates with the Brownian motion $B$ that how to apply theorem 3.1 in computation. First, we can prove the following result:
定理3.4：若$u(t,x)$是一个关于$t$和$x$的多项式，且
Theorem 3.4: If $u(t,x)$ is a polynomial of $t$ and $x$, and
$$\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2} = 0;$$
则$u(t,B_t)$是一个鞅。
then $u(t,B_t)$ is a martingale.
证明：首先，通过分部积分可以证明$\frac{\partial}{\partial t}E_xu(t+s,B_t) = 0$对任意$s\in\mathbb{R}^+$成立。然后，根据布朗运动的马尔科夫性，即可证得$u(t,B_t)$的鞅性：对任意$t > s$，有
Proof: First, we can prove through integration by parts that $\frac{\partial}{\partial t}E_xu(t+s,B_t) = 0$ for any $s\in\mathbb{R}^+$. Then, according to the Markov property of Brownian motions, we can show for any $t > s$ that
$$E_x(u(t,B_t)|\mathcal{F}_s) = E_x(u(t,B_{t-s})\circ \theta_s|\mathcal{F}_s) = E_{B_s}u(t,B_{t-s}) = u(s,B_s).$$
现在，考虑停时$T = \inf \{t:B_t\notin (-a,a)\}$，其中$a > 0$。根据定理3.4可知，$B_t^2 - t$和$B_t^4 - 6B_t^2t + 3t^2$都是鞅。将$t\wedge T$代入这两式中，并令$t\to\infty$；应用定理3.1和控制收敛定理即得
Now, consider the stopping time $T = \inf \{t:B_t\notin (-a,a)\}$, where $a > 0$. According to theorem 3.4, $B_t^2 - t$ and $B_t^4 - 6B_t^2t + 3t^2$ are both martingales. Substituting $t\wedge T$ into these two equations, letting $t\to\infty$, and applying theorem 3.1 and the dominated convergence theorem lead us to
$$E_0 T = a^2, \ E_0 T^2 = \frac{5}{3}a^4.$$
马尔科夫链 Markov Chains
在最后这一节中，我将介绍连续时间场景下马尔科夫链的定义及一些基本结论。在后续讨论中，我将多次引用本系列第三篇文章（以下简记为“前述文章”）中的离散马尔科夫链的相关结果。
In this section, I will introduce the definition of Markov chains under the continuous-time setting and some basic conclusions. In the following discussion, I will make frequent reference to the results in the third post in this series (abbreviated as “the previous post” subsequently).
在离散时间场景中，我们通过转移概率这一概念来对马尔科夫链进行定义，并且指出：对于任意转移概率$p$，可以构造出相应的离散马尔科夫链。下面我们将这种思路延拓到连续时间场景中。假设给定可测状态空间$(S,\mathcal{S})$，并记相应序列空间为$(\Omega,\mathcal{F}) = (S^{\mathbb{R}^+},\mathcal{S}^{\mathbb{R}^+})$。一个定义在该序列空间上的连续时间随机过程$X$被称为马尔科夫链，若存在一族转移概率$\{p_t\}_{t\geq 0}$使得对任意$t,s\geq 0, A\in \mathcal{S}$，以下条件成立：
In the discrete-time setting, we define the Markov chain with respect to the concept of transition probability, and show that for any transition probability $p$, we can construct a corresponding Markov chain. Here, we extend this idea to the continuous-time setting. Suppose a measurable state space $(S,\mathcal{S})$ is given, and we denote the corresponding sequence space as $(\Omega,\mathcal{F}) = (S^{\mathbb{R}^+}, \mathcal{S}^{\mathbb{R}^+})$. A continuous-time stochastic process $X$ defined on this sequence space is called a Markov chain, and if there exists a family of transition probabilities $\{p_t\}_{t\geq 0}$ such that for any $t,s\geq 0, A\in \mathcal{S}$, the following conditions hold:
$$P(X_{t+s}\in A|\mathcal{F}_s) = p_t(X_s,A). \tag{4.1}$$
对式(4.1)应用全期望公式可知，$\{p_t\}$需要满足以下相容性条件（Chapman-Kolmogorov方程）：对任意$s,t \geq 0$，$x\in S,A\in \mathcal{S}$，有
Applying iterative expectations to equation (4.1) shows that $\{p_t\}$ has to satisfy the following consistent conditions (Chapman-Kolmogorov equation): for any $s,t \geq 0$, $x\ In S,A\in \mathcal{S}$,
$$p_{s+t}(x,A) = \int p_t(y,A)p_s(x,dy). \tag{4.2}$$
读者应能注意到式(4.2)与前述文章中例子2.1的相似性。与此同时，容易验证在该条件下，对于任意初始分布$X_0\sim \mu$，相应的有限维分布满足Kolmogorov延拓定理的相容性条件。因此，与离散时间场景中一样，当状态空间$(S,\mathcal{S})$满足一定的正则性时，我们可以通过Kolmogorov延拓定理对任意初始分布$\mu$和相容的转移概率族$\{p_t\}$构造出一个概率测度$P_{\mu}$，使得序列空间$(\Omega,\mathcal{F})$中的坐标映射$X$在该测度下是个以$\{p_t\}$为转移概率族的马尔科夫链。
The reader should notice the similarity between equation (4.2) and example 2.1 in the previous post. At the same time, we can verify that under this condition, for any initial distribution $X_0\sim \mu$ the corresponding finite-dimensional distributions satisfy the consistent condition of the Kolmogorov extension theorem. Therefore, as in the discrete-time setting, when the state space $(S,\mathcal{S})$ is nice, we can apply the Kolmogorov extension theorem on any initial distribution $\mu$ and consistent transition probability family $\{p_t\}$ to construct a probability measure $P_{\mu}$, with which the coordinate map $X$ in the sequence space $(\Omega,\mathcal{F})$ is a Markov chain with transition probabilities $\{p_t\}$.
接下来，我们讨论$X$的马尔科夫性质。假设初始分布$\mu$已经给定。考虑$X$对应的自然滤子$\{\mathcal{F}_t\}$，并将序列空间$\Omega$上的平移算子记作$\theta_t$。通过运用与离散时间场景中一样的技术，我们可以证明：
Next, we will discuss the Markov property of $X$. Suppose the initial distribution $\mu$ is known. Consider the natural filtration $\{\mathcal{F}_t\}$ of $X$, and denote the shift operator on the sequence space $\Omega$ as $\theta_t$. By applying the technique in the discrete-time setting, we can prove:
定理4.1（马尔科夫性）：对任意序列空间$(\Omega,\mathcal{F})$上的可测有界函数$Y: \Omega \to \mathbb{R}$和时间$t\in\mathbb{R}^+$，有
Theorem 4.1 (Markov property): For any measurable bounded function $Y: \Omega \to \mathbb{R}$ and time $t\in\mathbb{R}^+$ on the sequence space $(\Omega,\mathcal{F})$,
$$E_{\mu}(Y\circ\theta_t|\mathcal{F}_t) = E_{X_t}Y. \tag{4.3}$$
但是，离散时间场景下的技术无法直接推导出强马尔科夫性。由于强马尔科夫性的结论中涉及到停时，根据前面几节的讨论可知此时我们需要$X$的右连续性。而对于强马尔科夫性，我们还需要一个新的条件。
However, the technique in the discrete-time setting along cannot derive the strong Markov property. Because the statement of the strong Markov property involves stopping times, from the discussions in the previous sections we can infer that the right-continuity of $X$ is needed. And for the strong Markov property, we need just one more condition.
定理4.2（强马尔科夫性）：假设马尔科夫链$X$右连续，且对任意有界连续函数$f: S\to \mathbb{R}$和时间$t$，映射$x\to E_xf(X_t)$是有界连续的。则对任意乘积空间$(\mathbb{R}^+,\mathcal{B})\times(\Omega,\mathcal{F})$上的可测有界函数$Y: \mathbb{R}^+\times\Omega \to \mathbb{R}$和停时$T$，有
Theorem 4.2 (Strong Markov property): Suppose the Markov chain $X$ is right-continuous, and for any bounded continuous function $f: S\to \mathbb{R}$ and time $t$, the mapping $x\to E_xf(X_t)$ is bounded and continuous. Then, for any measurable bounded function $Y: \mathbb{R}^+\times\Omega \to \mathbb{R}$ and stooping time $T$ on the product space $(\mathbb{R}^+,\mathcal{B})\times(\Omega,\mathcal{F})$,
$$E_{\mu}(Y_T\circ\theta_T|\mathcal{F}_T) = E_{X_T}Y_T \text{ on } \{T < \infty\}. \tag{4.4}$$
下面，我将以布朗运动为例子看几个马尔科夫性的应用。首先，容易验证布朗运动满足强马尔科夫性中的条件。因此我们可以得到以下推论：
Next, I will introduce several examples that illustrate with the Brownian motion $B$ the usage of Markov properties. First, we can verify that the Brownian motion satisfies the conditions in the strong Markov property. So we have the following corollary:
推论4.3（布朗运动的强马尔科夫性）：假设函数$Y:\mathbb{R}^+\times\Omega\to\mathbb{R}$有界可测。若$T$是一个停时，则对任意初值$x\in\mathbb{R}$，有
Corollary 4.3 (Strong Markov property of Brownian motions): Suppose the function $Y:\mathbb{R}^+\times\Omega\to\mathbb{R}$ is bounded and measurable. If $T$ is a stopping time, then for any initial value $x\in\mathbb{R}$,
$$E_x(Y_T \circ \theta_T\cdot 1_{\{T < \infty\}}|\mathcal{F}_T) = E_{B_T}Y_{T} \cdot 1_{\{T < \infty\}}. \tag{4.5}$$
通过在推论4.3中令$T\downarrow t$且$Y_t \equiv Y$，我们立刻可得
Letting $T\downarrow t$ and $Y_t \equiv Y$ in corollary 4.3, we immediately know
$$E_x(Y \circ \theta_t|\mathcal{F}_{t+}) = E_{B_t}Y. \tag{4.6}$$
通过比较式(4.3)和式(4.6)可知式(4.6)的结论更强。特别地，在式(4.6)中令$t=0,Y=1_A,A\in \mathcal{F}_{0+}$，可得
推论4.4（Blumenthal 0-1律）：对于任意$A\in \mathcal{F}_{0+}$和$x\in\mathbb{R}$，有$P_x(A)=1_A \in\{0,1\}$。
By comparing equations (4.3) and (4.6), we know that equation (4.6) is stronger. In particular, letting $t=0, Y=1_A, A\in \mathcal{F}_{0+}$ in equation (4.6), we have
Corollary 4.4 (Blumenthal’s 0-1 law): For any $A\in \mathcal{F}_{0+}$ and $x\in\mathbb{R}$, $P_x( A)=1_A \in\{0,1\}$.
最后，我们将随机游走中的反射原理（前述文章中例子2.2）延拓到布朗运动上。
Finally, we extend the reflection principle of random walks (example 2.2 in the previous post) to Brownian motions.
推论4.5（反射原理）：考虑停时$T = \inf \{s:B_s = a\}$，其中$a > 0$。则
Corollary 4.5 (Reflection Principle): Consider the stopping time $T = \inf \{s:B_s = a\}$, where $a > 0$. Then
$$P_0(T < t) = 2P_0(B_t \geq a).$$
证明：令$S = \inf\{s < t: B_s = a\} = T\cdot 1_{\{T < t\}} +\infty \cdot 1_{\{T \geq t\}}$。如果进一步定义$Y_s(\omega) = 1_{ \{s < t, \omega(t-s) > a\} }$，则有$Y_S\circ\theta_S (\omega) = 1_{ \{S < t, B_t > a\} } = 1_{ \{T < t, B_t > a\} }$。从而根据推论4.3，
Proof: Let $S = \inf\{s < t: B_s = a\} = T\cdot 1_{\{T < t\}} +\infty \cdot 1_{\{T \geq t\}}$. If we further define $Y_s(\omega) = 1_{ \{s < t, \omega(t-s) > a\} }$, then $Y_S\circ\theta_S (\omega) = 1_{ \{S < t, B_t > a\} } = 1_{ \{T < t, B_t > a\} }$. Thus, according to corollary 4.3,
$$P_0(B_t > a) = P_0(T < t, B_t > a) = E_0(Y_S\circ \theta_S \cdot 1_{\{S < \infty\} }) = E_0(E_{B_S}Y_S \cdot 1_{\{S < \infty\} }) = E_0(1/2\cdot 1_{\{T < t\} }) = 1/2\cdot P_0(T < t).$$
其中在上式中使用了结论$E_aY_s \equiv 1/2$。
Note that in the above equation we use the fact $E_aY_s \equiv 1/2$.