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1 The Definition of a Stochastic Process
1 随机过程的定义

Suppose that ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) is a probability space, and that X : Ω → R X : Ω → R X:Omega rarrRX: \Omega \rightarrow \mathbb{R} is a random variable. Recall that this means that Ω Ω Omega\Omega is a space, F F F\mathcal{F} is a σ σ sigma\sigma-algebra of subsets of Ω , P Ω , P Omega,P\Omega, \mathbb{P} is a countably additive, non-negative measure on ( Ω , F ) ( Ω , F ) (Omega,F)(\Omega, \mathcal{F}) with total mass P ( Ω ) = 1 P ( Ω ) = 1 P(Omega)=1\mathbb{P}(\Omega)=1, and X X XX is a measurable function, i.e., X − 1 ( B ) = { ω ∈ Ω : X ( ω ) ∈ B } ∈ F X − 1 ( B ) = { ω ∈ Ω : X ( ω ) ∈ B } ∈ F X^(-1)(B)={omega in Omega:X(omega)in B}inFX^{-1}(B)=\{\omega \in \Omega: X(\omega) \in B\} \in \mathcal{F} for every Borel set B ∈ B ( R ) B ∈ B ( R ) B inB(R)B \in \mathcal{B}(\mathbb{R}).
假设 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 是一个概率空间，而 X : Ω → R X : Ω → R X:Omega rarrRX: \Omega \rightarrow \mathbb{R} 是一个随机变量。回想一下，这意味着 Ω Ω Omega\Omega 是一个空间， F F F\mathcal{F} 是一个 σ σ sigma\sigma 子集的代数， Ω , P Ω , P Omega,P\Omega, \mathbb{P} 是 ( Ω , F ) ( Ω , F ) (Omega,F)(\Omega, \mathcal{F}) 上一个可数可加的非负度量，总质量为 P ( Ω ) = 1 P ( Ω ) = 1 P(Omega)=1\mathbb{P}(\Omega)=1 ，而 X X XX 是一个可测函数，即 X − 1 ( B ) = { ω ∈ Ω : X ( ω ) ∈ B } ∈ F X − 1 ( B ) = { ω ∈ Ω : X ( ω ) ∈ B } ∈ F X^(-1)(B)={omega in Omega:X(omega)in B}inFX^{-1}(B)=\{\omega \in \Omega: X(\omega) \in B\} \in \mathcal{F} 对于每个伯尔集合 B ∈ B ( R ) B ∈ B ( R ) B inB(R)B \in \mathcal{B}(\mathbb{R}) 都是可测函数。
A stochastic process is simply a collection of random variables indexed by time. It will be useful to consider separately the cases of discrete time and continuous time. We will even have occasion to consider indexing the random variables by negative time. That is, a discrete time stochastic process X = { X n , n = 0 , 1 , 2 , … } X = X n , n = 0 , 1 , 2 , … X={X_(n),n=0,1,2,dots}X=\left\{X_{n}, n=0,1,2, \ldots\right\} is a countable collection of random variables indexed by the non-negative integers, and a continuous time stochastic process X = { X t , 0 ≤ t < ∞ } X = X t , 0 ≤ t < ∞ X={X_(t),0 <= t < oo}X=\left\{X_{t}, 0 \leq t<\infty\right\} is an uncountable collection of random variables indexed by the non-negative real numbers.
随机过程是以时间为索引的随机变量集合。分别考虑离散时间和连续时间的情况是有用的。我们甚至有机会考虑以负时间为索引的随机变量。也就是说，离散时间随机过程 X = { X n , n = 0 , 1 , 2 , … } X = X n , n = 0 , 1 , 2 , … X={X_(n),n=0,1,2,dots}X=\left\{X_{n}, n=0,1,2, \ldots\right\} 是以非负整数为索引的可数随机变量集合，而连续时间随机过程 X = { X t , 0 ≤ t < ∞ } X = X t , 0 ≤ t < ∞ X={X_(t),0 <= t < oo}X=\left\{X_{t}, 0 \leq t<\infty\right\} 是以非负实数为索引的不可数随机变量集合。
In general, we may consider any indexing set I ⊂ R I ⊂ R I subRI \subset \mathbb{R} having infinite cardinality, so that calling X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} a stochastic process simply means that X α X α X_(alpha)X_{\alpha} is a random variable for each α ∈ I α ∈ I alpha in I\alpha \in I. (If the cardinality of I I II is finite, then X X XX is not considered a stochastic process, but rather a random vector.)
一般来说，我们可以考虑任何具有无限卡片性的索引集 I ⊂ R I ⊂ R I subRI \subset \mathbb{R} ，因此称 X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 为随机过程仅仅意味着 X α X α X_(alpha)X_{\alpha} 是每个 α ∈ I α ∈ I alpha in I\alpha \in I 的随机变量。(如果 I I II 的卡片数是有限的，那么 X X XX 就不能视为随机过程，而应视为随机向量）。
There are two natural questions that one might ask.
人们自然会提出两个问题。
(1) How can we construct a probability space on which a stochastic process is defined?
(1) 如何构建一个概率空间，并在其上定义随机过程？
(2) Is it possible to define a stochastic process by specifying, say, its finite dimensional distributions only?
(2) 是否有可能只指定随机过程的有限维分布来定义随机过程？
Instead of immediately addressing these (rather technical) questions, we assume the existence of an appropriate probability space, and carefully define a stochastic process on that space. In fact, we will defer answering these questions for some time.
我们不会立即讨论这些（相当技术性的）问题，而是假设存在一个适当的概率空间，并仔细定义该空间上的随机过程。事实上，我们将推迟一段时间来回答这些问题。
Definition 1.1. Suppose that ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) is a probability space, and that I ⊂ R I ⊂ R I subRI \subset \mathbb{R} is of infinite cardinality. Suppose further that for each α ∈ I α ∈ I alpha in I\alpha \in I, there is a random variable X α : Ω → R X α : Ω → R X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} defined on ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). The function X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} defined by X ( α , ω ) = X α ( ω ) X ( α , ω ) = X α ( ω ) X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) is called a stochastic process with indexing set I I II, and is written X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\}.
定义 1.1.假设 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 是一个概率空间，且 I ⊂ R I ⊂ R I subRI \subset \mathbb{R} 的卡方数为无限。再假设每个 α ∈ I α ∈ I alpha in I\alpha \in I 都有一个定义在 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的随机变量 X α : Ω → R X α : Ω → R X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} 。由 X ( α , ω ) = X α ( ω ) X ( α , ω ) = X α ( ω ) X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) 定义的函数 X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} 称为具有索引集 I I II 的随机过程，并被写为 X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 。
Remark. We will always assume that the cardinality of I I II is infinite, either countable or uncountable. If I = Z + I = Z + I=Z^(+)I=\mathbb{Z}^{+}, then we called X X XX a discrete time stochastic process, and if I = [ 0 , ∞ ) I = [ 0 , ∞ ) I=[0,oo)I=[0, \infty), then X X XX is said to be a continuous time stochastic processes.
备注我们将始终假设 I I II 的心数是无限的，可以是可数的，也可以是不可数的。如果 I = Z + I = Z + I=Z^(+)I=\mathbb{Z}^{+} ，那么我们称 X X XX 为离散时间随机过程；如果 I = [ 0 , ∞ ) I = [ 0 , ∞ ) I=[0,oo)I=[0, \infty) ，那么我们称 X X XX 为连续时间随机过程。
At first, this definition might seem a little complicated since we are regarding the stochastic process X X XX as a function of two variables defined on the product space I × Ω I × Ω I xx OmegaI \times \Omega. However, this is necessary since we do not always want to view the stochastic process X X XX as a collection of random variables. Sometimes, it is more advantageous to consider X X XX as the (random) function α ↦ X ( α , ω ) α ↦ X ( α , ω ) alpha|->X(alpha,omega)\alpha \mapsto X(\alpha, \omega) which is called the sample path (or trajectory) of X X XX at ω ω omega\omega (and is also written X ( ω ) ) X ( ω ) ) X(omega))X(\omega)). We will need to require X X XX as a function of α α alpha\alpha to have certain regularity properties such as continuity or measurability; as will be shown in Example 3.10 below, these properties do not come for free!
起初，这个定义似乎有点复杂，因为我们将随机过程 X X XX 视为定义在乘积空间 I × Ω I × Ω I xx OmegaI \times \Omega 上的两个变量的函数。然而，这是必要的，因为我们并不总是希望将随机过程 X X XX 视为随机变量的集合。有时，将 X X XX 视为（随机）函数 α ↦ X ( α , ω ) α ↦ X ( α , ω ) alpha|->X(alpha,omega)\alpha \mapsto X(\alpha, \omega) 更为有利，该函数被称为 X X XX 在 ω ω omega\omega 处的样本路径（或轨迹）（也可写为 X ( ω ) ) X ( ω ) ) X(omega))X(\omega)) 。我们需要要求 X X XX 作为 α α alpha\alpha 的函数具有某些规则性属性，例如连续性或可测性；正如下面的例 3.10 所示，这些属性并不是免费的！
Notation. A word should be said about notation. We have defined a stochastic process as a single function X X XX of two variables. That is, X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} is defined by specifying ( α , ω ) ↦ X ( α , ω ) ( α , ω ) ↦ X ( α , ω ) (alpha,omega)|->X(alpha,omega)(\alpha, \omega) \mapsto X(\alpha, \omega) which mimics the notation from multi-variable calculus. However, we are also viewing a stochastic process as a collection of random variables, one random variable for each α α alpha\alpha in the indexing set I I II. That is, if the random variable X α : Ω → R X α : Ω → R X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} is defined by specifying ω ↦ X α ( ω ) ω ↦ X α ( ω ) omega|->X_(alpha)(omega)\omega \mapsto X_{\alpha}(\omega), then the stochastic process X X XX is defined as X ( α , ω ) = X α ( ω ) X ( α , ω ) = X α ( ω ) X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega). In fact, we will often say for brevity that X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} is a stochastic process on ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). Because of this identification, when there is no chance of ambiguity we will use both X ( α , ω ) X ( α , ω ) X(alpha,omega)X(\alpha, \omega) and X α ( ω ) X α ( ω ) X_(alpha)(omega)X_{\alpha}(\omega) to describe the stochastic process. If the dependence on ω ω omega\omega is unnecessary, we will simply write X α X α X_(alpha)X_{\alpha} or even X ( α ) X ( α ) X(alpha)X(\alpha). The sample path of X X XX at ω ω omega\omega will be written as either α ↦ X α ( ω ) α ↦ X α ( ω ) alpha|->X_(alpha)(omega)\alpha \mapsto X_{\alpha}(\omega) or just X ( ω ) X ( ω ) X(omega)X(\omega).
记号先说一下符号。我们将随机过程定义为两个变量的单一函数 X X XX 。也就是说， X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} 是通过指定 ( α , ω ) ↦ X ( α , ω ) ( α , ω ) ↦ X ( α , ω ) (alpha,omega)|->X(alpha,omega)(\alpha, \omega) \mapsto X(\alpha, \omega) 来定义的，这模仿了多变量微积分的符号。不过，我们也将随机过程视为随机变量的集合，索引集 I I II 中的每个 α α alpha\alpha 都有一个随机变量。也就是说，如果通过指定 ω ↦ X α ( ω ) ω ↦ X α ( ω ) omega|->X_(alpha)(omega)\omega \mapsto X_{\alpha}(\omega) 来定义随机变量 X α : Ω → R X α : Ω → R X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} ，那么随机过程 X X XX 的定义就是 X ( α , ω ) = X α ( ω ) X ( α , ω ) = X α ( ω ) X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) 。事实上，为了简洁起见，我们经常会说 X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 是 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的随机过程。由于这种识别，在没有歧义的情况下，我们将同时使用 X ( α , ω ) X ( α , ω ) X(alpha,omega)X(\alpha, \omega) 和 X α ( ω ) X α ( ω ) X_(alpha)(omega)X_{\alpha}(\omega) 来描述随机过程。如果不需要依赖 ω ω omega\omega ，我们将简单地写成 X α X α X_(alpha)X_{\alpha} ，甚至 X ( α ) X ( α ) X(alpha)X(\alpha) 。 X X XX 在 ω ω omega\omega 处的样本路径将写成 α ↦ X α ( ω ) α ↦ X α ( ω ) alpha|->X_(alpha)(omega)\alpha \mapsto X_{\alpha}(\omega) 或只写成 X ( ω ) X ( ω ) X(omega)X(\omega) 。
Example 1.2. Perhaps the simplest example of a stochastic process is what may be termed i.i.d. noise. Suppose that X 1 , X 2 , … X 1 , X 2 , … X_(1),X_(2),dotsX_{1}, X_{2}, \ldots are independent and identically distributed random variables on a probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) each having mean zero and variance one. The stochastic process X : N × Ω → R X : N × Ω → R X:Nxx Omega rarrRX: \mathbb{N} \times \Omega \rightarrow \mathbb{R} defined by X ( n , ω ) = X n ( ω ) X ( n , ω ) = X n ( ω ) X(n,omega)=X_(n)(omega)X(n, \omega)=X_{n}(\omega) is called i.i.d. noise and serves as the building block for other more complicated stochastic processes. For example, define the stochastic process S : N × Ω → R S : N × Ω → R S:Nxx Omega rarrRS: \mathbb{N} \times \Omega \rightarrow \mathbb{R} by setting
例 1.2.随机过程最简单的例子可能就是所谓的 i.i.d. 噪声。假设 X 1 , X 2 , … X 1 , X 2 , … X_(1),X_(2),dotsX_{1}, X_{2}, \ldots 是概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上独立且同分布的随机变量，每个变量的均值为零，方差为一。由 X ( n , ω ) = X n ( ω ) X ( n , ω ) = X n ( ω ) X(n,omega)=X_(n)(omega)X(n, \omega)=X_{n}(\omega) 定义的随机过程 X : N × Ω → R X : N × Ω → R X:Nxx Omega rarrRX: \mathbb{N} \times \Omega \rightarrow \mathbb{R} 称为 i.i.d. 噪声，是其他更复杂随机过程的基础。例如，定义随机过程 S : N × Ω → R S : N × Ω → R S:Nxx Omega rarrRS: \mathbb{N} \times \Omega \rightarrow \mathbb{R} 时，设置
S ( n , ω ) = S n ( ω ) = ∑ i = 1 n X i ( ω ) S ( n , ω ) = S n ( ω ) = ∑ i = 1 n   X i ( ω ) S(n,omega)=S_(n)(omega)=sum_(i=1)^(n)X_(i)(omega)S(n, \omega)=S_{n}(\omega)=\sum_{i=1}^{n} X_{i}(\omega)
The stochastic process S S SS is called a random walk and will be studied in greater detail later.
随机过程 S S SS 被称为随机漫步，稍后将对其进行更详细的研究。
The following section discusses some examples of continuous time stochastic processes.
下一节将讨论一些连续时间随机过程的例子。

2 Examples of Continuous Time Stochastic Processes
2 连续时间随机过程实例

We begin by recalling the useful fact that a linear transformation of a normal random variable is again a normal random variable.
我们首先回顾一个有用的事实：正态随机变量的线性变换也是正态随机变量。
Proposition 2.1. Suppose that Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1). If a , b ∈ R a , b ∈ R a,b inRa, b \in \mathbb{R} and Y = a Z + b Y = a Z + b Y=aZ+bY=a Z+b, then Y ∼ N ( b , a 2 ) Y ∼ N b , a 2 Y∼N(b,a^(2))Y \sim \mathcal{N}\left(b, a^{2}\right).
命题 2.1.假设 Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1) 。如果 a , b ∈ R a , b ∈ R a,b inRa, b \in \mathbb{R} 和 Y = a Z + b Y = a Z + b Y=aZ+bY=a Z+b ，那么 Y ∼ N ( b , a 2 ) Y ∼ N b , a 2 Y∼N(b,a^(2))Y \sim \mathcal{N}\left(b, a^{2}\right) 。
Proof. By assumption, the characteristic function of Z Z ZZ is ϕ Z ( u ) = E ( e i u Z ) = exp ⁡ ( − u 2 / 2 ) ϕ Z ( u ) = E e i u Z = exp ⁡ − u 2 / 2 phi_(Z)(u)=E(e^(iuZ))=exp(-u^(2)//2)\phi_{Z}(u)=\mathbb{E}\left(e^{i u Z}\right)=\exp \left(-u^{2} / 2\right). Thus, the characteristic function of Y Y YY is
证明。根据假设， Z Z ZZ 的特征函数为 ϕ Z ( u ) = E ( e i u Z ) = exp ⁡ ( − u 2 / 2 ) ϕ Z ( u ) = E e i u Z = exp ⁡ − u 2 / 2 phi_(Z)(u)=E(e^(iuZ))=exp(-u^(2)//2)\phi_{Z}(u)=\mathbb{E}\left(e^{i u Z}\right)=\exp \left(-u^{2} / 2\right) 。因此， Y Y YY 的特征函数为
ϕ Y ( u ) = E ( e i u Y ) = e i u b E ( e i u a Z ) = e i u b ϕ Z ( a u ) = e i u b e − ( a u ) 2 2 = exp ⁡ ( i u b − a 2 u 2 2 ) ϕ Y ( u ) = E e i u Y = e i u b E e i u a Z = e i u b ϕ Z ( a u ) = e i u b e − ( a u ) 2 2 = exp ⁡ i u b − a 2 u 2 2 phi_(Y)(u)=E(e^(iuY))=e^(iub)E(e^(iuaZ))=e^(iub)phi_(Z)(au)=e^(iub)e^(-((au)^(2))/(2))=exp(iub-(a^(2)u^(2))/(2))\phi_{Y}(u)=\mathbb{E}\left(e^{i u Y}\right)=e^{i u b} \mathbb{E}\left(e^{i u a Z}\right)=e^{i u b} \phi_{Z}(a u)=e^{i u b} e^{-\frac{(a u)^{2}}{2}}=\exp \left(i u b-\frac{a^{2} u^{2}}{2}\right)
which is the characteristic function of a N ( b , a 2 ) N b , a 2 N(b,a^(2))\mathcal{N}\left(b, a^{2}\right) random variable.
是 N ( b , a 2 ) N b , a 2 N(b,a^(2))\mathcal{N}\left(b, a^{2}\right) 随机变量的特征函数。
Note that if a = 0 a = 0 a=0a=0, then Y Y YY reduces to the constant random variable Y = b Y = b Y=bY=b. Also note that if Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1), then − Z ∼ N ( 0 , 1 ) − Z ∼ N ( 0 , 1 ) -Z∼N(0,1)-Z \sim \mathcal{N}(0,1). For this reason, many authors restrict to a ≥ 0 a ≥ 0 a >= 0a \geq 0.
请注意，如果 a = 0 a = 0 a=0a=0 ，那么 Y Y YY 就会简化为常数随机变量 Y = b Y = b Y=bY=b 。还要注意，如果 Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1) ，那么 − Z ∼ N ( 0 , 1 ) − Z ∼ N ( 0 , 1 ) -Z∼N(0,1)-Z \sim \mathcal{N}(0,1) 。因此，许多作者限制了 a ≥ 0 a ≥ 0 a >= 0a \geq 0 。
Exercise 2.2. Show that if Z 1 ∼ N ( μ 1 , σ 1 2 ) Z 1 ∼ N μ 1 , σ 1 2 Z_(1)∼N(mu_(1),sigma_(1)^(2))Z_{1} \sim \mathcal{N}\left(\mu_{1}, \sigma_{1}^{2}\right) and Z 2 ∼ N ( μ 2 , σ 2 2 ) Z 2 ∼ N μ 2 , σ 2 2 Z_(2)∼N(mu_(2),sigma_(2)^(2))Z_{2} \sim \mathcal{N}\left(\mu_{2}, \sigma_{2}^{2}\right) are independent, then Z 1 + Z 2 ∼ N ( μ 1 + μ 2 , σ 1 2 + σ 2 2 ) Z 1 + Z 2 ∼ N μ 1 + μ 2 , σ 1 2 + σ 2 2 Z_(1)+Z_(2)∼N(mu_(1)+mu_(2),sigma_(1)^(2)+sigma_(2)^(2))Z_{1}+Z_{2} \sim \mathcal{N}\left(\mu_{1}+\mu_{2}, \sigma_{1}^{2}+\sigma_{2}^{2}\right).
练习 2.2.证明如果 Z 1 ∼ N ( μ 1 , σ 1 2 ) Z 1 ∼ N μ 1 , σ 1 2 Z_(1)∼N(mu_(1),sigma_(1)^(2))Z_{1} \sim \mathcal{N}\left(\mu_{1}, \sigma_{1}^{2}\right) 和 Z 2 ∼ N ( μ 2 , σ 2 2 ) Z 2 ∼ N μ 2 , σ 2 2 Z_(2)∼N(mu_(2),sigma_(2)^(2))Z_{2} \sim \mathcal{N}\left(\mu_{2}, \sigma_{2}^{2}\right) 是独立的，那么 Z 1 + Z 2 ∼ N ( μ 1 + μ 2 , σ 1 2 + σ 2 2 ) Z 1 + Z 2 ∼ N μ 1 + μ 2 , σ 1 2 + σ 2 2 Z_(1)+Z_(2)∼N(mu_(1)+mu_(2),sigma_(1)^(2)+sigma_(2)^(2))Z_{1}+Z_{2} \sim \mathcal{N}\left(\mu_{1}+\mu_{2}, \sigma_{1}^{2}+\sigma_{2}^{2}\right) .
Both of the following relatively straightforward examples of continuous time stochastic processes illustrate the two points-of-view that we are taking of a stochastic process, namely (i) a collection of random variables, and (ii) a random function of the index (called a trajectory).
下面两个相对简单的连续时间随机过程的例子说明了我们对随机过程的两种观点，即 (i) 随机变量集合和 (ii) 指数的随机函数（称为轨迹）。
Example 2.3. Consider the probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}), and let Z Z ZZ be a random variable with P ( Z = 1 ) = P ( Z = − 1 ) = 1 / 2 P ( Z = 1 ) = P ( Z = − 1 ) = 1 / 2 P(Z=1)=P(Z=-1)=1//2\mathbb{P}(Z=1)=\mathbb{P}(Z=-1)=1 / 2. Define the continuous time stochastic process X = X = X=X= { X t , t ≥ 0 } X t , t ≥ 0 {X_(t),t >= 0}\left\{X_{t}, t \geq 0\right\} by setting
例 2.3.考虑概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) ，设 Z Z ZZ 是一个具有 P ( Z = 1 ) = P ( Z = − 1 ) = 1 / 2 P ( Z = 1 ) = P ( Z = − 1 ) = 1 / 2 P(Z=1)=P(Z=-1)=1//2\mathbb{P}(Z=1)=\mathbb{P}(Z=-1)=1 / 2 的随机变量。定义连续时间随机过程 X = X = X=X= { X t , t ≥ 0 } X t , t ≥ 0 {X_(t),t >= 0}\left\{X_{t}, t \geq 0\right\} ，设
X t ( ω ) = Z ( ω ) sin ⁡ t for all t ≥ 0 X t ( ω ) = Z ( ω ) sin ⁡ t  for all  t ≥ 0 X_(t)(omega)=Z(omega)sin t" for all "t >= 0X_{t}(\omega)=Z(\omega) \sin t \text { for all } t \geq 0
The two sections of this stochastic process can be described as follows.
这个随机过程的两个部分可以描述如下。
  • For fixed t t tt, the section X ( t , ⋅ ) : Ω → R X ( t , ⋅ ) : Ω → R X(t,*):Omega rarrRX(t, \cdot): \Omega \rightarrow \mathbb{R} is the random variable X t X t X_(t)X_{t} which has distribution given by
    对于固定的 t t tt ，截面 X ( t , ⋅ ) : Ω → R X ( t , ⋅ ) : Ω → R X(t,*):Omega rarrRX(t, \cdot): \Omega \rightarrow \mathbb{R} 是随机变量 X t X t X_(t)X_{t} ，其分布式为
P ( X t = sin ⁡ t ) = P ( X t = − sin ⁡ t ) = 1 / 2 P X t = sin ⁡ t = P X t = − sin ⁡ t = 1 / 2 P(X_(t)=sin t)=P(X_(t)=-sin t)=1//2\mathbb{P}\left(X_{t}=\sin t\right)=\mathbb{P}\left(X_{t}=-\sin t\right)=1 / 2
For instance, when t = 3 π / 4 t = 3 π / 4 t=3pi//4t=3 \pi / 4, the random variable X 3 π / 4 X 3 π / 4 X_(3pi//4)X_{3 \pi / 4} has distribution
例如，当 t = 3 π / 4 t = 3 π / 4 t=3pi//4t=3 \pi / 4 时，随机变量 X 3 π / 4 X 3 π / 4 X_(3pi//4)X_{3 \pi / 4} 的分布是
P ( X 3 π / 4 = 1 2 ) = P ( X 3 π / 4 = − 1 2 ) = 1 / 2 . P X 3 π / 4 = 1 2 = P X 3 π / 4 = − 1 2 = 1 / 2 . P(X_(3pi//4)=(1)/(sqrt2))=P(X_(3pi//4)=-(1)/(sqrt2))=1//2.\mathbb{P}\left(X_{3 \pi / 4}=\frac{1}{\sqrt{2}}\right)=\mathbb{P}\left(X_{3 \pi / 4}=-\frac{1}{\sqrt{2}}\right)=1 / 2 .
  • For fixed w w ww, the section X ( ⋅ , ω ) : [ 0 , ∞ ) → R X ( ⋅ , ω ) : [ 0 , ∞ ) → R X(*,omega):[0,oo)rarrRX(\cdot, \omega):[0, \infty) \rightarrow \mathbb{R} describes the trajectory, or sample path, of X X XX at ω ω omega\omega. Note that there are two possible trajectories for X X XX. Namely, if ω ω omega\omega is such that Z ( ω ) = 1 Z ( ω ) = 1 Z(omega)=1Z(\omega)=1, then the trajectory is t ↦ sin ⁡ t t ↦ sin ⁡ t t|->sin tt \mapsto \sin t, and if ω ω omega\omega is such that Z ( ω ) = − 1 Z ( ω ) = − 1 Z(omega)=-1Z(\omega)=-1, then the trajectory is t ↦ − sin ⁡ t t ↦ − sin ⁡ t t|->-sin tt \mapsto-\sin t. Of course, each trajectory occurs with probability 1 / 2 1 / 2 1//21 / 2.
    对于固定的 w w ww ， X ( ⋅ , ω ) : [ 0 , ∞ ) → R X ( ⋅ , ω ) : [ 0 , ∞ ) → R X(*,omega):[0,oo)rarrRX(\cdot, \omega):[0, \infty) \rightarrow \mathbb{R} 部分描述了 X X XX 在 ω ω omega\omega 处的轨迹或采样路径。请注意， X X XX 有两种可能的轨迹。也就是说，如果 ω ω omega\omega 使得 Z ( ω ) = 1 Z ( ω ) = 1 Z(omega)=1Z(\omega)=1 ，那么轨迹就是 t ↦ sin ⁡ t t ↦ sin ⁡ t t|->sin tt \mapsto \sin t ；如果 ω ω omega\omega 使得 Z ( ω ) = − 1 Z ( ω ) = − 1 Z(omega)=-1Z(\omega)=-1 ，那么轨迹就是 t ↦ − sin ⁡ t t ↦ − sin ⁡ t t|->-sin tt \mapsto-\sin t 。当然，每条轨迹出现的概率都是 1 / 2 1 / 2 1//21 / 2 。
Example 2.4. Suppose that Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1), and define the continuous time stochastic process X = { X t , t ≥ 0 } X = X t , t ≥ 0 X={X_(t),t >= 0}X=\left\{X_{t}, t \geq 0\right\} by setting
例 2.4.假设 Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1) ，并定义连续时间随机过程 X = { X t , t ≥ 0 } X = X t , t ≥ 0 X={X_(t),t >= 0}X=\left\{X_{t}, t \geq 0\right\} ，设
X t ( ω ) = Z ( ω ) sin ⁡ t for all t ≥ 0 X t ( ω ) = Z ( ω ) sin ⁡ t  for all  t ≥ 0 X_(t)(omega)=Z(omega)sin t" for all "t >= 0X_{t}(\omega)=Z(\omega) \sin t \text { for all } t \geq 0
In this example, the two sections of the stochastic process are slightly more complicated.
在这个例子中，随机过程的两个部分稍微复杂一些。
  • If Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1), then it follows from Proposition 2.1 that ( sin ⁡ t ) ⋅ Z ∼ N ( 0 , sin 2 ⁡ t ) ( sin ⁡ t ) ⋅ Z ∼ N 0 , sin 2 ⁡ t (sin t)*Z∼N(0,sin^(2)t)(\sin t) \cdot Z \sim \mathcal{N}\left(0, \sin ^{2} t\right). Therefore, for fixed t t tt, the section X ( t , ⋅ ) : Ω → R X ( t , ⋅ ) : Ω → R X(t,*):Omega rarrRX(t, \cdot): \Omega \rightarrow \mathbb{R} is the random variable X t X t X_(t)X_{t} which has the N ( 0 , sin 2 ⁡ t ) N 0 , sin 2 ⁡ t N(0,sin^(2)t)\mathcal{N}\left(0, \sin ^{2} t\right) distribution. For instance, if t = 3 π / 4 t = 3 π / 4 t=3pi//4t=3 \pi / 4, then X 3 π / 4 ∼ N ( 0 , 1 / 2 ) X 3 π / 4 ∼ N ( 0 , 1 / 2 ) X_(3pi//4)∼N(0,1//2)X_{3 \pi / 4} \sim \mathcal{N}(0,1 / 2).
    如果 Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1) ，那么根据命题 2.1 可以得出 ( sin ⁡ t ) ⋅ Z ∼ N ( 0 , sin 2 ⁡ t ) ( sin ⁡ t ) ⋅ Z ∼ N 0 , sin 2 ⁡ t (sin t)*Z∼N(0,sin^(2)t)(\sin t) \cdot Z \sim \mathcal{N}\left(0, \sin ^{2} t\right) 。因此，对于固定的 t t tt ，截面 X ( t , ⋅ ) : Ω → R X ( t , ⋅ ) : Ω → R X(t,*):Omega rarrRX(t, \cdot): \Omega \rightarrow \mathbb{R} 是具有 N ( 0 , sin 2 ⁡ t ) N 0 , sin 2 ⁡ t N(0,sin^(2)t)\mathcal{N}\left(0, \sin ^{2} t\right) 分布的随机变量 X t X t X_(t)X_{t} 。例如，如果 t = 3 π / 4 t = 3 π / 4 t=3pi//4t=3 \pi / 4 ，那么 X 3 π / 4 ∼ N ( 0 , 1 / 2 ) X 3 π / 4 ∼ N ( 0 , 1 / 2 ) X_(3pi//4)∼N(0,1//2)X_{3 \pi / 4} \sim \mathcal{N}(0,1 / 2) 。
  • As in the previous example, for fixed w w ww, the section X ( ⋅ , ω ) : [ 0 , ∞ ) → R X ( ⋅ , ω ) : [ 0 , ∞ ) → R X(*,omega):[0,oo)rarrRX(\cdot, \omega):[0, \infty) \rightarrow \mathbb{R} describes the trajectory of X X XX at ω ω omega\omega. This time, however, since Z Z ZZ is allowed to assume uncountably many values, there will be infinitely many possible trajectories. Each trajectory t ↦ X t t ↦ X t t|->X_(t)t \mapsto X_{t} will simply be a standard sine curve with amplitude Z ( ω ) Z ( ω ) Z(omega)Z(\omega). That is, once Z Z ZZ is realized so that the number Z ( ω ) Z ( ω ) Z(omega)Z(\omega) is known, the trajectory is t ↦ Z ( ω ) sin ⁡ ( t ) t ↦ Z ( ω ) sin ⁡ ( t ) t|->Z(omega)sin(t)t \mapsto Z(\omega) \sin (t). For instance, if Z ( ω ) = 0.69834 Z ( ω ) = 0.69834 Z(omega)=0.69834Z(\omega)=0.69834, then the trajectory is t ↦ 0.69834 sin ⁡ t t ↦ 0.69834 sin ⁡ t t|->0.69834 sin tt \mapsto 0.69834 \sin t. Of course, since P ( Z = z ) = 0 P ( Z = z ) = 0 P(Z=z)=0\mathbb{P}(Z=z)=0 for any z ∈ R z ∈ R z inRz \in \mathbb{R}, any given trajectory occurs with probability 0 .
    与前面的示例一样，对于固定的 w w ww ， X ( ⋅ , ω ) : [ 0 , ∞ ) → R X ( ⋅ , ω ) : [ 0 , ∞ ) → R X(*,omega):[0,oo)rarrRX(\cdot, \omega):[0, \infty) \rightarrow \mathbb{R} 部分描述了 X X XX 在 ω ω omega\omega 处的轨迹。但这次，由于允许 Z Z ZZ 取不可计数的值，因此将有无限多条可能的轨迹。每条轨迹 t ↦ X t t ↦ X t t|->X_(t)t \mapsto X_{t} 都将是一条振幅为 Z ( ω ) Z ( ω ) Z(omega)Z(\omega) 的标准正弦曲线。也就是说，一旦实现了 Z Z ZZ ，从而知道了数字 Z ( ω ) Z ( ω ) Z(omega)Z(\omega) ，轨迹就是 t ↦ Z ( ω ) sin ⁡ ( t ) t ↦ Z ( ω ) sin ⁡ ( t ) t|->Z(omega)sin(t)t \mapsto Z(\omega) \sin (t) 。例如，如果 Z ( ω ) = 0.69834 Z ( ω ) = 0.69834 Z(omega)=0.69834Z(\omega)=0.69834 ，那么轨迹就是 t ↦ 0.69834 sin ⁡ t t ↦ 0.69834 sin ⁡ t t|->0.69834 sin tt \mapsto 0.69834 \sin t 。当然，由于对于任何 z ∈ R z ∈ R z inRz \in \mathbb{R} 都是 P ( Z = z ) = 0 P ( Z = z ) = 0 P(Z=z)=0\mathbb{P}(Z=z)=0 ，因此任何给定轨迹出现的概率都是 0 。
    Notice that in both of the previous two examples, the trajectories of the stochastic process X X XX were continuous. That is, the trajectories were all of the form t ↦ Z sin ⁡ t t ↦ Z sin ⁡ t t|->Z sin tt \mapsto Z \sin t for some constant Z Z ZZ, and Z sin ⁡ t Z sin ⁡ t Z sin tZ \sin t, as a function of t t tt, is continuous.
    请注意，在前面两个例子中，随机过程 X X XX 的轨迹都是连续的。也就是说，对于某个常数 Z Z ZZ ，轨迹的形式都是 t ↦ Z sin ⁡ t t ↦ Z sin ⁡ t t|->Z sin tt \mapsto Z \sin t ，而作为 t t tt 的函数， Z sin ⁡ t Z sin ⁡ t Z sin tZ \sin t 是连续的。
The next example is also of a continuous time stochastic process whose trajectories are continuous. It is, however, significantly more complicated.
下一个例子也是轨迹连续的连续时间随机过程。不过，这个例子要复杂得多。
Example 2.5. Consider a collection of random variables { B t , t ≥ 0 } B t , t ≥ 0 {B_(t),t >= 0}\left\{B_{t}, t \geq 0\right\}, having the following properties:
例 2.5.考虑具有以下性质的随机变量 { B t , t ≥ 0 } B t , t ≥ 0 {B_(t),t >= 0}\left\{B_{t}, t \geq 0\right\} 集合：
  • B 0 = 0 B 0 = 0 B_(0)=0B_{0}=0,
  • for 0 ≤ s < t < ∞ , B t − B s ∼ N ( 0 , t − s ) 0 ≤ s < t < ∞ , B t − B s ∼ N ( 0 , t − s ) 0 <= s < t < oo,B_(t)-B_(s)∼N(0,t-s)0 \leq s<t<\infty, B_{t}-B_{s} \sim \mathcal{N}(0, t-s), 为 0 ≤ s < t < ∞ , B t − B s ∼ N ( 0 , t − s ) 0 ≤ s < t < ∞ , B t − B s ∼ N ( 0 , t − s ) 0 <= s < t < oo,B_(t)-B_(s)∼N(0,t-s)0 \leq s<t<\infty, B_{t}-B_{s} \sim \mathcal{N}(0, t-s) 、
  • for 0 ≤ s < t < ∞ , B t − B s 0 ≤ s < t < ∞ , B t − B s 0 <= s < t < oo,B_(t)-B_(s)0 \leq s<t<\infty, B_{t}-B_{s} is independent of B s B s B_(s)B_{s},
    的 0 ≤ s < t < ∞ , B t − B s 0 ≤ s < t < ∞ , B t − B s 0 <= s < t < oo,B_(t)-B_(s)0 \leq s<t<\infty, B_{t}-B_{s} 与 B s B s B_(s)B_{s} 无关、
  • the trajectories t ↦ B t t ↦ B t t|->B_(t)t \mapsto B_{t} are continuous.
    t ↦ B t t ↦ B t t|->B_(t)t \mapsto B_{t} 的轨迹是连续的。
The stochastic process B = { B t , t ≥ 0 } B = B t , t ≥ 0 B={B_(t),t >= 0}B=\left\{B_{t}, t \geq 0\right\} is called Brownian motion and is of fundamental importance in both the theory, and applications, of probability.
随机过程 B = { B t , t ≥ 0 } B = B t , t ≥ 0 B={B_(t),t >= 0}B=\left\{B_{t}, t \geq 0\right\} 被称为布朗运动，在概率理论和应用中都具有根本性的重要意义。
It is actually a very deep result that there exists a stochastic process having these properties (continuous trajectories is the tough part). One way to prove the existence of Brownian motion is to take an appropriate limit of appropriately scaled simple symmetric random walks. (This concept of a scaling limit is of fundamental importance in modern probability research.)
实际上，存在一个具有这些特性（连续轨迹是困难的部分）的随机过程是一个非常深奥的结果。证明布朗运动存在的一种方法是对适当比例的简单对称随机游走进行适当的极限。(缩放极限这一概念在现代概率研究中具有根本性的重要意义）。
Remark. The history of Brownian motion is fascinating. In the summer of 1827, the Scottish botanist Robert Brown observed that microscopic pollen grains suspended in water move in an erratic, highly irregular, zigzag pattern. Following Brown’s initial report, other scientists verified the strange phenomenon. Brownian motion was apparent whenever very small particles were suspended in a fluid medium, for example smoke particles in air. It was eventually determined that finer particles move more rapidly, that their motion is stimulated by heat, and that the movement is more active when the fluid viscosity is reduced.
备注布朗运动的历史引人入胜。1827 年夏天，苏格兰植物学家罗伯特-布朗（Robert Brown）观察到，悬浮在水中的微小花粉粒呈不规则、极不规则的 "之 "字形运动。布朗首次报告后，其他科学家也验证了这一奇怪现象。只要极小的颗粒悬浮在流体介质中，例如空气中的烟雾颗粒，布朗运动就会显现出来。最终确定，较细的颗粒运动得更快，它们的运动受热的刺激，当流体粘度降低时，运动更加活跃。
However, it was only in 1905 that Albert Einstein, using a probabilistic model, could provide a satisfactory explanation of the Brownian motion. He asserted that the Brownian motion originates in the continual bombardment of the pollen grains by the molecules of the surrounding water, with successive molecular impacts coming from different directions and contributing different impulses to the particles. As a result of the continual collisions, the particles themselves had the same average kinetic energy as the molecules. Thus, he showed that Brownian motion provided a solution (in a certain sense) to the famous partial differential equation u t = u x x u t = u x x u_(t)=u_(xx)u_{t}=u_{x x}, the so-called heat equation.
然而，直到 1905 年，阿尔伯特-爱因斯坦才利用概率模型对布朗运动做出了令人满意的解释。他断言，布朗运动源于周围水分子对花粉粒的持续轰击，连续的分子撞击来自不同的方向，对颗粒产生不同的冲力。由于不断碰撞，颗粒本身具有与分子相同的平均动能。因此，他证明了布朗运动为著名的偏微分方程 u t = u x x u t = u x x u_(t)=u_(xx)u_{t}=u_{x x} ，即所谓的热方程提供了一种解（在一定意义上）。
Note that in 1905, belief in atoms and molecules was far from universal. In fact, Einstein’s “proof” of Brownian motion helped provide convincing evidence of atomic existence. Einstein had a busy 1905, also publishing seminal papers on the special theory of relativity and the photoelectric effect. In fact, his work on the photoelectric effect won him a Nobel prize. Curiously, though, history has shown that the photoelectric effect is the least monumental of his three 1905 triumphs. The world at that time simply could not accept special relativity!
请注意，在 1905 年，人们对原子和分子的信仰还远未普及。事实上，爱因斯坦对布朗运动的 "证明 "为原子的存在提供了令人信服的证据。爱因斯坦在 1905 年忙得不可开交，他还发表了关于狭义相对论和光电效应的开创性论文。事实上，他在光电效应方面的研究为他赢得了诺贝尔奖。但奇怪的是，历史表明，光电效应是他 1905 年三项成就中最不具纪念意义的一项。当时的世界根本无法接受狭义相对论！
Since Brownian motion described the physical trajectories of pollen grains suspended in water, Brownian paths must be continuous. But they were seen to be so irregular that the French physicist Jean Perrin believed them to be non-differentiable. (The German mathematician Karl Weierstrass had recently discovered such pathological functions do exist. Indeed the continuous function
由于布朗运动描述的是悬浮在水中的花粉粒的物理轨迹，因此布朗路径必须是连续的。但是，人们发现它们是如此不规则，以至于法国物理学家让-佩兰（Jean Perrin）认为它们是不可分的。(德国数学家卡尔-魏尔斯特拉斯（Karl Weierstrass）最近发现这种病态函数确实存在。事实上，连续函数
g ( x ) = ∑ n = 1 ∞ b n cos ⁡ ( a n π x ) g ( x ) = ∑ n = 1 ∞   b n cos ⁡ a n π x g(x)=sum_(n=1)^(oo)b^(n)cos(a^(n)pi x)g(x)=\sum_{n=1}^{\infty} b^{n} \cos \left(a^{n} \pi x\right)
where a a aa is odd, b ∈ ( 0 , 1 ) b ∈ ( 0 , 1 ) b in(0,1)b \in(0,1), and a b > 1 + 3 π / 2 a b > 1 + 3 π / 2 ab > 1+3pi//2a b>1+3 \pi / 2 is nowhere differentiable.) Perrin himself worked to show that colliding particles obey the gas laws, calculated Avogadro’s number, and won the 1926 Nobel prize.
其中 a a aa 为奇数， b ∈ ( 0 , 1 ) b ∈ ( 0 , 1 ) b in(0,1)b \in(0,1) 和 a b > 1 + 3 π / 2 a b > 1 + 3 π / 2 ab > 1+3pi//2a b>1+3 \pi / 2 无处可微分）。佩林本人致力于证明碰撞粒子遵守气体定律，计算出了阿伏加德罗数，并获得了 1926 年的诺贝尔奖。
Finally, in 1923, the mathematician Norbert Wiener established the mathematical existence of Brownian motion by verifying the existence of a stochastic process with the given properties.
最后，在 1923 年，数学家诺伯特-维纳通过验证具有给定性质的随机过程的存在，确立了布朗运动的数学存在性。
Exercise 2.6. Deduce from the definition of Brownian motion that for each t t tt, the random variable B t B t B_(t)B_{t} is normally distributed with mean 0 and variance t t tt. Why does this implies that E ( B t 2 ) = t E B t 2 = t E(B_(t)^(2))=t\mathbb{E}\left(B_{t}^{2}\right)=t ?
练习 2.6.根据布朗运动的定义推导，对于每个 t t tt ，随机变量 B t B t B_(t)B_{t} 是正态分布，均值为 0，方差为 t t tt 。为什么这意味着 E ( B t 2 ) = t E B t 2 = t E(B_(t)^(2))=t\mathbb{E}\left(B_{t}^{2}\right)=t ?
Exercise 2.7. Deduce from the definition of Brownian motion that for 0 ≤ s < t < ∞ 0 ≤ s < t < ∞ 0 <= s < t < oo0 \leq s<t<\infty, the distribution of the random variable B t − B s B t − B s B_(t)-B_(s)B_{t}-B_{s} is the same as the distribution of the random variable B t − s B t − s B_(t-s)B_{t-s}.
练习 2.7.根据布朗运动的定义推导，对于 0 ≤ s < t < ∞ 0 ≤ s < t < ∞ 0 <= s < t < oo0 \leq s<t<\infty ，随机变量 B t − B s B t − B s B_(t)-B_(s)B_{t}-B_{s} 的分布与随机变量 B t − s B t − s B_(t-s)B_{t-s} 的分布相同。
Exercise 2.8. Show that Cov ( B t , B s ) = min ( s , t ) Cov B t , B s = min ( s , t ) Cov(B_(t),B_(s))=min(s,t)\operatorname{Cov}\left(B_{t}, B_{s}\right)=\min (s, t). (Hint: Write B s B t = ( B s B t − B s 2 ) + B s 2 B s B t = B s B t − B s 2 + B s 2 B_(s)B_(t)=(B_(s)B_(t)-B_(s)^(2))+B_(s)^(2)B_{s} B_{t}=\left(B_{s} B_{t}-B_{s}^{2}\right)+B_{s}^{2}, take expectations, and then use the third part of the definition of Brownian motion and Exercise 2.6.)
练习 2.8.证明 Cov ( B t , B s ) = min ( s , t ) Cov B t , B s = min ( s , t ) Cov(B_(t),B_(s))=min(s,t)\operatorname{Cov}\left(B_{t}, B_{s}\right)=\min (s, t) 。(提示：写出 B s B t = ( B s B t − B s 2 ) + B s 2 B s B t = B s B t − B s 2 + B s 2 B_(s)B_(t)=(B_(s)B_(t)-B_(s)^(2))+B_(s)^(2)B_{s} B_{t}=\left(B_{s} B_{t}-B_{s}^{2}\right)+B_{s}^{2} ，求期望值，然后使用布朗运动定义的第三部分和练习 2.6）。

3 Regularity Properties of Continuous Time Stochastic Processes
3 连续时间随机过程的正则特性

We now define what it means for a continuous time stochastic process to be continuous.
现在我们来定义连续时间随机过程的连续性。
Definition 3.1. A continuous time stochastic process X = { X t , t ≥ 0 } X = X t , t ≥ 0 X={X_(t),t >= 0}X=\left\{X_{t}, t \geq 0\right\} is said to be stochastically continuous (or continuous in probability) for every t ≥ 0 t ≥ 0 t >= 0t \geq 0, and for every ϵ > 0 ϵ > 0 epsilon > 0\epsilon>0,
定义 3.1.对于每个 t ≥ 0 t ≥ 0 t >= 0t \geq 0 和每个 ϵ > 0 ϵ > 0 epsilon > 0\epsilon>0 而言，连续时间随机过程 X = { X t , t ≥ 0 } X = X t , t ≥ 0 X={X_(t),t >= 0}X=\left\{X_{t}, t \geq 0\right\} 称为随机连续（或概率连续）、
lim s → t P ( | X t − X s | > ϵ ) = 0 lim s → t   P X t − X s > ϵ = 0 lim_(s rarr t)P(|X_(t)-X_(s)| > epsilon)=0\lim _{s \rightarrow t} \mathbb{P}\left(\left|X_{t}-X_{s}\right|>\epsilon\right)=0
Definition 3.2. A continuous time stochastic process X = { X t , t ≥ 0 } X = X t , t ≥ 0 X={X_(t),t >= 0}X=\left\{X_{t}, t \geq 0\right\} is said to be continuous, or to have continuous sample paths, if t ↦ X t ( ω ) t ↦ X t ( ω ) t|->X_(t)(omega)t \mapsto X_{t}(\omega) is continuous for all ω ω omega\omega.
定义 3.2.如果 t ↦ X t ( ω ) t ↦ X t ( ω ) t|->X_(t)(omega)t \mapsto X_{t}(\omega) 对于所有 ω ω omega\omega 都是连续的，则称连续时间随机过程 X = { X t , t ≥ 0 } X = X t , t ≥ 0 X={X_(t),t >= 0}X=\left\{X_{t}, t \geq 0\right\} 是连续的，或具有连续的样本路径。
Remark. The definitions of a continuous time stochastic process having right-continuous, or left-continuous, sample paths are analogous.
备注具有右连续或左连续样本路径的连续时间随机过程的定义是类似的。
Next we consider what it means for two stochastic processes to be equal. In the presence of a probability measure, there are three natural, but related, ways to define sameness.
接下来，我们要考虑两个随机过程相等的含义。在存在概率度量的情况下，有三种自然但相关的方法来定义同一性。
Definition 3.3. Suppose that X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} and Y = { Y α , α ∈ I } Y = Y α , α ∈ I Y={Y_(alpha),alpha in I}Y=\left\{Y_{\alpha}, \alpha \in I\right\} are two stochastic processes defined on a common probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). We say that Y Y YY is a version of X X XX if for every α ∈ I α ∈ I alpha in I\alpha \in I, we have
定义 3.3.假设 X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 和 Y = { Y α , α ∈ I } Y = Y α , α ∈ I Y={Y_(alpha),alpha in I}Y=\left\{Y_{\alpha}, \alpha \in I\right\} 是定义在共同概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的两个随机过程。如果对每个 α ∈ I α ∈ I alpha in I\alpha \in I 都有以下条件，我们就说 Y Y YY 是 X X XX 的一个版本
P ( X α = Y α ) = P ( { ω ∈ Ω : X α ( ω ) = Y α ( ω ) } ) = 1 P X α = Y α = P ω ∈ Ω : X α ( ω ) = Y α ( ω ) = 1 P(X_(alpha)=Y_(alpha))=P({omega in Omega:X_(alpha)(omega)=Y_(alpha)(omega)})=1\mathbb{P}\left(X_{\alpha}=Y_{\alpha}\right)=\mathbb{P}\left(\left\{\omega \in \Omega: X_{\alpha}(\omega)=Y_{\alpha}(\omega)\right\}\right)=1
We say that X X XX and Y Y YY are indistinguishable if
如果出现以下情况，我们就说 X X XX 和 Y Y YY 是不可区分的
P ( X α = Y α ∀ α ∈ I ) = P ( { ω ∈ Ω : X α ( ω ) = Y α ( ω ) ∀ α ∈ I } ) = 1 P X α = Y α ∀ α ∈ I = P ω ∈ Ω : X α ( ω ) = Y α ( ω ) ∀ α ∈ I = 1 P(X_(alpha)=Y_(alpha)AA alpha in I)=P({omega in Omega:X_(alpha)(omega)=Y_(alpha)(omega)AA alpha in I})=1\mathbb{P}\left(X_{\alpha}=Y_{\alpha} \forall \alpha \in I\right)=\mathbb{P}\left(\left\{\omega \in \Omega: X_{\alpha}(\omega)=Y_{\alpha}(\omega) \forall \alpha \in I\right\}\right)=1
In other words, X X XX and Y Y YY are indistinguishable if
换句话说，如果出现以下情况， X X XX 和 Y Y YY 是不可区分的
{ ω ∈ Ω : X α ( ω ) ≠ Y α ( ω ) for some α ∈ I } ω ∈ Ω : X α ( ω ) ≠ Y α ( ω )  for some  α ∈ I {omega in Omega:X_(alpha)(omega)!=Y_(alpha)(omega)" for some "alpha in I}\left\{\omega \in \Omega: X_{\alpha}(\omega) \neq Y_{\alpha}(\omega) \text { for some } \alpha \in I\right\}
is a P P P\mathbb{P}-null set. 是一个 P P P\mathbb{P} 空集。
Remark. Some texts use the term modification in place of version.
备注有些文本用修改一词代替版本。
If two processes are indistinguishable, then they are trivially versions of each other. However, the distinction between version and indistinguishable can be subtle since two processes can be versions of each other, yet have completely different sample paths. This is illustrated with the following example and exercises.
如果两个进程无法区分，那么它们就是彼此的版本。然而，版本和无差别之间的区别可能很微妙，因为两个进程可以是彼此的版本，但样本路径却完全不同。下面的例子和练习可以说明这一点。
Example 3.4. Suppose that Z Z ZZ is an absolutely continuous random variable on the probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). (For example, suppose Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1).) Define the stochastic processes X X XX and Y Y YY on the product space [ 0 , ∞ ) × Ω [ 0 , ∞ ) × Ω [0,oo)xx Omega[0, \infty) \times \Omega by setting X t = 0 X t = 0 X_(t)=0X_{t}=0 and
例 3.4.假设 Z Z ZZ 是概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的绝对连续随机变量。(例如，假设 Z ∼ N ( 0 , 1 ) Z ∼ N ( 0 , 1 ) Z∼N(0,1)Z \sim \mathcal{N}(0,1) 。）通过设置 X t = 0 X t = 0 X_(t)=0X_{t}=0 和 Y Y YY 来定义乘积空间 [ 0 , ∞ ) × Ω [ 0 , ∞ ) × Ω [0,oo)xx Omega[0, \infty) \times \Omega 上的随机过程 X X XX 和 Y Y YY 。
Y t = { 0 , if t ≠ | Z | 1 , if t = | Z | Y t = 0 ,       if  t ≠ | Z | 1 ,       if  t = | Z | Y_(t)={[0","," if "t!=|Z|],[1","," if "t=|Z|]:}Y_{t}= \begin{cases}0, & \text { if } t \neq|Z| \\ 1, & \text { if } t=|Z|\end{cases}
We see that Y Y YY is a version of X X XX since for every t ≥ 0 t ≥ 0 t >= 0t \geq 0,
我们看到， Y Y YY 是 X X XX 的一个版本，因为对于每一个 t ≥ 0 t ≥ 0 t >= 0t \geq 0 来说、
P ( X t ≠ Y t ) = P ( | Z | = t ) = P ( Z = t ) + P ( Z = − t ) = 0 P X t ≠ Y t = P ( | Z | = t ) = P ( Z = t ) + P ( Z = − t ) = 0 P(X_(t)!=Y_(t))=P(|Z|=t)=P(Z=t)+P(Z=-t)=0\mathbb{P}\left(X_{t} \neq Y_{t}\right)=\mathbb{P}(|Z|=t)=\mathbb{P}(Z=t)+\mathbb{P}(Z=-t)=0
giving P ( X t = Y t ) = 1 P X t = Y t = 1 P(X_(t)=Y_(t))=1\mathbb{P}\left(X_{t}=Y_{t}\right)=1. On the other hand, X X XX and Y Y YY are not indistinguishable since
得出 P ( X t = Y t ) = 1 P X t = Y t = 1 P(X_(t)=Y_(t))=1\mathbb{P}\left(X_{t}=Y_{t}\right)=1 。另一方面， X X XX 和 Y Y YY 并非不可区分，因为
P ( X t = Y t ∀ t ≥ 0 ) = 0 P X t = Y t ∀ t ≥ 0 = 0 P(X_(t)=Y_(t)AA t >= 0)=0\mathbb{P}\left(X_{t}=Y_{t} \forall t \geq 0\right)=0
A similar example is provided by the next exercise.
下一个练习也提供了类似的例子。
Exercise 3.5. Suppose that Ω = [ 0 , 1 ] , F Ω = [ 0 , 1 ] , F Omega=[0,1],F\Omega=[0,1], \mathcal{F} are the Borel sets of [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1], and P P P\mathbb{P} is the uniform probability (i.e., Lebesgue measure) on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1], and assume that F F F\mathcal{F} is complete with respect to P P P\mathbb{P}. For t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1], and ω ∈ [ 0 , 1 ] ω ∈ [ 0 , 1 ] omega in[0,1]\omega \in[0,1], define X t ( ω ) = 0 X t ( ω ) = 0 X_(t)(omega)=0X_{t}(\omega)=0 and Y t ( ω ) = 1 { t = ω } Y t ( ω ) = 1 { t = ω } Y_(t)(omega)=1{t=omega}Y_{t}(\omega)=\mathbb{1}\{t=\omega\}𝟙. Show that X X XX and Y Y YY are versions of each other, but that they are not indistinguishable.
练习 3.5.假设 Ω = [ 0 , 1 ] , F Ω = [ 0 , 1 ] , F Omega=[0,1],F\Omega=[0,1], \mathcal{F} 是 [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1] 的 Borel 集， P P P\mathbb{P} 是 [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1] 上的均匀概率（即 Lebesgue 度量），并假设 F F F\mathcal{F} 关于 P P P\mathbb{P} 是完全的。对于 t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1] 和 ω ∈ [ 0 , 1 ] ω ∈ [ 0 , 1 ] omega in[0,1]\omega \in[0,1] ，定义 X t ( ω ) = 0 X t ( ω ) = 0 X_(t)(omega)=0X_{t}(\omega)=0 和 Y t ( ω ) = 1 { t = ω } Y t ( ω ) = 1 { t = ω } Y_(t)(omega)=1{t=omega}Y_{t}(\omega)=\mathbb{1}\{t=\omega\}𝟙 。证明 X X XX 和 Y Y YY 是彼此的版本，但它们并非不可区分。
Thus we see that if two processes are indistinguishable, then they will necessarily have a.s. indistinguishable sample paths. However, the same is not true if Y Y YY is only a version of X X XX. In both of the previous two instances, t ↦ X t t ↦ X t t|->X_(t)t \mapsto X_{t} has constant sample paths, but t ↦ Y t t ↦ Y t t|->Y_(t)t \mapsto Y_{t} has discontinuous sample paths.
因此，我们可以看到，如果两个过程是不可区分的，那么它们必然会有一个不可区分的样本路径。但是，如果 Y Y YY 只是 X X XX 的一个版本，情况就不一样了。在前两种情况下， t ↦ X t t ↦ X t t|->X_(t)t \mapsto X_{t} 的样本路径都是恒定的，而 t ↦ Y t t ↦ Y t t|->Y_(t)t \mapsto Y_{t} 的样本路径却是不连续的。
The following exercise gives a partial converse.
下面的练习给出了部分反义词。
Exercise 3.6. Suppose that Y Y YY is a version of X X XX, and that both X X XX and Y Y YY have rightcontinuous sample paths. Show that X X XX and Y Y YY are indistinguishable.
练习 3.6.假设 Y Y YY 是 X X XX 的一个版本，并且 X X XX 和 Y Y YY 都有右连续的样本路径。证明 X X XX 和 Y Y YY 是无差别的。
In order to define both version and indistinguishable, it was necessary that both X X XX and Y Y YY be defined on the same probability space. For our final definition of sameness, this is not necessary.
为了同时定义版本和无差别， X X XX 和 Y Y YY 必须定义在同一个概率空间上。对于我们最终的同一性定义来说，这并非必要。
Definition 3.7. Suppose that X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} is a stochastic processes defined on a probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}), and that Y = { Y α , α ∈ I } Y = Y α , α ∈ I Y={Y_(alpha),alpha in I}Y=\left\{Y_{\alpha}, \alpha \in I\right\} is a stochastic processes defined on a probability space ( Ω ~ , F ~ , P ~ ) ( Ω ~ , F ~ , P ~ ) ( tilde(Omega), tilde(F), tilde(P))(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{\mathbb{P}}). We say that X X XX and Y Y YY have the same finite dimensional distributions if for any integer n ≥ 1 n ≥ 1 n >= 1n \geq 1; for any distinct indices α 1 , α 2 , … , α n α 1 , α 2 , … , α n alpha_(1),alpha_(2),dots,alpha_(n)\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}, each α i ∈ I α i ∈ I alpha_(i)in I\alpha_{i} \in I; and for every Borel set A ∈ B ( R ) A ∈ B ( R ) A inB(R)A \in \mathcal{B}(\mathbb{R}), we have
定义 3.7.假设 X = { X α , α ∈ I } X = X α , α ∈ I X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 是定义在概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的随机过程，而 Y = { Y α , α ∈ I } Y = Y α , α ∈ I Y={Y_(alpha),alpha in I}Y=\left\{Y_{\alpha}, \alpha \in I\right\} 是定义在概率空间 ( Ω ~ , F ~ , P ~ ) ( Ω ~ , F ~ , P ~ ) ( tilde(Omega), tilde(F), tilde(P))(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{\mathbb{P}}) 上的随机过程。如果对于任何整数 n ≥ 1 n ≥ 1 n >= 1n \geq 1 ；对于任何不同的指数 α 1 , α 2 , … , α n α 1 , α 2 , … , α n alpha_(1),alpha_(2),dots,alpha_(n)\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} ，每个 α i ∈ I α i ∈ I alpha_(i)in I\alpha_{i} \in I ；以及对于每个伯尔集合 A ∈ B ( R ) A ∈ B ( R ) A inB(R)A \in \mathcal{B}(\mathbb{R}) ，我们都说 X X XX 和 Y Y YY 具有相同的有限维分布，那么
P ( ( X α 1 , … , X α n ) ∈ A ) = P ~ ( ( Y α 1 , … , Y α n ) ∈ A ) P X α 1 , … , X α n ∈ A = P ~ Y α 1 , … , Y α n ∈ A P((X_(alpha_(1)),dots,X_(alpha_(n)))in A)= tilde(P)((Y_(alpha_(1)),dots,Y_(alpha_(n)))in A)\mathbb{P}\left(\left(X_{\alpha_{1}}, \ldots, X_{\alpha_{n}}\right) \in A\right)=\tilde{\mathbb{P}}\left(\left(Y_{\alpha_{1}}, \ldots, Y_{\alpha_{n}}\right) \in A\right)
Exercise 3.8. Suppose that X X XX and Y Y YY are stochastic processes both defined on a common probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). Show that if Y Y YY is a version of X X XX, then X X XX and Y Y YY have the same finite dimensional distributions.
练习 3.8.假设 X X XX 和 Y Y YY 都是定义在共同概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的随机过程。证明如果 Y Y YY 是 X X XX 的一个版本，那么 X X XX 和 Y Y YY 具有相同的有限维分布。
In order to further study the sample path properties of a stochastic process, it will be convenient for it to have some joint measurability properties.
为了进一步研究随机过程的样本路径特性，需要使随机过程具有一些联合可测性。
Definition 3.9. Suppose that ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) is a probability space. The stochastic process X = X = X=X= { X α , α ∈ I } X α , α ∈ I {X_(alpha),alpha in I}\left\{X_{\alpha}, \alpha \in I\right\}, defined by X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} where X ( α , ω ) = X α ( ω ) X ( α , ω ) = X α ( ω ) X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega), is said to be measurable if for each ω ∈ Ω ω ∈ Ω omega in Omega\omega \in \Omega the section X ( ⋅ , ω ) : I → R X ( ⋅ , ω ) : I → R X(*,omega):I rarrRX(\cdot, \omega): I \rightarrow \mathbb{R} is a measurable function (called the sample path or trajectory of X X XX at ω ω omega\omega ).
定义 3.9.假设 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 是一个概率空间。由 X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} 定义的随机过程 X = X = X=X= { X α , α ∈ I } X α , α ∈ I {X_(alpha),alpha in I}\left\{X_{\alpha}, \alpha \in I\right\} ，其中 X ( α , ω ) = X α ( ω ) X ( α , ω ) = X α ( ω ) X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) 如果对于每个 ω ∈ Ω ω ∈ Ω omega in Omega\omega \in \Omega 部分 X ( ⋅ , ω ) : I → R X ( ⋅ , ω ) : I → R X(*,omega):I rarrRX(\cdot, \omega): I \rightarrow \mathbb{R} 都是可测函数（称为 X X XX 在 ω ω omega\omega 处的样本路径或轨迹），则称该过程为可测过程。
Remark. In other words, a stochastic process is measurable if the function
备注换句话说，如果函数
X : ( I × Ω , B ( I ) ⊗ F ) → ( R , B ( R ) ) X : ( I × Ω , B ( I ) ⊗ F ) → ( R , B ( R ) ) X:(I xx Omega,B(I)oxF)rarr(R,B(R))X:(I \times \Omega, \mathcal{B}(I) \otimes \mathcal{F}) \rightarrow(\mathbb{R}, \mathcal{B}(R))
is measurable, where B ( I ) ⊗ F = σ ( B ( I ) , F ) B ( I ) ⊗ F = σ ( B ( I ) , F ) B(I)oxF=sigma(B(I),F)\mathcal{B}(I) \otimes \mathcal{F}=\sigma(\mathcal{B}(I), \mathcal{F}) is the product σ σ sigma\sigma-algebra. Recall from [5, Theorem 10.2] that if X X XX is (jointly) measurable, then each of its sections is necessarily measurable.
是可测的，其中 B ( I ) ⊗ F = σ ( B ( I ) , F ) B ( I ) ⊗ F = σ ( B ( I ) , F ) B(I)oxF=sigma(B(I),F)\mathcal{B}(I) \otimes \mathcal{F}=\sigma(\mathcal{B}(I), \mathcal{F}) 是乘积 σ σ sigma\sigma 代数。回顾 [5, 定理 10.2]，如果 X X XX 是（共同）可测的，那么它的每个部分必然是可测的。
For the benefit of the reader, we summarize the definition of measurable stochastic process emphasizing the measurability of the sections. That is, if X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} is a measurable stochastic process with indexing set I I II on the probability space ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}), then
为了方便读者，我们总结了可测随机过程的定义，强调了各部分的可测性。也就是说，如果 X : I × Ω → R X : I × Ω → R X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} 是概率空间 ( Ω , F , P ) ( Ω , F , P ) (Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上具有索引集 I I II 的可测随机过程，那么
  • for each α ∈ I α ∈ I alpha in I\alpha \in I the section X ( α , ⋅ ) : Ω → R X ( α , ⋅ ) : Ω → R X(alpha,*):Omega rarrRX(\alpha, \cdot): \Omega \rightarrow \mathbb{R} is a random variable, and
    对于每个 α ∈ I α ∈ I alpha in I\alpha \in I 部分， X ( α , ⋅ ) : Ω → R X ( α , ⋅ ) : Ω → R X(alpha,*):Omega rarrRX(\alpha, \cdot): \Omega \rightarrow \mathbb{R} 是一个随机变量，并且
  • for each ω ∈ Ω ω ∈ Ω omega in Omega\omega \in \Omega the section X ( ⋅ , ω ) : I → R X ( ⋅ , ω ) : I → R X(*,omega):I rarrRX(\cdot, \omega): I \rightarrow \mathbb{R} is a measurable function (called the sample path or trajectory of X X XX at ω ) ω ) omega)\omega).
    对于每个 ω ∈ Ω ω ∈ Ω omega in Omega\omega \in \Omega ， X ( ⋅ , ω ) : I → R X ( ⋅ , ω ) : I → R X(*,omega):I rarrRX(\cdot, \omega): I \rightarrow \mathbb{R} 部分是一个可测函数（称为 X X XX 在 ω ) ω ) omega)\omega) 处的样本路径或轨迹）。
    The first condition above is simply a restatement of the definition that a stochastic process is a collection of random variables. It is the second condition that gives the definition of measurable stochastic process its substance.
    上述第一个条件只是重申了随机过程是随机变量集合的定义。第二个条件才是可测量随机过程定义的实质。
Not all stochastic processes are measurable, and the following example shows that the measurability of a stochastic process cannot be taken for granted.
并非所有的随机过程都是可测量的，下面的例子说明，不能想当然地认为随机过程是可测量的。
Example 3.10. Suppose that for each t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1], the random variable X t X t X_(t)X_{t} is uniformly distributed on [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1]. Suppose further that the collection of random variables { X t , t ∈ [ 0 , 1 ] } X t , t ∈ [ 0 , 1 ] {X_(t),t in[0,1]}\left\{X_{t}, t \in[0,1]\right\} is independent. We now show that X = { X t , t ∈ [ 0 , 1 ] } X = X t , t ∈ [ 0 , 1 ] X={X_(t),t in[0,1]}X=\left\{X_{t}, t \in[0,1]\right\} is not measurable. To the contrary, suppose that X X XX is measurable and for t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1], define Y t Y t Y_(t)Y_{t} by
例 3.10.假设对于每个 t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1] ，随机变量 X t X t X_(t)X_{t} 在 [ − 1 , 1 ] [ − 1 , 1 ] [-1,1][-1,1] 上均匀分布。再假设随机变量集合 { X t , t ∈ [ 0 , 1 ] } X t , t ∈ [ 0 , 1 ] {X_(t),t in[0,1]}\left\{X_{t}, t \in[0,1]\right\} 是独立的。我们现在证明 X = { X t , t ∈ [ 0 , 1 ] } X = X t , t ∈ [ 0 , 1 ] X={X_(t),t in[0,1]}X=\left\{X_{t}, t \in[0,1]\right\} 不可测。相反，假设 X X XX 是可测的，对于 t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1] ，通过以下方式定义 Y t Y t Y_(t)Y_{t}
Y t = ∫ 0 t X s d s Y t = ∫ 0 t   X s d s Y_(t)=int_(0)^(t)X_(s)dsY_{t}=\int_{0}^{t} X_{s} d s
It is not too difficult to show that each Y t Y t Y_(t)Y_{t} is a random variable. Furthermore, the function t ↦ Y t ( ω ) t ↦ Y t ( ω ) t|->Y_(t)(omega)t \mapsto Y_{t}(\omega) is continuous for all ω ω omega\omega. Now, if P ( Y t ≠ 0 ) = 0 P Y t ≠ 0 = 0 P(Y_(t)!=0)=0\mathbb{P}\left(Y_{t} \neq 0\right)=0 for all t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1], then Y t = 0 Y t = 0 Y_(t)=0Y_{t}=0 for all rationals a.s., and therefore Y t = 0 Y t = 0 Y_(t)=0Y_{t}=0 for all t t tt a.s. by path continuity. We therefore conclude that X t = 0 X t = 0 X_(t)=0X_{t}=0 for almost all t t tt a.s., and hence a.s. for all t t tt. Thus, there must exist a t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1] such that P ( Y t ≠ 0 ) > 0 P Y t ≠ 0 > 0 P(Y_(t)!=0) > 0P\left(Y_{t} \neq 0\right)>0. This implies that E ( Y t 2 ) > 0 E Y t 2 > 0 E(Y_(t)^(2)) > 0\mathbb{E}\left(Y_{t}^{2}\right)>0. However,
要证明每个 Y t Y t Y_(t)Y_{t} 都是随机变量并不难。此外，函数 t ↦ Y t ( ω ) t ↦ Y t ( ω ) t|->Y_(t)(omega)t \mapsto Y_{t}(\omega) 对于所有 ω ω omega\omega 都是连续的。现在，如果所有 t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1] 都是 P ( Y t ≠ 0 ) = 0 P Y t ≠ 0 = 0 P(Y_(t)!=0)=0\mathbb{P}\left(Y_{t} \neq 0\right)=0 ，那么所有有理数都是 Y t = 0 Y t = 0 Y_(t)=0Y_{t}=0 ，因此根据路径连续性，所有 t t tt 都是 Y t = 0 Y t = 0 Y_(t)=0Y_{t}=0 。因此，我们得出结论：几乎所有 t t tt 的 a.s. 都是 X t = 0 X t = 0 X_(t)=0X_{t}=0 ，因此所有 t t tt 的 a.s. 都是 t t tt。因此，一定存在一个 t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] t in[0,1]t \in[0,1] ，使得 P ( Y t ≠ 0 ) > 0 P Y t ≠ 0 > 0 P(Y_(t)!=0) > 0P\left(Y_{t} \neq 0\right)>0 存在。这意味着 E ( Y t 2 ) > 0 E Y t 2 > 0 E(Y_(t)^(2)) > 0\mathbb{E}\left(Y_{t}^{2}\right)>0 。但是
E ( Y t 2 ) = E ( ∫ 0 t ∫ 0 t X s X r d s d r ) = ∫ 0 t ∫ 0 t E ( X s X r ) d s d r = E ( X 1 2 ) ∫ 0 t ∫ 0 t 1 { s = r } d s d r = 0 , E Y t 2 = E ∫ 0 t   ∫ 0 t   X s X r d s d r = ∫ 0 t   ∫ 0 t   E X s X r d s d r = E X 1 2 ∫ 0 t   ∫ 0 t   1 { s = r } d s d r = 0 , {:[E(Y_(t)^(2))=E(int_(0)^(t)int_(0)^(t)X_(s)X_(r)dsdr)=int_(0)^(t)int_(0)^(t)E(X_(s)X_(r))dsdr],[=E(X_(1)^(2))int_(0)^(t)int_(0)^(t)1{s=r}dsdr],[=0","]:}\begin{aligned} \mathbb{E}\left(Y_{t}^{2}\right)=\mathbb{E}\left(\int_{0}^{t} \int_{0}^{t} X_{s} X_{r} d s d r\right) & =\int_{0}^{t} \int_{0}^{t} \mathbb{E}\left(X_{s} X_{r}\right) d s d r \\ & =\mathbb{E}\left(X_{1}^{2}\right) \int_{0}^{t} \int_{0}^{t} \mathbb{1}\{s=r\} d s d r \\ & =0, \end{aligned}𝟙
which is a contradiction to the assumption that X X XX is measurable.
这与 X X XX 是可测的这一假设相矛盾。
We end this section with an important result which tell us when a stochastic process has a measurable version, and a continuous version. Suppose that X X XX is a continuous time stochastic process with indexing set I = [ 0 , ∞ ) I = [ 0 , ∞ ) I=[0,oo)I=[0, \infty), and let T > 0 T > 0 T > 0T>0. For every δ > 0 δ > 0 delta > 0\delta>0, the modulus of continuity of the sample path X ( ω ) X ( ω ) X(omega)X(\omega) on [ 0 , T ] [ 0 , T ] [0,T][0, T] is
在本节的最后，我们将讨论一个重要的结果，它告诉我们一个随机过程何时具有可测版本和连续版本。假设 X X XX 是一个连续时间随机过程，其索引集为 I = [ 0 , ∞ ) I = [ 0 , ∞ ) I=[0,oo)I=[0, \infty) ，设 T > 0 T > 0 T > 0T>0 。对于每个 δ > 0 δ > 0 delta > 0\delta>0 ， [ 0 , T ] [ 0 , T ] [0,T][0, T] 上的样本路径 X ( ω ) X ( ω ) X(omega)X(\omega) 的连续性模为
m T ( X ( ω ) , δ ) = sup { | X t ( ω ) − X s ( ω ) | : | s − t | ≤ δ , 0 ≤ s , t ≤ T } m T ( X ( ω ) , δ ) = sup X t ( ω ) − X s ( ω ) : | s − t | ≤ δ , 0 ≤ s , t ≤ T m_(T)(X(omega),delta)=s u p{|X_(t)(omega)-X_(s)(omega)|:|s-t| <= delta,quad0 <= s,t <= T}m_{T}(X(\omega), \delta)=\sup \left\{\left|X_{t}(\omega)-X_{s}(\omega)\right|:|s-t| \leq \delta, \quad 0 \leq s, t \leq T\right\}
Theorem 3.11. Suppose that X X XX is a continuous time stochastic process.
定理 3.11.假设 X X XX 是连续时间随机过程。
  • If X X XX is stochastically continuous, then X X XX has a measurable version.
    如果 X X XX 是随机连续的，那么 X X XX 就有一个可测量的版本。
  • X X XX has a continuous version if and only if (i) X X XX is stochastically continuous, and (ii) m T ( X ( ω ) , δ ) → 0 m T ( X ( ω ) , δ ) → 0 m_(T)(X(omega),delta)rarr0m_{T}(X(\omega), \delta) \rightarrow 0 in probability as δ → 0 δ → 0 delta rarr0\delta \rightarrow 0.
    当且仅当 (i) X X XX 是随机连续的，且 (ii) m T ( X ( ω ) , δ ) → 0 m T ( X ( ω ) , δ ) → 0 m_(T)(X(omega),delta)rarr0m_{T}(X(\omega), \delta) \rightarrow 0 在概率上为 δ → 0 δ → 0 delta rarr0\delta \rightarrow 0 时， X X XX 才有连续版本。
Further Reading. Further foundational results about stochastic processes in continuous time, including the proof of Theorem 3.11, may be found in [1, Chapter 6].
进一步阅读关于连续时间随机过程的更多基本结果，包括定理 3.11 的证明，可参阅[1，第 6 章]。

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