Probability Measure on the Path Space

Stochastic Processes and Paths

Definition 1: Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space, and let $(X_t)_{t\in T}$ be a family of random variables on $(\Omega,\mathscr{F},\mathbb{P})$, that is, for every $t\in T$, we have
$$
X_t: (\Omega,\mathscr{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))
$$
measurable. Then
$$
\begin{aligned}
X:T\times\Omega&\to \mathbb{R}\\\
(t,\omega)&\mapsto X_t(\omega)
\end{aligned}
$$
is called a stochastic process. For fixed $\omega\in \Omega$, we call
$$
\begin{aligned}
X_\omega:T&\to \mathbb{R}\\\
t&\mapsto X(t,\omega)
\end{aligned}
$$
a path, and call the collection of all paths
$$
\mathbb{D} := \{X_\omega\ ;\ \omega\in \Omega\}
$$
the path space.

Remark: Each element $X_\omega$ of the path space is a mapping from $T$ to $\mathbb{R}$.

How to Define a Measure on the Path Space?

First, a measure is defined on a $\sigma$-algebra. Therefore, we first need to define a suitable $\sigma$-algebra on the path space $\mathbb{D}$. But how should this be defined?

To this end, let us first recall the definition of the $\sigma$-algebra generated by a family of mappings.

Definition 2: Let $\mathcal{H}$ be a collection of mappings from a set $G$ to a measurable space $(E,\mathcal{E})$. Then
$$
\sigma(f,f\in \mathcal{H}): = \sigma(\{\bigcup_{f\in \mathcal{H}} f^{-1}(B)\ ; B\in \mathcal{E}\})
$$
is called the $\sigma$-algebra generated by $\mathcal{H}$. It is the smallest $\sigma$-algebra with respect to which every $f\in \mathcal{H}$ is measurable.

Based on this, let us look at the path space $\mathbb{D}$. For each path $X_\omega$, there is a corresponding projection mapping
$$
\begin{aligned}
\pi^{\omega}_t: \mathbb{D}&\to \mathbb{R}\\\
X_\omega&\mapsto X_\omega(t) = X(t,\omega)
\end{aligned}
$$
The $\sigma$-algebra we define on $\mathbb{D}$ should at least guarantee that all $\pi_t^\omega:\mathbb{D}\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ are measurable. Thus we define
$$
\mathscr{D} = \sigma(\pi^\omega_t,\omega\in\Omega,t\in T)
$$
to be the $\sigma$-algebra on $\mathbb{D}$.

With the $\sigma$-algebra $\mathscr{D}$ on $\mathbb{D}$, we can define a measure on $\mathscr{D}$. It should clearly have some relation to the probability measure on $(\Omega,\mathscr{F},\mathbb{P})$:
$$
\begin{aligned}
\mu: \mathscr{D}&\to \mathbb{R}\\\
\mu(A)& = \mathbb{P}(\{\omega\in \Omega\ ;\ X_\omega\in A\})
\end{aligned}
$$
It is not difficult to verify that $\mu$ is indeed a probability measure on $(\mathbb{D},\mathscr{D})$, and $\mu$ is called the law of the stochastic process $X$.

At this point, we have defined a measure on the path space. But why is it called the law of the stochastic process? To explain this more intuitively, I suddenly thought of an example while having dinner tonight.

Example

We know that the convergence of some iterative algorithms is affected by the choice of the initial value, for example, Newton’s method. Thus I regard an iterative method as a stochastic process and construct the following example:

Consider the probability space $(\Omega,\mathscr{F},\mathbb{P})$, where $\Omega = [0,1]$, $\mathscr{F} = [0,1]\cap \mathcal{B}(\mathbb{R})$, namely, the $\sigma$-algebra consisting of all Borel measurable sets on the closed interval $[0,1]$, and $\mathbb{P}$ is the restriction of the Lebesgue measure on $\mathbb{R}$ to $[0,1]$. We have an iterative algorithm $\mathcal{A}$, which satisfies the following “law”:

• when the initial value $x_0\in [0,\frac{1}{3}]$, the iterative algorithm $\mathcal{A}$ converges;
• when the initial value $x_0\in (\frac{1}{3},1]$, the iterative algorithm $\mathcal{A}$ diverges.

Clearly, if one chooses an initial value arbitrarily in $\Omega$, then the probability that the iterative algorithm $\mathcal{A}$ converges is $\frac{1}{3}$.

But what exactly is this probability? (That is, on what measurable space is it defined?) And what is the relation between this so-called “law” and the law of a stochastic process defined above?

First, we regard the iterative algorithm $\mathcal{A}$ as a discrete stochastic process on $(\Omega,\mathscr{F},\mathbb{P})$: let $T$ be the set of all natural numbers, and let $\{x_t\}_{t\in T}$ be the iterative sequence generated by the algorithm $\mathcal{A}$ with initial value $x_0$. Then

$$
\begin{aligned}
\mathcal{A}: T\times \Omega&\to \mathbb{R}\\\
(t,x_0)&\mapsto x_t
\end{aligned}
$$

Then the path with respect to the initial value $x_0$ is

$$
\begin{aligned}
\mathcal{A}_{x_0}: T&\to \mathbb{R}\\\
t&\mapsto x_t
\end{aligned}
$$

At this time, the path space $\mathbb{D}$ is precisely the collection of all possible iterative sequences starting from $[0,1]$. According to the preceding steps, we can define the $\sigma$-algebra $\mathscr{D}$ on $\mathbb{D}$

such that all projections

$$
\begin{aligned}
\pi_t^{x_0}: \mathbb{D}&\to \mathbb{R}\\\
\mathcal{A}_{x_0}&\mapsto x_t
\end{aligned}
$$

are measurable. According to the following result:

Lemma 1: Let $f_n:(\Omega,\mathscr{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ be a sequence of measurable functions. Then

$$
\{\omega\in \Omega\ ;\ \lim_{n\to\infty} f_n(\omega)\text{ exists}\}\in \mathscr{F}
$$

Proof: This proof is left as an exercise.

Hence we have

Proposition 1: The set

$$
C = \{ \mathcal A_{x_0}; \lim_{t\to\infty} A(t,x_0) \text{ exists} \}\in \mathscr{D}.
$$

That is, the set consisting of all convergent paths belongs to the $\sigma$-algebra $\mathscr{D}$ on $\mathbb{D}$.

Proof: Since $\mathcal{A}(t,x_0) = \pi_t^{x_0}( \mathcal{A}_{x_0})$, by Lemma 1,

$$
C = \{\mathcal A_{x_0}\ ;\ \lim\limits_{t\to\infty}A(t,x_0)\text{ exists}\} = \{\mathcal A_{x_0}\ ;\ \lim\limits_{t\to\infty}\pi_t^{x_0}( \mathcal A_{x_0})\text{ exists}\}\in \mathscr{D}.
$$

Therefore, according to the definition of the measure $\mu$ on the $\sigma$-algebra $\mathscr{D}$ above, we have

$$
\mu(A) =\mathbb{P}(\{x_0\in [0,1]\ ;\ \mathcal{A}_{x_0}\in A\}),\quad \forall A\in \mathscr{D}
$$

In particular, for the set $C$ consisting of all convergent paths, we have

$$
\mu(C) =\mathbb{P}(\{x_0\in [0,1]\ ;\ \mathcal{A}_{x_0}\in C\}) =\mathbb{P}([0,\frac{1}{3}]) = \frac{1}{3}.
$$

That is, the probability measure of the event “the iterative method converges” is $\frac{1}{3}$, which is consistent with the “law” assumed above for the iterative algorithm $\mathcal{A}$. Hence it is called the law of the stochastic process.

The cover image of this article was taken on a sightseeing boat in Lucerne, Switzerland.