📂Probability Theory

Projection Mapping in Stochastic Processes

Definition

Let $S$ be a space that is both a metric space $( S , \rho)$ and a measurable space $(S,\mathcal{B}(S))$, and consider $k \in \mathbb{N}$.

1. Discrete Projection Mapping: For (discrete time) $N = \left\{ n \in \mathbb{N}: n \le \xi, \xi \in [0,\infty] \right\}\subset \mathbb{N}$ and an element $x:= (x_{1} , x_{2} , \cdots )$ of $\displaystyle S^{\sup N}:= \prod_{n \in N} S$ in $S$, the following defined $\pi_{k}: S^{\sup N} \to S^{k}$ is called a discrete projection mapping. $$\pi_{k} (x) = (x_{1} , x_{2} , \cdots , x_{k})$$
2. Continuous Projection Mapping: For (continuous time) $T \subset [0,\infty]$ and an element $x_{t} = x(t)$ of $\displaystyle S^{T}:= \prod_{t \in T} S$ with a finite set $T_{k}:=\left\{ t_{1} , t_{2} , \cdots , t_{k} \right\} \subset T$, the following defined $\pi_{T_{k}}: S^{T} \to S^{k}$ is called a continuous projection mapping. $$\pi_{T_{k}} (x) = (x_{t_{1}} , x_{t_{2}} , \cdots , x_{t_{k}})$$

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• The expression like $\displaystyle X = \prod_{\alpha \in \mathscr{A}} X_{\alpha}$ denotes the Cartesian product of spaces.

Description

Such mappings are generally referred to as projection mappings in mathematics, and their essential concept usually involves reducing dimensions, regardless of what their properties are. This is also introduced for the same purpose in stochastic process theory and used in the same manner. Reducing dimensions in probability theory means considering a separating class to discuss the equality of random variables and the convergence of random variables. It would be beneficial if one could examine and judge only the finite portion, whether the stochastic process continues indefinitely, discrete, or continuous.

It’s normal if the definitions alone do not make things clear, so let’s understand them with the examples below. Since there are no proofs, focus on what role $\pi$ plays and how a separating class is formed. Any space is firstly a separable space and a complete space, making it a Polish space. A space being a Polish space means that the defined probability measure is tight, making it an easier space for us to handle:

• (1) Multivariate random variables: From the viewpoint of stochastic process theory, even multivariate random variables are nothing but a finite sequence of random variables. There is no place for projection mapping to intervene if it’s merely finite $k$ dimensions. If a Euclidean distance $\rho = d_{2}$ is given for $S = \mathbb{R}^{k}$, then $\left( \mathbb{R}^{k} , \rho \right)$ exists such that it satisfies both separability and completeness, and the following set becomes a separating class. $$\left\{ (-\infty,x_{1}] \times \cdots \times (-\infty,x_{k}]: (x_{1} , \cdots , x_{k}) \in \mathbb{R}^{k} \right\}$$
• (2) Stochastic processes: If the distance $\rho$ between two elements $x:= (x_{1} , x_{2}, \cdots )$ and $y:= (y_{1} , y_{2}, \cdots )$ of $S = \mathbb{R}^{\infty}$ is defined as follows, $(\mathbb{R}^{\infty},\rho)$ becomes a metric space. $$\rho (x,y):= \sum_{i=1}^{\infty} {{ 1 } \over { 2^{i} }} \left( 1 \land \left| x_{i} - y_{i} \right| \right)$$ Sequences are functions, and such function spaces can be easily shown to have completeness. Similarly, $\mathbb{Q}^{\infty}$ exists proving separability in the same manner as for finite dimensions. Now, let’s define class $\mathcal{C}$ as follows. $$\mathcal{C}:= \left\{ \pi_{k}^{-1}(H): H \in \mathcal{B} \left( \mathbb{R}^{k} \right) \right\}$$ To get a feel for what these $\pi_{k}^{-1}(H) \subset \mathbb{R}^{\infty}$ are like, let’s look at a few examples:
  • (2)-1. $k=1$, $H = \left\{ 3 \right\}$ $$\pi_{1}^{-1}(H) = \left\{ (3, \times , \cdots), (3, \times , \cdots) , \cdots \right\}$$
  • (2)-2. $k=1$, $H = \left\{ 3, 5 \right\}$ $$\pi_{1}^{-1}(H) = \left\{ (5, \times , \cdots), (3, \times , \cdots), (3, \times , \cdots) , \cdots \right\}$$
  • (2)-3. $k=1$, $H = \left\{ (x_{1} , x_{2}): x_{1} = 1 , x_{2} \in [0,2] \right\}$ $$\pi_{2}^{-1}(H) = \left\{ (1, 0, \times , \cdots), (1, 1.24, \times , \cdots), (1, 1, \times , \cdots) , \cdots \right\}$$ For all $k\in \mathbb{N}$ and Borel sets $H \in \mathcal{B}\left( \mathbb{R}^{k} \right)$, $\mathcal{C}$ is a π-system and the fact that $\sigma ( \mathcal{C}) = \mathcal{B} \left( \mathbb{R}^{\infty} \right)$ is used to prove that $\mathcal{C}$ is a separating class.
• (3) Stochastic Paths: Consider the space $S = C[a,b]$ consisting of continuous functions with a closed interval $[a,b]$ as their domain. If the distance $\rho$ between two continuous functions $x,y \in C[a,b]$ is defined as follows, $(\mathbb{R}^{\infty},\rho)$ becomes not only a metric space but also a Banach space with completeness. $$\rho (x,y):= \sup_{t \in [a,b]} \left| x(t) - y(t) \right|$$ Demonstrating separability is not much different from other examples. The set of continuous functions $D_{m}$, whose function value at a finitely many points $m$ within $[a,b]$ is rational and linearly interpolated between, along with $\displaystyle D = \bigcup_{m \in \mathbb{N}} D_{m}$, guarantees that $C[a,b]$ is separable. Now, let’s define class $\mathcal{C}$ as follows. $$\mathcal{C}:= \left\{ \pi_{t_{k}}^{-1}(H): H \in \mathcal{B} \left( \mathbb{R}^{k} \right) \right\}$$ This is a collection of all the continuous functions that pass through a specific region depending on $H$ at some point in time $t_{i}$. It might sound complicated, but it becomes easier to understand with a diagram:
  • (3)-1. $T_{1}= \left\{ t_{1} \right\}$, $H = \left\{ 3 \right\}$

$\pi_{T_{1}}^{-1}(H)$ is a collection of continuous functions that pass through a point $(t_{1},3)$ as shown in the picture above.

• (3)-2. $T_{1}= \left\{ t_{1} \right\}$, $H = \left\{ 3, 5 \right\}$

$\pi_{T_{1}}^{-1}(H)$ is a collection of continuous functions that pass through two points $(t_{1},3)$ or $(t_{1},5)$ as shown in the picture above.

• (3)-3. $T_{2}= \left\{ t_{1} , t_{2} \right\}$, $H = \left\{ (x_{1} , x_{2}): x_{1} = 1 , x_{2} \in [0,2] \right\}$

$\pi_{T_{2}}^{-1}(H)$ is a collection of continuous functions that pass through an interval $[0,2]$ at points $(t_{1},1)$ and $t_{2}$ as shown in the picture above.