1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
|
---
title: "Stochastic process"
sort_title: "Stochastic process"
date: 2021-11-07
categories:
- Mathematics
- Stochastic analysis
- Measure theory
layout: "concept"
---
A **stochastic process** $$X_t$$ is a time-indexed
[random variable](/know/concept/random-variable/),
$$\{ X_t : t > 0 \}$$, i.e. a set of (usually correlated)
random variables, each labelled with a unique timestamp $$t$$.
Whereas "ordinary" random variables are defined on
a probability space $$(\Omega, \mathcal{F}, P)$$,
stochastic process are defined on
a **filtered probability space** $$(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$$.
As before, $$\Omega$$ is the sample space,
$$\mathcal{F}$$ is the event space,
and $$P$$ is the probability measure.
The **filtration** $$\{ \mathcal{F}_t : t \ge 0 \}$$
is a time-indexed set of [$$\sigma$$-algebras](/know/concept/sigma-algebra/) on $$\Omega$$,
which contains at least all the information generated
by $$X_t$$ up to the current time $$t$$,
and is a subset of $$\mathcal{F}_t$$:
$$\begin{aligned}
\mathcal{F}
\supseteq \mathcal{F}_t
\supseteq \sigma(X_s : 0 \le s \le t)
\end{aligned}$$
In other words, $$\mathcal{F}_t$$ is the "accumulated" $$\sigma$$-algebra
of all information extractable from $$X_t$$,
and hence grows with time: $$\mathcal{F}_s \subseteq \mathcal{F}_t$$ for $$s < t$$.
Given $$\mathcal{F}_t$$, all values $$X_s$$ for $$s \le t$$ can be computed,
i.e. if you know $$\mathcal{F}_t$$, then the present and past of $$X_t$$ can be reconstructed.
Given any filtration $$\mathcal{H}_t$$, a stochastic process $$X_t$$
is said to be *"$$\mathcal{H}_t$$-adapted"*
if $$X_t$$'s own filtration $$\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t$$,
meaning $$\mathcal{H}_t$$ contains enough information
to determine the current and past values of $$X_t$$.
Clearly, $$X_t$$ is always adapted to its own filtration.
Filtration and their adaptations are very useful
for working with stochastic processes,
most notably for calculating [conditional expectations](/know/concept/conditional-expectation/).
## References
1. U.H. Thygesen,
*Lecture notes on diffusions and stochastic differential equations*,
2021, Polyteknisk Kompendie.
|
