Learning Notes

Stochastic Process

Definition

Stochastic Process

A stochastic process X_T is defined to be a function that maps the set T to random variable R_v.

\displaystyle X_T : T \mapsto R_v

State Space

Let S and B be the values of the random variables and B be its Borel field. State space is denoted by (S, B)

σ-Algebra/σ-Field

Let  \sigma be the \sigma-field of some set \mathcal{F}, it then must follow

  1. \emptyset \in \sigma
  2. \text{if } A \in \sigma , \text{then} A^c \in \sigma
  3. \text{if } A_i \subset A, \text{then} \bigcup_i A_i = A

E.g. \mathcal{F} = {a, b, c, d, e} then an example of the sigma-algebra generated from \mathcal{F} is \sigma(\mathcal{F})=\{\emptyset, \{a, b\}, \{c, d, e\}, \{a\}, \{b, c, d, e\}, \{a, b, c, d, e\}\} .

Borel Field

Borel Field or Borel-sigma-algebra generated from \mathcal{R} is defined as the sigma-algebra of all open subsets of \mathcal{R} .

Probability Space

( \Omega, \mathcal{F} , P) defines a probability space

\Omega \text{ is the sample space} \\ \mathcal{F} \text{ is the } \sigma \text{-algebra subsets of event in sample space } \Omega \\ P \text{ is the probability measures}

Probability Measure

Probability measure P is a function which associates a number P(A) to each set A\in \mathcal{F} with the following properties:

  1. P(A) \in [0, 1]
  2. P(\Omega) = 1
  3. Associative properties on disjoint sets A_i \in \mathcal{F} , that is

\displaystyle P\left(\bigcup_i A_i \right) = \sum_{i} P(A_i)

Note: See P(\mathcal{F}) as the probability that the events \mathcal{F} occurs.

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