Stochastic Process

• \((\Omega, \mathcal{F}, P, \{\mathcal{F}_{t}\}_{0 \le t})\),
  • probablity space with filtration
• \((\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}))\),
  • $d$-dimentional borrel space,

Definition. stochastic process

The collection of random variables \(X := \{X_{t} \mid 0 \le t < \infty\}\) which take value on \((\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}))\) is called a stochastic process.,

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Definition. sample path

• $\omega \in \Omega$,
  • given
• $X$,
  • a stochastic process

The map $t \mapsto X_{t}(\omega)$ is called a sample path/realizatoin, trajectory of the process $X$ associated with $\omega$.

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Definition. equality

• $X, Y$
  • stochastic process

$X$ and $Y$ are said to be same if

\[\forall t \in [0, \infty), \ \forall \omega \in \Omega, \ X_{t}(\omega) = Y_{t}(\omega) .\]

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Definition. modification

• $X, Y$,
  • stochastic process

$Y$ is said to be modification of $X$ if

\[\forall t \in [0, \infty), \ P( X_{t} = Y_{t} ) = 1 .\]

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Definition. equality of finite dimensional distribution

• $X, Y$,
  • stochastic process

$X$ and $Y$ are said to have the same finite-dimensional distributions if

\[\forall n \in \mathbb{N}, \ t_{1} < t_{2} < \cdots < t_{n}, \ \forall A \in \mathcal{B}(\mathbb{R}^{nd}), \ P( (X_{t_{1}}, \ldots, X_{t_{n}}) \in A ) = P( (Y_{t_{1}}, \ldots, Y_{t_{n}}) \in A ) .\]

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Definition. indistinguishable

• $X, Y$,
  • stochastic process

$X$ and $Y$ are said to be indistinguishable if

\[P( X_{t} = Y_{t} \mid t \in [0, \infty) ) = 1 .\]

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Definition. measurable

• $X$,
  • stochastic process

$X$ is said to be measurable if

\[(t, \omega) \mapsto X_{t}(\omega) : \left( [0, \infty) \times \Omega, \mathcal{B}([0, \infty) \otimes \mathcal{F}) \right) \mapsto (\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}))\]

is measurable.

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Definition. RCLL

• $X$,
  • stochastic process
• $\omega \in \Omega$,

The sample path of $X$ associated with $\omega$ is said to be càdlàg/RCLL if

\[\lim_{t \searrow s} X_{t} = X_{s}, \ \lim_{t \nearrow s} X_{t} < \infty .\]

(i.e. right-continuous on $[0, \infty)$ with finite left-hand limits on $(0, \infty)$

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Definition. adapted

• $X$,
  • stochastic process

The stochastic process $X$ is said to be adapted to the filtration \(\{\mathcal{F}_{t}\}\) if for every $t \ge 0$, $X_{t}$ is $\mathcal{F}_{t}$-measurable.

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Definition. progressively measurable

• $X$,
  • stochastic process

The stochastic process $X$ is said to be progressively measurable with respect to the filtration \(\{\mathcal{F}_{t}\}\) if for every $t \ge 0$

\[(s, \omega) \mapsto X_{s}(\omega) : \left( [0, t] \times \Omega, \mathcal{B}([0, t] \otimes \mathcal{F}_{t}) \right) \mapsto (\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}))\]

is measurable.

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Reference