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Blumenthal's zero–one law

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In the mathematical theory of probability, Blumenthal's zero–one law, named after Robert McCallum Blumenthal, is a statement about the nature of the beginnings of right continuous Feller process. Loosely, it states that any right continuous Feller process on [ 0 , ∞ ) {\displaystyle [0,\infty )} starting from deterministic point has also deterministic initial movement.

Statement

Suppose that X = ( X t : t ≥ 0 ) {\displaystyle X=(X_{t}:t\geq 0)} is an adapted right continuous Feller process on a probability space ( Ω , F , { F t } t ≥ 0 , P ) {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )} such that X 0 {\displaystyle X_{0}} is constant with probability one. Let F t X := σ ( X s ; s ≤ t ) , F t + X := ⋂ s > t F s X {\displaystyle {\mathcal {F}}_{t}^{X}:=\sigma (X_{s};s\leq t),{\mathcal {F}}_{t^{+}}^{X}:=\bigcap _{s>t}{\mathcal {F}}_{s}^{X}} . Then any event in the germ sigma algebra Λ ∈ F 0 + X {\displaystyle \Lambda \in {\mathcal {F}}_{0+}^{X}} has either P ( Λ ) = 0 {\displaystyle \mathbb {P} (\Lambda )=0} or P ( Λ ) = 1. {\displaystyle \mathbb {P} (\Lambda )=1.}

Generalization

Suppose that X = ( X t : t ≥ 0 ) {\displaystyle X=(X_{t}:t\geq 0)} is an adapted stochastic process on a probability space ( Ω , F , { F t } t ≥ 0 , P ) {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )} such that X 0 {\displaystyle X_{0}} is constant with probability one. If X {\displaystyle X} has Markov property with respect to the filtration { F t + } t ≥ 0 {\displaystyle \{{\mathcal {F}}_{t^{+}}\}_{t\geq 0}} then any event Λ ∈ F 0 + X {\displaystyle \Lambda \in {\mathcal {F}}_{0+}^{X}} has either P ( Λ ) = 0 {\displaystyle \mathbb {P} (\Lambda )=0} or P ( Λ ) = 1. {\displaystyle \mathbb {P} (\Lambda )=1.} Note that every right continuous Feller process on a probability space ( Ω , F , { F t } t ≥ 0 , P ) {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )} has strong Markov property with respect to the filtration { F t + } t ≥ 0 {\displaystyle \{{\mathcal {F}}_{t^{+}}\}_{t\geq 0}} .

References

  1. Blumenthal, Robert M. (1957), "An extended Markov property", Transactions of the American Mathematical Society, 85 (1): 52–72, doi:10.1090/s0002-9947-1957-0088102-2, JSTOR 1992961, MR 0088102, Zbl 0084.13602
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