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σ-Algebra of τ-past

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The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probability theory.[1][2]

Definition

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Let be a stopping time on the filtered probability space . Then the σ-algebra

is called the σ-algebra of τ-past.[1][2]

Properties

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Monotonicity

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If are two stopping times and

almost surely, then

Measurability

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A stopping time is always -measurable.

Intuition

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The same way is all the information up to time , is all the information up time . The only difference is that is random. For example, if you had a random walk, and you wanted to ask, “How many times did the random walk hit −5 before it first hit 10?”, then letting be the first time the random walk hit 10, would give you the information to answer that question.[3]

References

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  1. ^ a b Karandikar, Rajeeva (2018). Introduction to Stochastic Calculus. Indian Statistical Institute Series. Singapore: Springer Nature. p. 47. doi:10.1007/978-981-10-8318-1. ISBN 978-981-10-8317-4.
  2. ^ a b Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 193. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  3. ^ "Earnest, Mike (2017). Comment on StackExchange: Intuition regarding the σ algebra of the past (stopping times)".