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[edit]

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[edit]

Monotonicity[edit]

Is are two stopping times and

almost surely, then

Measurability[edit]

A stopping time is always -measurable

References[edit]

  1. ^ a b Karandikar, Rajeeva (2018). Introduction to Stochastic Calculus. 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.