Abstract
The σ-algebra generated by a random variable X, denoted σ(X), is the σ-algebra by the collection of {X ∈ B} (i.e. {ω : X(ω) ∈ B}), where B is any interval in \({\mathbb {R}}\). Let \(\mathcal {G}\) be a σ-algebra on Ω. Then X is said to be \(\mathcal {G}\)-measurable if \(\sigma (X) \subseteq \mathcal {G}\).
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Notes
- 1.
Left continuous and \(\mathcal {F}_t\)-adapted.
- 2.
To be specific, any Borel measurable function.
References
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1988)
Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, New York (2003)
Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, New York (2005)
Shreve, S.E.: Stochastic Calculus for Finance II: Continuous-Time Models, 1st edn. Springer, New York (2004)
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Chan, R.H., Guo, Y.Z., Lee, S.T., Li, X. (2019). Stochastic Calculus Part I. In: Financial Mathematics, Derivatives and Structured Products. Springer, Singapore. https://doi.org/10.1007/978-981-13-3696-6_10
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DOI: https://doi.org/10.1007/978-981-13-3696-6_10
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