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This presentation by Ting-Hsuan Long explores the concept of stopping times in the context of American derivative securities. It covers definitions, discrete and continuous time frameworks, and provides an insightful example of first passage time for continuous processes. The presentation details the criteria that a stopping time must satisfy, illustrated through a structured proof. The discussion emphasizes the importance of these concepts in financial modeling and risk assessment, equipping participants with fundamental knowledge necessary for advanced studies in derivatives.
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American Derivative Securities Presenter:Ting-Hsuan Long
8.2 Stopping time • Discrete time • Continuous time should be in F(t)
Definition8.2.1 • A stopping time is a r.v taking values in [0,∞] and satisfying (8.2.1)
Example8.2.3(First passage time for a continuous process) • :adapted process with continuous paths • Show that is a stopping time. Let be given. We need to show thatis in F(t).
Proof: Case1: depending on whether . In either case,
Case2: step(1) Suppose In this interval, . is in the set A= We have shown that A
step(2) If Let and X has a continuous path, we see that . It follows that . We have shown that Under these two step,
step(3) Because there are only countably many rational numbers q in [0,t], they can be arranged in a sequence, and the union is really a union of a sequence of sets in F(t). We conclude that A
