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American Derivative Securities

American Derivative Securities. Presenter:Ting-Hsuan Long. 8.1 Introduction. 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). Remark8.2.2.

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American Derivative Securities

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  1. American Derivative Securities Presenter:Ting-Hsuan Long

  2. 8.1 Introduction

  3. 8.2 Stopping time • Discrete time • Continuous time should be in F(t)

  4. Definition8.2.1 • A stopping time is a r.v taking values in [0,∞] and satisfying (8.2.1)

  5. Remark8.2.2

  6. 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).

  7. Proof: Case1: depending on whether . In either case,

  8. Case2: step(1) Suppose In this interval, . is in the set A= We have shown that A

  9. step(2) If Let and X has a continuous path, we see that . It follows that . We have shown that Under these two step,

  10. 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

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