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1. Let B = (B₁)tzo be a standard Brownian motion started at zero, and let (FB)zo denote the natural filtration of B. (1.1) State the definition of B. (1.2) Determine whether (B₁/√t) defines a standard Brownian motion. Explain your answer. t>0 (1.3) Show that T = inf {t≥0: B = 1+sin(t)} is a stopping time with respect to (FB) >0.

1. Let B = (B₁)tzo be a standard Brownian motion started at zero, and let (FB)zo denote the natural filtration of B. (1.1) State the definition of B. (1.2) Determine whether (B₁/√t) defines a standard Brownian motion. Explain your answer. t>0 (1.3) Show that T = inf {t≥0: B = 1+sin(t)} is a stopping time with respect to (FB) >0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Let B = (B₁)tzo be a standard Brownian motion started at zero, and let (FB)zo denote the natural filtration of B. (1.1) State the definition of B. (1.2) Determine whether (B₂/√t) defines a standard Brownian motion. Explain your answer. t>0 (1.3) Show that 7 = inf{t 20: B = 1+sin(t)} is a stopping time with respect to (FB)t20. (1.4) Show that (B3-3tBt+e6B-3) 20 is a martingale with respect to (FB) t>0. (1.5) Set M₁ = B3-3tB₁+eV6B-3t for t20. Compute E(M.) and E(OB) when o= inf {t≥ 0:|B₁|= √5}.
Transcribed Image Text:1. Let B = (B₁)tzo be a standard Brownian motion started at zero, and let (FB)zo denote the natural filtration of B. (1.1) State the definition of B. (1.2) Determine whether (B₂/√t) defines a standard Brownian motion. Explain your answer. t>0 (1.3) Show that 7 = inf{t 20: B = 1+sin(t)} is a stopping time with respect to (FB)t20. (1.4) Show that (B3-3tBt+e6B-3) 20 is a martingale with respect to (FB) t>0. (1.5) Set M₁ = B3-3tB₁+eV6B-3t for t20. Compute E(M.) and E(OB) when o= inf {t≥ 0:|B₁|= √5}.
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