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Let {Sn. n ≥ 0} be a simple random walk with So= 0 and S₂ = X₁ + ... + X₁, for n ≥ 1, where ,i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1-p for 21. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive egers. Define T = min{n: Sn = -a or Sn=b}.

Let {Sn. n ≥ 0} be a simple random walk with So= 0 and S₂ = X₁ + ... + X₁, for n ≥ 1, where ,i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1-p for 21. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive egers. Define T = min{n: Sn = -a or Sn=b}.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.CR: Chapter 12 Review
Problem 84CR
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5. Let {S, n ≥ 0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X₁, for n ≥ 1, where X₁, i = 1, 2, ... are independent random variables with P(X; = 1) = p, P(X¡ = − 1) = q = 1-p for i > 1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T= min{n: S₁ = -a or Sn=b}. Define Zn = (q/p), n ≥ 0. Prove that {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0.
Transcribed Image Text:5. Let {S, n ≥ 0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X₁, for n ≥ 1, where X₁, i = 1, 2, ... are independent random variables with P(X; = 1) = p, P(X¡ = − 1) = q = 1-p for i > 1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T= min{n: S₁ = -a or Sn=b}. Define Zn = (q/p), n ≥ 0. Prove that {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0.
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