Let {Sn, n ≥ 0} be a simple random walk with So= 0 and Sn = X₁ + ... + Xn, for n ≥ 1, where i,i=1,2,... are independent random variables with P(X; = 1) = p, P(X₂ = -1) = q = 1-p for 1. Assume pq. Put Fo= {2,0}, Fn = 0(X₁, X2, Xn), n>1. Let b, a be two fixed positive tegers. Define T = min{n: Sn = -a or Sn=b}. me P(T **** Explain why one can use Doob Optional Stopping Theorem to conclu
Let {Sn, n ≥ 0} be a simple random walk with So= 0 and Sn = X₁ + ... + Xn, for n ≥ 1, where i,i=1,2,... are independent random variables with P(X; = 1) = p, P(X₂ = -1) = q = 1-p for 1. Assume pq. Put Fo= {2,0}, Fn = 0(X₁, X2, Xn), n>1. Let b, a be two fixed positive tegers. Define T = min{n: Sn = -a or Sn=b}. me P(T **** Explain why one can use Doob Optional Stopping Theorem to conclu
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![5. Let {S, n>0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X, 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 S₁₁=b}.
Assume P(T<∞) = 1. Explain why one can use Doob's Optional Stopping Theorem to conclude
that E[ZT] = 1.
We can use the conclusion that:
T is a stopping time with respect to the o-fields Fn, n ≥ 0.
A)
B)
Define Z₁ =
(q/p) Sn, n ≥ 0.
{Zn, n ≥ 0} is a martingale with respect to the o-fields
Fn, n ≥ 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd14927de-c618-4747-bb5c-8414c571fc63%2Fce26b25d-3a4a-416f-a1a4-a0de0af12569%2Fmvnvo_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let {S, n>0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X, 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 S₁₁=b}.
Assume P(T<∞) = 1. Explain why one can use Doob's Optional Stopping Theorem to conclude
that E[ZT] = 1.
We can use the conclusion that:
T is a stopping time with respect to the o-fields Fn, n ≥ 0.
A)
B)
Define Z₁ =
(q/p) Sn, n ≥ 0.
{Zn, n ≥ 0} is a martingale with respect to the o-fields
Fn, n ≥ 0.
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