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(a) Under which condition is the process (Sn)^o a martingale ? (b) We consider the case where pn = 1/2 for all n (symmetric random walk). Is the process (S2) a martingale ? (If so, prove it. If not, explain why.) (c) We let Y = S2 - n. Prove that if pn = 1/2 for all n, (Y) is a martingale. n n=0 We consider a sequence (X^)^-1 of independent random variables such that P(X=1)=1- P(Xn = −1) = Pn. n For all n < N, let So = 0 and S₁ == > X₂. We also consider the filtration (F)-1 given by i=1 Fn=σ(X1,..., Xn).

(a) Under which condition is the process (Sn)^o a martingale ? (b) We consider the case where pn = 1/2 for all n (symmetric random walk). Is the process (S2) a martingale ? (If so, prove it. If not, explain why.) (c) We let Y = S2 - n. Prove that if pn = 1/2 for all n, (Y) is a martingale. n n=0 We consider a sequence (X^)^-1 of independent random variables such that P(X=1)=1- P(Xn = −1) = Pn. n For all n < N, let So = 0 and S₁ == > X₂. We also consider the filtration (F)-1 given by i=1 Fn=σ(X1,..., Xn).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 27T
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(a) Under which condition is the process (Sn)^o a martingale ? (b) We consider the case where pn = 1/2 for all n (symmetric random walk). Is the process (S2) a martingale ? (If so, prove it. If not, explain why.) (c) We let Y = S2 - n. Prove that if pn = 1/2 for all n, (Y) is a martingale. n n=0
Transcribed Image Text:(a) Under which condition is the process (Sn)^o a martingale ? (b) We consider the case where pn = 1/2 for all n (symmetric random walk). Is the process (S2) a martingale ? (If so, prove it. If not, explain why.) (c) We let Y = S2 - n. Prove that if pn = 1/2 for all n, (Y) is a martingale. n n=0
We consider a sequence (X^)^-1 of independent random variables such that P(X=1)=1- P(Xn = −1) = Pn. n For all n < N, let So = 0 and S₁ == > X₂. We also consider the filtration (F)-1 given by i=1 Fn=σ(X1,..., Xn).
Transcribed Image Text:We consider a sequence (X^)^-1 of independent random variables such that P(X=1)=1- P(Xn = −1) = Pn. n For all n < N, let So = 0 and S₁ == > X₂. We also consider the filtration (F)-1 given by i=1 Fn=σ(X1,..., Xn).
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