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We consider a sequence (X)-1 of i.i.d. random variables such that P(X₁ = 1) = P(X₁ = −1) = 1/2. Consider the filtration (F)1 given by = Fn(X1,..., Xn). For all n N, let Sn == Xbe the random walk. We have seen in class that the process (f(Sn)) is a martingale if and only if f satisfies D2f = 0, where n=0 D2f(x) = f(x + 1) − 2f(x) + f(x − 1). - (a) Use the result above to prove again that (en) No is not a martingale. n=0 (b) We consider a process (Y)-0 given by Y₁ = f(n, Sn). Show that (Yn) is a martin- gale provided that - f(n+1, x + 1) − 2f(n, x) + f(n + 1, x − 1) = 0, - for all n, x. (c) Let Y = pen with p = ete. Use (b) to prove that (Y) is a martingale. 2 (d) Since (Y) is a martingale, use the Martingale Representation Theorem guarantees that there exists a previsible process (Zn) such that Y₁ = In = (Z · S)n. Determine Zn and prove that it is previsible. 1

We consider a sequence (X)-1 of i.i.d. random variables such that P(X₁ = 1) = P(X₁ = −1) = 1/2. Consider the filtration (F)1 given by = Fn(X1,..., Xn). For all n N, let Sn == Xbe the random walk. We have seen in class that the process (f(Sn)) is a martingale if and only if f satisfies D2f = 0, where n=0 D2f(x) = f(x + 1) − 2f(x) + f(x − 1). - (a) Use the result above to prove again that (en) No is not a martingale. n=0 (b) We consider a process (Y)-0 given by Y₁ = f(n, Sn). Show that (Yn) is a martin- gale provided that - f(n+1, x + 1) − 2f(n, x) + f(n + 1, x − 1) = 0, - for all n, x. (c) Let Y = pen with p = ete. Use (b) to prove that (Y) is a martingale. 2 (d) Since (Y) is a martingale, use the Martingale Representation Theorem guarantees that there exists a previsible process (Zn) such that Y₁ = In = (Z · S)n. Determine Zn and prove that it is previsible. 1

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.4: Hyperbolas
Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
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We consider a sequence (X)-1 of i.i.d. random variables such that P(X₁ = 1) = P(X₁ = −1) = 1/2. Consider the filtration (F)1 given by = Fn(X1,..., Xn). For all n N, let Sn == Xbe the random walk. We have seen in class that the process (f(Sn)) is a martingale if and only if f satisfies D2f = 0, where n=0 D2f(x) = f(x + 1) − 2f(x) + f(x − 1). - (a) Use the result above to prove again that (en) No is not a martingale. n=0 (b) We consider a process (Y)-0 given by Y₁ = f(n, Sn). Show that (Yn) is a martin- gale provided that - f(n+1, x + 1) − 2f(n, x) + f(n + 1, x − 1) = 0, - for all n, x. (c) Let Y = pen with p = ete. Use (b) to prove that (Y) is a martingale. 2 (d) Since (Y) is a martingale, use the Martingale Representation Theorem guarantees that there exists a previsible process (Zn) such that Y₁ = In = (Z · S)n. Determine Zn and prove that it is previsible. 1
Transcribed Image Text:We consider a sequence (X)-1 of i.i.d. random variables such that P(X₁ = 1) = P(X₁ = −1) = 1/2. Consider the filtration (F)1 given by = Fn(X1,..., Xn). For all n N, let Sn == Xbe the random walk. We have seen in class that the process (f(Sn)) is a martingale if and only if f satisfies D2f = 0, where n=0 D2f(x) = f(x + 1) − 2f(x) + f(x − 1). - (a) Use the result above to prove again that (en) No is not a martingale. n=0 (b) We consider a process (Y)-0 given by Y₁ = f(n, Sn). Show that (Yn) is a martin- gale provided that - f(n+1, x + 1) − 2f(n, x) + f(n + 1, x − 1) = 0, - for all n, x. (c) Let Y = pen with p = ete. Use (b) to prove that (Y) is a martingale. 2 (d) Since (Y) is a martingale, use the Martingale Representation Theorem guarantees that there exists a previsible process (Zn) such that Y₁ = In = (Z · S)n. Determine Zn and prove that it is previsible. 1
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