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0.1 Exercises

# 0.1 Exercises - 0.1 Exercises 2 1 Let V ar(X |F = E(X 2...

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0.1 Exercises 1. Let V ar ( X |F ) = E ( X 2 |F ) - ( E ( X |F )) 2 . Show that V ar ( X ) = E ( V ar ( X |F )) + V ar ( E ( X |F )) . Proof. RHS = E ( V ar ( X |F )) + V ar ( E ( X |F )) = E E ( X 2 |F ) - ( E ( X |F )) 2 ·· + E ( E ( X |F )) 2 - ( EE ( X |F )) 2 · = EE ( X 2 |F ) - ( EE ( X |F )) 2 = E ( X 2 ) - ( EX ) 2 = LHS 2. Show that if X and Y are r.v.’s with E ( Y |F ) = X and EX 2 = EY 2 < , then X = Y a.s. (i.e. P ( X = Y ) = 1). Proof. Note that X = E ( Y |F ) ∈ F , hence E ( X - Y ) 2 = EX 2 + EY 2 - 2 EXY = 2 EX 2 - 2 EE ( XY |F ) = 2 EX 2 - 2 E [ XE ( Y |F )] = 2 EX 2 - 2 EX 2 = 0 , implying that P ( X - Y = 0) = 0. 3. Let X i be independent with EX i = 0 and σ 2 i = V ar ( X i ) < , and let S 2 n = n i =1 X 2 i and B 2 n = ES 2 n = n i =1 σ 2 i . Show that S 2 n - B 2 n is a martingale (w.r.t. the natural filtration.) Proof. Clearly, E | S 2 n | < , and S 2 n ∈ F n := σ ( X 1 , ..., X n ). Also, E ( S 2 n +1 |F n ) = S 2 n + E ( X 2 n +1 - σ n +1 |F n ) = S 2 n + E ( X 2 n +1 - σ 2 n +1 ) = S 2 n . 4. Let X i be non-negative i.i.d. r.v.’s with EX i = 1 and P ( X i = 1) < 1. Show that T n = Q n i =1 X i is a martingale (w.r.t. the natural filtration.) Proof. Clearly, E | T n | = Q n i =1 EX i = 1 < , and T n ∈ F n := σ ( X 1 , ..., X n ). Also, E ( T n +1 |F n ) = T n E ( X n +1 |F n ) = T n E ( X

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0.1 Exercises - 0.1 Exercises 2 1 Let V ar(X |F = E(X 2...

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