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2. Let Xo = 0, X₂ = k=1&k, where (k)k21 is a sequence of independent and identically distributed random variables such that P[E = ±1] = 1. Let M and N be two positive integers and define T: min{n > 0: Xn = -N or X₂ = M}. (a) Prove that 7 is an F-stopping time, where F is the natural filtration generated by X. (b) Assume that 7 < +∞0 a.s., prove that P[X, & {-N, M}] = 1. (c) Under the condition of (b), compute E[X] and P[X, = -N]. Hint: Let X be a martingale and 7 be a stopping time with respect to a filtration F, and if 7 <∞ and the process (X^n) n20 is uniformly bounded. Then E[X₂] = E[Xo]. =

2. Let Xo = 0, X₂ = k=1&k, where (k)k21 is a sequence of independent and identically distributed random variables such that P[E = ±1] = 1. Let M and N be two positive integers and define T: min{n > 0: Xn = -N or X₂ = M}. (a) Prove that 7 is an F-stopping time, where F is the natural filtration generated by X. (b) Assume that 7 < +∞0 a.s., prove that P[X, & {-N, M}] = 1. (c) Under the condition of (b), compute E[X] and P[X, = -N]. Hint: Let X be a martingale and 7 be a stopping time with respect to a filtration F, and if 7 <∞ and the process (X^n) n20 is uniformly bounded. Then E[X₂] = E[Xo]. =

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2. Let Xo = 0, X₂ = Ek-1 Sk, where (Sk) k21 is a sequence of independent and identically distributed random variables such that P[$k = ±1] =. Let M and N be two positive integers and define 7:= min{n 20: X₂ = -N or X₂ = M}. (a) Prove that 7 is an F-stopping time, where F is the natural filtration generated by X. (b) Assume that 7 < +∞o a.s., prove that P[X, & {-N, M}] = 1. (c) Under the condition of (b), compute E[X,] and P[X, = -N]. Hint: Let X be a martingale and 7 be a stopping time with respect to a filtration F, and if 7 <∞ and the process (X^n) n20 is uniformly bounded. Then E[X₂] = E[Xo].
Transcribed Image Text:2. Let Xo = 0, X₂ = Ek-1 Sk, where (Sk) k21 is a sequence of independent and identically distributed random variables such that P[$k = ±1] =. Let M and N be two positive integers and define 7:= min{n 20: X₂ = -N or X₂ = M}. (a) Prove that 7 is an F-stopping time, where F is the natural filtration generated by X. (b) Assume that 7 < +∞o a.s., prove that P[X, & {-N, M}] = 1. (c) Under the condition of (b), compute E[X,] and P[X, = -N]. Hint: Let X be a martingale and 7 be a stopping time with respect to a filtration F, and if 7 <∞ and the process (X^n) n20 is uniformly bounded. Then E[X₂] = E[Xo].
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