5. Let {S, n ≥ 0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X₁, for n ≥ 1, whereX₁, i = 1, 2, ... are independent random variables with P(X; = 1) = p, P(X¡ = − 1) = q = 1-p fori > 1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positiveintegers. DefineT= min{n: S₁ = -a or Sn=b}.Define Zn = (q/p), n ≥ 0. Prove that {Zn, n ≥ 0} is a martingale with respect to the o-fieldsFn, n ≥ 0.We can use the conclusion that:T is a stopping time with respect to the o-fields Fn, n ≥ 0.

Given information: It is given that {Sn ; n > 0} is a simple random walk with S0 = 0 and Sn = X1…

Answered: Let {Sn. n ≥ 0} be a simple random walk…

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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Let X1, X2, and.X3 be independent and normally distributed random variables with E(X1) 4, E(X2) = 3, E(X3) = 2, Var(X1) = 1, Var(X2) = 5, Var(X3) = 2. Let Y = 2X1 + X2 – 3X3. Find 2. the distribution of Y.
Many people believe that the daily change in price of a company's stock in the stock market is a random variable with mean 0 and variance o² during the middle 2 months between 2 earning report dates when there is not much news released. That is, if Yn represents the closing price of the stock on the nth trading day of that period, then Yn = Yn-1+Xn, for 1 ≤ n ≤ 60, where X₁, X2,..., X60 are independent and identically distributed random variables with mean 0 and variance o². Suppose that the stock price at the beginning of the 2-month period is 100. If o = 2, what is the approximate probability that the stock's closing price on the 30th trading day will exceed 110, i.e., what is P(Y30 > 110)
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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.
Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).
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