By Sheldon M. Ross
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Exercises 17 An important property of conditional expectation is that the expected value of X is a weighted average of the conditional expectation of X given that Y = y. 1 E[X |Y = y]P(Y = y) E[X ] = y Proof. E[X |Y = y]P(Y = y) = y x P(X = x|Y = y)P(Y = y) y x y x x P(X = x, Y = y) = = P(X = x, Y = y) x x y x P(X = x) = x = E[X ] Let E[X |Y ] be that function of the random variable Y which, when Y = y, is defined to equal E[X |Y = y]. 15. 18 Probability What is the probability that the typist makes (a) at least four errors; (b) at most two errors?
That is, the value of W is determined by a knowledge of the values of X (s), 0 ≤ s ≤ t. Then, 2 2 2 E μ [W ] = e−μ t/2σ E 0 [W eμX (t)/σ ] Proof. 1, which states that, given X (t) = x, the conditional distribution of the process up to time t (and thus the conditional distribution of W ) is the same for all values μ. 2). 1 If X (t), t ≥ 0 is a Brownian motion process with drift parameter μ and variance parameter σ 2 for which X (0) = 0, show that −X (t), t ≥ 0 is a Brownian motion process with drift parameter −μ and variance parameter σ 2 .
We now verify that the preceding model becomes Brownian motion as we let become smaller and smaller. To begin, let Xi = 1, −1, if the change at time i if the change at time i is an increase is a decrease Hence, if X (0) is the process value at time 0, then its value after n changes is √ (X 1 + . . + X n ) X (n ) = X (0) + σ 36 Brownian Motion and Geometric Brownian Motion Because there would have been n = t/ changes by time t, this gives that √ t/ Xi X (t) − X (0) = σ i=1 Because the X i , i = 1, .
An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross