Homework Help is Here – Start Your Trial Now!
Math
Statistics
Recall the set-up to derive the density function for the maximum of a Brownian motion: Start with the Brownian motion: W(0) = 0 dW satisfies: E[dW] = 0 We immediately derive that: W(T) is normal. VAR[DW] = dt E[W(T)] = 0 dW's are all normal, i.i.d. VAR[W(T)] = T M(T) = maximum of W(t) over [0, T] = MAX {W(t)|t ε [0, T]} The pdf for M(T) equals the right-hand side of the pdf for W(T) multiplied by 2. [A] Write down an integral for E[M(T)]. No need to do the integral. [B] Recall that we defined the Brownian motion's stopping time random variable, denoted Ta, as the time that the Brownian motion first crosses the value a. Here's how to sample Ta : Run the Brownian motion (do the experiment!). Wait until the first time you cross a, and write down how long you had to wait. That's the sample value of Ta for that “run” of the experiment. Recall this clever relationship for each future time t: Prob{ Ta> t } = Prob { M(t) <= a }

Recall the set-up to derive the density function for the maximum of a Brownian motion: Start with the Brownian motion: W(0) = 0 dW satisfies: E[dW] = 0 We immediately derive that: W(T) is normal. VAR[DW] = dt E[W(T)] = 0 dW's are all normal, i.i.d. VAR[W(T)] = T M(T) = maximum of W(t) over [0, T] = MAX {W(t)|t ε [0, T]} The pdf for M(T) equals the right-hand side of the pdf for W(T) multiplied by 2. [A] Write down an integral for E[M(T)]. No need to do the integral. [B] Recall that we defined the Brownian motion's stopping time random variable, denoted Ta, as the time that the Brownian motion first crosses the value a. Here's how to sample Ta : Run the Brownian motion (do the experiment!). Wait until the first time you cross a, and write down how long you had to wait. That's the sample value of Ta for that “run” of the experiment. Recall this clever relationship for each future time t: Prob{ Ta> t } = Prob { M(t) <= a }

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
See similar textbooks
icon
Related questions
Question
Recall the set-up to derive the density function for the maximum of a Brownian motion: Start with the Brownian motion: W(0) = 0 dW satisfies: E[dW] = 0 We immediately derive that: W(T) is normal. E[W(T)] VAR[dW] = dt = 0 VAR[W(T)] = T M(T) = maximum of W(t) over [0, T] = MAX {W(t) | t ɛ [0, T]} The pdf for M(T) equals the right-hand side of the pdf for W(T) multiplied by 2. [A] Write down an integral for E[M(T)]. No need to do the integral. [B] Recall that we defined the Brownian motion's stopping time random variable, denoted Ta, as the time that the Brownian motion first crosses the value a. Here's how to sample Ta : Run the Brownian motion (do the experiment!). Wait until the first time you cross a, and write down how long you had to wait. That's the sample value of Ta for that “run” of the experiment. Recall this clever relationship for each future time t: dW's are all normal, i.i.d. Prob{ Ta > t } = Prob {M(t) <= a } Let H(u) denote the cumulative distribution function of Ta (dummy variable: u.) [A] Explain why the left-hand-side above equals 1 - H(t). [B] Using what we know about the density function of M(t) .. can you write an expression for H(t) as an integral? [C] How can we get from H(u) to h(u), the pdf for Ta ? [D] Write an expression for E[ Ta ].
Transcribed Image Text:Recall the set-up to derive the density function for the maximum of a Brownian motion: Start with the Brownian motion: W(0) = 0 dW satisfies: E[dW] = 0 We immediately derive that: W(T) is normal. E[W(T)] VAR[dW] = dt = 0 VAR[W(T)] = T M(T) = maximum of W(t) over [0, T] = MAX {W(t) | t ɛ [0, T]} The pdf for M(T) equals the right-hand side of the pdf for W(T) multiplied by 2. [A] Write down an integral for E[M(T)]. No need to do the integral. [B] Recall that we defined the Brownian motion's stopping time random variable, denoted Ta, as the time that the Brownian motion first crosses the value a. Here's how to sample Ta : Run the Brownian motion (do the experiment!). Wait until the first time you cross a, and write down how long you had to wait. That's the sample value of Ta for that “run” of the experiment. Recall this clever relationship for each future time t: dW's are all normal, i.i.d. Prob{ Ta > t } = Prob {M(t) <= a } Let H(u) denote the cumulative distribution function of Ta (dummy variable: u.) [A] Explain why the left-hand-side above equals 1 - H(t). [B] Using what we know about the density function of M(t) .. can you write an expression for H(t) as an integral? [C] How can we get from H(u) to h(u), the pdf for Ta ? [D] Write an expression for E[ Ta ].
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS