Question 1 geometric Brownian motion. This question is about the Wiener process and the (a) The random process Y(t), t> 0, is defined by Y(t) = W(t)², where W(t), t> 0, is the standard Wiener process. (i) Compute Cov(Y;, Y.) for t>s and derive from this result the expression for the variance of Y. Hint. Note that if t>s then W = [(W-W,) +W.j = (W- W.)+ 2(W - W.)W, + W;, where (Wt - W.) and W(s) are independence random variables. (ii) Compute the expectation of the product (Y - Y,)Y,, where t>s > 0. (iii) Does the process Y(t) have independent increments? (b) Consider the geometric Brownian motion of the form S(t) = eW(t) Compute the expectation of the product S(t)S(2t)S(3t).
Question 1 geometric Brownian motion. This question is about the Wiener process and the (a) The random process Y(t), t> 0, is defined by Y(t) = W(t)², where W(t), t> 0, is the standard Wiener process. (i) Compute Cov(Y;, Y.) for t>s and derive from this result the expression for the variance of Y. Hint. Note that if t>s then W = [(W-W,) +W.j = (W- W.)+ 2(W - W.)W, + W;, where (Wt - W.) and W(s) are independence random variables. (ii) Compute the expectation of the product (Y - Y,)Y,, where t>s > 0. (iii) Does the process Y(t) have independent increments? (b) Consider the geometric Brownian motion of the form S(t) = eW(t) Compute the expectation of the product S(t)S(2t)S(3t).
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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part b
Transcribed Image Text:Question 1
geometric Brownian motion.
This question is about the Wiener process and the
(a) The random process Y(t), t> 0, is defined by
Y(t) = W(t),
where W(t), t> 0, is the standard Wiener process.
(i) Compute Cov(Y1, Ys) for t> s and derive from this result the expression for
the variance of Yt.
Hint. Note that if t > s then
W; = [(W – W,) +W,J² = (W – W.)? + 2(Wt – W,)W, + W;, where
(W - W.) and W(s) are independence random variables.
(ii) Compute the expectation of the product (Yt - Y)Y,, where t >s > 0.
(iii) Does the process Y(t) have independent increments?
(b) Consider the geometric Brownian motion of the form S(t) = eW(t). Compute the
expectation of the product S(t)S(2t)S(3t).
%3D
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