Let (W)to be a Brownian Motion. (a) Let 1 if Wt > 0, Xt=sgn(W)= 0 t if W₁ = 0, - 1 if Wt < 0. Check that (X+) is an Itô-integrable process. (b) Let Y₁ = [X, W. Find E[Y₁] and Var(Y;). 0 (c) Check that (W2) is an Itô-integrable process. (d) Consider the function f : [0,T] → R defined by f(t) = ta, where a € R. For which f(s) dW, well-defined? When it is, find E[Z₁] values of a is the process Z₁ = and Var(Z). Bonus question: In part (c), consider a partition 0 = to < ··· < tn = T. Prove that n-1 the simple process W221 +) (+) converges to W2 in the L²-sense. i=0 1
Let (W)to be a Brownian Motion. (a) Let 1 if Wt > 0, Xt=sgn(W)= 0 t if W₁ = 0, - 1 if Wt < 0. Check that (X+) is an Itô-integrable process. (b) Let Y₁ = [X, W. Find E[Y₁] and Var(Y;). 0 (c) Check that (W2) is an Itô-integrable process. (d) Consider the function f : [0,T] → R defined by f(t) = ta, where a € R. For which f(s) dW, well-defined? When it is, find E[Z₁] values of a is the process Z₁ = and Var(Z). Bonus question: In part (c), consider a partition 0 = to < ··· < tn = T. Prove that n-1 the simple process W221 +) (+) converges to W2 in the L²-sense. i=0 1
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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i need the solution of part d and bonus. THANK YOU
![Let (W)to be a Brownian Motion.
(a) Let
1
if Wt > 0,
Xt=sgn(W)=
0
t
if W₁ = 0,
-
1
if Wt < 0.
Check that (X+) is an Itô-integrable process.
(b) Let Y₁ = [X, W. Find E[Y₁] and Var(Y;).
0
(c) Check that (W2) is an Itô-integrable process.
(d) Consider the function f : [0,T] → R defined by f(t) = ta, where a € R. For which
f(s) dW, well-defined? When it is, find E[Z₁]
values of a is the process Z₁ =
and Var(Z).
Bonus question: In part (c), consider a partition 0 = to < ··· < tn = T. Prove that
n-1
the simple process W221 +) (+) converges to W2 in the L²-sense.
i=0
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a1c925e-790f-448e-9276-e5adcf0e8758%2F46f5eab8-b863-4757-a771-a8c45d06ad87%2Frep8eku_processed.png&w=3840&q=75)
Transcribed Image Text:Let (W)to be a Brownian Motion.
(a) Let
1
if Wt > 0,
Xt=sgn(W)=
0
t
if W₁ = 0,
-
1
if Wt < 0.
Check that (X+) is an Itô-integrable process.
(b) Let Y₁ = [X, W. Find E[Y₁] and Var(Y;).
0
(c) Check that (W2) is an Itô-integrable process.
(d) Consider the function f : [0,T] → R defined by f(t) = ta, where a € R. For which
f(s) dW, well-defined? When it is, find E[Z₁]
values of a is the process Z₁ =
and Var(Z).
Bonus question: In part (c), consider a partition 0 = to < ··· < tn = T. Prove that
n-1
the simple process W221 +) (+) converges to W2 in the L²-sense.
i=0
1
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