BOOK: Aitrage Theory in Continuous Time, Third Edition, Tomas Bjork 8 Questions: exercise: 4.1, 4.3,...

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BOOK: Aitrage Theory in Continuous Time, Third Edition, Tomas Bjork 8 Questions: exercise: 4.1, 4.3, 4.4, and 4.5; exercise: 5.6, and 5.12; exercise: 7.4, and 7.5 Handwriting solution is NOT acceptable. Please must use MS-word like ?√?2??+ ∑ ????=1 .HINT Exercise 5.5 and Exercise 4.4

Banasree · Accepted Answer

Ans. 4.1
a)
dz(t) = α  dt + αdW(t)
whereas, Z satisfies SDE.
dZ(t) = αZ(t)dt+αZ(t)dW(t)
Z(0) = 1
Integral form,
Z(t) = 1 + α  + α 
After moving the expectation within the integral sign in the ds-integral and defining m by m(t) = Z(t)
Then,
m(t) = 1 + α
m(0) = 1
solving these
Z(t) = m(t) = 
b)
Z(t) = W^2(t)
dZ(t)=dt+2g(t)dW(t)
Integrated form
W^2(t) = t+ 2 
 = 
c)
dz(t) = α  dt + αdW(t)
whereas, Z satisfies SDE.
dZ(t) = αZ(t)dt+αZ(t)dW(t)
Z(0) = 1
Integral form,
Z(t) = 1 + α  + α 
After moving the expectation within the integral sign in the ds-integral and defining m by m(t) = Z(t)
Then,
m(t) = 1 + α
m(0) = 1
solving these
Z(t) = m(t) = = 
d)
we know that 
X(t) = x0 +  + 
W is a wiener process. So rewriting the above equ.
dX(t) = µ(t)dt+σ(t)dW(t)
Where X = Stochastic differential.

BOOK: Arbitrage Theory in Continuous Time, Third Edition, Tomas Bjork 8 Questions: exercise: 4.1, 4.3, 4.4, and 4.5; exercise: 5.6, and 5.12; exercise: 7.4, and 7.5 Handwriting solution is NOT...

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