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MATH4091/7091: Financial calculus

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MATH4091/7091: Financial calculus

MATH4091/7091: Financial calculus

 

 

Notation: “Lx.y” refers to [Lecture x, Slide y]

1.Recall the definition on L2.34


Show that if a random variable X 2 L2( ) then X 2 L1( ).

 

Solution: Use Cauchy-Schwarz inequality.

2. Suppose the non-negative random variable X takes countably many values x0, x1, : : :. Show that

 

1

X

EP[X] =         xkPfX = xkg:

k=0

Hint: use the Monotone Convergence Theorem.

Solution: Let Ak = fX = xkg, so that X can be written as

 

 

 

1

 

 

X =

xkIAk :

 

 

 

=0

 

 

Xk

 

n

, and we have

 

 

Define Xn = Pk=0 xkIAk

 

 

 

0  X1   X2   :::;

 

and surely

 

 

 

 

lim Xn = X:

 

Next, we have

n!1

 

 

n

 

 

 

 

 

 

X

 

 

E[Xn] =

xkPfX = xkg:

 

k=0

Taking the limit on both sides as n ! 1, and using the Monotone Convergence Theorem, we obtain

 

n

1

 

E[X] = nlim EXn =

Xk

X

 

nlim

xkPfX = xkg =    xkPfX = xkg:

 

!1

!1

 

 

 

=0

k=0

 

3. For each positive integer n, define fn to be the normal density with mean zero and variance n, i.e

(a) What is the function f(x) = limn!1 fn(x)?

 

Solution: Since jfn(x)

p

1

, we have f(x) = limn!1 fn(x) = 0.

 

 

2n

 

(b) What is the limit limn!1 R 1 fn(x)dx?

 

 

 

 

Solution: It can be shown that

Z 1

 

 

 

so

fn(x)dx = 1

 

 

 

1

 

 

 

 

 

n!1 Z

 

 

 

 

 

fn(x)dx = 1:

 

 

 

 

lim

 

 

 

 

(c) Verify that

1

 

 

1

 

 

 

 

n!1 Z

fn(x)dx 6= Z

 

 

 

 

lim

 

 

 

f(x)dx:

 

 

 

MATH4091/7091: Financial calculus ,financial calculus exercises,financial calculus problems,financial calculus help,financial calculus guide



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