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# 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:

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