Risk Analysis in Finance and Insurance, 2nd Edition

Appendix A

Problems

A.1 Probability theory and elements of stochastic anal-

ysis

Problem A.1.1 Consider probability space



[0, 1],B(0, 1),m



,wherem is the

Lebesgue measure. Find E(ξ|η) for random variables

ξ(ω)=2ω

2

and η(ω)=

⎧

⎨

⎩

0,ω∈ [0, 1/3]

2,ω∈ (1/3, 2/3]

1,ω∈ (2/3, 1]

.

Problem A.1.2 Consider a sequence of independent random variables

(ξ

n

)

n=1,...,N

such that each ξ

n

takes two values: e =2.71828 ... and −2e with

probabilities (2e +1)/3e and (e − 1)/3e, respectively. Define

X

n

=

n



k=1

ξ

k

, F

n

= σ{ξ

1

,...,ξ

n

}.

Is sequence (X

n

, F

n

)

n=1,...,N

a martingale?

Problem A.1.3 Let (ξ

n

)

n=1,...,N

be a sequence of independent random vari-

ables taking values −1 and +1 with probabilities ...