Risk Analysis in Finance and Insurance, 2nd Edition

270 Risk Analysis in Finance and Insurance

is a martingale with respect to the same filtration (F

n

)

n≤N

.

Defining the probability



P (A)=E



Z

N

I

A



on F

N

, prove that, for indepen-

dent increments ΔV

n

, this probability has the form



P (A)=E

!

I

A

exp{α

N

ΔV

N

}

E



exp{α

N

ΔV

N

}



"

.

Problem A.1.27 Investigate the martingale property of the stochastic se-

quence (Y

n

, F

n

),where

Y

n

= αX

2

n

+ βX

n

+ γ,

α, β, γ are real numbers and (X

n

) is a martingale on a stochastic basis

(Ω, F, (F

n

)

n=0,1,...

,P).

Problem A.1.28 Let (X

n

)

n≥1

be a sequence of identically distributed inde-

pendent random variables with the density function

f(x)=

⎧

⎪

⎨

⎪

⎩

1

8

if x ∈ [0, 2),

y if x ∈ [2, 4),

0 outside of the interval ...