Risk Analysis in Finance and Insurance, 2nd Edition

260 Risk Analysis in Finance and Insurance

Clearly, for any finite interval, we have

P



{ω :



R(s) ≥ 0 for all 0 ≤ s ≤ t}



= P



{ω : R(s) ≥ 0 for all 0 ≤ s ≤ t}



= ϕ(x, t) ,

since processes



R(t)andR(t) are positive multiples of each other.

Then we have the following estimate from below.

Theorem 8.2 For al l R such that

f(R, t)

=exp





t

0



λ

1

+ λ −λ

1

E



exp{−Rc

i

e

−rs

}



− λE



exp{RX

i

e

−rs

}





ds

%

< ∞,

and for all t ≥ 0, the process e

−R

e

R(t)

*

f(R, t) is a martingale and

ϕ(x, t) ≥ 1 − f(R, t) e

−Rx

.

Proof Denote g(



R(t),t)=e

−R

e

R(t)

, and compute

E



g(x, t +Δt)



=



1 − (λ

1

+ λ)Δt



E



g(x, t)



+λ

1

Δt



∞

0

g(x + νe

−rt

,t) dG(ν)

+λ Δt



∞

0

g(x − ue

−rt

,t) dF (u)+o(Δt) .

Hence, we obtain the following ...