II.2 Planteamiento del modelo de daño por corrosión
II.2.2 Versión simple de un modelo estocástico estacionario de daño acumulado
7.3.1. The Ordinary Case
The first problem is to find the analogue of the Lundberg exponent for the Ammeter risk model. In the classical case and in the renewal case we constructed an expo-nential martingale. Here, as in the renewal case, the process {Ct} is not a Markov process anymore. But for each k ∈ IIN the process {Ct+k∆− Ck∆} is independent of Fk∆. Let us therefore only consider the time points 0, ∆, 2∆, . . ..
Lemma 7.2. Let {Ct} be an Ammeter risk model. For any r ∈ IR such that ML(∆(MY(r) − 1)) < ∞ let θ(r) be the unique solution to
e−(θ(r)+cr)∆
ML(∆(MY(r) − 1)) = 1 . Then the discrete time process
{exp{−rCk∆− θ(r)k∆}}
is a martingale. Moreover, θ(r) is strictly convex, θ(0) = 0 and θ0(0) = IIE[L1]µ−c <
0.
Proof. Considering the process at the time points 0, ∆, 2∆, . . . we find Ck∆= Ck∆∆ . These points are exactly the claim times of the renewal risk process {Ck∆∆ }. The
assertion follows from Lemma 5.1 and from (5.3).
Because θ(r) is a convex function and θ(0) = 0 there may exist a second solution R to the equation θ(r) = 0. R is, if it exists, unique and strictly positive. R is then called the Lundberg exponent or adjustment coefficient of the Ammeter risk model.
Next we proof Lundberg’s inequality.
Theorem 7.3. Let {Ct} be an Ammeter risk model and assume that the Lundberg exponent R exists. Then
ψ(u) < ecR∆e−Ru.
If there exists an r > R such that ML(∆(MY(r) − 1)) < ∞ then R is the right exponent in the sense that there exists a constant C > 0 such that
ψ(u) ≥ Ce−Ru.
Remark. The constant ecR∆ might be too large, especially if ∆ is large. If possible, one should therefore try to find an alternative for the constant. An upper bound, based on the martingale (7.2), is found in [35] and [66]. However, one has to accept that the constant might be larger than 1. Recall that this was also the case for Lundberg’s inequality in the general case of the renewal risk model. Proof. Using Lemma 7.1 the proof becomes almost trivial. From Theorem 5.5 it follows that ψ∆(u) < e−Ru. Thus
ψ(u) ≤ ψ∆(u − c∆) < e−R(u−c∆) = ecR∆e−Ru.
If there exist an r > R such that ML(∆(MY(r) − 1)) < ∞ then by Theorem 5.7 there exists a constant C > 0 such that ψ∆(u) ≥ Ce−Ru. Thus also
ψ(u) ≥ ψ∆(u) ≥ Ce−Ru.
The constant ecR∆ seems to increase to infinity as ∆ tends to infinity. But R also dependends on ∆. At least in some cases ecR∆ remains bounded as ∆ → ∞.
Proposition 7.4. Assume that R exists and that there exists a solution r0 6= 0 to ML(µr)e−cr = 1 .
Then
ecR∆≤ ecr0. (7.1)
In particular, the condition will be fulfilled if R exists and if there exists `∞ ≤ ∞ such that ML(r) < ∞ ⇐⇒ r < `∞.
Remark. Note that r0 depends on the claim size distribution via µ only. Proof. The function log(ML(µr)) − cr has second derivative
µ2 ML00(µr)
ML(µr) −ML0(µr) ML(µr)
2
> 0 (see Lemma 1.9) and is therefore strictly convex. Because
µML0(0) ML(0)
− c = µIIE[Li] − c < 0 r0 has to be strictly positive. In particular, if
ML(µr)e−cr = exp{log(ML(µr)) − cr} ≤ 1
then r ≤ r0. If ML(r) < ∞ ⇐⇒ r < `∞ then by monotone convergence lim
r↑`∞
ML(r) = ML(`∞) = ∞ .
Thus by the convexity a strictly positive solution to log(ML(µr)) − cr = 0 has to exist.
Recall that ML(r) is increasing and that MY(r) − 1 ≥ rMY0 (0) = rµ. Since 1 = e−cR∆ML(∆(MY(R) − 1)) ≥ e−cR∆ML(∆Rµ)
it follows from the consideration above that R∆ ≤ r0.
Example 7.5. Assume that Li ∼ Γ(γ, α). The case ∆ = 1 was the case considered in Ammeter’s original paper [2]. In order to obtain the Lundberg exponent we have to solve the equation
e−cr∆ α
α − ∆(MY(r) − 1)
γ
= 1 or equivalently
e−cr∆/γ α
α − ∆(MY(r) − 1) = 1 . This can be written as
[α − ∆(MY(r) − 1)] ecr∆/γ = α .
It follows immediately that the above equation admits only a solution in closed form for special cases. For example, if Yi = 1 is deterministic then one has to solve
∆e(1+c∆/γ)r− (α + ∆)e(c∆/γ)r+ α = 0 .
One can only hope for a solution in closed form if c∆/γ ∈ Q. For instance if c∆/γ = 1 then R = log(α/∆) = log(αc/γ).
Let us now consider (7.1). r0 is the strictly positive solution to
α
α − rµ
γ
e−cr = 1 or
(α − rµ)ecr/γ = α .
The latter equation also has to be solved numerically. From Theorem 7.3 we find that
C ≤ lim
u→∞
ψ(u)eRu ≤ lim
u→∞ψ(u)eRu ≤ ecR∆. The question is: exists there a constant C such that
u→∞lim ψ(u)eRu = C ,
i.e. does the Cram´er-Lundberg approximation hold? This question is answered in the following theorem.
Theorem 7.6. Let {Ct} be an Ammeter risk model. Assume that R exists and that there exists an r > R such that ML(∆(MY(r) − 1)) < ∞. Then there exists a constant 0 < C ≤ ecR∆ such that
u→∞lim ψ(u)eRu = C .
Proof. This is a special case of Theorem 2 of [71]. We next want to derive an alternative proof of Lundberg’s inequality. For this reason we have to find an exponential martingale in continuous time. In applications it is usually rather easy to find a martingale if one deals with Markov processes. But we have remarked before that {Ct} is not a Markov process. The future of the process is dependent on the present level λt of the intensity. Let us therefore consider the process {(Ct, λt)}. But this is not a homogeneous Markov process neither. The future of the process is also dependent on time because the future intensity level is known until the next change of the intensity. Instead of considering {(Ct, λt, t)}
we consider {(Ct, λt, Vt)} where Vt is the time remaining till the next change of the intensity, i.e.
Vt= k∆ − t, if (k − 1)∆ ≤ t < k∆.
Remarks.
i) The process {λt} cannot be observed directly. The only information on λt are the number of claims occured in the interval [t + Vt− ∆, t]. Thus, provided Li is not deterministic, we can, in the best case, find the distribution of λt given FtC, the information generated by {Cs : s ≤ t} up to time t. But for our purpose it is no problem to work with the filtration Ft generated by {(Ct, λt, Vt)}. The results will not depend on the chosen filtration.
ii) As an alternative we could consider the time elapsed since the last change of the intensity Wt= ∆ − Vt. For the present choice of the Markovization see also
Section 8.2.
We find the following martingale.
Lemma 7.7. Let r such that ML(∆(MY(r) − 1)) < ∞. Then the process
Mtr = e−rCte(λt(MY(r)−1)−cr−θ(r))Vte−θ(r)t (7.2) is a martingale with respect to the natural filtration of {(Ct, λt, Vt)}.
Proof. Let first (k − 1)∆ ≤ s < t < k∆. Then λt= λs and IIE[Mtr| Fs] = IIE e−rCte(λt(MY(r)−1)−cr−θ(r))Vte−θ(r)t
Fs
= MsrIIE e−r(Ct−Cs)
Fs e−(λt(MY(r)−1)−cr)(t−s)
= Msre(λt(MY(r)−1)−cr)(t−s)
e−(λt(MY(r)−1)−cr)(t−s)
= Msr. Let now (k − 1)∆ ≤ s < t = k∆. Then, using the definition of θ(r),
IIE[Mtr | Fs] = MsrIIE e−r(Ct−Cs)
Fs IIE e(Lk+1(MY(r)−1)−cr−θ(r))∆
×e−(λs(MY(r)−1)−cr−θ(r))(t−s)e−θ(r)(t−s)
= Msre(λs(MY(r)−1)−cr)(t−s)ML(∆(MY(r) − 1))e−(cr+θ(r))∆
e−(λs(MY(r)−1)−cr)(t−s)
= Msr.
The assertion follows by iteration.
From the martingale stopping theorem we find
e−Ru = IIE[M0R] = IIE[Mτ ∧tR ] ≥ IIE[MτR; τ ≤ t] . By monotone convergence it follows that
e−Ru ≥ IIE[MτR; τ < ∞] = IIE[MτR| τ < ∞]IIP[τ < ∞] , from which
ψ(u) ≤ e−Ru IIE[MτR | τ < ∞]
follows. Note that Cτ < 0 on {τ < ∞}. Thus
IIE[MτR | τ < ∞] > IIE[e(λτ(MY(R)−1)−cR)Vτ | τ < ∞] > IIE[e−cRVτ | τ < ∞] > e−cR∆. We have recovered Lundberg’s inequality
ψ(u) < ecR∆e−Ru.
7.3.2. The General Case
The second proof of Lundberg’s inequality is in particular useful if the process does not start at a point where the intensity changes. For instance, the premium for the next year has to be determined before the end of the year. In this case the actuary has some information about the present intensity via the number of claims occured
in the first part of the year. Let us therefore assume that (λ0, V0) has the joint distribution function F1(`, v). We assume that IIP[V0 ≤ ∆] = 1. By allowing V0 to be stochastic we can also treat cases where no information on the intensity changing times is available. This might be a little bit strange from a practical point of view, but it is nevertheless interesting from a theoretical point of view. One special case is the case of a stationary intensity process where
F1(`, v) =v
∆1I0≤v≤∆+ 1Iv>∆ H(`) . We get the following version of Lundberg’s inequality.
Theorem 7.8. Let {Ct} be an Ammeter risk model. Assume that the Lundberg exponent R exists and that
IIE[eλ0(MY(R)−1)V0] < ∞ . Then
ψ(u) < ecR∆IIE[e(λ0(MY(R)−1)−cR)V0]e−Ru.
Remark. The condition
IIEeλ0(MY(R)−1)V0 < ∞
will be fulfilled in all reasonable situations. For instance, if (λ0, V0) is distributed such that {λt} becomes stationary, then
IIE[eλ0(MY(R)−1)V0] = 1
∆ Z ∆
0
ML(v(MY(R) − 1)) dv < ML(∆(MY(R) − 1)) < ∞ .
Proof. From the stopping theorem it follows that
e−RuIIEe(λ0(MY(R)−1)−cR)V0 = IIE[M0R] = IIE[Mτ ∧tR ] ≥ IIE[MτR; τ ≤ t] .
The assertion follows now as before.
For the Cram´er-Lundberg approximation we need the following
Lemma 7.9. Assume that an r > R exists such that ML(∆(MY(r) − 1)) < ∞ and that
IIE[eλ0(MY(r)−1)V0] < ∞ . Then
u→∞lim IIP[τ ≤ V0 | C0 = u]eRu= 0 . Proof. Again using the stopping theorem we get
e−ruIIE[e(λ0(MY(r)−1)−cr−θ(r))V0] = IIE[M0r] = IIE[Mτ ∧Vr
0] ≥ IIE[Mτr; τ ≤ V0] . As before it follows that
IIP[τ ≤ V0] ≤ e(rc+θ(r))∆IIE[e(λ0(MY(r)−1)−cr−θ(r))V0]e−ru
< e(rc+θ(r))∆IIE[eλ0(MY(r)−1)V0]e−ru
where we used that θ(r) > 0. Multiplying with eRu and then letting u → ∞ yields
the assertion.
The Cram´er-Lundberg approximation in the general case can now be proved.
Theorem 7.10. Let {Ct} be an Ammeter risk model. Assume that R exists and that there exists an r > R such that both ML(∆(MY(r) − 1)) < ∞ and
IIE[eλ0(MY(r)−1)V0] < ∞ . Then
u→∞lim ψ(u)eRu = IIE[e(λ0(MY(R)−1)−cR)V0] C where C is the constant obtained in Theorem 7.6.
Proof. Let ψo(u) denote the ruin probability in the ordinary case and define B(x; `, v) := IIP[Cv− u ≤ x | λ0 = `, V0 = v] .
Then
ψ(u) = Z ∞
`=0
Z ∆ v=0
Z cv y=−u
ψo(u + y)IIP[τ > v | λ0 = `, V0 = v, Cv = u + y]
× B(dy; `, v)F1(d`, dv) + IIP[τ ≤ V0] .
We multiply the above equation by eRu. The last term tends to 0 by Lemma 7.9.
The first term on the right hand side can be written as Z ∞
`=0
Z ∆ v=0
Z cv y=−u
ψo(u + y)eR(u+y)IIP[τ > v | λ0 = `, V0 = v, Cv = u + y]
× e−RyB(dy; `, v)F1(d`, dv) .
Noting that by Theorem 7.6 ψo(u + y)eR(u+y) is uniformly bounded by a constant C we see that¯
Z ∞
`=0
Z ∆ v=0
Z cv y=−u
ψo(u + y)eR(u+y)IIP[τ > v | λ0 = `, V0 = v, Cv = u + y]
× e−RyB(dy; `, v)F1(d`, dv)
≤ ¯CIIE[e−R(CV0−u)] = ¯CIIE[e(λ0(MY(R)−1)−cR)V0] < ∞ .
We can therefore interchange integration and limit. Because ψo(u + y)eR(u+y) tends to C and IIP[τ > v | λ0 = `, V0 = v, Cv = u + y] tends to 1 as u → ∞ the desired limit is
u→∞lim ψ(u)eRu = C Z ∞
`=0
Z ∆ v=0
Z cv y=−∞
e−RyB(dy; `, v)F1(d`, dv)
= IIE[e(λ0(MY(R)−1)−cR)V0] C .