2. THE MODEL

The classical risk model can be described as follows:

where U(t) denotes the surplus at time t, u denotes the initial surplus, c denotes the premium rate in a unit time, X_i denotes the amount of the ith claim, and N(t) denotes the number of claims occurring in the time interval (0, t] . It is assumed that {N(t); t > 0} is a homogeneous Poisson process with an intensity of λ. It is also assumed that the claim amounts are independent of the claim number process and are positive and mutually independent and identically distributed (i.i.d.) with common distribution function F, where F satisfies F(0) = 0. Moreover, we assume that F has a mean p₁ and

We assume that

where θ is called security loading. We assume that θ > 0 in the first part (up to Section 6) of this article, and we discuss the case where θ ≤ 0 in Section 7.

In the literature, there are many articles dealing with the ruin problems for the classical risk model. However, in the real world, investment income plays a very important role. In this article, we consider the insurance risk model where the interest rate is random. Let T_i denote the ith claim time and let W_t denote the discount factor. The discounted value of total claim amounts occurring in the time interval (0, t] can then be described as

S = {S_t} is called a filtered Poisson process. Using the properties of the filtered Poisson process (see Deng and Liang [6] or Snyder [12]), we have

where σ(W_s : 0 ≤ s < t) denotes the σ-field generated by {W_s : 0 ≤ s < t}.

Assuming that the interest rate is random (equivalently, the discount factor W_t is random), similar to (2.1), the surplus process of an insurance company can then be described by the following model:

where U_t^(W) denotes the discounted value of the surplus (discounted from time t to time 0) and all the other notations remain the same as earlier. We call model (2.5) the compound filtered Poisson model.

In this article, we make the following assumptions:

The first assumption is a common one in actuarial science. Although, recently, there have been some works on the dependent risk in the actuarial literature, we do not consider this issue here. The second assumption states that the interest rate cannot be positive infinity. Our model here is very general. In practice, we usually assume that the interest rate is nonnegative at any time. This corresponds to W_t > 0 and W_t is nonincreasing for all t > 0.

3. CONVERGENCE OF U_t^(W)

In this section, we will prove a convergence result. This convergence result will then be used in later sections to prove some useful results in the insurance risk theory. In this section, we assume that the discount factor W_t satisfies the following assumption:

Under the above assumption, we have the following result.

Theorem 1: Under assumption (3.1), there exists an integrable random variable U_∞^(W) such that

and

where a.e. means almost everywhere with respect to probability measure P. Moreover, the characteristic function of U_∞^W is given by

Proof: Let

Then, M_t is a martingale with a mean of zero, and M_t is independent of the discount factor W_t. Using this notation, the discounted surplus can be rewritten as

where

is also a martingale with a mean of zero.

First, we will prove the convergence of

. Since W_s is nonnegative for all s ≥ 0, the monotone convergence theorem implies

because

The random variable

is P − a.s. finite, which proves the convergence of

. Next, we will prove the convergence of H_t. Note that {H_t} is an L₂-bounded martingale; the result follows from a standard result. Therefore, there exists a finite random variable H_∞ such that

From the above results, we see that there exists a finite random variable U_∞^(W) such that

The characteristic function of U_∞^W is obtained by straightforward calculation. █

Remark 1: It is well known that for the classical compound Poisson model, if θ > 0, the surplus process will move toward +∞ with a probability 1 as the time t → ∞. This means that in the classical compound Poisson model, if we let the surplus process continue even when the ruin occurs, the surplus process will eventually become very large (tend to infinity with probability 1 as time tends to infinity). See Dassios and Embrechts [4], for further detailed discussions on this issue. Theorem 1 tells us that if we include the interest effect in the model, this is not the case.

4. AN EXPRESSION OF RUIN PROBABILITY

Ruin probability has been one of the most important research topics for almost a century, ever since the pioneering work of Lundberg [7,8]. Elegant mathematics has been developed on this topic.

In this section, we will use the result of Theorem 1 to give an expression for the ruin probability. Let T^(u) = inf{t; U_t^(W) < 0} and ψ(u) denote the ruin probability of model (2.5). Then,

Let

; then,

Let

then,

Notice that U_t has stationary and independent increments and that it is independent of W_t. If we assume

then we can obtain that

where

Remark 2: Because we do not require that W_t be nonincreasing in this article, any Levy processes will satisfy assumption (*).

Lemma 1: The process {U_t^(W)}, condition on σ(W_s : 0 ≤ s < t), has an independent increment, and under assumption (*),

where

means that both the left-hand side and the right-hand side of (4.4) have the same distribution.

Proof: For t > s > 0,

Notice that {W_t} is independent of {N(t) : t ≥ 0} and {X_i : i = 1,2,…}, so

has independent increments and is independent of W_s. Therefore, this lemma holds. █

From the above results, we have the following theorem.

Theorem 2: Under assumption (*), let F_∞(x) be the distribution function of U_∞^(W) − u. Then,

Proof:

Here, we have used the notation

. From this, the result is proved. █

We say that a distribution function F(x) is a new worse than used (NWU) distribution if F(x) is a distribution function (d.f.) of a nonnegative random variable and ÝF(x) = 1 − F(x) satisfies that ÝF(x)ÝF(y) ≤ ÝF(x + y) for x ≥ 0 and y ≥ 0. We say that a distribution function F(x) is a new better than used (NBU) distribution if F(x) is a d.f. of a nonnegative random variable and satisfies that ÝF(x)ÝF(y) ≥ ÝF(x + y) for x ≥ 0 and y ≥ 0.

Proposition 1: Under assumptions (3.1) and (*), we have the following:

Proof: We only give the proof for part 1 because part 2 can be proved by the same method, and part 3 can be obtained from parts 1 and 2 because the exponential distribution is both NWU and NBU.

where T^(u)− means the time just before ruin.

When U_∞^(W) − u > 0, we can get from the distribution of X_i is NBU, that the above probability is less than P{X_i > U_∞^(w) − u}—that is, the value E(F_∞(X_i)). When U_∞^(w) − u < 0, the above probability is 1 and the value P{X_i > U_∞^(w) − u} also equals one.

After all, when the distribution of X_i is NBU, we have

so we can get part 1. █

Remark 3:

5. CONSTANT DISCOUNT RATE AND EXPONENTIAL CLAIM SIZE

We consider a special case in which the discount rate is a constant δ and the claim size is an exponentially distributed random variable. This case was treated in Sundt and Teugels [13]. We will obtain this result as a special case of our results in the previous sections. In this section, we will assume that W_t = e^−δt and X_i is exponentially distributed with a mean of p₁. We will first give Lemma 2.

Lemma 2: The discounted value of all of the claim amounts occurring in the time interval (0, t],

, converges to a random variable S(+∞). Furthermore, S(+∞) has a gamma distribution with parameters (λ/δ, p₁⁻¹).

Proof: According to the properties of the filtered Poisson process, we can derive the characteristic function of S(t) by (see Snyder [12])

Let t → +∞; then,

The characteristic function of S(+∞) is then

This is the characteristic function of a gamma-distributed random variable with parameters (λ/δ, p₁⁻¹). Therefore, the lemma holds. █

As explained in Remark 4, part 2 that follows, the following theorem gives the same result as that in Sundt and Teugels [13].

Theorem 3: Under the assumptions in this section, the ruin probability ψ(u) of model (2.5) has the following expression:

where F_{Γ(λδ⁻¹, p1⁻¹)}(·) denotes the distribution function of the Γ random variable with the parameters (λδ⁻¹, p₁⁻¹).

Proof: Under the assumptions in this section, model (2.5) becomes

From Lemma 2, we know that

where S(+∞) is gamma-distributed with the parameters (λ/δ, p₁⁻¹), (i.e., u + [(1 + θ)/δ] p₁λ − U_t^(W) converges to a Γ random variable with the parameters (λ/δ, p₁⁻¹)). Then, using Theorem 2, the result is proved. █

Remark 4:

6. UPPER BOUNDS FOR RUIN PROBABILITY

In this section, we will derive the upper bounds for ruin probability. The main results are given in the following two theorems. The techniques used in the proof are similar to those in Yang [14].

Theorem 4: Suppose that G(·) is a nonnegative measurable function, U_t^(W) is the discounted surplus of the insurance company at time t, and G(U_t^(W)) is a supermartingale with respect to σ-field

. Then,

If lim_t→∞ E(G(U_t^(W))|T^(u) > t) = 0 and G(U_t^(W)) is an

martingale, then

where T^(u) is the same as earlier.

Proof: T^(u) is an

stopping time. For ∀t > 0, T^(u) ∧ t is a bound

stopping time. Using Doob's bounded stopping time theorem, we have

Let t → +∞; by the positive of G(·), we have

Therefore, (6.1) holds.

Moreover, if limt→+∞ E(G(UT(u)(W))|T(u) > t) = 0 and G(Ut(W)) is an

martingale, then (6.2) also holds. █

Theorem 5: If, for all t ≥ 0, 0 < W_t < 1 a.e., assume that there exists R > 0 such that

Then,

Proof: For all x ∈ (0,1] , y > 0,

holds; that is,

Let h(r) = E(e^rX_i − 1). Because for all t > 0, 0 < W_t < 1 a.e, we then have

Let

Using a similar argument as that in Deng and Liang [6, pp.329–332] (also see Snyder [12]) and after some calculations, we obtain

so

Therefore, N_t is an

supermartingale. From Theorem 4, we have

where R > 0 and UT(u)(W) < 0. Note that E [exp(−RUT(u)(W))|T(u) < ∞] > 1. Therefore, the result of the theorem holds. █

7. UPPER BOUNDS FOR RUIN PROBABILITY IN THE CASE OF NEGATIVE LOADING

It is well known that, in the classical model, if the loading is negative, the ruin becomes certain. We will show below that this is not the case in the model with stochastic interest rate. We first introduce the following sets:

From the definition, we can easily see that D is an interval of the form (0, r) or (0, r] . It is not difficult to see that for 0 < t < ∞, we have D_t = D if D = (0, r] , and if D = (0, r) with r < ∞, then D_t = (0, r) or (0, r] . Furthermore, if 0 < t and u < ∞, then D_t = D_u.

It is not difficult to prove that the assumption

implies that D_∞ = D_t.

The following theorem provides us with an upper bound for ruin probability in the case of negative safety loading.

Theorem 6: Assume that the safety loading θ ≤ 0 and 0 < t < ∞. Then, we have the following results:

Proof: For r ∈ D_t, let h(r) = E [exp(rX₁)] − 1 and define

With the same arguments as in the proof of Theorem 5, it is not difficult to check that M_t is a martingale with respect to

, where

Applying Doob's bounded stopping time theorem to the martingale {M_t : t ∈ R₊} for the stopping time T ∧ t, 0 < t < ∞, we have

where I_{A} denotes the indicator function of set A, and we have used the fact that, at the ruin time T,

.

Hence, if r ≥ 0,

Therefore,

From the inequality e^x − 1 ≥ x, θ ≤ 0, and that W_s and X₁ are positive random variables, we have that

Hence, (7.6) can be rewritten as

From this, the theorem follows. █

Notice that the upper bounds in theorem 6 only make sense if the right-hand sides of (7.6) and (7.7) are smaller than one. We will address this problem now.

For 0 < t ≤ ∞, let

Proposition 2: Suppose that 0 < t ≤ ∞ and the safety loading θ ≤ 0. Φ_t(r) is then a convex mapping on R₊ and is finite on D_t.

Proof: Similar to Lemma 4.5 of Delbaen and Haezendonck [5], we can prove the following:

If

then

If

then

From this, it follows that Φ_t(r) is a convex mapping on R₊. It is obvious that Φ_t(r) is finite on D_t.

The above analysis indicates that inf_r≥0 Φ_t(r) < 1 if the right derivative of Φ_t(r) at r = 0 is strictly negative. However,

where the last equality holds because X₁ and W_s are independent. This leads to the following proposition.

Proposition 3: If 0 < t < ∞ and

there exists a real number r_t > 0 such that

Furthermore, if

there exists a real number r_∞ > 0, such that

We conclude this section by giving two special cases. First, we consider the classical compound Poisson model (W_s = 1). In this case, if the risk loading θ is negative, the ultimate ruin probability will be one. The upper bound for the ruin probability at a finite time, say before time t, is nontrivial if

The second case is the situation with a constant interest force. In this case, W_s = e^−δs. A detailed discussion on this case can be found in Boogaert and Crijins [3].