Asymptotic ruin probabilities for discrete time risk models with heavy-tailed claims

Xiao Wei and Yijun Hu

School of Mathematics and Statistics Wuhan University Hubei, Wuhan 430072 People’s Republic of China

Email: smitty weixiao@yahoo.com.cn (Xiao Wei)

Email: yijunhu@public.wh.hb.cn (Yijun Hu)

1. The models

• Consider a discrete time risk model:

U_{n} = (U_{n−1} + X_{n})(1 + I_{n}) − Y_{n}, n = 1, 2, ⋯, (1)

U_{0} = M > 0: the initial surplus;

I_{n}: the rate of interest during the nth period, i.e. from time n − 1 to time n;

Y_{n}: the amount of claims during the nth period, and is paid at the end of the nth period, i.e. at time n;

X_{n}: the amount of premiums during the nth period, and is received at the beginning of the nth period, i.e. at time n − 1;

U_{n}: the reserve of an insurance company at time n.

• Assumptions:

(1) {X_{n}; n ≥ 1}, {I_{n}; n ≥ 1} and {Y_{n}; n ≥ 1} are three sequences of independent and identically distributed (i.i.d.) nonnegative random variables.

(2) We assume that the sequences {X_{n}}, {I_{n}} and {Y_{n}} are independent.

• The time of ruin in model (1)

τ_{M} := inf{n ≥ 1; U_{n} < 0},  (2)

and the corresponding ultimate ruin probability

ψ(M) := P (τ_{M} < ∞).

• Some existing results about the quantity ψ(M):

Case I: When Y_{1} is light tailed, i.e. there is a positive β such that E(exp(βY_{1})) < ∞.

• Yang (1999): the Lundberg type inequality for ψ(M) when {I_{n}} are identical constants;

• Cai (2002): the Lundberg type inequality for ψ(M) when {I_{n}} are assumed to have a dependent autoregressive structure of order 1.

Case II: When Y_{1} is heavy tailed, i.e. E(exp(βY_{1})) = ∞ for every β > 0.

• It is well known that the Lundberg type inequality for ψ(M) is no longer available.

• Our Gaol:

The asymptotic behavior of ψ(M) as M → ∞ when Y_{1} is heavy tailed.

• Our main objective:

To give reasons for the asymptotic estimate ψ(M) ≈ M^{−ω} where ω is a specific positive parameter. More precisely, for model (1), we will prove that, under suitable conditions,

lim_{M→∞} \frac{1}{log M} log P (τ_{M} < ∞) = −ω.

2. Main results

• Two parameters:

\overline{λ} := − lim sup_{t→∞} \frac{1}{t} log P (log (\frac{Y_{1}}{1 + I_{1}}) > t) ∈ [0, +∞],

and

\underline{λ} := − lim inf_{t→∞} \frac{1}{t} log P (log (\frac{Y_{1}}{1 + I_{1}}) > t) ∈ [0, +∞].

• Remark 1:

Obviously, \overline{λ} ≤ \underline{λ}. The parameters are finite and equal with the common value λ if and only if for every ε > 0,

\frac{1}{t^{λ+ϵ}} ≤ P (\frac{Y_{1}}{1 + I_{1}} > t) ≤ \frac{1}{t^{λ−ϵ}}

for large t.

• Remark 2:

One may have \overline{λ} < \underline{λ}. A further useful fact is that

\overline{λ} = sup\{λ ≥ 0; E (\frac{Y_{1}}{1 + I_{1}})^{λ}) < ∞\}.

• Theorem 1: Assume that esssup(\frac{1}{1+I_{1}}) < 1, and that 0 < \overline{λ} ≤ \underline{λ} < ∞, and E(X_{1}^{t}) < ∞ for every 0 < t < \underline{λ}. Then

lim sup_{M→∞} \frac{1}{log M} log P (τ_{M} < ∞) ≤ −\overline{λ},

and

lim inf_{M→∞} \frac{1}{log M} log P (τ_{M} < ∞) ≥ −\underline{λ},

where τ_{M} is as in (2).

• Corollary 1:

Suppose that all the conditions in Theorem 1 hold. Assume that there is a 0 < λ < ∞ such that for every ϵ > 0,

\frac{1}{t^{λ+ϵ}} ≤ P (\frac{Y_{1}}{1 + I_{1}} > t) ≤ \frac{1}{t^{λ−ϵ}}  (3)

for large t. Then

lim_{M→∞} \frac{1}{log M} log P (τ_{M} < ∞) = −λ.  (4)

• Remark 3: (4) says that

P (τ_{M} < ∞) ≈ M^{−λ},

where λ = sup\{t ≥ 0; E (\frac{Y_{1}}{1+I_{1}})^{t} < ∞\}.

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• Remark 4:

The judgment of the conditions in Theorem 1 and Corollary 1:

♣ If Y_{1} is of the so-called Pareto distribution with index α > 0, i.e. P (Y_{1} > t) = c^{α}t^{−α} for some fixed c > 0 and t ≥ c, and I_{1} is bounded away from zero, then (3) is satisfied with λ = α and esssup (\frac{1}{1+I_{1}}) < 1.

♣ If X_{1} is of the exponential distribution with index β > 0, i.e. P (X_{1} > t) = e^{−βt} for t ≥ 0, then E(X_{1}^{t}) < ∞ for every t > 0.

3. Keys of proofs

• Generalization of model (1) ( H. Nyrhinen (1999):)

S_{n} = B_{1} + A_{1}B_{2} + A_{1}A_{2}B_{3} + ⋯ + A_{1} ⋯ A_{n−1}B_{n}, (5)

n = 1, 2, · · · . {A_{n}; n = 1, 2, ⋯} : i.i.d. random variables, strictly positive; {B_{n}; n = 1, 2, ⋯} : i.i.d. random variables.

• The time of ruin in model (5)

T_{M} := inf{n ≥ 1; S_{n} > M},  (6)

where M is a positive real number.

• Nyrhinen (1999):

Under the assumption that the sequences {A_{n}} and {B_{n}} are independent, Nyrhinen (1999) gave reasons for the asymptotic estimate

P (T_{M} < ∞) ≈ M^{−θ},  (7)

where θ is a positive parameter.

• Here:

Without the assumption of independence between sequence {A_{n}} and {B_{n}}. We prove that, under some mild conditions,

lim sup_{M→∞} \frac{1}{log M} log P (T_{M} < ∞) ≤ −\overline{α},  (8)

and

lim inf_{M→∞} \frac{1}{log M} log P (T_{M} < ∞) ≥ −\underline{α},  (9)

where T_{M} is as in (6), and

\overline{α} := − lim sup_{t→∞} \frac{1}{t} log P (log(B_{1}I_{[B_{1}>0]}) > t) ∈ [0, +∞],

and

\underline{α} := − lim inf_{t→∞} \frac{1}{t} log P (log(B_{1}I_{[B_{1}>0]}) > t) ∈ [0, +∞].

• Proof of Theorem 1:

(8) + (9)+ Total expectation rule ⟹ Theorem 1.

Bibliography

[1] Cai, J., (2002) Ruin probabilities with depedent rates of interest. Journal of Applied Probability 39, 312-323.

[2] Nyrhinen, H., (1999) On the ruin probabilities in a general economic environment, Stachastic Process. Appl. 83, 319-330.

[3] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., (1999). Stochastic Processes for Insurance and Finance. Wiley and Sons, Chichester.

[4] Hu, Yijun, Nyrhinen, H., (2003) Large deviations viewpionts for heavy tailed random walks. To appear in J. Theoretical Probability.

[5] Yang, H., (1999) Non-exponential bounds for ruin probability with interest effect included. Scand. Actuarial J. 1, 66-79.

Thank you!

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