Risk processes with dependence and premium adjusted to solvency targets

Abstract

This paper considers risk processes with various forms of dependence between waiting times and claim amounts. The standing assumption is that the increments of the claims process possess exponential moments so that variations of the Lundberg upper bound for the probability of ruin are in reach. The traditional point of view in ruin theory is reversed: rather than studying the probability of ruin as a function of the initial reserve under fixed premium, the problem is to adjust the premium dynamically so as to obtain a given ruin probability (solvency requirement) for a fixed initial reserve (the financial capacity of the insurer). This programme is carried through in various models for the claims process, ranging from Cox processes with i.i.d. claim amounts, to conditional renewal (Sparre Andersen) processes.

Appendix: A brief introduction to Dirichlet processes

We recapitulate the definition of Ferguson’s [11] Dirichlet process and some of its properties. A d-dimensional random vector $(P_1,\ldots,P_d)$ is said to be Dirichlet distributed with parameter $({\alpha}_1,\ldots,{\alpha}_d) \in (0,\infty)^d$, written

$$ (P_1,\ldots,P_d) \sim \hbox{Dir}_d({\alpha}_1,\ldots,{\alpha}_d),$$

if its joint density is

$$ {\frac{{\Upgamma} \left(\sum_{i=1}^d {\alpha}_i \right)} {{\prod_{i=1}^d {\Upgamma}({\alpha}_i)}}} \prod_{i=1}^d p_i^{{\alpha}_i-1}, $$

$(p_1,\ldots,p_d) \in [0,1]^d, \sum_{i=1}^d p_i = 1$ (essentially a (d − 1)-dimensional distribution). The expected value of P _j is

$$ {\mathbb {E}} \, P_j = \frac{{\alpha}_j}{\sum_{i=1}^d {\alpha}_i}. $$

(32)

In particular, Dir₂(α₁, α₂) = Beta(α₁, α₂), the Beta distribution with parameters α₁ and α₂.

Let $({\mathbb {A}},{\mathcal {A}},{\alpha})$ be a measure space, the measure α being finite. A random measure P on $({\mathbb {A}},{\mathcal {A}})$ is said to be a Dirichlet process with parameter α, and we write

$$ P \sim \hbox{Dir}({\alpha}), $$

if for any measurable partition $(A_1,\ldots,A_d)$ of ${\mathbb{A}}$ we have

$$ (P(A_1),\ldots,P(A_d)) \sim \hbox{Dir}_d ({\alpha}(A_1),\ldots,{\alpha}(A_d)). $$

Let Y be a random element with values in ${\mathbb {A}}$ such that the conditional distribution of Y, given P, is P itself:

$$ Y|_P \sim P. $$

The marginal distribution of Y is obtained by norming α to a probability measure (essentially a consequence of (32)):

$$ Y \sim P_0 = \frac{1}{{\alpha}({\mathbb {A}})} {\alpha}. $$

(33)

Let $h: {\mathbb {A}} \mapsto {\mathbb {R}}$ be a measurable function. When it exists, the expected value of h(Y) is

$$ {\mathbb {E}} \,h(Y) = \int h(y) \, P_{0}(dy). $$

(34)

Let $(Y_1,\ldots,Y_n)$ be random elements in ${\mathbb{A}}$ such that, conditional on $P, Y_{1},\ldots,Y_n$ are i.i.d. replicates of the generic Y:

$$ (Y_1,\ldots,Y_n)|_P \sim P \times \cdots \times P. $$

Then the posterior distribution of P is also Dirichlet:

$$ P|_{(Y_1,\ldots,Y_n)} \sim \hbox{Dir} \left( \sum_{i=1}^n \varepsilon_{{Y_i}} + {\alpha} \right) . $$

(35)

One says that the Dirichlet process is a natural conjugate prior for the nonparametric class of distributions on $({\mathbb{A}},{\mathcal{A}})$. Denoting the posterior distribution by P _n , we can recast (35) as

$$ P_n = z_n \,\hat{P}_n + (1-z_n) P_0, $$

(36)

where $\hat{P}_n$ is the empirical distribution of the observations,

$$ \hat{P}_n = \frac{1}{n} \sum_{i=1}^n \varepsilon_{{Y_i}}, $$

and

$$ z_n = \frac{n}{n + {\alpha}({\mathbb {A}})}. $$

Thus, the posterior probability distribution is a weighted average of the empirical distribution and the prior distribution, the weight on the former increasing with the number of the observations and decreasing with ${{\alpha}({\mathbb {A}})}$, the latter therefore being a measure of the a priori certainty about P. Upon combining (34) and (36), we see that also posterior expected values are weighted averages of empirical means and prior means:

$$ {\mathbb {E}} [h(Y)|Y_1,\ldots,Y_n] = z_n \int h(y)\, \hat{P}_n(dy) + (1 - z_n) \int h(y)\, P_0(dy) $$

(37)

$$ = \frac{\sum_{i=1}^n h(Y_i) + \int_{{\mathbb {A}}} h(y) \,{\alpha}(dy)}{n + {\alpha}({\mathbb {A}})}. $$

(38)

In actuarial science formulas like (37) are called credibility formulas, and the weight z _n would be called a credibility since it measures the amount of “credence” given to the observations.

Finally, observe that the posterior distribution in (36) converges in distribution to P almost surely as n tends to infinity (z _n tends to 1 and $\hat{P}_n$ converges in distribution to P almost surely). Thus, if h is continuous, then the posterior expected value in (37) converges almost surely to $\int h(y)\, P(dy)$.