2. Definitions and Siegmund duality

First, let us impose assumptions on our model (2). Recall that the wealth of the insurance company is modeled by the right-continuous process with left limits $X = (X(t),\, t \ge 0)$ , governed by the integral equation

(7) \begin{align}X(t) = u + \int_0^tp(X(s))\,{\mathrm{d}} s + \int_0^t{\sigma}(X(s))\,{\mathrm{d}} W(s) - L(t)\end{align}

or, equivalently, by the stochastic differential equation (SDE) with initial condition $X(0) = u$ , given by (2). We say that X is driven by the Brownian motion W and Lévy process L. A function $f \colon {\mathbb{R}} \to {\mathbb{R}}$ , or $f \colon {\mathbb{R}}_+ \to {\mathbb{R}}$ , is Lipschitz-continuous (or simply Lipschitz) if there exists a constant K such that $|\,f(x) - f(\,y)| \le K|x-y|$ for all x, y.

Assumption 1 is not too restrictive as it allows us to consider classical risk process such as (a) the compound Poisson risk process when $p(x)=p$ and $\sigma(x)=0$ , or (b) the compound Poisson risk process under constant interest force when $p(x)=p+ix$ and $\sigma(x)=0$ . However, the regime-switching premium rate when the surplus hits some target is not covered. Assumption 2 allows the study of the compound Poisson risk process perturbed by a diffusion when $p(x)=p$ and $\sigma(x)=\sigma$ , extensively discussed in the paper by Dufresne and Gerber [Reference Dufresne and Gerber8], as well as the Lévy-driven risk processes defined for example by Furrer [Reference Furrer9] or Morales and Schoutens [Reference Morales and Schoutens24]. It is known from the standard theory (see e.g. [Reference Karatzas and Shreve13, Section 6.2]) that the Lévy measure of this process is a measure $\mu$ on ${\mathbb{R}}_+$ which satisfies

(8) \begin{equation}\int_0^{\infty}(1\wedge x)\,\mu({\mathrm{d}} x) < \infty.\end{equation}

Therefore, for all $c > 0$ we have

\begin{equation*}\mu[c, \infty) < \infty\quad \text{and}\quad \int_0^cx\,\mu(\mathrm{d}x) < \infty.\end{equation*}

When $\mu(\mathbb R_+) = \infty$ , there are infinitely many jumps on any finite-time interval. If $\mu(\mathbb R_+) < \infty$ and thus the Lévy measure is finite, then there are finitely many jumps in a finite-time interval, and we simply have a compound Poisson process: times between jumps form i.i.d. exponential random variables with rates $\mu(\mathbb R_+)$ , and the displacement during each jump is distributed according to the normalized measure $[\mu(\mathbb R_+)]^{-1}\mu(\!\cdot\!)$ . From Assumption 2, we have

(9) \begin{equation}{\mathbb E} \,{\mathrm{e}}^{-\lambda L(t)} = \exp (t\kappa(-\lambda)) \quad \text{for every}\ t, \lambda \ge 0,\end{equation}

where $\kappa(\lambda)$ is the Lévy exponent:

\begin{equation*} \kappa(\lambda) := \int_0^{\infty} [{\mathrm{e}}^{\lambda x} - 1] \mu({\mathrm{d}} x),\quad \lambda \in {\mathbb{R}}.\end{equation*}

As shown in [Reference Appelbaum1, Theorem 3.3.15], applied to the case $b = 0$ and $X(t) \equiv t$ (in the notation of that book), under Assumption 2, L is a Feller-continuous strong Markov process, with generator

(10) \begin{equation}\mathcal N f(x) = \int_0^{\infty} [\,f(x+y) - f(x)]\,\mu({\mathrm{d}} y)\end{equation}

for $f \in C^{\infty}({\mathbb{R}})$ with a compact support. We impose an additional assumption.

Assumption 3. The measure $\mu$ has finite exponential moment: for some $\lambda_0 > 0$ ,

(11) \begin{equation}\int_1^{\infty}\,{\mathrm{e}}^{\lambda_0 x}\,\mu({\mathrm{d}} x) < \infty.\end{equation}

Under Assumption 3, we can combine (8) and (11) to get

\begin{equation*}\kappa(\lambda) < \infty \quad \text{for}\ \lambda \in [0, \lambda_0).\end{equation*}

Proofs of the following technical lemmas are postponed to the Appendix.

Lemma 1. Under Assumption 3, we can extend the formula (10) for functions $f \in C^{\infty}({\mathbb{R}})$ which satisfy

(12) \begin{equation}C_f := \sup_{x \ge 0}\,{\mathrm{e}}^{-\lambda x}|\,f(x)| < \infty\quad {for\ some}\ {\lambda} \in (0, {\lambda}_0).\end{equation}

Lemma 2. Under Assumptions 2 and 3, the following quantity is finite:

(13) \begin{equation}m(\mu) := \int_0^{\infty}x\,\mu({\mathrm{d}} x) < \infty.\end{equation}

The following lemma can be proved by a classic argument, a version of which can be found in any textbook on stochastic analysis; see for example [Reference Karatzas and Shreve13, Section 5.2] or [Reference Sato31, Chapter 6]. For the sake of completeness, we give the proof in the Appendix.

Lemma 3. Under Assumptions 1 and 2, for every initial condition $X(0) = u$ there exists (in the strong sense, i.e. on a given probability space) a pathwise unique version of (2), driven by the given Brownian motion W and Lévy process L. This is a Markov process, with generator

(14) \begin{equation}{\mathcal L} f(x) := p(x)f'(x) + \dfrac12{\sigma}^2(x)f''(x) + \int_0^{\infty}[\,f(x-y) - f(x)]\,\mu({\mathrm{d}} y)\end{equation}

for $f \in C^2({\mathbb{R}})$ with a compact support. Under Assumption 3, this expression (14) is also valid for functions $f \in C^2({\mathbb{R}})$ satisfying (12) with $f(-x)$ instead of f(x).

Define the ruin probability in finite- and infinite-time horizons as in (4) and (3). We are interested in finding an estimate of the form

\begin{equation*}0 \le \psi(u) - \psi(u, T) \le C\,{\mathrm{e}}^{-kT},\quad u, T \ge 0,\end{equation*}

for some constants $C,\,k > 0$ . Recall the concept of Siegmund duality.

Definition 1. Two Markov processes $X = (X(t),\, t \ge 0)$ and $Y = (Y(t),\, t \ge 0)$ on ${\mathbb{R}}_+$ are called Siegmund dual if, for all $t, x, y \ge 0$ ,

\begin{equation*}{\mathbb P}_x(X(t) \ge y) = {\mathbb P}_y(Y(t) \le x).\end{equation*}

Here, the indices x and y refer to initial conditions $X(0) = x$ and $Y(0) = y$ .

Siegmund duality allows us to reduce the ruin problem to a convergence problem of a reflected jump-diffusion $Y=\{Y(t), \, t\geq0\}$ toward stationarity. There is a vast literature on Siegmund duality and a more general concept of functional duality of stochastic processes, both in discrete and continuous time. The earliest example of Siegmund duality was [Reference Lévy17, page 210]: absorbed and reflected Brownian motions on ${\mathbb{R}}_+$ are Siegmund dual. In a more general case, duality between absorbed and reflected processes was noted in [Reference Siegmund32]. Siegmund duality was studied for diffusions, jump-diffusions, and their absorbed and reflected versions, in [Reference Kolokoltsev15], [Reference Kolokoltsev and Lee16], [Reference Sigman and Ryan33], and for continuous-time discrete-space Markov chains in [Reference Zhao37]. The paper [Reference Sturm and Swart34] deals with Siegmund duality for general partially ordered spaces. See also survey [Reference Jansen and Kurt11] and references therein.

Take some functions $p_{\ast},\, \sigma_{\ast} \colon \mathbb{R}_+\to\mathbb{R}$ .

Definition 2. Consider an ${\mathbb{R}}_+$ -valued process $Y = (Y(t),\, t \ge 0)$ with right-continuous trajectories with left limits, which satisfies the following SDE:

(15) \begin{equation}Y(t) = Y(0) + \int_0^tp_{\ast}(Y(s))\,{\mathrm{d}} s + \int_0^t\sigma_{\ast}(Y(s))\,{\mathrm{d}} W(s) + L(t) + R(t),\end{equation}

where $R = (R(t),\, t \ge 0)$ is a non-decreasing right-continuous process with left limits, which starts from $R(0) = 0$ and can increase only when $Y(t) = 0$ . Then the process Y is called a reflected jump-diffusion on the half-line, with drift coefficient $p_{\ast}$ , diffusion coefficient ${\sigma}_{\ast}$ , and driving jump process L with Lévy measure $\mu$ .

The following result is the counterpart of Lemma 3 for the process $Y=\{Y(t), \, t\geq0\}$ .

Lemma 4. If $p_{\ast}$ and ${\sigma}_{\ast}$ are Lipschitz, then for every initial condition $Y(0) = y$ , there exists in the strong sense a pathwise unique version of (15). This is a Markov process with generator ${\mathcal A}$ , such that, for $f \in C^2({\mathbb{R}}_+)$ with compact support and $f'(0) = 0$ ,

(16) \begin{equation}{\mathcal A} f(x) = p_{\ast}(x)f'(x) + \dfrac12{\sigma}_{\ast}^{2}(x)f''(x) + \int_0^{\infty} [\,f(x +y) - f(x)]\,\mu({\mathrm{d}} y).\end{equation}

The proof, which is similar to that of Lemma 3, is provided in the Appendix. It was shown in [Reference Siegmund32] that a Markov process on ${\mathbb{R}}_+$ has a (Siegmund) dual process if and only if it is stochastically ordered.

This equivalence follows from [Reference Kamae, Krengel and Brien12]. Now, consider the process (2), stopped at hitting 0. The following result is known in the literature. A proof is provided in the Appendix for the sake of completeness.

Our main convergence theorem relies on the following result. We have not found this exact result in the literature. A proof is given in the Appendix.

Lemma 6. Under Assumptions 1 and 2, the Siegmund dual process for the jump-diffusion (2), absorbed at zero, is the reflected jump-diffusion on ${\mathbb{R}}_+$ from (15), starting at $Y(0)=0$ , with drift and diffusion coefficients

\begin{equation*}p_{\ast}(x) = -p(x) + {\sigma}(x){\sigma}'(x),\quad {\sigma}_{\ast}(x) = {\sigma}(x).\end{equation*}

We summarize this section as follows. We have shown that under Assumptions 1, 2, and 3, the wealth process is a stochastically ordered Markov process that admits as a Siegmund dual process a Markov process defined as a reflected jump-diffusion process. Therefore the rate of convergence for ruin probabilities coincides with that of the dual process $Y=\{Y(t), \, t\geq0\}$ , associated to the risk process $X=\{X(t), \, t\geq0\}$ , toward stationarity.

3. Main results

A common method for proving an exponential rate of convergence toward the stationary distribution is to construct a Lyapunov function.

Definition 3. Let $V\colon \mathbb{R}_+\to[1,\infty)$ be a continuous function and assume there exists $b,k,z>0$ such that

\begin{equation*} {\mathcal A} V(x)\leq-kV(x)+b{1}_{[0,z]}(x),\quad x\in\mathbb{R}_+.\end{equation*}

Then V is called a Lyapunov function.

We shall build a Lyapunov function for the Markov process Y in the form $V_{\lambda}(x)={\mathrm{e}}^{\lambda x}$ for $\lambda>0$ . This choice appears to be suitable to tackle the rate of convergence problem for the reflected jump-diffusion process as the generator acts on it in a simple way. Under Assumption 3, consider the function

\begin{equation*}\phi({\lambda}, x) := p_*(x){\lambda} + \dfrac12{\sigma}^2(x){\lambda}^2 + \kappa({\lambda}),\quad {\lambda} \in[0,\lambda_0),\ x \in {\mathbb{R}}{.}\end{equation*}

For a signed measure $\nu$ on $\mathbb{R}_+$ and a function $f\colon \mathbb{R}_+\to\mathbb{R}$ , we let $(\nu, f)=\int f\mathrm{d}\nu$ . Further, for a function $f\colon \mathbb{R}_+\to [1,+\infty)$ , define the norm ${\lVert{\nu}\rVert}_f:= \sup_{|g| \le f}|(\nu,g)|$ . If $f\equiv1$ , then ${\lVert{\cdot}\rVert}_f$ is the total variation norm. Define

\begin{equation*} \Phi(\lambda)=\inf_{x \ge 0}({-}\phi({\lambda}, x)) = -\sup_{x \ge 0}\phi({\lambda}, x).\end{equation*}

Theorem 2. Under Assumptions 1, 2, and 3, suppose

(17) \begin{equation}\Phi(\lambda) > 0\quad {for\ some}\ {\lambda} \in (0, {\lambda}_0).\end{equation}

Then there exists a unique stationary distribution $\pi$ for the reflected jump-diffusion Y. Take a ${\lambda} \in (0, {\lambda}_0)$ such that $k = \Phi(\lambda) > 0$ . This stationary distribution satisfies $(\pi, V_{{\lambda}})< \infty$ . The transition function $Q^t(x, \cdot)$ of the process Y satisfies

\begin{equation*} {\lVert{Q^t(x, \cdot) - \pi(\!\cdot\!)}\rVert}_{V_\lambda} \le [V_{\lambda}(x) + (\pi,V_{\lambda})]\,{\mathrm{e}}^{-kt}.\end{equation*}

The proof of Theorem 2 is postponed to Section 5. The central result of this paper is a corollary of Theorem 2, a direct consequence of the duality link established between the processes X and Y.

Corollary 1. Under Assumptions 1, 2, and 3, and condition (17),

(18) \begin{equation}0 \le \psi(u) - \psi(u, T) \le [1 + (\pi, V_{{\lambda}})]\,{\mathrm{e}}^{-kT},\quad u, T \ge 0.\end{equation}

Proof. By virtue of Siegmund duality,

(19) \begin{equation} \psi(u) - \psi(u, T)={\mathbb P}(Y(\infty) \ge u) - {\mathbb P}(Y(T) \ge u),\end{equation}

where $Y=(Y(t) , \,t\geq0)$ is a reflected jump-diffusion on $\mathbb{R}_{+}$ , starting at $Y(0)=0$ , and $Y(\infty)$ is a random variable distributed as $\pi$ . We may rewrite (19) as $\psi(u) - \psi(u, T)=\pi([u,\infty)) - Q^{T}(0,[u,\infty))$ . Then the inequality (18) follows immediately from the application of Theorem 2.

In the space-homogeneous case $p(x) \equiv p$ and ${\sigma}(x) \equiv {\sigma}$ , the quantity $\phi({\lambda}, x)$ is independent of x, and condition (17) means that there exists a ${\lambda} > 0$ such that $\phi({\lambda}) < 0$ . Then $p_{\ast} = p$ and $\phi'(0) = -p + \psi'(0) = -p + m(\mu)$ . It is easy to show that $\phi(\!\cdot\!)$ is a convex function with $\phi(0) = 0$ . Therefore condition (17) holds if and only if $\phi'(0) < 0$ or, equivalently,

\begin{equation*} p > m(\mu).\end{equation*}

4. Explicit rate of exponential convergence calculation

In this section we study the rate k of exponential convergence depending on the parameters of the risk model. In the examples we assume a constant premium rate that satisfies the net benefit condition with

\begin{equation*}p=(1+\eta)\cdot\mathbb{E}[L(1)],\quad \eta>0,\end{equation*}

and a constant diffusion parameter $\sigma$ around the premium rate. Under these settings and for finite measure $\nu$ (when L is the compound Poisson process), the rate of exponential convergence derived in this work has been shown to be optimal in a certain sense; see [Reference Sarantsev29, Section 6]. However, we absolutely do not claim that for general (non-constant) premium rate p and diffusion parameter $\sigma$ , this exponential rate is optimal. Let us remark that the concept of optimal rate of convergence could be understood in various ways; at the very least, it depends on the distance used.

4.1. Compound Poisson risk model perturbed by a diffusion

In this subsection the risk process $X=(X(t), \, t\geq0)$ is defined as

\begin{equation*}X(t)=u+pt+\sigma W(t)-\sum_{k=1}^{N(t)}U_k,\end{equation*}

where $u\geq0$ denotes the initial capital and p corresponds to the premium rate. The process $W=(W(t), \, t\geq0)$ is a standard Brownian motion allowing us to capture the volatility around the premium rate encapsulated in the parameter $\sigma>0$ . The process $N=(N(t), \, t\geq0)$ is a homogeneous Poisson process with intensity $\beta>0$ , independent of the claims $U_1,U_2,\ldots$ i.i.d. with distribution function B. The premium rate satisfies the net benefit condition: $p=(1+\eta)\beta\mathbb{E}(U)$ , where $\eta>0$ is safety loading.

We can study the rate of exponential convergence of ruin probabilities; specifically, how it depends on the parameters of the model: (a) the diffusion coefficient $\sigma$ in front of the perturbation term; (b) the safety loading $\eta$ ; (c) the shape of the claim size distribution. The function $\phi({\lambda}, x)$ for this risk process is given by

\begin{equation*}\phi({\lambda}, x) = -p{\lambda} + \dfrac12{\sigma}^2{\lambda}^2 + \beta [\widehat{B}(\lambda)-1 ],\quad {\lambda} \ge 0,\ x \in {\mathbb{R}},\end{equation*}

where $\widehat{B}(\lambda)=\mathbb{E}({\mathrm{e}}^{\lambda U})$ denotes the moment generating function (MGF) of the claim amount distribution. As the expression of $\phi({\lambda}, x)$ does not in fact depend on x, then

\begin{equation*}\inf_{x \ge 0}({-}\phi({\lambda}, x))=\Phi(\lambda)= p{\lambda} - \dfrac12{\sigma}^2{\lambda}^2 - \beta [\widehat{B}(\lambda)-1],\quad {\lambda} \ge 0,\ x \in {\mathbb{R}}.\end{equation*}

Based on Corollary 18, we define the rate of exponential convergence by

\begin{equation*}k=\max\{\Phi(\Lambda)\mid {\lambda}\geq0 \,;\; \widehat{B}(\lambda)<\infty\}.\end{equation*}

The function $\lambda\mapsto \Phi(\lambda)$ is strictly concave as

\begin{equation*}\Phi''(\lambda)=-\sigma^{2}-\beta\widehat{B}''(\lambda)<0\quad \text{for all ${\lambda}\in\{{\lambda}\geq0;\; \widehat{B}(\lambda)<\infty\}$}.\end{equation*}

It follows that

\begin{equation*} {\lambda}_{\ast}:= \underset{\{{\lambda}\geq0;\; \widehat{B}(\lambda)<\infty\}}{\text{argmax}}\,\Phi(\lambda)\end{equation*}

is a solution of the equation $p-\sigma^{2}{\lambda}-\beta\widehat{B}'(\lambda)=0$ under the constraint

\begin{equation*} {\lambda}^{\ast}\in\{\lambda\geq0;\; \widehat{B}(\lambda)<\infty\}. \end{equation*}

The rate of exponential convergence is then given by

\begin{equation*}k=\Phi({\lambda}_{\ast})=p{\lambda}_{\ast} - \dfrac12{\sigma}^2{\lambda}_{\ast}^2 - \beta [\widehat{B}({\lambda}_{\ast})-1].\end{equation*}

In this example, we compare the rate of convergence k for three claim size distributions: the gamma distribution $\text{Gamma}(\alpha, \delta)$ with probability density function

\begin{equation*}p(x;\,\; \alpha, \beta) = \dfrac{\delta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}\,{\mathrm{e}}^{-\delta x}\quad \text{for}\ x>0.\end{equation*}

Define the exponential distribution $\text{Exp}(\delta) = \text{Gamma}(1, \delta)$ , and the mixture of exponential distributions $\text{MExp}(\kern1pt p,\delta_1,\delta_2)$ with associated probability density function

\begin{equation*}p(x;\,\; p,\delta_1,\delta_2) = p\delta_1\,{\mathrm{e}}^{-\delta_1 x}+(1-p)\delta_2\,{\mathrm{e}}^{-\delta_2 x},\quad x > 0.\end{equation*}

Let the claim size be distributed as $\text{Gamma}(2,1)$ . Table 1 gives the rate of exponential convergence for various combinations of values for the safety loading and the volatility. For a given value of the safety loading, the rate of convergences decreases when the volatility increases. Conversely, for a given volatility level, the rate of convergence increases with the safety loading. The first row of Table 1 contains the rates of convergence when $\sigma=0$ , associated to the compound Poisson risk model. Figure 1 displays the rates of exponential convergence depending on the volatility level for different values of the safety loading: $\eta=0.1,0.2,0.3$ .

Table 1. Rate of exponential convergence in the compound Poisson risk model perturbed by a diffusion, with Gamma(2,1) claim size

Figure 1. Rate of exponential convergence in the compound Poisson risk model perturbed by a diffusion depending on the volatility, for $\eta=0.1,0.2,0.3$ .

Table 2. Rate of exponential convergence in the compound Poisson risk model perturbed by a diffusion for different claim size distribution

Figure 2. Rate of exponential convergence in the compound Poisson risk model perturbed by a diffusion for different claim size distributions.

Remark 2. Consider the compound Poisson risk model perturbed by a diffusion under constant interest force $i>0$ by assuming that $p(x)=p+ix$ ; the function $\phi({\lambda},x)$ then becomes

\begin{equation*}\phi({\lambda}, x) = -(\kern1pt p+ix){\lambda} + \dfrac12{\sigma}^2{\lambda}^2 + \beta [\widehat{B}(\lambda)-1],\quad {\lambda} \ge 0,\ x \in {\mathbb{R}}.\end{equation*}

Although the function $\phi({\lambda},x)$ depends on x, it is easily seen that

\begin{equation*}\inf_{x \ge 0}({-}\phi({\lambda}, x))=\Phi(\lambda)= p{\lambda} - \dfrac12{\sigma}^2{\lambda}^2 - \beta [\widehat{B}(\lambda)-1],\quad {\lambda} \ge 0,\ x \in {\mathbb{R}}.\end{equation*}

The maximization problem is the same as for the compound Poisson risk model perturbed by a diffusion and will lead to the same rate of convergence.

Let us turn to the study of rate of convergence for different claim size distributions. We assume the claims are exponentially distributed $\text{Exp}(1/2)$ , or gamma-distributed $\text{Gamma}(2,1)$ , or a mixture of exponentially distributed $\text{MExp}(1/4,3/4,1/4,3/4)$ . The mean associated with claim size distributions is the same but the variance differs:

\begin{equation*}\text{Var}[\text{Gamma}(2,1)]<\text{Var}[ \text{Exp}(1/2)]<\text{Var}[ \text{MExp}(3/4,3/4,1/4)].\end{equation*}

Table 2 contains the values of the rate of exponential convergence over the three claim size distributions. The fastest convergence occurs in the gamma case. The slowest convergence occurs in the exponential-mixture case. Figure 2 displays the evolution of the rate of exponential convergence depending on the safety loading and the diffusion parameter for the different assumptions on the claim size. In the wake of this numerical study, we may conclude that the speed of convergence depends on the variance of the process. Increasing the variance via the claim size distribution or the diffusion component makes convergence slower.

4.2. Lévy-driven risk process

In this subsection we compare the rate of exponential convergence of the ruin probabilities when the liability of the insurance company is modeled by a gamma process and an inverse Gaussian Lévy process. The Lévy measure of a gamma process, $\text{GammaP}(\alpha,\beta)$ , is given by

\begin{equation*} \mu(\mathrm{d}x)=\dfrac{\alpha \,{\mathrm{e}}^{-\beta x}}{x}\quad \text{for }x>0,\end{equation*}

where $\alpha,\,\beta>0$ . Its Lévy exponent is

\begin{equation*} \kappa(\lambda)=\alpha\ln\biggl(\dfrac{\beta}{\beta-{\lambda}}\biggr)\quad \text{for }{\lambda}\in[0,\beta).\end{equation*}

Note that in this case an explicit expression for finite-time and infinite-time ruin probabilities can be found in [Reference Meyn and Tweedie22]. The function $\Phi(\!\cdot\!)$ is strictly concave as

\begin{equation*}\Phi''(\lambda)=-\sigma^{2}-\dfrac{\alpha}{(\beta-\lambda)^2}<0.\end{equation*}

It follows that ${\lambda}_{\ast}$ is the solution of the equation

\begin{equation*}p-\sigma^{2}{\lambda}-\dfrac{\alpha}{\beta-\lambda}=0.\end{equation*}

The rate of exponential convergence is then given by

\begin{equation*}k=\Phi({\lambda}_{\ast})=p{\lambda}_{\ast} - \dfrac12{\sigma}^2{\lambda}_{\ast}^2 - \alpha\ln\biggl(\dfrac{\beta}{\beta-{\lambda}_\ast}\biggr).\end{equation*}

The Lévy measure of the inverse Gaussian Lévy process, $\text{IGP}(\gamma)$ , is defined as

\begin{equation*} \mu(\mathrm{d}x)=\dfrac{1}{\sqrt{2\pi}x^{3/2}}\,{\mathrm{e}}^{-x\gamma^{2}/2}\quad \text{for }x>0 {,}\end{equation*}

where $\gamma>0$ . Its Lévy exponent is

\begin{equation*} \kappa(\lambda)=\gamma-\sqrt{\gamma^{2}-2{\lambda}}\quad \text{for }{\lambda}\in[0,\gamma^{2}/2).\end{equation*}

The function $\Phi$ is strictly concave as $\Phi''(\lambda)=-\sigma^{2}-(\gamma^{2}-2\lambda)^{-3/2}<0$ . It follows that ${\lambda}_{\ast}$ is the solution of the equation

\begin{equation*}p-\sigma^{2}{\lambda}-\dfrac{1}{\sqrt{\gamma^{2}-2\lambda}}=0{.}\end{equation*}

The rate of exponential convergence is then given by

\begin{equation*}k=\Phi({\lambda}_{\ast})=p{\lambda}_{\ast} - \dfrac12{\sigma}^2{\lambda}_{\ast}^2 - \gamma+\sqrt{\gamma^2-2{\lambda}_\ast}.\end{equation*}

We set $\gamma=1$ , $\alpha=1/2$ , $\beta=1/2$ , to match the first moment of the liabilities in both risk models at time $t=1$ . The premium rate is then given by

\begin{equation*} p=(1+\eta)\mathbb{E}(L(1))=(1+\eta)\dfrac{\alpha}{\beta}=(1+\eta)\dfrac{1}{\gamma}.\end{equation*}

Table 3 shows the value of the exponential rate of convergence when the liability of the insurance company is governed by a gamma process or an inverse Gaussian Lévy process depending on the safety loading and the volatility of the diffusion. Figure 3(a) displays the rates of exponential convergence for the considered Lévy-driven risk models. We observe that the impact of the volatility and the safety loading on the convergence rate remains the same as in the compound Poisson case. The rate of exponential convergence is noticeably greater when the liability of the insurance company follows an inverse Gaussian Lévy process.

Table 3. Rate of exponential convergence in Lévy-driven risk models

5. Proof of Theorem 2

If Y were a reflected jump-diffusion with a.s. finitely many jumps in finite time, and with positive diffusion coefficient, then we could directly apply [Reference Sarantsev29, Theorem 4.1, Theorem 4.3], and complete the proof of Theorem 2. We can have: (a) ${\sigma}(x) = 0$ for some x; (b) infinite Lévy measure $\mu$ .

In the proof of [Reference Sarantsev29, Theorem 3.2], we used the following property: for all $t > 0$ , $x \in {\mathbb{R}}_+$ , and $A \subseteq {\mathbb{R}}_+$ of positive Lebesgue measure, we have $Q^t(x, A) > 0$ . This property might not hold for the case ${\sigma}(x) = 0$ for some $x \in {\mathbb{R}}_+$ . We bypass this difficulty by approximating the reflected jump-diffusion Y by a ‘regular’ reflected jump-diffusion, where ${\sigma}(x) > 0$ for $x \in {\mathbb{R}}_+$ , and the Lévy measure is finite.

Figure 3. Rate of exponential convergence for Lévy-driven risk processes.

For an ${\varepsilon} > 0$ , let $Y_{{\varepsilon}} = (Y_{{\varepsilon}}(t),\, t \ge 0)$ be the reflected jump-diffusion on ${\mathbb{R}}_+$ , with drift coefficient $p_*$ , diffusion coefficient ${\sigma}_{{\varepsilon}}(\!\cdot\!) = {\sigma}(\!\cdot\!) + {\varepsilon}$ , and jump measure $\mu_{{\varepsilon}}(\!\cdot\!) = \mu(\cdot\cap[{\varepsilon}, {\varepsilon}^{-1}])$ . Note that this is a reflected jump-diffusion with positive diffusion coefficient ${\sigma}_{{\varepsilon}}(\,y) > 0$ for all $y \in {\mathbb{R}}_+$ , and with finite Lévy measure $\mu_{{\varepsilon}}({\mathbb{R}}_+) < \infty$ . Therefore we can apply the results of [Reference Sarantsev29] to this process. For $x \in {\mathbb{R}}_+$ , let

\begin{equation*}\phi_{{\varepsilon}}(x, {\lambda}) := p_{\ast}(x){\lambda} + \dfrac12{\sigma}^2_{{\varepsilon}}(x){\lambda}^2 + \int_{{\varepsilon}}^{{\varepsilon}^{-1}} ({\mathrm{e}}^{{\lambda} y} - 1)\,\mu_{{\varepsilon}}({\mathrm{d}} y).\end{equation*}

For every $x \ge 0$ , we have

(20) \begin{equation}\phi(x, {\lambda}) -\phi_{{\varepsilon}}(x, {\lambda}) = -\Big[{\varepsilon}{\sigma}_{{\varepsilon}}(x) + \dfrac12{\varepsilon}^2\Big]{\lambda}^2 + \biggl(\int_0^{{\varepsilon}} + \int_{{\varepsilon}^{-1}}^{\infty}\biggr) ({\mathrm{e}}^{{\lambda} y} - 1 )\,\mu({\mathrm{d}} y).\end{equation}

Recall also that

(21) \begin{equation}\int_0^{\infty} ({\mathrm{e}}^{{\lambda} y} - 1 )\,\mu({\mathrm{d}} y) < \infty.\end{equation}

Combining (20) with (21) and the boundedness of ${\sigma}$ from Assumption 1, we have

(22) \begin{equation}\sup_{x \ge 0} |\phi_{{\varepsilon}}(x, {\lambda}) - \phi(x, {\lambda})| \to 0,\quad {\varepsilon} \downarrow 0.\end{equation}

By our assumptions, $\sup_{x \ge 0}\phi(x, {\lambda}) = -\Phi({\lambda}) < 0$ . From (22),

\begin{equation*}-\sup_{x \ge 0}\phi_{{\varepsilon}}(x, {\lambda}) =: \Phi_{{\varepsilon}}({\lambda}) \to \Phi({\lambda}).\end{equation*}

Thus we conclude that there exists an ${\varepsilon}_0 > 0$ such that for ${\varepsilon} \in [0,{\varepsilon}_0]$ , $\Phi_{{\varepsilon}}({\lambda}) > 0$ . Apply [Reference Sarantsev29, Theorem 4.3] to prove the statement of Theorem 2 for the process $Y_{{\varepsilon}}$ . For consistency of notation, let $Y_0 := Y$ . There exists a unique stationary distribution $\pi_{{\varepsilon}}$ for $Y_{{\varepsilon}}$ , which satisfies $(\pi_{{\varepsilon}}, V_{{\lambda}}) < \infty$ ; its transition kernel $Q_{{\varepsilon}}^t(x, \cdot)$ satisfies

(23) \begin{equation}{\lVert{Q_{{\varepsilon}}^t(x, \cdot) - \pi_{{\varepsilon}}(\!\cdot\!)}\rVert}_{V_{{\lambda}}} \le [V_{{\lambda}}(x) + (\pi_{{\varepsilon}}, V_{{\lambda}})]\,{\mathrm{e}}^{-\Phi_{{\varepsilon}}({\lambda})t}.\end{equation}

We would like to take the limit ${\varepsilon} \downarrow 0$ in (23). To this end, let us introduce some new notation. Take a smooth non-decreasing $C^{\infty}$ function $\theta \colon {\mathbb{R}}_+ \to {\mathbb{R}}_+$ with

\begin{equation*}\theta(x) =\begin{cases}0 & x \le s_-,\\x & x \ge s_+,\end{cases} \quad \theta(x) \le x,\end{equation*}

for some fixed $s_+ > s_- > 0$ . The function $\theta$ is Lipschitz on ${\mathbb{R}}_+$ : there exists a constant $C(\theta) > 0$ such that

(24) \begin{equation}|\theta(s_1) - \theta(s_2)| \le C(\theta)|s_1 - s_2|\quad \text{for all}\ s_1, s_2 \in {\mathbb{R}}_+.\end{equation}

Next, define $\tilde{V}_{{\lambda}}(x) = V_{{\lambda}}(\theta(x)) = {\mathrm{e}}^{{\lambda}\theta(x)}$ . The process $Y_{{\varepsilon}}$ has the generator ${\mathcal L}_{{\varepsilon}}$ :

\begin{equation*}{\mathcal L}_{{\varepsilon}}f(x) = p_{\ast}(x)f'(x) + \dfrac12{\sigma}_{{\varepsilon}}^2(x)f''(x) + \int_{{\varepsilon}}^{{\varepsilon}^{-1}}[\,f(x+y) - f(x)]\,\mu({\mathrm{d}} y)\end{equation*}

for $f \in C^2({\mathbb{R}}_+)$ with $f'(0) = 0$ . Repeating calculations from [Reference Sarantsev29, Theorem 3.2] with minor changes, we get

(25) \begin{equation}\begin{split}{\mathcal L}_{{\varepsilon}}\tilde{V}_{{\lambda}}(x) & \le -\Phi_{{\varepsilon}}({\lambda})\tilde{V}_{{\lambda}}(x) + c_{{\varepsilon}}1_{[0, s_+]}(x),\quad x \in {\mathbb{R}}_+,\\c_{{\varepsilon}} & := \max_{x\in[0, s_+]} [{\mathcal L}_{{\varepsilon}}\tilde{V}_{{\lambda}}(x) + \phi_{{\varepsilon}}({\lambda},x)\tilde{V}_{{\lambda}}(x)].\end{split}\end{equation}

Proof. The functions $V_{{\lambda}}$ and $\tilde{V}_{{\lambda}}(x)$ are of the same order, in the sense that

(26) \begin{equation}0 < \inf_{x \ge 0}\dfrac{\tilde{V}_{{\lambda}}(x)}{V_{{\lambda}}(x)} \le \sup_{x \ge 0}\dfrac{\tilde{V}_{{\lambda}}(x)}{V_{{\lambda}}(x)} < \infty.\end{equation}

Therefore it suffices to show that

(27) \begin{equation}\varlimsup_{{\varepsilon} \downarrow 0}(\pi_{{\varepsilon}}, \tilde{V}_{{\lambda}}) < \infty.\end{equation}

Apply the measure $\pi_{{\varepsilon}}$ to both sides of the inequality (25). This probability measure is stationary and thus the left-hand side of (25) becomes $(\pi_{{\varepsilon}}, {\mathcal L}_{{\varepsilon}}\tilde{V}_{{\lambda}}) = 0$ . Therefore

\begin{equation*}-\Phi_{{\varepsilon}}({\lambda})(\pi_{{\varepsilon}}, \tilde{V}_{{\lambda}}) + c_{{\varepsilon}}(\pi_{{\varepsilon}}, 1_{[0, s_+]}) \ge 0.\end{equation*}

From $(\pi_{{\varepsilon}}, 1_{[0, s_+]}) = \pi_{{\varepsilon}}([0, s_+]) \le 1$ , we get

(28) \begin{equation}(\pi_{{\varepsilon}}, \tilde{V}_{{\lambda}}) \le \dfrac{c_{{\varepsilon}}}{\Phi_{{\varepsilon}}({\lambda})}.\end{equation}

By (28), to show (27), it suffices to show that $\varlimsup_{{\varepsilon}\downarrow 0}c_{{\varepsilon}} < \infty$ . This, in turn, would follow from (25) and the relation

(29) \begin{equation}{\mathcal L}_{{\varepsilon}}\tilde{V}_{{\lambda}}(x) \to {\mathcal L}\tilde{V}_{{\lambda}}(x)\ \text{uniformly on}\ [0, s_+].\end{equation}

We can express the difference of generators as

(30) \begin{equation} {\mathcal L}_{{\varepsilon}} \tilde{V}_{\lambda}(x) - {\mathcal L}\tilde{V}_{\lambda}(x) = \dfrac12 ({\sigma}_{{\varepsilon}}^2(x) - {\sigma}^2(x) )f''(x) - \biggl(\int_0^{{\varepsilon}}+\int_{{\varepsilon}^{-1}}^{\infty}\biggr)\bigl[\tilde{V}_{{\lambda}}(x+y) - \tilde{V}_{{\lambda}}(x)\bigr]\,\mu({\mathrm{d}} y).\end{equation}

The first term on the right-hand side of (30) is equal to $\frac12(2{\varepsilon}{\sigma}(x) + {\varepsilon}^2)\,f''(x)$ . Since ${\sigma}$ is bounded, this term converges to 0 as ${\varepsilon} \downarrow 0$ uniformly on $[0, s_+]$ . It suffices to prove that the second term converges to zero as well. For all $x, y \ge 0$ , using (24), we have

(31) \begin{align}0 &\le \tilde{V}_{{\lambda}}(x + y) - \tilde{V}_{{\lambda}}(x) \notag \\& = {\mathrm{e}}^{{\lambda}\theta(x+y)} - {\mathrm{e}}^{{\lambda}\theta(x)} \notag \\& = {\mathrm{e}}^{{\lambda}\theta(x)} [{\mathrm{e}}^{{\lambda}(\theta(x+y) - \theta(x))} - 1] \notag \\& \le \tilde{V}_{{\lambda}}(x) [{\mathrm{e}}^{{\lambda} C(\theta)y} - 1].\end{align}

Changing the parameter $s_-$ and letting $s_- \downarrow 0$ , we have $\theta(x) \to x$ uniformly on ${\mathbb{R}}_+$ . Therefore we can make the Lipschitz constant $C(\theta)$ as close to 1 as necessary. Also, note that for ${\lambda}'$ in some neighborhood of ${\lambda}$ , we have

(32) \begin{equation}\int_0^{\infty} ({\mathrm{e}}^{{\lambda}'x} - 1)\,\mu({\mathrm{d}} x) < \infty.\end{equation}

Combining (31) and (32), using that $\sup_{x \in [0, s_+]}\tilde{V}_{{\lambda}}(x) < \infty$ , and taking $C(\theta)$ close enough to 1, we complete the proof that the second term on the right-hand side of (30) tends to 0 as ${\varepsilon} \downarrow 0$ . This completes the proof of (29), and with it that of Lemma 7.

Now, we state a fundamental lemma, and complete the proof of Theorem 2 assuming that this lemma is proved. The proof is postponed to the Appendix.

Lemma 8. Take a version $\tilde{Y}_{{\varepsilon}}$ of the reflected jump-diffusion $Y_{{\varepsilon}}$ , starting from $y_{{\varepsilon}} \ge 0$ , for ${\varepsilon} \ge 0$ . If $y_{{\varepsilon}} \to y_0$ , then we can couple $\tilde{Y}_{{\varepsilon}}$ and $\tilde{Y}_0$ so that, for every $T \ge 0$ ,

\begin{equation*}\lim_{{\varepsilon} \downarrow 0}{\mathbb E}\sup_{0 \le t \le T} |\tilde{Y}_{{\varepsilon}}(t) - \tilde{Y}_{0}(t)|^2 = 0.\end{equation*}

Proof. Since $V_{{\lambda}}(\infty) = \infty$ , Lemma 7 implies tightness of the family $(\pi_{{\varepsilon}})_{{\varepsilon} \in (0, {\varepsilon}_0]}$ of probability measures. Now take a stationary version $\overline{Y}_{{\varepsilon}}$ of the reflected jump-diffusion $Y_{{\varepsilon}}$ : for every $t \ge 0$ , let $\overline{Y}_{{\varepsilon}}(t) \sim \pi_{{\varepsilon}}$ . Take a sequence $({\varepsilon}_n)_{n \ge 1}$ such that ${\varepsilon}_n \downarrow 0$ as $n \to \infty$ , and $\pi_{{\varepsilon}_n} {\Rightarrow} \pi_0$ (where ${\Rightarrow}$ stands for weak convergence) for some probability measure $\pi_0$ on ${\mathbb{R}}_+$ . It follows from Lemma 8 that, for every $t \ge 0$ , we have $\overline{Y}_{{\varepsilon}_n}(t) {\Rightarrow} \overline{Y}_0(t)$ as $n \to \infty$ , where $\overline{Y}_0$ is a stationary version of the reflected jump-diffusion $Y_0$ , that is, $\overline{Y}_0(t) \sim \pi_0$ for every $t \ge 0$ .

Next, take a measurable function $g \colon {\mathbb{R}}_+ \to {\mathbb{R}}$ such that $|g(x)| \le V_{{\lambda}}(x),\, x \in {\mathbb{R}}_+$ .

Proof. The function $\Phi$ is a supremum of a family of functions $-\phi(\cdot, x)$ , which are continuous in ${\lambda}$ . Thus $\Phi$ is lower semicontinuous, and the set $\{{\lambda} > 0\mid \Phi({\lambda}) > 0\}$ is open. Apply Lemma 7 to some ${\lambda}' > {\lambda}$ (which exists by the observation above). Then we get

\begin{equation*}\varlimsup_{{\varepsilon} \downarrow 0}(\pi_{{\varepsilon}_n}, V_{{\lambda}'}) < \infty.\end{equation*}

Note also that

\begin{equation*}|g(x)|^{{\lambda}'/{\lambda}} \le [V_{{\lambda}}(x)]^{{\lambda}'/{\lambda}} = V_{{\lambda}'}(x)\quad\text{for all $x \ge 0$.}\end{equation*}

Therefore the family $(\pi_{{\varepsilon}}g^{-1})_{{\varepsilon} \in (0, {\varepsilon}_0]}$ of probability distributions is uniformly integrable. Uniform integrability plus a.s. convergence imply convergence of expected values. Thus we complete the proof of Lemma 10.

For all ${\varepsilon} \ge 0$ , take a copy $Y^{{\varepsilon}}$ of $Y_{{\varepsilon}}$ starting from the same initial point $x \in {\mathbb{R}}_+$ .

Proof. Following calculations in the proof of [Reference Sarantsev29, Theorem 3.2], we get

(33) \begin{equation}{\mathbb E} \tilde{V}_{{\lambda}}(Y^{{\varepsilon}}(t)) - \tilde{V}_{{\lambda}}(x) \le \int_0^t\bigl[-\Phi_{{\varepsilon}}({\lambda})\tilde{V}_{{\lambda}}(Y^{{\varepsilon}}(s)) + c_{{\varepsilon}}1_{[0, s_+]}(s)\bigr]\,{\mathrm{d}} s \le c_{{\varepsilon}}t.\end{equation}

Therefore, from (33) we have $\varlimsup_{{\varepsilon}\downarrow 0}{\mathbb E} \tilde{V}_{{\lambda}}(Y^{{\varepsilon}}(t)) < \infty$ . Because of (26), this holds for $V_{{\lambda}}$ in place of $\tilde{V}_{{\lambda}}$ . This is also true for ${\lambda}' > {\lambda}$ slightly larger than ${\lambda}$ . Applying the same uniform integrability argument as in the proof of Lemma 10, we complete the proof of Lemma 11.

Finally, let us complete the proof of Theorem 2. From (23), we have

(34) \begin{equation}|{\mathbb E} g(Y^{{\varepsilon}}(t)) - (\pi_{{\varepsilon}}, g)| \le [V_{{\lambda}}(x) + (\pi_{{\varepsilon}}, V_{{\lambda}})]\,{\mathrm{e}}^{-\Phi_{{\varepsilon}}({\lambda})t}.\end{equation}

Taking ${\varepsilon} = {\varepsilon}_n$ and letting $n \to \infty$ in (34), we use Lemma 10 and 11 to conclude that

\begin{equation*} |{\mathbb E} g(Y^{0}(t)) - (\pi_{0}, g)| \le [V_{{\lambda}}(x) + (\pi_{0}, V_{{\lambda}})]\,{\mathrm{e}}^{-\Phi({\lambda})t}.\end{equation*}

Take the supremum over all functions $g \colon {\mathbb{R}}_+ \to {\mathbb{R}}$ which satisfy $|g(x)| \le V_{{\lambda}}(x)$ for all $x \in {\mathbb{R}}_+$ , and complete the proof of Theorem 2 for Lipschitz $p_*$ .

6. Concluding remarks

We showed that the convergence of ruin probabilities in a rather broad class of risk processes is achieved exponentially fast. This rate is easy to compute (at least in the examples considered in Section 4), and happens to be sharp when the premium rate and its variability are independent of the current wealth of the insurance company.

A natural question concerns the practical implication of having access to the value of the rate of exponential convergence, and in particular whether this leads to an numerical approximation of the finite-time ruin probability. This issue has been discussed by Asmussen [Reference Asmussen2]: the answer was negative. For constant premium rate and diffusion parameter, one may approximate the actual gap between the ruin probabilities using numerical integration techniques based on the paper by Michna, Palmowski, and Pistorius [Reference Michna, Palmowski and Pistorius23].

Another direction is to relax the condition on the tail of the claim size. It is of practical interest to let the claim size distribution be heavy-tailed. An extension of the early work of Asmussen and Teugels [Reference Asmussen and Teugels4] could be envisaged. For example, in the work of Tang [Reference Tang35], a compound Poisson risk model under constant interest force with sub-exponentially distributed claim size is considered. When comparing the asymptotics provided by Tang [Reference Tang35, (2.5), (3.2)], it seems that exponential convergence holds for large initial reserves.