Optimal excess-of-loss reinsurance for stochastic factor risk models

We study the optimal excess-of-loss reinsurance problem when both the intensity of the claims arrival process and the claim size distribution are influenced by an exogenous stochastic factor. We assume that the insurer's surplus is governed by a marked point process with dual-predictable projection affected by an environmental factor and that the insurance company can borrow and invest money at a constant real-valued risk-free interest rate $r$. Our model allows for stochastic risk premia, which take into account risk fluctuations. Using stochastic control theory based on the Hamilton-Jacobi-Bellman equation, we analyze the optimal reinsurance strategy under the criterion of maximizing the expected exponential utility of the terminal wealth. A verification theorem for the value function in terms of classical solutions of a backward partial differential equation is provided. Finally, some numerical results are discussed.


Introduction
In this paper we analyze the optimal excess-of-loss reinsurance problem from the insurer's point of view, under the criterion of maximizing the expected utility of the terminal wealth. It is well known that the reinsurance policies are very effective tools for risk management. In fact, by means of a risk sharing agreement, they allow the insurer to reduce unexpected losses, to stabilize operating results, to increase business capacity and so on. Among the most common arrangements, the proportional and the excess-of-loss contracts are of great interest. The former was intensively studied in [Irgens and Paulsen, 2004], [Liu and Ma, 2009], [Liang et al., 2011], [Liang and Bayraktar, 2014], [Zhu et al., 2015], [Brachetta and Ceci, 2019] and references therein. The latter was investigated in these articles: in [Zhang et al., 2007] and [Meng and Zhang, 2010], the authors proved the optimality of the excess-of-loss policy under the criterion of minimizing the ruin probability, with the surplus process described by a Brownian motion with drift; in [Zhao et al., 2013] the Cramér-Lundberg model is used for the surplus process, with the possibility of investing in a financial market represented by the Heston model; in [Sheng et al., 2014] and [Li and Gu, 2013] the risky asset is described by a Constant Elasticity of Variance (CEV) model, while the surplus is modelled by the Cramér-Lundberg model and its diffusion approximation, respectively; finally, in [Li et al., 2018] the authors studied a robust optimal strategy under the diffusion approximation of the surplus process.
The common ground of the cited works is the underlying risk model, which is the Cramér-Lundberg model (or its diffusion approximation) 1 . In the actuarial literature it is of great importance, because it is simple enough to perform calculations. In fact, the claims arrival process is described by a Poisson process with constant intensity (or a Brownian motion, in the diffusion model). Nevertheless, as noticed by many authors (e.g. [Grandell, 1991], [Hipp, 2004]), it needs generalization in order to take into account the so callled size fluctuations and risk fluctuations, i.e. variations of the number of policyholders and modifications of the underlying risk, respectively.
The main goal of our work is to extend the classical risk model by modelling the claims arrival process as a marked point process with dual-predictable projection affected by an exogenous stochastic process Y . More precisely, both the intensity of the claims arrival process and the claim size distribution are influenced by Y . Thanks to this environmental factor, we achieve a reasonably realistic description of any risk movement. For example, in automobile insurance Y may describe weather conditions, road conditions, traffic volume and so on. All these factors usually influence the accident probability as well as the damage size.
Some noteworthy attempts in that direction can be found in [Liang and Bayraktar, 2014] and [Brachetta and Ceci, 2019], where the authors studied the optimal proportional reinsurance. In the former, the authors considered a Markov-modulated compound Poisson process, with the (unobservable) stochastic factor described by a finite state Markov chain. In the latter, the stochastic factor follows a general diffusion. In addition, in [Brachetta and Ceci, 2019] the insurance and the reinsurance premia are not evaluated by premium calculation principles (see [Young, 2006]), because they are stochastic processes depending on Y . In our paper, we extend further the risk model, because the claim size distribution is influenced by the stochastic factor, which is described by a diffusion-type stochastic differential equation (SDE). In addition, we study a different reinsurance contract, which is the excess-of-loss agreement.
In our model the insurer is also allowed to lend or borrow money at a given interest rate r. During the last years, negative interest rates drew the attention of many authors. For example, since June 2016 the European Central Bank (ECB) fixed a negative Deposit facility rate, which is the interest banks receive for depositing money within the ECB overnight. Nowadays, it is −0.4%. As a consequence, in our framework r ∈ R. We point out that there is no loss of generality due to the absence of a risky asset, because as long as the insurance and the financial markets are independent (which is a standard hypothesis in non-life insurance), the optimal reinsurance strategy turns out to depend only on the risk-free asset (see [Brachetta and Ceci, 2019] and references therein). As a consequence, the optimal investment strategy can be eventually obtained using existing results in the literature.
The paper is organized as follows: in Section 2, we formulate the model assumptions and describe the maximization problem; in Section 3 we derive the Hamilton-Jacobi-Bellman (HJB) equation; in Section 4, we investigate the candidate optimal strategy, which is suggested by the HJB derivation; in Section 5, we provide the verification argument with a probabilistic representation of the value function; finally, in Section 6 we perform some numerical simulations.

Model formulation
Let (Ω, F, P, {F t } t∈[0,T ] ) be a complete probability space endowed with a filtration which satisfies the usual conditions, where T > 0 is the insurer's time horizon. We model the insurance losses through a marked point process {(T n , Z n )} n≥1 with local characteristics influenced by an environment stochastic factor Y .
The stochastic factor Y is defined as the unique strong solution to the following SDE: We will denote by {F Y t } t∈[0,T ] the natural filtration generated by the process Y . The random measure corresponding to the losses process {(T n , Z n )} n≥1 is given by where δ (t,x) denotes the Dirac measure located at point (t, x). We assume that its {F t } t∈[0,T ] -dual predictable projection ν(dt, dz) has the form is a strictly positive measurable function.
In the sequel, we will assume the following integrability conditions: which implies the following: According with the definition of dual predictable projection, for every nonnegative, {F t } t∈[0,T ]predictable and [0, +∞)-indexed process . (2.8) i.e. the claims arrival process N t = m((0, t] × [0, +∞)) = ∑ n≥1 1 {Tn≤t} is a point process with Now we give the interpretation of F (z, Y t ) as conditional distribution of the claim sizes 3 .
Proposition 2.1. ∀n = 1, . . . and ∀A ∈ B([0, +∞)) In particular, this implies that where F T − n is the strict past of the σ-algebra generated by the stopping time T n : This means that in our model both the claim arrival intensity and the claim size distribution are affected by the stochastic factor Y . This is a reasonable assumption; for example, in automobile insurance Y may describe weather, road conditions, traffic volume, and so on. For a detailed discussion of this topic see also [Brachetta and Ceci, 2019].
Moreover, the predictable covariation process of {M t } t∈[0,T ] is given by In this framework we define the cumulative claims up to time t ∈ [0, T ] as follows and the reserve process of the insurance is described by For these results and other related topics see e.g. [Bass, 2004].
where R 0 > 0 is the initial wealth and {c t } t∈[0,T ] is a non negative {F t } t∈[0,T ] -adapted process representing the gross insurance risk premium. In the sequel we assume < +∞. Now we allow the insurer to buy an excess-of-loss reinsurance contract. By means of this agreement, the insurer chooses a retention level α ∈ [0, +∞) and for any future claim the reinsurer is responsible for all the amount which exceeds that threshold α (e.g. α = 0 means full reinsurance). For any dynamic reinsurance strategy {α t } t∈[0,T ] , the insurer's surplus process is given by T ] -adapted process representing the reinsurance premium rate. In addition, we suppose that the following assumption holds true.
Assumption 2.1. (Excess.of-loss reinsurance premium) Let us assume that for any reinsurance strategy {α t } t∈[0,T ] the corresponding reinsurance premium process {q α t } t∈[0,T ] admits the following representation: because the cedant is not allowed to gain a profit without risk.
In the rest of the paper, ∂q(t,y,0) ∂α should be intended as a right derivative.
Assumption 2.1 formalizes the minimal requirements for a process {q α t } t∈[0,T ] to be a reinsurance premium. In the next examples we briefly recall the most famous premium calculation principles, because they are widely used in optimal reinsurance problems solving. In Appendix B the reader can find a rigorous derivation of the following formulas (2.9) and (2.10).
Example 2.1. The most famous premium calculation principle is the expected value principle (abbr. EVP) 5 . The underlying conjecture is that the reinsurer evaluates her premium in order to cover the expected losses plus a load which depends on the expected losses. In our framework, under the EVP the reinsurance premium is given by the following expression: for some safety loading θ > 0.
Example 2.2. Another important premium calculation principle is the variance premium principle (abbr. VP). In this case, the reinsurer's loading is proportional to the variance of the losses. More formally, the reinsurance premium admits the following representation: for some safety loading θ > 0.
From now on we assume the following condition: Furthermore, the insurer can lend or borrow money at a fixed interest rate r ∈ R. More precisely, every time the surplus is positive, the insurer lends it and earns interest income if r > 0 (or pays interest expense if r < 0); on the contrary, when the surplus becomes negative, the insurer borrows money and pays interest expense (or gains interest income if r < 0).
Under these assumptions, the total wealth dynamic associated with a given strategy α is described by the following SDE: It can be verified that the solution to (2.12) is given by the following expression: Our aim is to find the optimal strategy α in order to maximize the expected exponential utility of the terminal wealth, that is where η > 0 is the risk-aversion parameter and A is the set of all admissible strategies as defined below.
Definition 2.1. We denote by A the set of all admissible strategies, that is the class of all non negative {F t } t∈[0,T ] -predictable processes α t . With the notation A t we refer to the same class, restricted to the strategies starting from t ∈ [0, T ].
As usual in stochastic control problems, we focus on the corresponding dynamic problem:

HJB formulation
In order to solve the optimization problem (2.14), we introduce the value function v : This function is expected to solve the Hamilton-Jacobi-Bellman (HJB) equation: where L α denotes the Markov generator of the couple (X α t , Y t ) associated with a constant control α. In what follows, we denote by C 1,2 b the class of all bounded functions f (t, x 1 , . . . , x n ), with n ≥ 1, with bounded first order derivatives ∂f ∂t , ∂f ∂x1 , . . . , ∂f ∂xn and bounded second order derivatives with respect to the spatial variables ∂ 2 f The Markov generator of the stochastic process (X α t , Y t ) for all constant strategies α ∈ [0, +∞) is given by the following expression: Proof. For any f ∈ C 1,2 b , applying Itô's formula to the stochastic process f (t, X α t , Y t ), we get the following expression: where L α is defined in (3.3) and ) .
In order to complete the proof, we have to show that For the first term, we observe that because the partial derivative is bounded and using the assumption (2.2). For the second term, it is sufficient to use the boundedness of f and the condition (2.6).
Remark 3.1. Since the couple (X α t , Y t ) is a Markov process, any Markovian control is of the form α t = α(t, X α t , Y t ), where α(t, x, y) denotes a suitable function. The generator L α f (t, x, y) associated to a general Markovian strategy can be easily obtained by replacing α with α t in (3.3).
In order to simplify our optimization problem, we present a preliminary result.
By Remark 3.2, taking g(z) = e ηze r(T −t) − 1, we can rewrite the last integral in this more convenient way: Now we define Ψ α (t, y) by means of the equation (3.6), obtaining the following equivalent expression: Taking the infimum over α ∈ [0, +∞), by (3.5) we find out the PDE in (3.2). The terminal condition in (3.2) immediately follows by definition.
The previous result suggests to focus on the minimization of the function (3.6), that is the aim of the next section.
By the previous proposition, we observe that λ(t, y) is an important threshold for the insurer: as long as the marginal cost of the full reinsurance falls in the interval (0, λ(t, y)], the optimal choice is full reinsurance.
Unfortunately, it is not always easy to check whether Ψ α (t, y) is strictly convex in α ∈ [0, +∞) or not. In the next result such an hypothesis is relaxed, while the uniqueness of the solution to (4.3) is required.
The next result deals with the existence of a solution to (4.3). In particular, it is sufficient to require that the claim size distribution is heavy-tailed, which is a relevant case in non-life insurance (see [Rolski et al., 1999, Chapter 2]), plus a technical condition for the reinsurance premium.
Proof. The following property of heavy-tailed distributions is a well known implication of our assumption: lim z→+∞ e kzF (z, y) = +∞ ∀k > 0, y ∈ R.
Hence, by equation (4.5), for any (t, y) On the other hand, we know that As a consequence, ∂Ψ α (t,y) ∂α being continuous in α ∈ [0, +∞), there existsα(t, y) ∈ (0, +∞) such that ∂Ψα(t,y) ∂α = 0. Now we turn the attention to the other crucial hypothesis of Proposition 4.1, which is the convexity of Ψ α (t, y). The reader can easily observe that the reinsurance premium convexity plays a central role.
Proof. Recalling the expression (3.6), it is sufficient to prove the convexity of the following term: For this purpose, let us evaluate its second order derivative: ) .

Now the term in brackets is
The proof is complete.
By Proposition 2.1, the hypothesis on the claim sizes distribution above may be read as assuming that the claims are exponentially distributed conditionally to Y .

Expected value principle
Now we investigate the special case of the expected value principle introduced in Example 2.1.
Proposition 4.5. Under the EVP (see equation (2.9)), the optimal reinsurance strategy α * (t) ∈ [0, +∞) is given by Proof. Using Remark 3.2, we can rewrite the equation (2.9) as follows: As a consequence, we have that For α = 0, we have that hence A 0 = ∅ and by Proposition 4.1 the minimizer belongs to (0, +∞). Now we look for the stationary points, i.e. the solutions to the equation (4.3), that in this case reads as follows: (1 + θ)λ(t, y)F (α, y) = λ(t, y)e ηαe r(T −t)F (α, y). (4.8) Solving this equation, we obtain the unique solution given by (4.7). In order to prove that it coincides with the unique minimizer to (4.1), it is sufficient to show that For this purpose, observe that The proof is complete.
Remark 4.1. Formula (4.7) was found by [Zhao et al., 2013] (see equation 3.31, page 508). We point out that it is a completely deterministic strategy. This fact is crucially related to the use of the EVP rather than the underlying model; in fact, in [Zhao et al., 2013] the authors considered the Cramér-Lundberg model under the EVP 7 .
From the economic point of view, by equation (4.7) it is easy to show that the optimal retention level is decreasing with respect to the interest rate and the risk-aversion; on the contrary, it is increasing with respect to the reinsurer's safety loading. In addition, the sensitivity with respect to the time-to-maturity depends on the sign of r.
Another relevant aspect of (4.7) is that it is independent of the claim size distribution. To the authors this result seems quite unrealistic. In fact, any subscriber of an excess-of-loss contract is strongly worried about possibly extreme events, hence the claims distribution is expected to play an important role.

Variance premium principle
This subsection is devoted to derive an optimal strategy under the variance premium principle (see Example 2.2).
Under the VP (see equation (2.10)) the optimal reinsurance strategy α * (t, y) is the unique solution to the following equation: (4.10) Proof. The proof is based on Proposition (4.1). By equation (2.10) we get its derivative: It is clear that the set A 0 defined in (4.2) is empty, because for any (t, y) Hence the minimizer should coincide with the unique stationary point of Ψ α (t, y), i.e. the solution to (4.10). In order to prove it, we need to ensure the existence of a solution to (4.10). For this purpose, we notice that on the one hand ∂Ψ 0 (t, y) ∂α = −2θλ(t, y) On the other hand, for α → +∞, by (4.9) we get As a consequence, by the continuity of Ψ α (t, y) there exists a point α * ∈ (0, +∞) such that ∂Ψ α * (t,y) ∂α = 0. Such a solution is unique because Ψ α (t, y) is strictly convex by hypothesis.
Conversely to Proposition 4.5, the optimal retention level given in Proposition 4.6 is still dependent on the stochastic factor Y . Such a dependence is spread through the claim size distribution.
Remark 4.2. We observe that any heavy-tailed distribution (see the proof of Proposition 4.3) satisfies the condition (4.9) with l = +∞. Now we specialize the variance premium principle to conditionally exponentially distributed claims.
Contrary to the equation (4.7), the explicit formula (4.11) keeps the dependence on the stochastic factor Y . In addition, the following result holds true.
Remark 4.3. Suppose that F (z, y) = (1 − e −ζ(y)z )1 {z>0} for some function ζ(y) such that ζ(y) > 0 ∀y ∈ R. We consider two different reinsurance safety loadings θ EVP , θ VP > 0, referring to the EVP and VP, respectively. Moreover, let us denote by α * EVP (t) and α * VP (t, y) the optimal retention level under the EVP and VP, given in equations (4.7) and (4.11), respectively. It is easy to show that ∀t ∈ [0, T ] From the practical point of view, as long as the stochastic factor fluctuations result in a rate parameter ζ(y) higher than the threshold 2θVP θEVP , the optimal retention level evaluated through the expected value principle turns out to be larger than the variance principle.
= α * (t, Y t ) described in Proposition 4.1 is an optimal control. Proof. By Proposition 3.1, the function v(t, x, y) defined in equation (3.7) solves the HJB problem (3.2). Hence for any (t, x, y where {X α t,x (s)} s∈[t,T ] and {Y t,y (s)} s∈[t,T ] denote the solutions to (2.12) and (2.1) at time s ∈ [t, T ], starting from (t, x) ∈ [0, T ] × R and (t, y) ∈ [0, T ] × R, respectively. From Itô's formula we get v(T, X α t,x (T ), Y t,y (T )) = v(t, x, y) + The reader can easily check that {τ n } n∈N is a non decreasing sequence of stopping time such that lim n→+∞ τ n = +∞. For the diffusion term of M r , using the assumptions (5.1) and (2.2), we notice that where C n > 0 is a constant depending on n. For the jump term, by the condition (2.7) and Remark 2.1, we get withC n denoting a positive constant dependent on n. Thus {M r } r∈[t,T ] turns out to be an {F t } t∈[0,T ] -local-martingale and {τ n } n∈N is a localizing sequence for it. Now, taking the expectation of (5.2) with T ∧ τ n in place of T , we obtain that whereC > 0 is a constant. As a consequence, {v(T ∧ τ n , X α t,x (T ∧ τ n ), Y t,y (T ∧ τ n ))} n∈N is a sequence of uniformly integrable random variables. By classical results in probability theory, it converges almost surely. Using the monotonicity and the boundedness of {τ n } n∈N , together with the non explosion of {X α t,x (s)} s∈[t,T ] and {Y t,y (s)} s∈[t,T ] (see (2.13) and (2.3)), taking the limit for n → +∞ we conclude that As a byproduct, since α * (t, y) given in Proposition 4.1 realizes the infimum in (4.1), we have that L α * v(t, x, y) = 0 and, replicating the calculations above, we obtain the equality i.e. α * t .
= α * (t, Y t ) is an optimal control. By Theorem 5.1, the value function (3.1) can be characterized as a transformation of the solution to the partial differential equation (PDE) (3.5). Nevertheless, an explicit expression is not available, except for very special cases. The following result provides a probabilistic representation by means of the Feynman-Kac theorem.
to the Cauchy problem (3.5), such that the condition (5.1) is fulfilled. Then the value function (3.1) admits the following representation: where Ψ α (t, y) is the function defined in (3.6).
Proof. The thesis immediately follows by Theorem 5.1 and the Feynman-Kac representation of φ(t, y).
Remark 5.1. We refer to [Heath and Schweizer, 2000] for existence and uniqueness of a solution to the PDE (3.5).
The parameters are set according to Table 1 below. Parameter Value The SDEs are approximated through a classical Euler's scheme with steps length T N , while the expectations are evaluated by means of Monte Carlo simulations with parameter M .
In Figure 1 we show the dynamic strategies under EVP and VP, computed by the equations (4.7) and (4.11), respectively. In Figure 2 we start the sensitivity analysis investigating the effect of the risk aversion parameter on the optimal strategy at time t = 0. As expected, there is an inverse relationship. Notice that for high values of η the two strategies tend to the same level.   In Figure 4 we observe that the distance between the retention levels in the two cases is larger when r < 0 and it decreases as long as r increases. Nevertheless, even for positive values of the risk-free interest rate the distance is not negligible (see the pictures above, with r = 0.05). In Figure 5 we study the response of the optimal strategy to variations of the time horizon. The two cases exhibit the same behavior, which is strongly influenced by the sign of the interest rate. In fact, if r < 0 the retention level increases with the time horizon, while if r > 0 the optimal strategy decreases with T . Finally, thanks to Proposition 5.1 we are able to numerically approximate the value function by simulating the trajectories of Y . The graphical result (under VP) is shown in Figure 6 below. Denoting by ⟨M α ⟩ t the predictable covariance process of M α t , using Remark 2.1 we finally obtain var[C α t ] = E[⟨M α ⟩ t ] + var ] .
Extending this formula to the model formulated in Section 2, we obtain the expression (2.10). Of course, there will be an approximation error, because in our general model the intensity and the claim size distribution depend on the stochastic factor. Nevertheless, this is a common procedure in the actuarial literature.