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# Optimal Robust Reinsurance with Multiple Insurers

Emma Kroell  Department of Statistical Sciences, University of Toronto
<EMAIL_ADDRESS>Sebastian Jaimungal  Department of Statistical
Sciences, University of Toronto<EMAIL_ADDRESS>Silvana M.
Pesenti  Department of Statistical Sciences, University of Toronto
<EMAIL_ADDRESS>
###### Abstract

We study a reinsurer who faces multiple sources of model uncertainty. The
reinsurer offers contracts to $n$ insurers whose claims follow different
compound Poisson processes. As the reinsurer is uncertain about the insurers’
claim severity distributions and frequencies, they design reinsurance
contracts that maximise their expected wealth subject to an entropy penalty.
Insurers meanwhile seek to maximise their expected utility without ambiguity.
We solve this continuous-time Stackelberg game for general reinsurance
contracts and find that the reinsurer prices under a distortion of the
barycentre of the insurers’ models. We apply our results to proportional
reinsurance and excess-of-loss reinsurance contracts, and illustrate the
solutions numerically. Furthermore, we solve the related problem where the
reinsurer maximises, still under ambiguity, their expected utility and compare
the solutions.

Keywords— Stackelberg game, reinsurance, excess-of-loss reinsurance,
barycentre, Kullback-Leibler divergence

##  1 Introduction

Reinsurance is an integral part of the insurance industry, allowing smaller,
first-line insurers to take on a larger client base while passing on large
risks to reinsurance companies. Therefore, reinsurance companies play an
important stabilising role in financial systems — but they remain vulnerable
to events that cause large, industry-wide losses, such as natural disasters.
The cost of these events is increasing, with 100B USD in insured losses from
natural catastrophes in 2022.111https://www.swissre.com/press-release/Insured-
losses-from-natural-catastrophes-break-through-USD-100-billion… Furthermore,
while the reinsurance industry is large, it is also concentrated — reinsurance
premia of the top 50 reinsurance players totalled 353B USD in 2021, with 24%
underwritten by the top
two.222https://www.insurancejournal.com/news/international/2022/08/17/680847.htm
Given these facts, we examine the pricing and risk-management behaviour of a
large reinsurance company who contracts with many different first-line
insurance companies but exhibits different degrees of ambiguities for each
insurance company. The insurers are insuring similar risk but have different
models for losses. We approach this problem as a Stackelberg game and examine
the optimal behaviour of the large reinsurer and the individual insurance
companies.

The problem of optimal reinsurance has been studied extensively in the one-
period setting, dating back to [7]. We focus on optimal reinsurance in
continuous time, where the literature has typically taken the insurer’s
perspective. Examples include [14], [17], and [20] who looked at an insurer
purchasing reinsurance to maximise their wealth or utility and [24, 25] who
consider an insurer minimising their probability of ruin. Recent studies
allowed for ambiguity-aversion of the insurer ([26]; [19]). However, these
studies leave open the question of whether the reinsurer would accept the
individual insurer’s optimal contract. Recent years have seen considerable
work aimed at closing this gap by modelling the behaviour of the reinsurer as
a continuous Stackelberg game. This was first introduced by [11], who solve
the optimal proportional reinsurance problem when the insurer and reinsurer’s
surpluses follow diffusion processes and both agents maximise exponential
utility functions. [12] consider the mean-variance problem for a general
reinsurance treaty, and find that proportional reinsurance is optimal under a
variance premium while excess-of-loss reinsurance is optimal with an expected
value premium.

Ambiguity-aversion in the Stackelberg game setting is introduced by [15] for
proportional reinsurance contracts, where the insurer’s process is
approximated by a diffusion. They allow for uncertainty in the drift of the
insurer’s claims process, and show that the ambiguity-averse reinsurer expects
higher claims and sets higher reinsurance prices, which decreases the demand
for reinsurance. In the Cramér-Lundberg setting, [16] allow for ambiguity of
the reinsurer about the intensity of the insurer’s compound Poisson process,
and show that the ambiguity-averse reinsurer expects a higher intensity of the
compound Poisson process. Furthermore, for exponential claim distributions,
they show that this causes the reinsurer to charge a higher premium, and the
insurer to purchase less reinsurance for both proportional and excess-of-loss
reinsurance contracts. The framework is further extended by [13], who study
the excess-of-loss reinsurance problem and allow for investment in a risky
asset by both the insurer and reinsurer, and show that the behaviour of the
agents at equilibrium depends on each other’s ambiguity aversions. Most
recently, [8, 9] study the related expected wealth reinsurance problem with
squared deviation and general divergence ambiguity penalties, respectively.
They show that the insurer’s and reinsurer’s probability distortions are
increasing in the loss, and that proportional reinsurance is optimal under
squared-error divergence, but not necessarily under a general divergence
measure.

While the optimal reinsurance problem with more than a single insurer or
reinsurer has been well-studied in the one-period setting (see, e.g., [6] and
[4, 5]), to our knowledge, its study in continuous time is limited. An
exception is [10], who consider the reinsurance problem with multiple
reinsurers, and compare different configurations of reinsurance agreements.
[3] consider the case of two competing insurers in a hybrid Stackelberg game,
who maximise their expected wealth with delay and show that their competition
reduces their demand for reinsurance.

This work is distinct from the aforementioned literature in several ways.
First, our focus is on the behaviour of an ambiguity-averse reinsurer who
takes on potentially different contracts from _multiple_ insurers, each having
different loss models. The reinsurer allows for distorting both the intensity
and severity of the loss distributions, and moreover accounts for different
degrees of ambiguity for each insurer’s loss models using a weighted Kullback-
Leibler divergence penalty. Second, rather than considering a specific
reinsurance contract, we solve the Stackelberg game for a general reinsurance
contract, encompassing proportional and excess-of-loss reinsurance. Our
reinsurance contracts take the form of a _retention function_
$r(z,\alpha_{k})$, where $z$ is the loss and $\alpha_{k}$ is an aspect of the
contract chosen by the insurer.333For an overview of reinsurance forms, see,
e.g., [1, Chapter 2] or [2, Chapter I.4] We consider a broad class of
retention functions which include most common reinsurance contracts, including
proportional reinsurance and excess-of-loss reinsurance, with or without a
policy limit. Third, we compare the wealth- and exponential utility-
maximisation cases, bringing together these two distinct approaches in the
literature.

We derive the Stackelberg equilibrium for a general reinsurance contract, and
find that the reinsurer’s optimal pricing model is closely related to the
barycentre of the insurers’ models. We show that the optimal reinsurance
prices, demand, and the reinsurer’s pricing model are independent of time.
Furthermore, we show that when the reinsurer is ambiguity-averse, their
optimal pricing model depends on the size of their aggregate loss. We further
consider the Stackelberg equilibrium for specific reinsurance contracts —
proportional reinsurance and excess-of-loss reinsurance, with or without a
policy limit — and illustrate the solution numerically in the case of two
insurers. Finally, we compare our solution to an ambiguity-averse reinsurer
who maximises a CARA utility, thus generalising known results in the
literature to account for ambiguity. We find that the safety loadings charged
to the insurers are increasing in the reinsurer’s risk aversion parameter.

The rest of the paper is structured as follows. Section 2 details the wealth
evolution of the insurers and the reinsurer and defines the general
reinsurance contract. Section 3 provides the solution to the insurer’s
problem. Section 4 details and solves the reinsurer’s optimisation problem,
and presents the Stackelberg equilibrium. In Section 5, we examine the
Stackelberg equilibrium for different reinsurance contract types, namely,
proportional reinsurance and excess-of-loss reinsurance with and without a
policy limit. We provide numerical illustrations of the equilibrium and its
key features for the case of two insurers with given severity distributions.
Finally, in Section 6 we solve the related utility-maximisation problem for
the reinsurer under ambiguity, and compare it to the wealth-maximisation
approach.

##  2 Reinsurance market

We consider an reinsurance market with $n\in\mathbb{N}$ non-life insurance
companies and a single reinsurer over a finite time horizon $[0,T]$. Each
insurer provides non-life insurance policies to policyholders and receives
regular premium payments. They further aim to transfer a part of their
potential losses to a reinsurance company. Each insurer decides their optimal
level of reinsurance based on the reinsurance price and the contract terms,
with the goal of maximising their expected utility. On the other hand, the
reinsurer determines the prices at which they offer reinsurance to each
individual insurer, aiming to optimise their expected wealth subject to model
uncertainty. We formalise this as a Stackelberg game, where the reinsurer is
the leader and the insurers are the followers.

We consider a complete and filtered measurable space
$(\Omega,{\mathcal{F}},\mathbb{F}=({\mathcal{F}}_{t})_{t\in[0,T]})$ and $n$
equivalent probability measures ${\mathbb{P}}_{1},\ldots,{\mathbb{P}}_{n}$.
For $k\in{\mathcal{N}}:=\\{1,\ldots,n\\}$, the probability measure
${\mathbb{P}}_{k}$ encapsulates the $k$-th insurer’s beliefs about their
losses. We refer to an individual insurer as insurer-$k$ or $k$-th insurer,
$k\in{\mathcal{N}}$. Throughout, unless stated otherwise, we assume the
filtration $\mathbb{F}$ is such that all processes defined below are
progressively measurable with respect to it.

###  2.1 Insurers’ wealth evolution

We suppose that each insurer’s loss process follows a Cramér-Lundberg model.
Thus, the claims of insurer-$k$ arrive according to a Poisson process with
constant intensity $\lambda_{k}\in{\mathds{R}}_{+}:=(0,\infty)$ and the claim
severity has a cumulative distribution function $F_{k}(\cdot)$ with support on
${\mathds{R}}_{+}$ and mean $\mu_{k}\in{\mathds{R}}_{+}$. The insurer sets a
constant premium rate $p_{k}^{I}$; for example the expected value principle
with fixed safety loading $\theta_{k}>0$, i.e.
$p_{k}^{I}=(1+\theta_{k})\,\mu_{k}\,\lambda_{k}$. Each insurer purchases
reinsurance on their aggregate claim amount. The available reinsurance
contracts are characterised by a retention function $r$. Insurer-$k$ chooses
an aspect of the reinsurance contract
$a_{k}\in{\mathcal{A}}\subseteq{\mathds{R}}$, which we refer to as
insurers-$k$’s control. The interpretation of $a_{k}$ and the set of possible
controls ${\mathcal{A}}$ will vary depending on the form of the retention
function, as shown in the examples below. For a choice of retention function
$r$ and insurer-$k$’s control $a_{k}$, the reinsurer charges a reinsurance
premium $p^{R}$ with safety loading $c_{k}$.

######  Definition 2.1 (Reinsurance Contract).

A reinsurance contract is characterised by a retention function
$r\colon{\mathds{R}}_{+}\times{\mathcal{A}}\to{\mathds{R}}_{+}$, which is non-
decreasing in the first argument, satisfying $0<r(z,a)\leq z$, for all
$z\in{\mathds{R}}_{+}$, $a\in{\mathcal{A}}$, and a corresponding reinsurance
premium $p^{R}\colon{\mathcal{A}}\times{\mathds{R}}_{+}\to{\mathds{R}}_{+}$.

For a tuple $(a,c)\in{\mathcal{A}}\times{\mathds{R}}_{+}$, the reinsurer
agrees to cover $z-r(z,a)$ for a premium $p^{R}(a,c)$, where $c$ is the
reinsurer’s safety loading.

The bounds on the retention function guarantee that the insurer must retain
some of the losses. Typically the inequality, $r(z,a)\leq z$, is strict for
some $z\in{\mathds{R}}_{+}$.

######  Assumption 2.2.

We consider retention functions that are continuous in the loss $z$ and
increasing and almost everywhere differentiable in the control $a$.

Of particular interest are piecewise linear retention functions, as these are
the reinsurance contracts encountered in practice. Examples of such retention
functions include the following:

######  Example 2.3.

Proportional reinsurance: $r(z,a)=a\hskip 1.00006ptz$, $a\in(0,1]$. Here, the
insurer chooses the proportion of the loss to retain.

######  Example 2.4.

Excess-of-loss insurance: $r(z,a)=\min\\{a,\,z\\}$, $a\in{\mathds{R}}_{+}$.
Here, the insurer chooses the retention limit, beyond which the reinsurer
covers any excess losses.

######  Example 2.5.

Excess-of-loss insurance with a fixed policy limit $\ell$:
$r(z,a)=\min\\{a,z\\}\,{\mathds{1}}_{z\leq
a+\ell}+(z-\ell)\,{\mathds{1}}_{z>a+\ell}$, $a\in{\mathds{R}}_{+}$, $\ell>0$.
Here, the insurer chooses the retention limit, beyond which the reinsurer
covers any excess losses. If however the losses exceed the limit $a+\ell$, the
losses are borne by the insurer. We assume that the insurer cannot choose
$\ell$ as the policy limit is in practice set by the reinsurer, e.g. by the
purchase of further reinsurance (see, e.g., [22, Chapter 3.4]).

The insurer’s control of the retention function may vary over time, thus we
denote by $\alpha_{k}:=(\alpha_{t,k})_{t\in[0,T]}$ the insurer-$k$’s control.
Further, we denote by ${\mathfrak{A}}$ the set of square-integrable,
$\mathbb{F}$-predictable processes taking values in ${\mathcal{A}}$. We
require that $\alpha_{k}\in{\mathfrak{A}}$. Thus, the $k$-th insurers’ wealth
process $X_{k}:=(X_{t,k})_{t\in[0,T]}$ evolves according to the stochastic
differential equation (SDE)

$X_{t,k}=X_{0,k}+\int_{0}^{t}\\!\\!\left[p_{k}^{I}-p^{R}_{k}(\alpha_{u,k},c_{k})\right]du-\int_{0}^{t}\int_{0}^{\infty}r(z,\alpha_{u,k})N(dz,du)\,,$
(1)

where $p_{k}^{R}$ is the reinsurance premium for insurer-$k$ with safety
loading $c_{k}\in{\mathds{R}}_{+}$. The first integral is the difference of
the premia the insurer collects from their policyholders and the premium they
pay to the reinsurer. In the second integral, $N(dz,dt)$ is a Poisson random
measure (PRM) and insurer-$k$ uses a probability measure ${\mathbb{P}}_{k}$
such that $N(dz,dt)$ has ${\mathbb{P}}_{k}$-compensator

$\nu_{k}(dz,dt)=\lambda_{k}\,F_{k}(dz)\,dt.$ (2)

That is, $\int_{0}^{t}\int_{0}^{\infty}z\left[N(dz,du)-\nu_{k}(dz,du)\right]$
is a ${\mathbb{P}}_{k}$-martingale. We assume the compensator admits a density
$v_{k}(z)$ such that $\nu_{k}(dz,dt)=v_{k}(z)\,dz\,dt$.

###  2.2 Reinsurer’s wealth evolution

The reinsurer offers reinsurance contracts to each insurer, where they cover
the losses not retained by insurer-$k$, i.e., $z-r(z,a_{k})$, and charge a
continuous premium rate $p^{R}_{k}(a_{k},c_{k})$. As the reinsurer knows how
each insurer will respond to the price, their premium depends on the insurer’s
control $\alpha_{k}$. We assume that the reinsurer sets the reinsurance
premium rate for insurer-$k$ using the expected value principle with
deterministic safety loading $\eta_{k}:=(\eta_{t,k})_{t\in[0,T]}$, so the
premium charged to insurer-$k$ is

$p^{R}_{t,k}=p^{R}_{k}(\alpha_{k},\eta_{t,k}):=(1+\eta_{t,k})\,\lambda_{k}\int_{0}^{\infty}\\!\\!\\!\left[z-r(z,\alpha_{t,k})\right]F_{k}(dz)\,.$

We further assume that $\eta:=(\eta_{k})_{k\in{\mathcal{N}}}$ is restricted to
the space ${\mathfrak{C}}$ consisting of non-negative square integrable
deterministic processes. Summing over the premia charged to each insurer and
the losses taken on from each insurer, the reinsurer’s wealth process
$Y:=(Y_{t})_{t\in[0,T]}$ evolves according to the SDE

$Y_{t}=Y_{0}+\sum_{k\in{\mathcal{N}}}\int_{0}^{t}p^{R}_{k}(\alpha_{u,k},\eta_{u,k})\,du-\int_{0}^{t}\int_{0}^{\infty}L(z,{\boldsymbol{\alpha}}_{u})\;N(dz,du)\,,$
(3)

where for ${\boldsymbol{\alpha}}_{u}:=(\alpha_{u,1},\ldots,\alpha_{u,n})$, an
$n$-dimensional vector taking values in ${\mathcal{A}}^{n}$, the aggregate
total loss of the reinsurer is

$L(z,{\boldsymbol{\alpha}}_{u}):=\sum_{k\in{\mathcal{N}}}\,\big{[}z-r(z,\alpha_{u,k})\big{]}\,.$
(4)

With the evolution of the wealth of the insurers and reinsurer at hand, we
next discuss the optimal strategy of each individual insurer, who maximises
their utility.

##  3 Insurers’ utility maximisation strategy

Each insurer has their own model for their losses, given by the compensating
measure $\nu_{k}(dz,dt)=\lambda_{k}F_{k}(dz)dt$. One may view this as follows:
each insurer has a model ${\mathbb{P}}_{k}$ for the losses such that the
compensating measure for $N(dz,dt)$ under ${\mathbb{P}}_{k}$ is
$\nu_{k}(dz,dt)$. Taking this viewpoint, we write ${\mathbb{P}}_{k}$ for the
probability measure associated to insurer-$k$’s beliefs. Given their beliefs
about the loss, each insurer seeks to maximise their expected utility. We
assume that insurer-$k$ has an exponential, or constant absolute risk aversion
(CARA), utility function $u_{k}(x)=-\frac{1}{\gamma_{k}}e^{-\gamma_{k}x}$ with
risk aversion parameter $\gamma_{k}>0$.

######  Optimization Problem 3.1.

Insurer-$k$ seeks the contract parameter that attains the supremum

$\sup_{\alpha_{k}\in{\mathfrak{A}}}{\mathbb{E}}^{{\mathbb{P}}_{k}}\left[\,u_{k}(X_{T,k})\,\right]\,,$
($P_{I}$)

where ${\mathbb{E}}^{{\mathbb{P}}_{k}}[\cdot]$ denotes the expected value
under ${\mathbb{P}}_{k}$.

The proposition below characterises the optimal choice of the insurer’s
control $\alpha_{k}$. The condition may appear strong, however, as far as we
are aware, the conditions are satisfied by all examples in the extant
literature.

######  Proposition 3.2.

For $k\in{\mathcal{N}}$, $c\in{\mathds{R}}_{+}$, and a retention function $r$,
consider the following non-linear equation for $a\in{\mathcal{A}}$

$\int_{0}^{\infty}\partial_{a}r(z,a)\left\\{(1+c)-e^{\gamma_{k}r(z,a)}\right\\}F_{k}(dz)=0\,$
(5)

and denote by $\alpha_{k}^{\dagger}[c]$ its solution, if it exists. If
furthermore,

$\int_{0}^{\infty}\partial_{a}^{2}r(z,a)\Big{|}_{a=\alpha_{k}^{\dagger}[c]}\left((1+c)-e^{\gamma_{k}r(z,\alpha_{k}^{\dagger}[c])}\right)\;F_{k}(dz)\leq
0\,,\quad\forall\;t\in[0,T]\,,$ (6)

then the process $\alpha_{t,k}^{*}:=\alpha^{\dagger}_{k}[\eta_{t,k}]$,
$t\in[0,T]$, is insurer-$k$’s optimal control in feedback form of problem
($P_{I}$).

###### Proof.

Insurer-$k$’s value function is

$J_{k}(t,x):=\sup_{\alpha_{k}\in{\mathfrak{A}}}{\mathbb{E}}^{{\mathbb{P}}_{k}}_{t,x}\left[-\tfrac{1}{\gamma_{k}}e^{-\gamma_{k}X_{T,k}}\right],$

where ${\mathbb{E}}^{{\mathbb{P}}_{k}}_{t,x}$ denotes the conditional
${\mathbb{P}}_{k}$-expectation given that the insurer’s wealth process
satisfies $X_{t-,k}=x$. The dynamic programming principle gives that the value
function satisfies the following Hamilton-Jacobi-Bellman (HJB) equation

$\displaystyle 0$
$\displaystyle=\partial_{t}J_{k}(t,x)+\sup_{a_{k}\in{\mathcal{A}}}\Bigg{\\{}\Big{[}p_{k}^{I}-(1+c)$
$\displaystyle\,\lambda_{k}\int_{0}^{\infty}\left[z-r(z,a_{k})\right]F_{k}(dz)\Big{]}\,\partial_{x}J_{k}(t,x)$
$\displaystyle+\int_{0}^{\infty}\left[J_{k}(t,x-r(z,a_{k}))-J_{k}(t,x)\right]\,\nu_{k}(dz,dt)\Bigg{\\}},$
$\displaystyle J_{k}(T,x)$
$\displaystyle=-\tfrac{1}{\gamma_{k}}\,e^{-\gamma_{k}\,x}.$

Given the functional form of the terminal condition, we make the Ansatz

$J_{k}(t,x)=-\tfrac{1}{\gamma_{k}}\,e^{-\gamma_{k}\,[f(t)+g(t)x]}\,,$ (7)

for deterministic functions $f(t)$ and $g(t)$ satisfying terminal conditions
$f(T)=0$ and $g(T)=1$, respectively. Upon substituting (7) into the HJB
equation, and after simplifying, we obtain

$\displaystyle
0=f^{\prime}(t)+\gamma_{k}\,g^{\prime}(t)x+\sup_{a_{k}\in{\mathcal{A}}}\Bigg{\\{}\gamma_{k}\,g(t)\Big{[}p^{I}_{k}-(1+c)$
$\displaystyle\,\lambda_{k}\int_{0}^{\infty}\left[z-r(z,a_{k})\right]\,F_{k}(dz)\Big{]}$
(8) $\displaystyle-$
$\displaystyle\,\lambda_{k}\int_{0}^{\infty}\left[e^{\gamma_{k}g(t)r(z,a_{k})}-1\right]F_{k}(dz)\Bigg{\\}}\,.$

As the above must hold for all $x$, it follows that $g(t)\equiv 1$, for all
$t\in[0,T]$. The first order conditions (in $a_{k}$) imply that the supremum
satisfies Equation 5. Furthermore the condition in Equation 6 ensures that
$(\alpha_{t,k}^{*})_{t\in[0,T]}$ is indeed a local maximiser. ∎

Furthermore, Proposition 3.2 implies that the insurer reduces their demand for
reinsurance (retains more of the aggregate loss) as the price increases.

######  Corollary 3.3.

The insurer’s optimal response to the reinsurer’s price,
$\alpha_{k}^{\dagger}[c]$, is an increasing function of the price $c$.

###### Proof.

Taking the derivative of Equation 5 gives that

$\displaystyle\frac{\partial\alpha_{k}}{\partial c}$
$\displaystyle=\frac{\int_{0}^{\infty}\partial_{a}r(z,a)F_{k}(dz)}{\gamma_{k}\int_{0}^{\infty}\left(\partial_{a}r(z,a)\right)^{2}e^{\gamma_{k}r(z,a)}F_{k}(dz)-\int_{0}^{\infty}\partial_{a}^{2}r(z,a)\left((1+c)-e^{\gamma_{k}r(z,a)}\right)\;F_{k}(dz)},$

which is positive by Equation 6 and the fact that the retention function $r$
is non-decreasing in $a$. ∎

Next, we examine the insurer’s optimal response when the reinsurance contract
is of a specific type. For proportional reinsurance and excess-of-loss
reinsurance, we recover the results of [16], where these problems are
considered separately. Furthermore, if the insurer chooses the retention level
of an excess-of-loss reinsurance contract, then Equation 5 in Proposition 3.2
simplifies to an explicit expression for $\alpha^{*}_{k}$. This is in contrast
to proportional reinsurance, where this is not the case.

######  Example 3.4 (Proportional reinsurance).

In the case of proportional reinsurance (Example 2.3), the non-linear equation
for $a\in{\mathcal{A}}$ in Proposition 3.2 simplifies to

$(1+c)\,\mu_{k}={\mathbb{E}}^{{\mathbb{P}}_{k}}\left[Z\,e^{\gamma_{k}aZ}\right]\,,$
(9)

where $Z$ is a random variable with ${\mathbb{P}}_{k}$-distribution function
$F_{k}(\cdot)$. If a solution $\alpha_{k}^{\dagger}[c]$ to the equation
exists, then it is insurer-$k$’s optimal control. Furthermore, it is a global
maximiser, since for proportional reinsurance, the expression in the curly
braces in Equation 8 is strictly convex in $a$.

######  Example 3.5 (Excess-of-loss reinsurance).

In the case of excess-of-loss reinsurance (Example 2.4), we have
$\partial_{a}r(z,a)={\mathds{1}}_{\\{a\leq z\\}}$, therefore the condition in
Proposition 3.2 reduces to

$(1+c)\int_{a}^{\infty}F_{k}(dz)=\int_{a}^{\infty}e^{\gamma_{k}a}\,F_{k}(dz)\,.$

Rearranging gives

$\alpha_{k}^{\dagger}[c]=\tfrac{1}{\gamma_{k}}\log(1+c)\,.$ (10)

Furthermore, $\alpha_{k}^{\dagger}[c]$ is a global maximiser as the derivative
of (8) with respect to $\alpha_{k}$ is increasing on
$(0,\alpha_{k}^{\dagger}[c])$, decreasing on
$(\alpha_{k}^{\dagger}[c],\overline{\alpha})$, and constant on
$(\overline{\alpha},\infty)$, where $\overline{\alpha}$ is the essential
supremum of the insurer-$k$’s claim severity distribution.

We further consider excess-of-loss with a policy limit, and find that the
limit does not affect the insurer’s optimal choice of retention level.

######  Example 3.6 (Excess-of-loss reinsurance with a policy limit).

In the case of excess-of-loss reinsurance with a policy limit (Example 2.5), a
similar calculation to the previous example shows that the condition in
Proposition 3.2 simplifies to

$\alpha_{k}^{\dagger}[c]=\tfrac{1}{\gamma_{k}}\log(1+c)\,,$ (11)

and $\alpha_{k}^{\dagger}[c]$ is a global maximiser.

We observe in all three examples, that $\alpha_{k}^{\dagger}$ is an increasing
function of $c$, the reinsurer’s safety loading. Recall that
$\alpha^{\dagger}$ is in Example 2.3 the percentage of the loss the insurer
retains, and in Examples 2.4 and 5.6 the level at which the insurer purchases
reinsurance. Thus $\alpha_{k}^{\dagger}$ being increasing in the reinsurer’s
safety loading implies that the insurer purchases less reinsurance if the
safety loading increases.

##  4 Reinsurer’s strategy under ambiguity

Next, we consider the reinsurance problem from the perspective of the
reinsurer. We assume that, unlike the insurer, the reinsurer does not have a
utility but rather seeks to maximise their expected wealth. They are, however,
ambiguity averse in the following sense: as their premia and losses depend on
the insurers’ loss models, the reinsurer must calculate reinsurance premia
that take into account the different insurers’ loss models, as well as
uncertainty about their accuracy. Using the insurers’ models
$\\{{\mathbb{P}}_{k}\\}_{k\in{\mathcal{N}}}$ as reference measures, the
reinsurer seeks to find a single probability measure to price reinsurance
contracts and estimate losses.

Specifically, the reinsurer seeks to maximise their wealth under a single
probability measure ${\mathbb{Q}}$ that accounts for the different insurers’
loss models as well as uncertainty about their accuracy. The reinsurer chooses
${\mathbb{Q}}$ from the set of measures that are absolutely continuous with
respect to the reference measures by choosing a compensator of the PRM under
${\mathbb{Q}}$, satisfying the following conditions.

######  Definition 4.1 (Reinsurer’s compensator).

An admissible compensator for the reinsurer is a nonnegative,
$\mathbb{F}$-predictable random field
$\varsigma=(\varsigma_{t}(\cdot))_{t\in[0,T]}$ such that, for all $t\in[0,T]$,
$\varsigma_{t}(\cdot):{\mathds{R}}_{+}\to{\mathds{R}}_{+}$ and

${\mathbb{E}}^{{\mathbb{P}}_{k}}\left[\exp\left(\int_{0}^{T}\int_{\mathds{R}}\left[\frac{1-\varsigma_{t}(z)}{v_{k}(z)}\right]^{2}N(dz,dt)\right)\right]<\infty,\qquad\forall
k\in{\mathcal{N}}\,,$ (12)

where $v_{k}(z)$ is the density of the PRM’s ${\mathbb{P}}_{k}$-compensator.
We denote by $\mathcal{V}$ the set of admissible compensators.

Next, ${\varsigma}\in{\mathcal{V}}$ induces a probability measure
${\mathbb{Q}}^{\varsigma}$, which can be characterised by the Radon-Nikodym
derivative from ${\mathbb{P}}_{k}$ to ${\mathbb{Q}}^{\varsigma}$,
$k\in{\mathcal{N}}$,

$\frac{d{\mathbb{Q}}^{\varsigma}}{d{\mathbb{P}}_{k}}:=\exp\left(\int_{0}^{T}\int_{\mathds{R}}\log\left(\frac{\varsigma_{t}(z)}{v_{k}(z)}\right)N(dz,dt)-\int_{0}^{T}\int_{\mathds{R}}\left[\frac{\varsigma_{t}(z)}{v_{k}(z)}-1\right]v_{k}(z)\,dz\,dt\right)\,.$
(13)

Then by Girsanov’s theorem (see, e.g., [23]), $N(dz,dt)$ has
${\mathbb{Q}}^{\varsigma}$-compensator
$\nu^{{\mathbb{Q}}^{\varsigma}}(dz,dt):={\varsigma}_{t}(z)\,dz\,dt$. For
future use, we define the set ${\mathcal{G}}$ as the set consisting of
functions $g:{\mathds{R}}_{+}\times{\mathds{R}}_{+}\to{\mathds{R}}_{+}$ such
that $\int_{{\mathds{R}}_{+}}g(t,z)\,dz<\infty$ for every
$t\in{\mathds{R}}_{+}$.

The reinsurer quantifies possible misspecification in ${\mathbb{P}}_{k}$,
$k\in{\mathcal{N}}$, with relative entropy penalties. The relative entropy, or
Kullback-Leibler (KL) divergence, from ${\mathbb{P}}_{k}$ to an absolutely
continuous candidate probability measure ${\mathbb{Q}}^{\varsigma}$ is given
by

$D_{\mathrm{KL}}({{\mathbb{Q}}^{\varsigma}}\,\|\,{{\mathbb{P}}_{k}}):={\mathbb{E}}^{{\mathbb{Q}}^{\varsigma}}\left[\log\left(\frac{d{\mathbb{Q}}^{\varsigma}}{d{\mathbb{P}}_{k}}\right)\right]\,,$
(14)

where ${\mathbb{E}}^{{\mathbb{Q}}^{\varsigma}}[\cdot]$ denotes the expectation
under ${\mathbb{Q}}^{\varsigma}$. The reinsurer then penalises a candidate
probability measure ${\mathbb{Q}}^{\varsigma}$ by the weighted average of the
KL divergences from the individual insurers’ probability measures to the
candidate one

$\sum_{k\in{\mathcal{N}}}\pi_{k}\,D_{\mathrm{KL}}({{\mathbb{Q}}^{\varsigma}}\,\|\,{{\mathbb{P}}_{k}})\,,$
(15)

where $\pi_{k}$ are weights satisfying $\pi_{k}\geq 0$ and
$\sum_{k\in{\mathcal{N}}}\pi_{k}=1$. Each weight $\pi_{k}$ represents the
reinsurer’s a priori belief in insurer-$k$’s loss model. For example if
$\pi_{k}=0$, this means that the reinsurer disregards insurer-$k$’s loss
model. A naive way for the reinsurer to model the losses would be to use a
weighted arithmetic mean of the insurers’ compensator. We shall see, however,
that the optimal behaviour of the reinsurer does relate to the weighted
geometric mean. Thus, we define the weighted arithmetic mean of the insurers’
compensators $v^{a}$ and the weighted geometric mean of the insurers’
compensators, $v^{g}$ as

$v^{a}(z):=\sum_{k\in{\mathcal{N}}}\pi_{k}\hskip
1.00006ptv_{k}(z)\,,\quad\text{and}\quad
v^{g}(z):=\prod_{k\in{\mathcal{N}}}v_{k}(z)^{\pi_{k}}\,.$

The reinsurer maximises their expected wealth, subject to a KL penalty, by
choosing safety loadings $\\{\eta_{k}\\}_{k\in{\mathcal{N}}}$ and compensator
$\varsigma$. As the reinsurer is the first-mover, they are able to anticipate
the insurers’ responses to their prices.

######  Optimization Problem 4.2.

Let insurer-$k$’s demand for reinsurance be parameterised by
$\alpha^{\dagger}[\eta_{t,k}]$. Then the reinsurer’s problem is

$\sup_{\eta\in{\mathfrak{C}}}\inf_{{\varsigma}\in{\mathcal{V}}}\,{\mathbb{E}}^{{\mathbb{Q}}^{\varsigma}}\left[Y_{T}+\frac{1}{\varepsilon}\sum_{k\in{\mathcal{N}}}\pi_{k}\,D_{\mathrm{KL}}({{\mathbb{Q}}^{\varsigma}}\,\|\,{{\mathbb{P}}_{k}})\right],$
($P_{R}$)

where $\varepsilon$ is a parameter representing the reinsurer’s overall
ambiguity aversion and we recall that $Y$ depends on $\eta$.

The next statement provides the optimal reinsurer’s compensator in feedback
form for problem ($P_{R}$). We observe that the optimal safety loadings of the
reinsurer are independent of time, which results in the insurers’ demand for
reinsurance also being constant over time. Furthermore, the reinsurer’s
optimal compensator is time independent, and thus the optimal probability
measure that the reinsurer uses to determine premia is constant over time. We
further observe that the optimal reinsurance compensator is proportional to
the weighted geometric mean of the compensators of the insurers, i.e. $v^{g}$.
Thus, if $\pi_{k}$ is small, then the insurer-$k$’s compensator only has a
small affect on the reinsurer’s compensator. Furthermore, if $\pi_{k}=0$, then
the reinsurer’s compensator, and thus the optimal probability measure, is
independent of insurer-$k$’s compensator. Moreover, the reinsurer’s
compensator scales exponentially with the ambiguity $\epsilon$.

######  Proposition 4.3.

For $\boldsymbol{c}:=(c_{1},\ldots,c_{n})\in{\mathds{R}}^{n}$, define the
following parametrised set of compensators:

$\varsigma^{\dagger}(z,\boldsymbol{c}):=v^{g}(z)\;\exp\left\\{\varepsilon\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\right\\}\,.$
(16)

Further, suppose that the system of non-linear equations in $\boldsymbol{c}$

$1+c_{k}=\frac{\displaystyle\int_{0}^{\infty}\left(z-r(z,\alpha_{k}^{\dagger}[c_{k}])\right)\nu_{k}(dz)+\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=c_{k}}\,\varsigma^{\dagger}(z,\boldsymbol{c})\,dz}{\displaystyle\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=c_{k}}\,\nu_{k}(dz)}\,,\qquad\forall
k\in{\mathcal{N}}$ (17)

has a solution, which we denote by $\boldsymbol{\eta}^{*}$. Then, the
reinsurer’s optimal compensator in feedback form of problem ($P_{R}$) is
$\varsigma^{*}(\cdot):=\varsigma^{\dagger}(\cdot,\boldsymbol{\eta}^{*})$. Note
that the optimal compensator is independent of time, thus we simply write
$\varsigma^{*}=\varsigma^{*}_{t}$.

###### Proof.

The reinsurer’s value function is

$\Phi(t,y)=\sup_{\eta\in{\mathfrak{C}}}\inf_{\varsigma\in\mathcal{V}}\,\mathbb{E}^{\mathbb{Q}^{\varsigma}}_{t,y}\left[Y_{T}+\frac{1}{\varepsilon}\sum_{k\in{\mathcal{N}}}\pi_{k}\int_{t}^{T}\int_{\mathds{R}}\left(\varsigma_{u}(z)\left[\log\left(\frac{\varsigma_{u}(z)}{v_{k}(z)}\right)-1\right]+v_{k}(z)\right)dz\,du\right]\,,$

where ${\mathbb{E}}^{{\mathbb{Q}}^{\varsigma}}_{t,y}$ denotes the conditional
${\mathbb{Q}}^{\varsigma}$-expectation given that the reinsurer’s wealth
process satisfies $Y_{t-}=y$. The value function may be written as

$\Phi(t,y)=\sup_{\eta\in{\mathfrak{C}}}\inf_{\varsigma\in\mathcal{V}}\,\mathbb{E}^{\mathbb{Q}^{\varsigma}}_{t,y}\left[Y_{T}+\frac{1}{\varepsilon}\int_{t}^{T}\int_{\mathds{R}}\left(\varsigma_{u}(z)\left[\log\left(\frac{\varsigma_{u}(z)}{v^{g}(z)}\right)-1\right]+v^{a}(z)\right)dz\,du\right]\,.$
(18)

As $\eta_{k,t}$ are deterministic, the optimal $\varsigma_{t}(z)$ does not
depend on the reinsurer’s wealth and is a deterministic field, denoted by
$\varsigma(t,z)$. In this case, the conditional expectation simplifies, and we
have that

$\displaystyle\Phi(t,y)=\sup_{\eta\in{\mathfrak{C}}}\inf_{\varsigma\in\mathcal{V}}\,\Bigg{\\{}y$
$\displaystyle+\int_{t}^{T}p_{k}^{R}\left(\alpha^{*}_{u,k},\eta_{u,k}\right)\,du$
$\displaystyle+\frac{1}{\varepsilon}\int_{t}^{T}\int_{\mathds{R}}\left(\varsigma(u,z)\left[\log\left(\frac{\varsigma(u,z)}{v^{g}(z)}\right)-1-\varepsilon\sum_{k\in{\mathcal{N}}}(z-r(z,\alpha^{*}_{u,k}))\right]+v^{a}(z)\right)dz\,du\Bigg{\\}}\,.$

By the dynamic programming principle, the value function $\Phi$ should satisfy
the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation:

$\displaystyle
0=\partial_{t}\Phi(t,y)+\sup_{\boldsymbol{c}\in{\mathds{R}}^{n}_{+}}\inf_{v\in{\mathcal{G}}}\Bigg{\\{}$
$\displaystyle\sum_{k\in{\mathcal{N}}}p_{k}^{R}\left(\alpha^{\dagger}_{k}[c_{k}],c_{k}\right)\partial_{y}\Phi(t,y)+\int_{\mathds{R}}\left[\Phi(t,y-L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}]))\,-\Phi(t,y)\right]v(t,z)\,dz$
$\displaystyle+\frac{1}{\varepsilon}\int_{\mathds{R}}\left(v(t,z)\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-1\right]+v^{a}(z)\right)dz\Bigg{\\}}\,$

with terminal condition $\Phi(T)=y$. Since the value function is linear in
$y$, we write $\Phi(t,y)=y+\phi(t)$, leading to the ordinary differential
equation (ODE) for $\phi$:

$\displaystyle
0=\phi^{\prime}(t)+\sup_{\boldsymbol{c}\in{\mathds{R}}^{n}_{+}}\inf_{v\in{\mathcal{G}}}\Bigg{\\{}$
$\displaystyle\sum_{k\in{\mathcal{N}}}p_{k}^{R}\left(\alpha^{\dagger}_{k}[c_{k}],_{k}\right)-\int_{\mathds{R}}L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\,v(t,z)\,dz$
$\displaystyle+\frac{1}{\varepsilon}\int_{\mathds{R}}\left(v(t,z)\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-1\right]+v^{a}(z)\right)dz\Bigg{\\}}\,,$

with terminal condition $\phi(T)=0$.

We first consider the inner optimisation, that is the infimum problem, where
we minimise the functional of $v$ given by

$\mathcal{L}[v]:=\int_{{\mathds{R}}_{+}}\\!\\!v(t,z)\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-1-\varepsilon\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\right]dz\,.$

Let $\delta>0$, $h$ be an arbitrary function such that $\delta
h+v\in{\mathcal{G}}$. Applying a variational first order condition, and using
$\log\left(\frac{v(t,z)+\delta
h(x)}{v^{g}(z)}\right)=\log\left(\frac{v(t,z)}{v^{g}(z)}\right)+o(\delta)$,
results in

$\displaystyle 0$ $\displaystyle=\lim_{\delta\to 0}\frac{\mathcal{L}[v+\delta
h]-\mathcal{L}[v]}{\delta}=\int_{{\mathds{R}}_{+}}h(z)\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-\varepsilon\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\right]dz\,.$

Since $h$ was arbitrary, this implies that

$0=\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-\varepsilon\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\right]\,.$

Solving the above equation for $v$ gives the optimal compensator in feedback
form, i.e. ${\varsigma}^{*}$:

$\varsigma^{*}_{t}(z,\boldsymbol{c})=v^{g}(z)\;\exp\left\\{\varepsilon\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\right\\}\,.$
(19)

Substituting this into the HBJI equation, we have that

$0=\phi^{\prime}(t)+\sup_{\boldsymbol{c}\in{\mathds{R}}^{n}_{+}}\Bigg{\\{}\sum_{k\in{\mathcal{N}}}p_{k}^{R}\left(\alpha^{\dagger}_{k}[c_{k}],c_{k}\right)+\frac{1}{\varepsilon}\int_{\mathds{R}}\left[v^{a}(z)-\varsigma^{*}_{t}(z,\boldsymbol{c})\right]dz\Bigg{\\}}\,.$
(20)

The derivative of the term in curly braces with respect to $c_{k}$, for each
$k\in{\mathcal{N}}$, is

$\displaystyle\frac{\partial}{\partial c_{k}}\\{\cdots\\}=$
$\displaystyle\lambda_{k}\left[\int_{0}^{\infty}\left[z-r(z,\alpha^{\dagger}_{k}[c_{k}])\right]F_{k}(dz)-(1+c_{k})\int_{0}^{\infty}\partial_{c_{k}}r(z,\alpha_{k}^{\dagger}[c_{k}])F_{k}(dz)\right]$
$\displaystyle+\int_{\mathds{R}}\varsigma^{*}_{t}(z,\boldsymbol{c})\,\partial_{c_{k}}r(z,\alpha_{k}^{\dagger}[c_{k}])\,dz\,.$

The first order conditions requires setting this expression to $0$ for all
$k\in{\mathcal{N}}$ and all $t\in[0,T]$. Hence, we obtain that the optimal
$\eta_{k,t}^{*}$ satisfies the non-linear equation

$\eta_{k,t}^{*}=\frac{\displaystyle\int_{0}^{\infty}\left[z-r(z,\alpha^{\dagger}_{k}[\eta_{k,t}^{*}])\right]\,F_{k}(dz)}{\displaystyle\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\Big{|}_{c=\eta_{k,t}^{*}}F_{k}(dz)}+\frac{\displaystyle\int_{0}^{\infty}\partial_{a}r(z,a)\Big{|}_{a=\alpha_{k}^{\dagger}[\eta_{k,t}^{*}]}\,\varsigma^{*}(z,\boldsymbol{c})\,dz}{\displaystyle\lambda_{k}\int_{0}^{\infty}\partial_{a}r(z,a)\Big{|}_{a=\alpha_{k}^{\dagger}[\eta_{k,t}^{*}]}F_{k}(dz)}-1.$

As there is no explicit time dependence in the above equation, the roots of
these non-linear equations are constant over time. ∎

The following theorem gives the Stackelberg equilibrium for the insurer-
reinsurer game. The proof follows by combining Proposition 3.2, which gives
the optimal behaviour of the insurers, and Proposition 4.3, which gives the
optimal behaviour of the reinsurer.

######  Theorem 4.4.

Under the assumptions of Proposition 3.2 and Proposition 4.3, the Stackelberg
equilibrium is
$({\boldsymbol{\alpha}}^{*},{\boldsymbol{\eta}}^{*},\varsigma^{*})$, where
${\boldsymbol{\alpha}}^{*}:=(\alpha_{1}^{*},\ldots,\alpha_{n}^{*})$, with
$\alpha_{k}^{*}:=\alpha^{\dagger}_{k}[\eta_{k}^{*}]$ for all
$k\in{\mathcal{N}}$,

$\varsigma^{*}(z,{\boldsymbol{\eta}}^{*})=v^{g}(z)\exp\left\\{\varepsilon\,L(z,{\boldsymbol{\alpha}}^{*})\right\\}\,,$

and ${\boldsymbol{\eta}}^{*}=(\eta_{1}^{*},\ldots,\eta_{n}^{*})$ satisfy

$\displaystyle 0$
$\displaystyle=\int_{0}^{\infty}\partial_{a}r(z,a)\big{|}_{a=\alpha_{k}^{*}}\left\\{(1+\eta_{k}^{*})-e^{\gamma_{k}r(z,\alpha_{k}^{*})}\right\\}\nu_{k}(dz)\,,\qquad\forall
k\in{\mathcal{N}}\,,$ $\displaystyle 1+\eta_{k}^{*}$
$\displaystyle=\frac{\displaystyle\int_{0}^{\infty}\left(z-r(z,\alpha_{k}^{*})\right)\nu_{k}(dz)+\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=\eta_{k}^{*}}\,\varsigma^{*}(z,{\boldsymbol{\eta}}^{*})\,dz}{\displaystyle\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=\eta_{k}^{*}}\,\nu_{k}(dz)}\,,\qquad\forall
k\in{\mathcal{N}}\,.$

One observes that the reinsurer’s optimal compensator is an exponential
distortion of the weighted geometric mean of the insurers’ compensators. For
small ambiguity-aversion of the reinsurer, $\varepsilon\to 0$, the compensator
approaches the weighted geometric mean. We see that the Stackelberg
equilibrium
$({\boldsymbol{\alpha}}^{*},{\boldsymbol{\eta}}^{*},\varsigma^{*})$ is
independent of time, as the premia, the reinsurance demand, and the
reinsurance optimal probability measure are all constant over time. Note that
in this general case, the optimal insurance demand ${\boldsymbol{\alpha}}^{*}$
is a function of the optimal reinsurance safety loadings
${\boldsymbol{\eta}}^{*}$ and the optimal reinsurance compensator
$\varsigma^{*}$. Thus, the amount of reinsurance each insurer buys depends on
the reinsurer’s ambiguity aversion $\varepsilon$ and their belief in each of
the insurer’s loss models, i.e. on $\pi_{k}$, $k\in{\mathcal{N}}$.

The connection between ${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$,
and $\varsigma^{*}$ is difficult to disentangle for general reinsurance
contracts. However, for concrete reinsurance contracts, e.g., proportional and
excess-of-loss, and a market with two insurers, we can illustrate the
Stackelberg equilibrium, as we do in the next section.

##  5 Application to reinsurance contract types

In this section, we calculate and illustrate the Stackelberg equilibrium for
specific reinsurance contracts. In particular, we consider proportional
reinsurance and excess-of-loss reinsurance, with and without a policy limit.
We further illustrate the solutions numerically in the case of two insurers.

###  5.1 Proportional reinsurance

First, we explicitly calculate the optimal proportional reinsurance contract.
Recall that for this type of contract, the insurer decides a proportion of the
loss to retain, $\alpha_{k}\in(0,1]$, while the reinsurer covers the remaining
proportion, $(1-\alpha_{k})$.

######  Proposition 5.1 (Proportional reinsurance).

Let ${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$ denote the solution
of the following system of equations:

$\displaystyle(1+\eta^{*}_{k})\,\lambda_{k}\,\mu_{k}$
$\displaystyle=\int_{0}^{\infty}z\,e^{\gamma_{k}\alpha_{k}^{*}z}\,\nu_{k}(dz)\,,\qquad\forall
k\in{\mathcal{N}}\,,$ $\displaystyle(1+\eta^{*}_{k})\,\lambda_{k}\,\mu_{k}$
$\displaystyle=(1-\alpha^{*}_{k})\,\gamma_{k}\,\int_{0}^{\infty}z^{2}\,e^{\gamma_{k}\alpha^{*}_{k}z}\,\nu_{k}(dz)+\int_{0}^{\infty}z\,e^{\varepsilon
z\sum_{j\in{\mathcal{N}}}(1-\alpha^{*}_{j})}\,v^{g}(z)\,dz\,,\qquad\forall
k\in{\mathcal{N}}\,.$

If such ${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$ exist, then the
Stackelberg equilibrium for proportional reinsurance is
$({\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*},\varsigma^{*})$, where

$\varsigma^{*}(z)=v^{g}(z)\exp\left({\varepsilon\hskip
1.00006ptz\sum_{k\in{\mathcal{N}}}(1-\alpha_{k}^{*})}\right)\,.$

To further analyse the Stackelberg equilibrium in the proportional reinsurance
case, we assume that there are two insurers whose losses are Gamma distributed
with different parameters.

######  Example 5.2.

Suppose there are two insurers with Gamma distributed losses with shape
parameters $m_{1},m_{2}>0$ and scale parameters $\xi_{1},\xi_{2}>0$, and
intensities $\lambda_{1},\lambda_{2}>0$, respectively. Let
$\xi_{k}\leq\min\\{\frac{1}{\gamma_{k}},\,\frac{1}{2\varepsilon}\\}$ and
define

$\displaystyle\tilde{m}$ $\displaystyle:=m_{1}\pi_{1}+\pi_{2}m_{2}\,,$
$\displaystyle\tilde{\xi}$
$\displaystyle:=\frac{\xi_{1}\xi_{2}}{\pi_{1}\xi_{2}+\pi_{2}\xi_{1}-\varepsilon\,\xi_{1}\xi_{2}(2-\alpha^{*}_{1}-\alpha^{*}_{2})}\,.$

Then, at the Stackelberg equilibrium, the reinsurer’s severity distribution is
a Gamma distribution with shape $\tilde{m}$ and scale $\tilde{\xi}$. This
means that the reinsurer calculates premia as if both insurers have a Gamma
distribution with parameters $\tilde{m}$ and $\tilde{\xi}$ and constant
intensity

$\Lambda:=\tilde{\xi}^{\tilde{m}}\,\Gamma(\tilde{m})\left(\frac{\lambda_{1}}{\Gamma(m_{1})\xi_{1}^{m_{1}}}\right)^{\pi_{1}}\\!\\!\left(\frac{\lambda_{2}}{\Gamma(m_{2})\xi_{2}^{m_{2}}}\right)^{\pi_{2}}\,,$

where $\Gamma(\cdot)$ is the Gamma function. The assumption
$\xi_{k}\leq\min\\{\frac{1}{\gamma_{k}},\,\frac{1}{2\varepsilon}\\}$ implies
that the insurer only underwrites losses that are in line with their absolute
risk aversion. Recall that the mean of a Gamma random variable is
$m_{k}\xi_{k}$, thus insurer-$k$ only considers risks with mean
$m_{k}\xi_{k}\leq\frac{m_{k}}{\gamma_{k}}$. Similarly, the reinsurer requires
that losses be in line with their ambiguity aversion, for example, they
require that the means of each of the insurers’ losses be bounded
proportionally to the reciprocal of their ambiguity, i.e.
$m_{1}\xi_{1}+m_{2}\xi_{2}\leq\frac{m_{1}+m_{2}}{2\varepsilon}$.

The insurers choose to reinsure proportions $\alpha_{k}^{*}$ of their losses,
where $\alpha^{*}_{k}$, $k=1,2$, is the solution of the following system of
equations:

$0=\frac{-1}{(1-\alpha^{*}_{k}\,\gamma_{k}\,\xi_{k})^{m_{k}+1}}+\frac{\gamma_{k}(1+m_{k})\,\xi_{k}\,(1-\alpha^{*}_{k})}{(1-\alpha_{k}^{*}\,\gamma_{k}\,\xi_{k})^{m_{k}+2}}+\frac{\tilde{m}\,\tilde{\xi}\,\Lambda}{m_{k}\,\xi_{k}\,\lambda_{k}}\,,\quad
k=1,2\,.$

The reinsurer charges safety loadings

$\eta^{*}_{k}=\frac{1}{(1-\alpha^{*}_{k}\hskip
1.00006pt\gamma_{k}\,\xi_{k})^{m_{k}+1}}-1\,,\quad k=1,2\,.$

In the numerical example, the insurers’ shape and scale parameters and
intensities are $m_{1}=1.5,\,\xi_{1}=1,\,\lambda_{1}=2$ and
$m_{2}=2,\,\xi_{2}=1.25,\,\lambda_{2}=2.5$, respectively. In particular,
insurer-2 has greater expected losses than insurer-1, as it has on average
more frequent claims, and the mean of their claim distribution is larger. The
insurers have the same risk aversion, i.e. $\gamma_{1}=\gamma_{2}=0.5$.

0.00.10.20.30.40.501234$z$$\varsigma^{*}(z)/\Lambda$$\varepsilon$00.0750.15
(a) Normalized compensator for different $\varepsilon$ and
$\pi_{1}=\pi_{2}=0.5$

0.00.10.20.30.40.501234$z$$\varsigma^{*}(z)/\Lambda$$\pi_{2}$00.250.50.751 (b)
Normalized compensator for different $\pi_{2}$ and $\varepsilon=0.1$

Figure 1: The severity distributions of insurer-1 (dot-dashed line), insurer-2
(dotted line), and the reinsurer (solid lines) for three values of
$\varepsilon$. The insurers have risk aversion parameters
$\gamma_{1}=\gamma_{2}=0.5$, intensities $\lambda_{1}=2,\,\lambda_{2}=2.5$,
and their severity distributions are Gamma distributed with scale parameters
$\xi_{1}=1,\,\xi_{2}=1.25$ and shape parameters $m_{1}=1.5,\,m_{2}=2$. In
panel (a), $\varepsilon$ is varied while the weights are fixed at
$\pi_{1}=\pi_{2}=0.5$. In panel (b), $\varepsilon=0.1$ and the weights are
varied.

Figure 1 plots the optimal severity distributions (normalised compensators) of
insurer-1 (dot-dashed line), insurer-2 (dotted line), and the reinsurer (solid
lines). Panel (a) shows the effect of the ambiguity aversion $\varepsilon$ on
the severity distribution. Here, the weights applied to the insurers’ loss
models are kept fixed at $\pi_{1}=\pi_{2}=0.5$. The salmon-coloured line gives
the reinsurer’s severity distribution when the reinsurer is not ambiguity-
averse ($\varepsilon=0$). In this case, the reinsurer’s severity distribution
is the normalised geometric mean of the insurers’ distributions. As the
reinsurer’s ambiguity aversion increases, they place more weight on larger
losses. This can be seen in the figure in the pink and purple lines
($\varepsilon=0.075,\,0.15$). Panel (b) shows the effect of the choice of the
weights, $\pi_{1}$ and $\pi_{2}$, on the severity distribution, when
$\varepsilon$ is kept fixed at 0.1. As the weights vary, the reinsurer
interpolates between the two insurers’ severity distributions; however, they
give a larger weight to large losses $z$ due to their ambiguity aversion.

2.22.42.62.80.00.10.20.3$\varepsilon$$\Lambda$

(a) Reinsurer’s total intensity for different $\varepsilon$

2.22.42.62.80.000.250.500.751.00$\pi_{2}$$\Lambda$

(b) Reinsurer’s total intensity for different $\pi_{2}$

Figure 2: The total intensity, $\Lambda$, of the reinsurer. In panel (a), the
reinsurer’s ambiguity aversion parameter, $\varepsilon$, is varied while the
weights are fixed at $\pi_{1}=\pi_{2}=0.5$. In panel (b), $\varepsilon=0.1$
and the weights applied to each insurer’s loss model are varied. The insurers
have risk aversion parameters $\gamma_{1}=\gamma_{2}=0.5$, intensities
$\lambda_{1}=2,\,\lambda_{2}=2.5$, and their severity distributions are Gamma
distributed with scale parameters $\xi_{1}=1,\,\xi_{2}=1.25$ and shape
parameters $m_{1}=1.5,\,m_{2}=2$.

Figure 2 shows the reinsurer’s total constant intensity, $\Lambda$, as a
function of the reinsurer’s ambiguity aversion, $\varepsilon$, (panel (a)) and
the weights applied to insurer-2’s loss model, $\pi_{2}$ (panel (b)). Panel
(a) shows that as the reinsurer becomes more ambiguity averse (increasing
$\varepsilon$), the rate of arrival of claims increases under their optimal
model. Panel (b) shows that the total intensity is also increasing as the
reinsurer puts more weight on the loss model with greater expected losses.

Figure 3 shows the safety loading charged to each insurer at equilibrium,
$\eta_{k}^{*}$, and the proportion of each loss they retain, $\alpha_{k}^{*}$,
as a function of $\pi_{2}$, the weight applied by the reinsurer to insurer-2’s
loss model. Three different values of the ambiguity aversion parameter,
$\varepsilon=0,\,0.05,\,0.1$, are shown. Consider first the solid line,
$\varepsilon=0$, where the reinsurer has no ambiguity aversion. We observe
that as $\pi_{2}$ increases, meaning that the reinsurer puts more weight onto
the model with greater expected losses, the safety loadings, $\eta^{*}_{k}$,
for both insurers are increasing. The amount retained by each insurer,
$\alpha^{*}_{k}$, increases as well.

In particular, we see that when both loss models are used to calculate the
insurance premium ($\pi_{2}\in(0,1)$), insurer-1 pays more than if only their
loss model were used ($\pi_{2}=0$). Meanwhile, insurer-2 benefits from the use
of both models, paying less when $\pi_{2}\in(0,1)$ than when $\pi_{2}=1$. We
further observe that insurer-1 is more sensitive to the pricing model than
insurer-2. Insurer-1 buys reinsurance for 34% of the loss when $\pi_{2}=0$ and
reduces their reinsurance demand to 20% for $\pi_{2}=1$, whereas insurer-2
only decreases their demand for reinsurance from 27% to 24%. When the
reinsurer is ambiguity-averse (the dashed lines, $\varepsilon=0.05,\,0.1$),
they increase the safety loadings for both insurers and consequently, both
insurer buy less reinsurance, i.e. the retain more, as $\alpha_{k}$ is
increasing with $\varepsilon$.

$\alpha_{k}^{*}$$\eta_{k}^{*}$0.000.250.500.751.000.000.250.500.751.00234560.700.750.80$\pi_{2}$$\varepsilon$00.050.1Insurer
($k$)12 Figure 3: The equilibrium safety loadings charged to insurer-$k$,
$\eta_{k}^{*}$, and the proportion of losses each insurer retains,
$\alpha_{k}^{*}$, $k=1,2$, as a function of the weight the reinsurer places on
insurer-2’s loss model, $\pi_{2}$. The insurers have risk aversion parameters
$\gamma_{1}=\gamma_{2}=0.5$, intensities $\lambda_{1}=2,\,\lambda_{2}=2.5$,
and their severity distributions are Gamma distributed with scale parameters
$\xi_{1}=1,\,\xi_{2}=1.25$ and shape parameters $m_{1}=1.5,\,m_{2}=2$. In
particular, insurer-2’s model has greater expected losses.

We note that one can extend this example to $n$ insurers, as long as
$\xi_{k}\leq\min\\{\frac{1}{\gamma_{k}},\,\frac{1}{n\varepsilon}\\}$ for
$k\in{\mathcal{N}}$. In this case one solves a similar $n$-dimensional system
of equations for ${\boldsymbol{\alpha}}$, the reinsurer’s severity
distribution remains Gamma distributed, and the equation to obtain
$\eta_{k}^{*}$ from $\alpha_{k}^{*}$ remains the same. The constraints on the
scale parameters $\xi_{k}$ mean that the insurer’s loss model must be
sufficiently light-tailed that they offer insurance according to their own
utility preferences as parameterised by $\gamma_{k}$, and sufficiently light-
tailed that the reinsurer offers reinsurance according to their ambiguity
aversion, $\varepsilon$.

###  5.2 Excess-of-loss reinsurance

Next, we determine the optimal excess-of-loss reinsurance contract. Recall
that for this type of contract, each insurer chooses a retention level,
$\alpha_{k}^{*}$, beyond which the reinsurer covers the losses.

######  Proposition 5.3 (Excess-of-loss reinsurance).

Let $(z-a)_{+}:=\min\\{z-a,\,0\\}$ for $a\in{\mathds{R}}$ and define
insurer-$k$’s survival function, $\overline{F}_{k}(z):=1-F_{k}(z)$. Let
${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$ denote the solution of
the following system of equations:

$\displaystyle\alpha_{k}^{*}$
$\displaystyle=\frac{1}{\gamma_{k}}\log\left(\frac{\int_{\alpha_{k}^{*}}^{\infty}{\varsigma}^{*}(z)dz}{\lambda_{k}\left[\overline{F}_{k}(\alpha_{k}^{*})-\gamma_{k}\int_{\alpha_{k}^{*}}^{\infty}\overline{F}_{k}(z)dz\right]}\right)\,,\qquad\forall
k\in{\mathcal{N}}\,,$ $\displaystyle\eta_{k}^{*}$
$\displaystyle=e^{\gamma_{k}\alpha^{*}_{k}}-1\,,\qquad\forall
k\in{\mathcal{N}}\,,$

where

$\varsigma^{*}(z)=v^{g}(z)\,\exp\left({\varepsilon\,\sum_{k\in{\mathcal{N}}}(z-\alpha^{*}_{k})_{+}}\right)\,.$

If such ${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$ exist, then the
solution to the Stackelberg game is
$({\boldsymbol{\alpha}}^{*},{\boldsymbol{\eta}}^{*},\varsigma^{*})$.

Recall that, for fixed ambiguity aversion $\varepsilon$, and fixed weights
$\pi_{k},\,k\in{\mathcal{N}}$, the reinsurance demand
$\alpha_{k}^{*}=\alpha_{k}^{\dagger}[\eta_{k}^{*}]$, is a function of the
optimal safety loadings ${\boldsymbol{\eta}}^{*}$. Interestingly, the safety
loading of the premia for insurer-$k$ depends on insurer-$k$’s absolute risk
aversion $\gamma_{k}$. Thus, an insurer with larger risk aversion is willing
to pay a larger markup for the same contract. Furthermore, as the reinsurer’s
compensator $\varsigma^{*}$ is a function of ${\boldsymbol{\alpha}}^{*}$ and
${\boldsymbol{\eta}}^{*}$, the way the reinsurer distorts the insurer’s loss
model depends both on the covered loss and the safety loadings.

To illustrate the solution, consider the case of two insurers, where each has
exponentially distributed losses, but with different means and intensities.

######  Example 5.4.

Suppose there are two insurers, each with exponentially distributed losses
with scale parameters
$\xi_{k}\in(0,\min\\{\frac{1}{\gamma_{k}},\,\frac{1}{2\varepsilon}\\})$, and
intensity $\lambda_{k}>0$, $k=1,2$. Then at the Stackelberg equilibrium, the
insurers choose retention limits $\alpha_{k}^{*}$, $k=1,2$, that solve the
following set of non-linear equations:

$\alpha_{k}^{*}=\frac{1}{\gamma_{k}}\,\log\left(\frac{\displaystyle\int_{\alpha_{k}^{*}}^{\infty}v^{g}(z)\,\exp\left({\varepsilon\,\left[(z-\alpha^{*}_{1})_{+}+(z-\alpha^{*}_{2})_{+}\right]}\right)dz}{\lambda_{k}\,e^{-\alpha_{k}^{*}/\xi_{k}}\left[1-\gamma_{k}\,\xi_{k}\right]}\right),\qquad
k=1,2\,.$

The reinsurer charges safety loadings

$\eta_{k}^{*}=e^{\gamma_{k}\alpha_{k}^{*}}-1\,,\quad k=1,2\,,$

and has a compensator under the optimal model given by

$\varsigma^{*}(z)=v^{g}(z)\,\exp\left({\varepsilon\,\left[(z-\alpha^{*}_{1})_{+}+(z-\alpha^{*}_{2})_{+}\right]}\right)\,.$
(21)

In the following numerical example, the insurers’ scale parameters and
intensities are $\xi_{1}=1,\,\lambda_{1}=2$ and
$\xi_{2}=1.25,\,\lambda_{2}=2.5$, respectively. In particular, insurer-2 has
greater expected losses than insurer-1, as it has on average more frequent
claims, and the mean of their claim distribution is larger. Both insurers have
the same risk aversion parameter $\gamma_{1}=\gamma_{2}=0.5$.

0.010.101.0001234$z$$\varsigma^{*}(z)/\Lambda$modelinsurer 1insurer
2reinsurer$\varepsilon$00.0750.15 Figure 4: The severity distributions on a
log scale of insurer-1, insurer-2, and the reinsurer for three values of
$\varepsilon$. The triangles mark the retention limits of each insurer,
$\alpha_{1}^{*}$ and $\alpha_{2}^{*}$. The insurers have risk aversion
parameters $\gamma_{1}=\gamma_{2}=0.5$, intensities
$\lambda_{1}=2,\,\lambda_{2}=2.5$, and their severity distributions are
exponential with scale parameters $\xi_{1}=1,\,\xi_{2}=1.25$. In particular,
insurer-2’s model has greater expected losses. The insurers’ models are
equally weighted with $\pi_{1}=\pi_{2}=0.5$.

Figure 4 plots the severity distributions of insurer-1 (dot-dashed line),
insurer-2 (dotted line), and the reinsurer (solid lines) on a log scale. We
assume that the insurers’ models are equally weighted by the reinsurer, i.e.,
$\pi_{1}=\pi_{2}=0.5$. The retention limits of each insurer, $\alpha_{1}^{*}$
and $\alpha_{2}^{*}$, are marked with triangles. We consider three levels of
ambiguity-aversion for the reinsurer: $\varepsilon=0,\,0.075,\,0.15$, where
the first case corresponds to the reinsurer being ambiguity-neutral. When
$\varepsilon=0$, the reinsurer’s severity distribution is the normalised
weighted geometric mean of the insurers’ severity distributions, which appears
as a straight line on the log scale as it is exponentially distributed.
Including ambiguity aversion shifts the tail of the reinsurer’s severity
distribution up, creating kinks at $z=\alpha_{1}^{*}$ and $z=\alpha_{2}^{*}$.
Thus, when the reinsurer is ambiguity-averse, they add weight to the parts of
the distribution that are past each insurer’s retention limit. Note that for
all $\varepsilon$, $\alpha_{2}^{*}<\alpha^{*}_{1}$, and with larger ambiguity
of the reinsurer, we observe that $\alpha_{k}^{*}$ shifts to the right,
indicating that both insurers buy less reinsurance.

Figure 5 shows the safety loading charged to each insurer at equilibrium,
$\eta_{k}^{*}$, and the quantile of their retention level,
$F_{k}(\alpha_{k}^{*})$, as a function of the weight applied by the reinsurer
to insurer-2’s loss model, $\pi_{2}$. Again, three different values of the
ambiguity aversion parameter, $\varepsilon$, are shown. Similar to the case of
proportional reinsurance (Example 5.2, in particular Figure 3), the safety
loading charged to each insurer, $\eta_{k}^{*}$, are increasing as the weight
applied to the distribution with greater expected losses, $\pi_{2}$ increases,
and the amount of reinsurance purchased by the insurers is decreasing
($\alpha_{k}^{*}$ is increasing). In particular, when $\varepsilon=0.1$ and
when the riskier loss model of insurer-2 is more heavily weighted ($\pi_{2}$
close to 1), then insurer-1 chooses to retain almost all of their losses
($F_{1}(\alpha_{1}^{*})$ is close to 1). When the reinsurer is more ambiguity
averse ($\varepsilon=0.05,0.1$, the dashed and dotted lines), the safety
loadings are increased and so is the amount retained by the insurers.

$F_{k}(\alpha_{k}^{*})$$\eta_{k}^{*}$0.000.250.500.751.000.000.250.500.751.002.55.07.510.00.60.70.80.91.0$\pi_{2}$$\varepsilon$00.050.1Insurer
($k$)12 Figure 5: The equilibrium safety loadings charged to insurer-$k$,
$\eta_{k}^{*}$, and the quantiles of their retention limits,
$F_{k}(\alpha_{k}^{*})$, $k=1,2$, as a function of the weight the reinsurer
places on insurer-2’s loss model, $\pi_{2}$. The insurers have risk aversion
parameters $\gamma_{1}=\gamma_{2}=0.5$, intensities
$\lambda_{1}=2,\,\lambda_{2}=2.5$, and their severity distributions are
exponential with scale parameters $\xi_{1}=1,\,\xi_{2}=1.25$. In particular,
insurer-2’s model has greater expected losses.

Next, we consider the case where the excess-of-loss reinsurance is capped at a
policy limit $\ell_{k}>0$, $k\in{\mathcal{N}}$.

######  Proposition 5.5 (Capped excess-of-loss reinsurance).

Let ${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$ denote the solution
of the following system of equations:

$\displaystyle\alpha_{k}^{*}$
$\displaystyle=\frac{1}{\gamma_{k}}\log\left(\frac{\displaystyle\int_{\alpha_{k}^{*}}^{\alpha_{k}^{*}+\ell_{k}}\\!\\!\\!\\!{\varsigma}^{*}(z)dz}{\displaystyle\lambda_{k}\left[F_{k}(\alpha_{k}^{*}+\ell_{k})-F_{k}(\alpha_{k}^{*})-\gamma_{k}\int_{\alpha_{k}^{*}}^{\alpha_{k}^{*}+\ell_{k}}\overline{F}_{k}(z)dz\right]}\right)\,,\qquad\forall
k\in{\mathcal{N}}\,,$ $\displaystyle\eta_{k}^{*}$
$\displaystyle=e^{\gamma_{k}\alpha_{k}^{*}}-1\,,\qquad\forall
k\in{\mathcal{N}}\,,$

where

$\varsigma^{*}(z)=v^{g}(z)\,\exp\left(\varepsilon\,\sum_{k\in{\mathcal{N}}}\min\left\\{(z-\alpha^{*}_{k})_{+},\ell_{k}\right\\}\right)\,.$

If such ${\boldsymbol{\alpha}}^{*}$, ${\boldsymbol{\eta}}^{*}$ exist, then the
Stackelberg game has the solution
$({\boldsymbol{\alpha}}^{*},{\boldsymbol{\eta}}^{*},\varsigma^{*})$.

We observe that as the policy limit gets large, $\ell_{k}\to\infty$, the
retention level of the insurers, $\alpha_{k}^{*}$, approaches the retention
level in Proposition 5.3, i.e., when there is no policy limit on the excess-
of-loss reinsurance. As the safety loadings ${\boldsymbol{\eta}}^{*}$ and the
optimal compensator ${\varsigma}^{*}$ depend on ${\boldsymbol{\alpha}}^{*}$,
the Stackelberg equilibrium from Proposition 5.3 (no policy limit) is the
limiting case of the capped excess-of-loss reinsurance equilibrium as the
policy limit goes to infinity. Furthermore, the premium charged by the
reinsurer when there is a policy limit,

$p^{R}_{k}=(1+\eta_{k}^{*})\,\lambda_{k}\int_{\alpha_{k}^{*}}^{\alpha_{k}^{*}+\ell_{k}}\overline{F}_{k}(z)dz\,,$

approaches the premium charged when the reinsurance coverage is not capped, as
$\ell_{k}\to\infty$.

######  Example 5.6.

We return to the previous example (Example 2.4) with the addition of policy
limits $\ell_{k}$. Figure 6 shows the reinsurer’s compensator on the log scale
when the policy limits are $\ell_{1}=\ell_{2}=1$, which corresponds to the
63rd and 55th quantiles of the insurer-1 and insurer-2’ loss distributions,
respectively, prior to purchasing reinsurance. We assume both insurers’ loss
models are equally weighted ($\pi_{1}=\pi_{2}=0.5$). Similar to Figure 4, at
the retention limits, $\alpha_{k}^{*}$ (marked with triangles), the
compensator shifts away from the normalised geometric mean when the reinsurer
is ambiguity averse ($\varepsilon>0$). However, once the policy limits of both
insurers are reached (at $\alpha_{k}^{*}+\ell_{k}$, $k=1,2$, marked with
squares), the compensator has again an exponential decay.

0.010.101.0001234$z$$\varsigma^{*}(z)/\Lambda$$\varepsilon$00.0750.15modelinsurer
1insurer 2reinsurerkey points$\alpha_{k}^{*}$$\alpha_{k}^{*}+\ell_{k}$ Figure
6: The severity distributions on a log scale of insurer-1, insurer-2, and the
reinsurer for three values of $\varepsilon$. The triangles mark the retention
limits of each insurer, $\alpha_{1}^{*}$ and $\alpha_{2}^{*}$, and the squares
mark the upper limit of the reinsurance coverage, $\alpha_{1}^{*}+\ell_{1}$
and $\alpha_{2}^{*}+\ell_{2}$. The insurers have risk aversion parameters
$\gamma_{1}=\gamma_{2}=0.5$, intensities $\lambda_{1}=2,\,\lambda_{2}=2.5$,
and their severity distributions are exponential with scale parameters
$\xi_{1}=1,\,\xi_{2}=1.25$. The insurers’ loss models are equally weighted
with $\pi_{1}=\pi_{2}=0.5$.

Figure 7 shows the quantiles of the insurers’ retention limits,
$F_{k}(\alpha_{k}^{*})$, their safety loadings, $\eta_{k}^{*}$, as well as the
premium charged, $p_{k}^{R}$, as a function of the policy limit, $\ell_{k}$.
The faint horizontal lines give the optimal values when there is no policy
limit. As expected, we observe that in all cases the variables approach the no
policy limit equilibrium as $\ell_{k}$ gets large — note that $\ell_{k}=6$
corresponds approximately to the 99th quantile of both insurers’ loss
distributions.

First, examine the equilibrium values for insurer-1 (red lines). Their
retention limit, $\alpha_{1}^{*}$, is increasing in $\ell_{1}$ for all levels
of ambiguity aversion, $\varepsilon$. Moreover, the safety loading,
$\eta_{1}^{*}$, is also increasing in $\ell_{1}$ for all $\varepsilon$. The
premium, $p^{R}_{1}$, however, is not monotone in $\ell_{1}$, as it depends on
both the reinsurance purchased and the safety loading. Indeed, we observe that
for large ambiguity, $\varepsilon=0.1$, the safety loading is steeply
increasing in $\ell_{1}$, which results that insurer-1’s demand of reinsurance
drastically decreases.

The equilibrium values for the riskier insurer (insurer-2, blue lines) have
different behaviour with increasing $\ell_{2}$: when the reinsurer is not
ambiguity-averse, insurer-2 decreases their retention limit as the policy
limit increases, purchasing more reinsurance. Moreover, the safety loading is
also decreasing in $\ell_{2}$, however, the combined result is that insurer-2
pays a larger premium, $p_{2}^{R}$, as $\ell_{k}$ increases. This behaviour is
dampened when the reinsurer is ambiguity-averse ($\varepsilon=0.1$), as the
safety loading $\eta_{2}^{*}$ becomes increasing in $\ell_{2}$.

$\alpha_{k}^{*}$$\eta_{k}^{*}$$p_{k}^{R}$0246024602460.00.51.01.52.01.01.52.02.53.01.52.02.5$\ell_{k}$$\varepsilon$00.050.1Insurer
($k$)12 Figure 7: The safety loading, $\eta_{k}^{*}$, quantile of the
retention limit, $F_{k}(\alpha_{k}^{*})$, and the premium, $p_{k}^{R}$,
charged to each insurer as a function of the policy limit, $\ell_{k}$. The
dotted horizontal lines give the corresponding values for the case of no
policy limit. The insurers have risk aversion parameters
$\gamma_{1}=\gamma_{2}=0.5$, intensities $\lambda_{1}=2,\,\lambda_{2}=2.5$,
and their severity distributions are exponential with scale parameters
$\xi_{1}=1,\,\xi_{2}=1.25$.

##  6 Reinsurer’s utility maximisation strategy under ambiguity

In the literature, the problem of a reinsurer who maximises their expected
utility, rather than their expected wealth, has been well-studied for the case
of one insurer. Examples include [26], [27], [15, 16], and [19]. In this
section, we compare our problem to this related utility-maximisation problem.
For this, we assume that the reinsurer, like the insurers, has a CARA utility
function $u(x)=-\frac{1}{m}e^{-mx}$ with risk aversion parameter $m>0$. To
obtain an analytical solution, we follow the convention in the literature and
adopt a state-dependent ambiguity preference term for the reinsurer. First
introduced by [21] one generalises the ambiguity aversion parameter
$\varepsilon$ to one that depends on time and reinsurer’s wealth, $(t,y)$, as
follows:

$\varphi(t,y):=-\frac{\varepsilon}{m\,\Phi(t,y)}\,,$

where $\Phi(\cdot,\cdot)$ is the value function of the reinsurer’s utility
maximisation problem. This type of ambiguity aversion has been used
extensively in reinsurance games in continuous time, and results that the
reinsurer’s ambiguity aversion scales inter-temporally with the value
function. These changes give the following modified value function for the
reinsurer:

$\begin{split}\Phi(t,y)&=\sup_{\eta\in{\mathfrak{C}}}\inf_{{\varsigma}\in\mathcal{V}}\,\mathbb{E}^{\mathbb{Q}^{\varsigma}}_{t,y}\left[u(Y_{T})+\sum_{k\in{\mathcal{N}}}\pi_{k}\int_{t}^{T}\int_{\mathds{R}}\frac{1}{\varphi(s,Y_{s-})}\left({\varsigma}_{s}(z)\left[\log\left(\frac{{\varsigma}_{s}(z)}{v_{k}(z)}\right)-1\right]+v_{k}(z)\right)dzds\right]\\\
&=\sup_{\eta\in{\mathfrak{C}}}\inf_{{\varsigma}\in\mathcal{V}}\,\mathbb{E}^{\mathbb{Q}^{\varsigma}}_{t,y}\left[u(Y_{T})-\frac{m}{\varepsilon}\sum_{k\in{\mathcal{N}}}\pi_{k}\int_{t}^{T}\int_{\mathds{R}}\Phi(s,Y_{s-})\left({\varsigma}_{s}(z)\left[\log\left(\frac{{\varsigma}_{s}(z)}{v_{k}(z)}\right)-1\right]+v_{k}(z)\right)dzds\right]\,.\end{split}$
(22)

The solution to the reinsurer’s problem with this modified value function is
given in the following proposition. We assume the insurers’ problem is
unchanged, so the solution to their problem is the one given in Proposition
3.2.

######  Proposition 6.1.

For $\boldsymbol{c}:=(c_{1},\ldots,c_{n})\in{\mathds{R}}^{n}$, define the
following parameterised set of compensators:

$\varsigma^{\dagger}(z,\boldsymbol{c}):=v^{g}(z)\exp\left(\frac{\varepsilon}{m}\left[e^{mL(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}-1\right]\right)\,.$
(23)

Further, suppose that the system of non-linear equations in $\boldsymbol{c}$

$1+c_{k}^{*}=\frac{\displaystyle\int_{0}^{\infty}\left(z-r(z,\alpha_{k}^{\dagger}[c_{k}])\right)\nu_{k}(dz)+\int_{0}^{\infty}e^{m\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}\,\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=c_{k}}\,\varsigma^{\dagger}(z,\boldsymbol{c})\,dz}{\displaystyle\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=\eta_{k}^{*}}\,\nu_{k}(dz)}\,,\qquad\forall
k\in{\mathcal{N}}$ (24)

has a solution, which we denote by $\boldsymbol{\eta}^{*}$. Then, the
reinsurer’s optimal compensator in feedback form for (22) is
$\varsigma^{*}(\cdot):=\varsigma^{\dagger}(\cdot,\boldsymbol{\eta}^{*})$.

The proof of this proposition is similar to the proof of Proposition 4.3 and
can be found in Appendix A. Combining this result with Proposition 3.2, we
obtain the Stackelberg equilibrium for the case where the reinsurer has
exponential utility and a scaled entropy penalty.

######  Theorem 6.2.

Under the assumptions of Proposition 3.2 and Proposition 6.1, the Stackelberg
equilibrium when the reinsurer maximises their exponential utility is
$({\boldsymbol{\alpha}}^{*},{\boldsymbol{\eta}}^{*},\varsigma^{*})$, where
${\boldsymbol{\alpha}}^{*}=(\alpha^{*}_{1},\ldots,\alpha^{*}_{n})$, with
$\alpha^{*}_{k}:=\alpha^{\dagger}_{k}[\eta_{k}^{*}]$, $k\in{\mathcal{N}}$,

$\varsigma^{*}(z)=v^{g}(z)\exp\left(\frac{\varepsilon}{m}\left[e^{mL(z,{\boldsymbol{\alpha}}^{*})}-1\right]\right)\,,$

and ${\boldsymbol{\eta}}^{*}=(\eta_{1}^{*},\ldots,\eta_{n}^{*})$ satisfy

$\displaystyle 0$
$\displaystyle=\int_{0}^{\infty}\partial_{a}r(z,a)\big{|}_{a=\alpha_{k}^{*}}\left\\{(1+\eta_{k}^{*})-e^{\gamma_{k}r(z,\alpha_{k}^{*})}\right\\}\nu_{k}(dz)\,,\qquad\forall
k\in{\mathcal{N}}\,,$ $\displaystyle 1+\eta_{k}^{*}$
$\displaystyle=\frac{\displaystyle\int_{0}^{\infty}\left(z-r(z,\alpha_{k}^{*})\right)\nu_{k}(dz)+\int_{0}^{\infty}e^{m\,L(z,{\boldsymbol{\alpha}}^{*})}\,\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=\eta_{k}^{*}}\,\varsigma^{*}(z)\,dz}{\displaystyle\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\big{|}_{c=\eta_{k}^{*}}\,\nu_{k}(dz)}\,,\qquad\forall
k\in{\mathcal{N}}\,.$

The main difference to the case where the reinsurer maximises their expected
wealth (Theorem 4.4) is in the reinsurer’s compensator. Here, the weighted
geometric mean of the insurers’ compensators is multiplied by a double
exponential term which depends on the aggregate total loss of the reinsurer,
as opposed to a single exponential term. In addition, the second set of
equations has an additional exponential term in the second integral in the
numerator. We observe that as $m\to 0$, that is diminishing absolute risk
aversion, the Stackelberg equilibrium approaches the one where the reinsurer
maximises their expected wealth (Theorem 4.4).

Comparing the compensators from the two problems, we observe that when the
reinsurer’s risk aversion parameter satisfies $m\geq 1$, the compensator in
the utility-maximisation case dominates that of the wealth-maximisation case
for any non-negative loss. When $m\in(0,1)$, the compensator in the wealth-
maximisation case dominates the compensator in the utility-maximisation case
until an aggregate loss level $L^{*}(z)$444$L^{*}(z)=-W(-me^{-m})/m-1$, where
$W$ is the Lambert $W$ function., after which the relationship reverses. Thus
for lower levels of risk aversion ($m<1$), the reinsurer up-weights larger
losses and down-weights smaller losses as compared to the wealth-maximisation
case. When the reinsurer is both risk-averse (CARA utility, $m>0$) and
ambiguity-averse ($\varepsilon>0$), they adopt a model that puts much more
weight on tail losses. Furthermore, as $L(z,{\boldsymbol{\alpha}})$ is a
function of $z$, their compensator is not integrable for many common types of
reinsurance contracts and common choices of loss distributions, for example
excess-of-loss reinsurance without a policy limit and proportional reinsurance
require the existence of the moment generating function of the random variable
$e^{Z}$, where $Z$ is the aggregate loss of the reinsurer. Thus, in these
cases, the reinsurance contracts cannot be priced for the case when the
reinsurer is both ambiguity- and risk- averse.

Our solution is consistent with the literature. For example, [16] allow for
measure changes which depend on time only and find that for an excess-of-loss
reinsurance contract, the optimal probability distortion is

$\varsigma^{*}=v(z)\exp\left(\frac{\varepsilon}{m}\left[\left(\int_{\alpha^{*}}^{\infty}e^{m(z-\alpha^{*})}-1\right)F(dz)\right]\right)$

(adjusting for notational differences and differences in investment behaviour
of the reinsurer). In our setting, allowing for time and state dependence, for
a single insurer and an excess-of-loss contract, we obtain

$\varsigma^{*}(z)=v(z)\exp\left(\frac{\varepsilon}{m}\left[e^{m(z-\alpha^{*})}-1\right]\right)\,.$

Thus, the difference is an expectation within the exponential. Allowing the
probability distortion to depend on the loss size (equivalently, allowing the
severity distribution to change under the optimal measure) generalises the
results of [16]. It also illustrates an important point: the problem where the
reinsurer is both risk-averse and ambiguity-averse is ill-posed when the
reinsurer’s probability distortion can depend on the size of the loss $z$, as
the reinsurer seeks to avoid risk to an extent to which they cannot price
their reinsurance contracts. Thus, it is preferable to consider the case of a
wealth-maximising reinsurer, as in this paper, to not penalise tail losses
twice.

Next, we illustrate the solution, when the reinsurer is both ambiguity- and
risk- averse and the aggregate loss of the reinsurer has bounded support. This
holds in particular, for excess-of-loss reinsurance contracts with a policy
limit, as shown in the next example.

######  Example 6.3.

We consider the equilibrium of an excess-of-loss reinsurance contract with a
premium limit between two insurers and a reinsurer, when the reinsurer
maximises their expected CARA utility. The model parameters are the same as in
Example 5.6, i.e., the insurers’ scale parameters and intensities are
$\xi_{1}=1,\,\lambda_{1}=2$ and $\xi_{2}=1.25,\,\lambda_{2}=2.5$,
respectively. The insurers have risk aversion parameters
$\gamma_{1}=\gamma_{2}=0.5$ and the policy limits are $\ell_{1}=\ell_{2}=1$.

Figure 8 shows the quantile of the insurer-$k$’s retention limit
$F_{k}(\alpha_{k}^{*})$ and the safety loading, $\eta_{k}^{*}$, as a function
of the reinsurer’s risk aversion parameter, $m$. The points mark the solution
in the case where the reinsurer is wealth-maximising (which corresponds to the
limiting case, $m\to 0$). As expected, the safety loadings, $\eta_{k}^{*}$, of
both insurers are increasing in $m$ for all levels of ambiguity $\varepsilon$,
and thus as $m$ increases, the insurers buy less reinsurance and retain a
larger part of the losses themselves.

As both insurers have a policy limit of 1, insurer-1 is insuring further into
the tail of the loss distribution than insurer-2. For example, when $m=0$ and
$\varepsilon=0$, insurer-2 insures from the 71st to the 87th quantile of their
loss distribution, whereas insurer-1 insures from the 84th to the 94th
quantile of their loss distribution. Given that the CARA utility increases the
reinsurer’s risk aversion for larger losses, this causes the reinsurer to
charge a much higher safety loading to insurer-1 than to insurer-2. This high
price pushes insurer-1 to retain nearly all losses ($F_{1}(\alpha_{1}^{*})$
approaches 1) for $m$ close to 1.

$F_{k}(\alpha_{k}^{*})$$\eta_{k}^{*}$0.000.250.500.751.000.000.250.500.751.005101520250.70.80.91.0$m$$\varepsilon$00.050.1Insurer
($k$)12 Figure 8: The equilibrium safety loadings charged to insurer-$k$,
$\eta_{k}^{*}$, and the quantiles of their retention limits,
$F_{k}(\alpha_{k}^{*})$, $k=1,2$, as a function of the reinsurer’s risk-
aversion parameter, $m$. The insurers have risk aversion parameters
$\gamma_{1}=\gamma_{2}=0.5$, intensities $\lambda_{1}=2,\,\lambda_{2}=2.5$,
and their severity distributions are exponential with scale parameters
$\xi_{1}=1,\,\xi_{2}=1.25$. The insurers’ loss models are equally weighted
with $\pi_{1}=\pi_{2}=0.5$.

##  7 Conclusion

We study optimal reinsurance in continuous time via a Stackelberg game.
Specifically, we consider the novel problem of an ambiguity-averse reinsurer
who interacts with multiple insurers, where each insurer has a different loss
model and utility function. Each insurer chooses the portion of their losses
to reinsure that optimises their expected utility, while the reinsurer sets
prices that maximise their expected wealth and simultaneously accounts for
ambiguity around the insurer’s loss models. In particular, the reinsurer has
(a) a priori beliefs in each insurer’s model, and then penalises each model
with the Kullback-Leibler divergence, and (b) an overall ambiguity factor
indicating uncertainty in the insurance market. We solve this Stackelberg game
for a unified framework of reinsurance contracts and allow the reinsurer’s
probability distortion to depend on time, claim frequency, and claim severity.
We find that at the Stackelberg equilibrium, the reinsurer sets prices under a
model which is closely related to the barycentre of the insurers’ loss models.
Indeed if the reinsurer is ambiguity neutral, they choose the barycentre
model, while if the reinsurer is ambiguity averse, they weight the tail of
their losses more heavily. We illustrate the Stackelberg equilibrium on
proportional reinsurance and excess-of-loss reinsurance with and without a
policy limit.

We further compare our solution to the case where the ambiguity-averse
reinsurer maximises their CARA utility, generalising known results in the
literature to account for ambiguity in the severity distribution. We find that
a solution to the utility maximisation case only exists in very restrictive
cases, such as when the losses covered by the reinsurer are bounded. In this
case, the safety loadings charged to the insurers are increasing in the
reinsurer’s risk aversion parameter.

Study of the optimal behaviour of a reinsurer that contracts with multiple
insurers opens up many possible directions for further research. Following
[18], who study reverse stress testing of compound Poisson models, one could
consider stress testing of the reinsurer’s model. For example, one could ask
what type of shock would exceed a given risk threshold for the reinsurer, or
threaten their solvency. Another fruitful direction would be to allow for more
heterogeneity in the insurers’ beliefs about their losses by relaxing the
assumption of the equivalence of the insurers’ probability measures, as [5] do
in the one-period setting.

## Acknowledgements

EK is supported by an NSERC Canada Graduate Scholarship-Doctoral. SJ and SP
would like to acknowledge support from the Natural Sciences and Engineering
Research Council of Canada (grants RGPIN-2018-05705, RGPAS-2018-522715, and
DGECR-2020-00333, RGPIN-2020-04289). SP also acknowledges support from the
Canadian Statistical Sciences Institute (CANSSI).

##  Appendix A Proof of Proposition 6.1

###### Proof.

We write the reinsurer’s value function as

$\Phi(t,y)=\sup_{\eta\in{\mathfrak{C}}}\inf_{{\varsigma}\in\mathcal{V}}\,\mathbb{E}^{\mathbb{Q}^{\varsigma}}_{t,y}\left[u(Y_{T})-\frac{m}{\varepsilon}\int_{t}^{T}\int_{\mathds{R}}\Phi(s,Y_{s-})\left({\varsigma}_{s}(z)\left[\log\left(\frac{{\varsigma}_{s}(z)}{v^{g}(z)}\right)-1\right]+v^{a}(z)\right)dz\,ds\right]\,.$

By the dynamic programming principle, the value function $\Phi$ should satisfy
the HJBI equation:

$\displaystyle
0=\partial_{t}\Phi(t,y)+\sup_{\boldsymbol{c}\in{\mathds{R}}^{n}_{+}}\inf_{v\in{\mathcal{G}}}\Bigg{\\{}$
$\displaystyle\sum_{k\in{\mathcal{N}}}p_{k}^{R}\big{(}\alpha^{\dagger}_{k}[c_{k}],c_{k}\big{)}\partial_{y}\Phi(t,y)+\int_{\mathds{R}}\left[\Phi(t,y-L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}]))\,-\Phi(t,y)\right]v(t,z)\,dz$
$\displaystyle+\Phi(t,y)\left(\frac{m}{\varepsilon}\right)\int_{\mathds{R}}\left(v(t,z)\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-1\right]+v^{a}(z)\right)dz\Bigg{\\}}\,,$

with terminal condition $\Phi(T,y)=-\tfrac{1}{m}e^{-my}$.

Given the functional form of the terminal condition, we make the Ansatz
$\Phi(t,y)=-\tfrac{1}{m}\,e^{-m[f(t)+g(t)y]}$ for deterministic functions
$f(t)$ and $g(t)$ satisfying terminal conditions $f(T)=0$ and $g(T)=1$. Upon
substituting the Ansatz into the HJBI equation and simplifying, we obtain the
ODE

$\displaystyle
0=mf^{\prime}(t)+\sup_{\boldsymbol{c}\in{\mathds{R}}^{n}_{+}}\inf_{v\in{\mathcal{G}}}\Bigg{\\{}$
$\displaystyle
m\sum_{k\in{\mathcal{N}}}p_{k}^{R}\left(\alpha^{\dagger}_{k}[c_{k}],c_{k}\right)-\int_{\mathds{R}}\left(e^{mL(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}-1\right)v(t,z)\,dz$
$\displaystyle+\frac{m}{\varepsilon}\int_{\mathds{R}}\left(v(t,z)\left[\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-1\right]+v^{a}(z)\right)dz\Bigg{\\}}\,,$

with terminal condition $f(T)=0$.

We first consider the inner optimisation, that is the infimum problem. We wish
to minimize the functional of $v$ given by

$\mathcal{L}[v]=\int_{{\mathds{R}}_{+}}\\!\\!v(t,z)\left[\frac{m}{\varepsilon}\left(\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-1\right)-e^{mL(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}+1\right]dz\,.$

Let $\delta>0$, $h$ be an arbitrary function such that $\delta
h+\nu\in{\mathcal{G}}$. Applying a variational first order condition, we
obtain

$\displaystyle 0$ $\displaystyle=\lim_{\delta\to 0}\frac{\mathcal{L}[v+\delta
h]-\mathcal{L}[v]}{\delta}=\int_{{\mathds{R}}_{+}}h(z)\left[\frac{m}{\varepsilon}\log\left(\frac{v(t,z)}{v^{g}(z)}\right)-e^{mL(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}+1\right]dz\,.$

Since $h$ was arbitrary, this implies that the term in the square brackets
must equal 0, and so the optimal compensator in feedback form,
${\varsigma}^{*}$:

$\varsigma^{*}_{t}(z,\boldsymbol{c})=v^{g}(z)\;\exp\left\\{\frac{\varepsilon}{m}\left(e^{mL(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}-1\right)\right\\}\,.$

Substituting this into the HBJI equation above, we have that

$0=mf^{\prime}(t)+\sup_{\boldsymbol{c}\in{\mathds{R}}^{n}_{+}}\Bigg{\\{}m\sum_{k\in{\mathcal{N}}}p_{k}^{R}\big{(}\alpha^{\dagger}_{k}[c_{k}],c_{k}\big{)}-\frac{m}{\varepsilon}\int_{\mathds{R}}\varsigma^{*}_{t}(z,\boldsymbol{c})\,dz+\frac{m}{\varepsilon}\sum_{k\in{\mathcal{N}}}\pi_{k}\lambda_{k}\mu_{k}\Bigg{\\}}\,.$

Next, the derivative of the term in curly braces with respect to $c_{k}$,
$k\in{\mathcal{N}}$ is

$\displaystyle\frac{\partial}{\partial c_{k}}\\{\cdots\\}=$ $\displaystyle
m\left[\int_{0}^{\infty}\left[z-r(z,\alpha^{\dagger}_{k}[c_{k}])\right]\nu_{k}(dz)-(1+c_{k})\int_{0}^{\infty}\partial_{c_{k}}r(z,\alpha_{k}^{\dagger}[c_{k}])\nu_{k}(dz)\right]$
$\displaystyle+m\int_{0}^{\infty}\partial_{c_{k}}r(z,\alpha_{k}^{\dagger}[c_{k}])\,\exp\left\\{\frac{\varepsilon}{m}\left[e^{m\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])}-1\right]+m\,L(z,{\boldsymbol{\alpha}}^{\dagger}[\boldsymbol{c}])\right\\}\,v^{g}(z)\,dz\,.$

The first order conditions require setting this expression to $0$ for all
$k\in{\mathcal{N}}$ and all $t\in[0,T]$. Hence, we obtain that the optimal
$\eta_{k,t}^{*}$ satisfies the non-linear equation

$\eta_{k,t}^{*}=\frac{\displaystyle\int_{0}^{\infty}\left[z-r(z,\alpha^{\dagger}_{k}[\eta_{k,t}^{*}])\right]\,\nu_{k}(dz)}{\displaystyle\int_{0}^{\infty}\partial_{c}\,r(z,\alpha_{k}^{\dagger}[c])\Big{|}_{c=\eta_{k,t}^{*}}\nu_{k}(dz)}+\frac{\displaystyle\int_{0}^{\infty}e^{m\hskip
0.70004ptL(z,{\boldsymbol{\alpha}}^{\dagger}[{\boldsymbol{\eta}}^{*}])}\,\partial_{a}r(z,a)\Big{|}_{a=\alpha_{k}^{\dagger}[\eta_{k,t}^{*}]}\,\varsigma_{t}^{*}(z,\boldsymbol{c})\,dz}{\displaystyle\int_{0}^{\infty}\partial_{a}r(z,a)\Big{|}_{a=\alpha_{k}^{\dagger}[\eta_{k,t}^{*}]}\nu_{k}(dz)}-1.$

Again, there is no explicit time dependence in the above equation, so the
roots of these non-linear equations are constant over time. ∎

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