# Impulse Control of Proportional Reinsurance with Constraints

Authors: Hui Meng; Tak Kuen Siu
Journal: International Journal of Stochastic Analysis
 (2011)

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## Abstract

We consider an insurance company whose surplus follows a diffusion process
with proportional reinsurance and impulse dividend control. Our objective is to
maximize expected discounted dividend payouts to shareholders of the company
until the time of bankruptcy. To meet some essential requirements of solvency
control (e.g., bankruptcy not soon), we impose some constraints on the insurance
company's dividend policy. Under two types of constraints, we derive the value
functions and optimal control policies of the company.


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## Body

## 1. Introduction

Reinsurance is an effective tool for insurance companies to manage and control their exposure to risk, and distributions of dividends are used by firms as a vehicle for distributing some of their profits to their shareholders. The problem of determining an optimal dividend policy can be formulated as a singular/regular stochastic control problem in absence of fixed transaction costs, or an impulse control consisting of lump sum dividends distributed at discrete moments of time with fixed transaction cost. For details, interested readers may refer to Gerber [1], Asmussen and Taksar [2], Paulsen [3], Benkherouf and Bensoussan [4], and Cadenillas et al. [5].Recently, optimizing dividends payouts with solvency constraints have received much attention. For example, Paulsen [6] and He et al. [7] studied optimal singular dividend problems under barrier constraints with no reinsurance and proportional reinsurance, respectively. Choulli et al. [8] investigated an optimal singular dividend problem under constrained proportional reinsurance. Bai et al. [9] and Ormeci et al. [10] considered optimal impulse dividend problems under different constraints. By these ideas we further discuss an optimal impulse control of an insurance company with proportional reinsurance policy under some different solvency constraints.The paper is organized as follows. In Section2 we establish optimal impulse control problems of the insurance company with proportional reinsurance policy and discrete dividends. In Section 3, we derive the value function and an optimal policy under some constraints of liquid reserves at impulse times. With some constraints of dividends amounts, we obtain the value function and an optimal policy in Section 4. The final section gives concluding remarks.


## 2. The Model

We fix a complete, filtered probability space(Ω,ℱ,{ℱt},ℙ) on which a real-valued, ({ℱt},ℙ)-standard Brownian motion is defined, where ℙ is a real-world probability as usual. Consider the following controlled process:(2.1)Xt=x+∫0tμu(s)ds+∫0tσu(s)dWs-∑n=1∞I{τn<t}ξn,
where μ>0, σ>0, u(t)∈[0,1], and {τi;i=1,2,…} is an increasing sequence of stopping times and {ξi;i=1,2,…} is a sequence of random variables, associated with amounts of the dividends paid to shareholders of an insurance company.Definition 2.1.
A pair(2.2)π≡{u;S}≡{u;τ1,τ2,…,τn,…;ξ1,ξ2,…,ξn,…}
is an admissible policy of an insurance company with initial capital x if it satisfies the following conditions:  (1)
for eachi=1,2,… and each t≥0, {τi≤t}∈ℱt and ξi∈ℱτi;(2)
the induced dividend process, sayQ, defined by
(2.3)Qt:=∑n=1∞I{τn<t}ξn
is {ℱt} adapted, increasing, and càdlàg;(3)
0<ξi≤Xτi-;(4)
the stochastic differential equation forX:={Xt∣t≥0} admits a unique strong solution;(5)
u(t)∈[0,1];(6)
P(limn→∞τn≤T)=0, for allT≥0. We write Π(x) for the space of these admissible policies.
For eachπ∈Π(x), we write  {Xtπ∣t≥0} for the surplus process of the insurance company associated with π. Here the superscript π is added to emphasize the dependence of the surplus process Xπ on the strategy π. The ruin time corresponding to π is defined as
(2.4)τπ:=inf⁡{t≥0:Xtπ≤0}
To simplify the notation, we suppress the superscript π and write τ:=τπ.
The goal of the insurance company is to select an optimal strategyπ∈Π(x) so as to maximize the expected present value of dividends before bankruptcy.
LetK, (K>0), be the fixed transaction cost attributed to the advisory and consulting fees and k, (0<k<1), the proportional transaction cost due to taxes on dividends. Then the optimization problem of the insurance company  is to select π∈Π(x) so as to maximize the following performance function:
(2.5)J(x,π):=E[∑n=1∞e-λτn(-K+kξn)I{τn≤τ}],
where λ is the impatient factor and λ>0, that is, to determine the value function
(2.6)V(x):=sup⁡{J(x,π);π∈Π(x)},
and the optimal strategy π* such that V(x)=J(x,π*).
The value functionV(x) is also called an optimal return function.
Without imposing any constraints, Cadenillas et al. [5] investigated the model under the performance function (2.5) and showed that the (s,S) dividend policy is optimal, that is, when the surplus reaches a barrier level S, it is reduced to s via a dividend payment, and the process continues. To meet some requirements of solvency control (e.g., bankruptcy not soon), we impose some constraints on the insurance company's dividend policy. Under solvency constraints Xτi-≥ζ¯,Xτi≥ζ̲, and 0≤ξi≤d, the optimal control problems are presented in Sections 3 and 4, respectively
In what follows, we still useV(x) to denote the value function different cases of constraints.


## 3. Case IXτi-≥ζ¯, Xτi≥ζ̲

Without reinsurance, the constraintsXτi-≥ζ¯ and Xτi≥ζ̲ were considered by Bai et al. [9]. To prove our main results, we first recall some results (Propositions 3.1–3.3) without constraints, see Cadenillas et al. [5].Proposition 3.1.
The optimal return functionV is a continuous, nondecreasing function in x satisfying V(0)=0.To simplify our notation, we define(3.1)γ=λ(μ2/2σ2)+λ,x0=(1-γ)σ2μ,θ+=-μ+μ2+2λσ2σ2,θ-=-μ-μ2+2λσ2σ2,a1=γx0γ-1-θ-x0γθ+-θ-,a2=θ+x0γ-γx0γ-1θ+-θ-.Proposition 3.2.
The functionν(x), subject to the linear growth condition ν(x)≤k(x+μ/c), is continuously differentiable on (0,∞) and is twice continuously differentiable on (0,x1C̃)∪(x1C̃,∞), where
(3.2)ν(x)={C̃xγ,0≤x<x0,C̃(a1eθ+(x-x0)+a2eθ-(x-x0)),x0≤x≤x1C̃,k(x-x̃C̃)+ν(x̃C̃)-K,x≥x1C̃.
Here C̃,x̃C̃, and x1C̃ satisfy the following conditions:
(3.3)ν′(x̃C̃)=ν′(x1C̃)=k,ν(x1C̃)-ν(x̃C̃)=k(x1C̃-x̃C̃)-K.Proposition 3.3.
The control(3.4)π*=(u*,T*,ξ*)=(u*;τ1*,τ2*,…,τn*,…;ξ1*,ξ2*,…,ξn*,…)
defined by
(3.5)u*(t)=u*(Xt*)={μσ2(1-γ)Xt*,0≤Xt*≤x0,1,Xt*≥x0,τ1*=inf⁡{t≥0:Xt*=x1C̃},ξ1*=x1C̃-x̃C̃,
and for every n≥2(3.6)τn*=inf⁡{t≥τn-1*:Xt*=x1C̃},ξn*=x1C̃-x̃C̃,
where X* is the solution of the stochastic differential equation
(3.7)Xt*=x+∫0tμu*(Xs*)ds+∫0tσu*(Xs*)dWs-(x1C̃-x̃C̃)∑n=1∞I{τn*≤t},
is the QVI control associated with the function ν(x) defined by (3.2). This control is optimal, and the function ν(x) coincides with the value function. That is,
(3.8)V(x)=ν(x)=J(x,π*)=J(x;u*,T*,ξ*).For a functionψ(x), we define the operator by(3.9)Luψ(x)=12σ2u2ψ′′(x)+μuψ′(x)-λψ(x).
Define, for ζ̲>x̃C̃,(3.10)νζ̲(x):={Cxγ,0≤x<x0,C(a1eθ+(x-x0)+a2eθ-(x-x0)),x0≤x≤ζ¯1,k(x-ζ̲)+νζ̲(ζ̲)-K,x>ζ¯1,
where C and ζ¯1 satisfy the following conditions:(3.11)νζ̲′(ζ¯1)=k,νζ̲(ζ¯1)-νζ̲(ζ̲)=k(ζ¯1-ζ̲)-K.
We can easily prove that νζ̲′(x) is convex on (0,ζ¯1) and ζ¯1>x1C̃.Lemma 3.4.
Forζ̲>x̃C̃, (a)
maxu∈[0,1]ℒuνζ̲(x)=0 for 0≤x≤ζ¯1 and maxu∈[0,1]ℒuνζ̲(x)<0 for x>ζ¯1,(b)
fory≥x≥ζ̲, one has(3.12)νζ̲(y)-νζ̲(x)≥k(y-x)-K,
and the equality holds when x=ζ̲ and y≥ζ¯1.Proof.
(a) Forx<x0,(3.13)u*(x)=Argmax⁡u∈[0,1][12σ2u2γ(γ-1)+μuγx-λx2]=-μxσ2(γ-1),max⁡u∈[0,1]⁡Luνζ̲(x)=Cxγ-2max⁡u∈[0,1]⁡[12σ2u2γ(γ-1)+μuγx-λx2]=Cxγ-2[12σ2u*2γ(γ-1)+μu*γx-λx2]=0.
Let
(3.14)x*=Arg⁡min⁡⁡νζ̲′(x).
That is, x* satisfies
(3.15)a1θ+2eθ+(x*-x0)+a2θ-2eθ-(x*-x0)=0.
For x0<x<x*,
(3.16)u¯(x)=Argmax⁡u∈[-∞,∞]Luνζ̲(x)=-μσ2a1θ+eθ+(x-x0)+a2θ-eθ-(x-x0)a1θ+2eθ+(x-x0)+a2θ-2eθ-(x-x0).
We can easily show that u¯(x) is an increasing function of x on (x0,x*) and u¯(x)≥u¯(x0)=1.
Forx*<x<ζ¯1, the function νζ̲′(x) is increasing and νζ̲′′(x)>0. Thus for x0<x<ζ¯1,
(3.17)u*(x)=Arg⁡max⁡u∈[0,1]⁡Luνζ̲(x)=1,max⁡u∈[0,1]⁡Luνζ̲(x)=12σ2C[a1θ+2eθ+(x-x0)+a2θ-2eθ-(x-x0)]+μC[a1θ+eθ+(x-x0)+a2θ-eθ-(x-x0)]-λC[a1eθ+(x-x0)+a2eθ-(x-x0)]=0.
From the above steps of the proof, we notice that (1/2)σ2νζ̲′′(ζ¯1-)+μk-λνζ̲(ζ¯1)=0. Thus, for x>ζ¯1,
(3.18)max⁡u∈[0,1]Luνζ̲(x)=max⁡u∈[0,1]{μuk-λ[k(x-ζ̲)+νζ̲(ζ̲)-K]}=μk-λ[k(x-ζ̲)+νζ̲(ζ̲)-K]≤μk-λνζ̲(ζ¯1)=-12σ2νζ̲′′(ζ¯1-)<0.
(b) By the observation,νζ̲′(x)≤k on (ζ̲,∞). Consequently,
(3.19)νζ̲(y)-νζ̲(x)-k(y-x)=∫xy(νζ̲′(z)-k)dz
is smallest for x=ζ̲ and y≥ζ¯1 with constrained condition y≥x≥ζ̲, and then it equals -K. This completes the proof.To consider the case whereXτi-≥ζ¯ and Xτi≥ζ̲, we first consider the case where Xτi≥ζ.Proposition 3.5.
Assume the dividend policy has to satisfyXτi≥ζ̲ for some positive ζ̲, that is, the surplus is not allowed to be less than ζ̲ immediately after the dividend payout. Then,(a)
ifζ̲≤x̃C̃, the optimal policy and the value function are as in Proposition 3.3;(b)
ifζ̲>x̃C̃, the optimal policy and the value function are(3.20)u*(t)=u*(Xt*)={μσ2(1-γ)Xt*,0≤Xt*<x0,1,x≥x0,τ1*=inf⁡{t≥0:Xt*=ζ¯1},ξ1*=ζ¯1-ζ̲,
and for every n≥2(3.21)τn*=inf⁡{t≥τn-1*:Xt*=ζ¯1},ξn*=ζ¯1-ζ̲,
and V(x)=νζ̲(x).Proof.
Part (a) is obvious since the optimal policy is feasible under the constraint.
The idea of the proof of Part (b) is similar to that of Corollary 2.2 of Alvarez and Lempa [11].
Sinceνζ̲(x) is not twice continuously differentiable at ζ¯1, we cannot use Ito's differentiation rule directly. However, we can show that there exists a sequence {νζ̲,n}n=1∞ of mappings νζ̲,n∈𝒞2(ℝ+) such that as n→∞(1)
νζ̲,n→νζ̲ uniformly on compact subsets of ℝ+;(2)
ℒu(νζ̲,n)→ℒu(νζ̲) uniformly on compact subsets of ℝ+∖𝒩, where 𝒩 is a subset of ℝ+ which of measure zero;(3)
{ℒu(νζ̲,n)}n=1∞ is locally bounded on ℝ+.
Applying Ito's differentiation rule to the mapping(t,x)↦e-λtνζ̲,n(x), conditioning on ℱτj, and reordering terms yield
(3.22)e-λτjνζ̲,n(Xτj)=EFτj[e-λτj+1νζ̲,n(Xτj+1-)]-EFτj[∫τjτj+1-e-λsLuνζ̲,n(Xs)ds],
where Xτj≥ζ̲.
Lettingn→∞, applying Fatou's theorem, and invoking the use of the variational inequality ℒaνζ̲(x)≤0 then result in the inequality
(3.23)e-λτjνζ̲(Xτj)≥EFτj[e-λτj+1νζ̲(Xτj+1-)].
Taking expectation in both sides, we have
(3.24)E[e-λτjνζ̲(Xτj)]≥E[e-λτj+1νζ̲(Xτj+1-)].
Letting τ0=0, summing over j, and applying the nonnegativity of the mapping νζ̲(x) give
(3.25)νζ̲(x)≥∑j=1nE[e-λτj(νζ̲(Xτj-)-νζ̲(Xτj))I{τj<τ}].
Since Xτj=Xτj--ξj, νζ̲(y)-νζ̲(x)≥k(y-x)-K with y≥x≥ζ̲, we find that
(3.26)νζ̲(x)≥E[∑j=1ne-λτj(kξj-K)I{τj<τ}].
Letting n→∞ and invoking the use of the dominated convergence then imply that
(3.27)νζ̲(x)≥E[∑j=1∞e-λτj(kξj-K)I{τj<τ}].
Using the strategy of Part (b) gives the equality
(3.28)νζ̲(x)=E[∑j=1∞e-λτj*(kξj*-K)I{τj*<τ}].Theorem 3.6.
Assume the dividend policy has to satisfyXτi-≥ζ¯ and Xτi≥ζ̲ for some positive ζ̲ and ζ¯. Then,(a)
ifζ̲≤x̃C̃ and ζ¯≤x1C̃, the optimal policy and value function are as in Proposition 3.3;(b)
ifζ̲≤x̃C̃ and ζ¯>x1C̃, the optimal policy and value function are(3.29)u*(t)=u*(Xt*)={μσ2(1-γ)Xt*,0≤Xt*<x0,1,Xt*≥x0,τ1*=inf⁡{t≥0:Xt*=ζ¯},ξ1*=ζ¯-x̃C̃,
and for every n≥2(3.30)τn*=inf⁡{t≥τn-1*:Xt*=ζ¯},ξn*=ζ¯-x̃C̃,V(x)=νx̃C̃,ζ¯(x):={Cxγ,0≤x<x0,C(a1eθ+(x-x0)+a2eθ-(x-x0)),x0≤x≤ζ¯,k(x-x̃C̃)+νx̃C̃,ζ¯(x̃C̃)-K,x>ζ¯,
and C satisfies the following conditions:
(3.31)νx̃C̃,ζ¯(ζ¯)-νx̃C̃,ζ¯(x̃C̃)=k(ζ¯-x̃C̃)-K;(c)
ifζ̲>x̃C̃ and ζ¯≤ζ¯1, the optimal policy and value function are as in Proposition 3.5;(d)
ifζ̲>x̃C̃ and ζ¯>ζ¯1, the optimal policy and value function are(3.32)u*(t)=u*(Xt*)={μσ2(1-γ)Xt*,0≤Xt*<x0,1,Xt*≥x0,τ1*=inf⁡{t≥0:Xt*=ζ¯},ξ1*=ζ¯-ζ̲,
and for every n≥2(3.33)τn*=inf⁡{t≥τn-1*:Xt*=ζ¯},ξn*=ζ¯-ζ̲,V(x)=νζ̲,ζ¯(x):={Cxγ,0≤x<x0,C(a1eθ+(x-x0)+a2eθ-(x-x0)),x0≤x≤ζ¯,k(x-ζ̲)+νζ̲,ζ¯(ζ̲)-K,x>ζ¯,
and C satisfies the following condition:
(3.34)νζ̲,ζ¯(ζ¯)-νζ̲,ζ¯(ζ̲)=k(ζ¯-ζ̲)-K.Proof.
Parts (a) and (c) are obvious since the optimal policy is feasible under the constraint. The proofs of parts (b) and (d) are similar to Theorem 3.2 in [9], so we state the result here without giving the proof.


## 4. Case II0≤ξi≤d

Without reinsurance, the constraint0≤ξi≤d was considered by Ormeci et al. [10].Ifd≤(K/k), by definition (2.5), we know that the optimal policy is τ1*=∞. So we only consider the case where d>K/k.Ifξn*=x1C̃-x̃C̃≤d, the control band policy that is optimal for the unconstrained problem is also optimal for the constrained problem. If ξn*=x1C̃-x̃C̃>d, we will prove that a control band policy is also an optimal policy for the constrained problem. To prove this result, we use the Lagrangian relaxation, that is, to introduce a Lagrange multiplier δ≥0. For each scalar δ≥0 and policy π, we define the Lagrangian function (4.1)J(x,π,δ)∶=E[∑n=1∞e-λτn[-K+kξn+δ(d-ξn)]I{τn≤τ}]=E[∑n=1∞e-λτn[-(K-δd)+(k-δ)ξi]I{τn≤τ}].The resulting unconstrained problem is equivalent to the original problem with parametersK-δd and k-δ and gives an upper bound on the objective function of the original constrained problem. That is,(4.2)J(x,π)≤J(x,π,δ),V(x)≤sup⁡0≤ξ≤dJ(x,π,δ),for0≤δ≤K/d.
By conditions 0≤δ≤K/d,d>K/k, we deduce that δ<k. In the following, we find a control band policy that achieves this bound thereby proving its optimality.Forδ∈[0,K/d], if there exists an x̃∈(0,x0] satisfying
(4.3)Cγx̃γ-1=k-δ,C(a1θ+eθ+(x̃+d-x0)+a2θ-eθ-(x̃+d-x0))=k-δ,
then x̃ satisfies
(4.4)γx̃γ-1=a1θ+eθ+(x̃+d-x0)+a2θ-eθ-(x̃+d-x0),(4.5)C=k-δγx̃γ-1.
Define
(4.6)f(δ):=(k-δ)d-(ν(x̃+d)-ν(x̃))=(k-δ)d-C(a1eθ+(x̃+d-x0)+a2eθ-(x̃+d-x0)-γx̃γ-1),
where C and ν(x) are given by (4.5) and (3.2), respectively.Obviously, we havef(K/d)>0. It can be shown that f(0)<I(C̃)=K, where I(C̃) can be seen in Equation (5.25) of Cadenillas et al. [5]. For f(δ) being a decreasing function of δ, the equation(4.7)f(δ)=K-δd
has a unique solution δ*∈(0,K/d).If there does not exist anx̃∈(0,x0] satisfying (4.3), then there must exist an x̃∈(x0,∞) such that(4.8)C(a1θ+eθ+(x̃-x0)+a2θ-eθ-(x̃-x0))=k-δ,C(a1θ+eθ+(x̃+d-x0)+a2θ-eθ-(x̃+d-x0))=k-δ,
which results in x̃ satisfying
(4.9)a1θ+eθ+(x̃-x0)+a2θ-eθ-(x̃-x0)=a1θ+eθ+(x̃+d-x0)+a2θ-eθ-(x̃+d-x0),C=k-δa1θ+eθ+(x̃-x0)+a2θ-eθ-(x̃-x0).
Define(4.10)g(δ):=(k-δ)d-(ν(x̃+d)-ν(x̃))=(k-δ)d-C[a1eθ+(x̃+d-x0)+a2eθ-(x̃+d-x0)-a1eθ+(x̃-x0)-a2eθ-(x̃-x0)],
where C and ν(x) are given by (4.11) and (3.2), respectively.By a similar analysis, we also can show that the equation(4.11)g(δ)=K-δd
has a unique solution δ*∈(0,K/d).Theorem 4.1.
Assume the dividend policy has to satisfy0≤ξn≤d. Then,(a)
ifx1C̃-x̃C̃≤d, the optimal policy and value function are as in Proposition 3.3;(b)
ifx1C̃-x̃C̃>d, the optimal policy and value function are(4.12)u*(t)=u*(Xt*)={μσ2(1-γ)Xt*,0≤Xt*<x0,1,Xt*≥x0,τ1*=inf⁡{t≥0:Xt*=x̃+d},ξ1*=d,
and for every n≥2(4.13)τn*=inf⁡{t≥τn-1*:Xt*=x̃+d},ξn*=d,V(x)=νd(x):={Cxγ,0≤x<x0,C(a1eθ+(x-x0)+a2eθ-(x-x0)),x0≤x≤x̃+d,(k-δ*)(x-x̃)+νd(x̃)-(K-δ*d),x>x̃+d,
and C and x̃ satisfy the following conditions:
(4.14)νd′(x̃)=k-δ*,νd′(x̃+d)=k-δ*,νd(x̃+d)-νd(x̃)=(k-δ*)d-(K-δ*d).Proof.
We only prove Part (b). By the above analysis and a similar proof of Cadenillas et al. [5],
(4.15)sup⁡0≤ξi<∞J(x,π,δ*)=E[∑n=1∞e-λτn*[-K+kξn*+δ*(d-ξn*)]I{τn*≤τ,ξn*=d}]=E[∑n=1∞e-λτn*[-K+kξn*]I{τn*≤τ,ξn*=d}]=J(x,π*).
However,
(4.16)J(x,π*)≤V(x)≤sup⁡0≤ξ≤dJ(x,π,δ*)≤sup⁡0≤ξi<∞J(x,π,δ*).
By (4.15) and (4.16), we obtain
(4.17)J(x,π*)=V(x),
which results in the claim of the theorem.


## 5. Conclusion

We have discussed some important issues about the combined optimal reinsurance and dividend problem in the presence of both fixed and proportional transaction costs. We supposed that the goal of the insurance company is to maximize the expected present value of dividends and formulated the problem into an optimal impulse control problem. Under some cases of constraints, we provided a detailed mathematical analysis for the solution of the problem and derived that the optimal dividend strategy is still an(s,S)-(band)-policy.



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*Source: 190603-2011-08-10.xml*