SCIENCE CHINA
Mathematics
. ARTICLES .
doi: 10.1007/s11425-015-0542-7
Time-consistent investment-reinsurance strategies
towards joint interests of the insurer and the
reinsurer under CEV models
ZHAO Hui1 , WENG ChengGuo2 , SHEN Yang3 & ZENG Yan4,∗
1School of Science, Tianjin University, Tianjin 300072, China;
of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N 2L 3G1, Canada;
3Department of Mathematics and Statistics, York University, Toronto, ON M 3J 1P 3, Canada;
4Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China
Email: [email protected], [email protected], [email protected], [email protected]
2Department
Received August 15, 2015; accepted April 18, 2016
Abstract The present paper studies time-consistent solutions to an investment-reinsurance problem under a
mean-variance framework. The paper is distinguished from other literature by taking into account the interests
of both an insurer and a reinsurer jointly. The claim process of the insurer is governed by a Brownian motion
with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the
expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is
distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated
on a weighted sum of the insurer’s and the reinsurer’s surplus processes. Upon a verification theorem, which
is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed
investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are provided
to demonstrate those derived results as well as the economic implications behind.
Keywords
investment-reinsurance problem, mean-variance analysis, time-consistent strategy, constant elasticity of variance model
MSC(2010)
60H30, 90C39, 91B30, 91G80
Citation: Zhao H, Weng C G, Shen Y, et al. Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models. Sci China Math, 2017, 60, doi: 10.1007/s11425015-0542-7
1
Introduction
The quest for optimal reinsurance design has remained a fascinating problem among insurers, reinsurers and academicians. An appropriate use of reinsurance could reduce the underwriting risk of an
insurer and thereby enhance its value. This explains why almost all the insurance companies throughout
the world have a reinsurance program. Reinsurance has also been used as an effective mechanism of
risk-sharing within an insurance group for various purposes, such as tax alleviation, stabilizing the profitability and satisfying external regulatory capital requirement. The optimal reinsurance design models
can primarily be divided into two classes, known as static models and dynamic models. In the static
models, the reinsurance agreement is reached in advance with fixed provisions to apply over a time horizon and therefore they are also called the single period models. Recent literature on the static models
include [8–10] and references therein.
∗ Corresponding
author
c Science China Press and Springer-Verlag Berlin Heidelberg 2016
math.scichina.com
link.springer.com
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In a dynamic model, the provisions of the reinsurance treaty are adjusted dynamically over time depending on the accumulated information. Typically, the model boils down to solve an optimal control
problem. In a dynamic setting, the literature commonly study the optimal reinsurance strategy and
investment strategy simultaneously, leading to the optimal investment-reinsurance problem. The optimality criteria adopted in the literature for such problems include minimizing the insurer’s ruin probability
(see [1,35]), maximizing the adjustment coefficient of the insurer’s risk process (see [23]), maximizing the
insurer’s expected utility (see [2, 33, 40]), and mean-variance (MV) optimization (see [3, 4, 13, 36, 37]).
In the present paper, we focus on the MV criterion to study the optimal reinsurance-investment
problem. The proportional reinsurance treaty is considered in our model, and moreover, both the insurer
and reinsurer are assumed to invest on a risky asset, which is distinct for each other, in additional
to a risk free asset. The dynamics of the risky assets for the insurer and the reinsurer to invest are
driven by a constant elasticity of variance (CEV) model, which has been widely applied due to its
parsimonious form and its ability in capturing the volatility smile of financial asset returns. The CEV
model has also been commonly advocated in actuarial literature, see, for example, [18, 21, 32] among
many others. Also, refer to [38] for a precommitment MV problem under the CEV model. To solve the
proposed reinsurance-investment model, we embed it into a more general dynamic control problem and
formally establish a verification theorem for the general problem, which may have potential applications
for other relevant problems. Upon the established verification theorem, explicit solutions are obtained
for the proposed reinsurance-investment model. Moreover, numerous sensitivity analysis is conducted,
both mathematically and numerically with in-depth explanations on the economic implications behind,
to explore how the obtained investment-reinsurance strategies depend on various exogenous parameters,
including weight parameter in the weighted sum process, the reinsurance premium rate, as well as those
associated with the investment assets.
Along with many others, three main features distinguish the present paper from those existing literature
on optimal investment-reinsurance problems. First, the interests of the insurer and the reinsurer are
jointly reflected in our optimal reinsurance model, while a majority of the existing literature study the
optimal reinsurance problem (in both static and dynamic settings) merely from the insurer’s perspective.
Indeed, to the best of our knowledge, [8, 15, 28, 29] are the only five papers with an optimal reinsurance
model jointly reflecting the interests of both parties, and only Li et al. [28, 29] studied the problems in
dynamic settings with the optimality criteria of maximizing the product of the utilities and the weighted
sum of objectives for both parties, respectively. In the present paper, we consider a weighted sum of the
insurer’s and the reinsurer’s surplus processes and study the optimal reinsurance and investment strategies
for both parties by a mean-variance analysis on the weighted process. The weight can be viewed as a
regularization parameter to stress the importance of each party on the contract in the process of decision,
and in the special case with the weight being equal to 1, our model reduces to the corresponding problem
for a mean-variance insurer only. Thus, our model is more general than other literature in this aspect.
Second, our model can also be viewed as a decision formulation for an insurance group. As we know, in
reality only the smallest insurers exist as a single corporation, and most major insurance companies actually exist as insurance groups, say, for example, Allianz SE, Zurich Insurance Group, Munich Reinsurance
Company among many others. Therefore, from the risk management point of view, a risk transfer among
an insurance group is a natural risk management mechanism for insurance companies, and such a risk
transfer is often realized via certain reinsurance contracts between the insurer and the other members
(or another member) within the group. While different member companies may have distinct capacity
to absorb risks of different types, all the counter-parties in the reinsurance contracts can be generally
viewed as a single identity in terms of absorbing the risk from the insurer. Therefore, the problem can be
abstracted to a framework for an insurance group consisting of only two members—one insurer and one
reinsurer. Throughout the paper, we shall frequently refer to such a framework to explain the economic
meanings of the our results.
Third, the MV analysis for the optimal reinsurance-investment strategies is a time-inconsistent problem
in that an optimal solution obtained at a time is no longer optimal as the time goes forward into a
future point. In the present paper, we aim to develop time-consistent reinsurance-investment strategy,
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whereas the solutions obtained in the most previous literature, including those reviewed previously, are
time-inconsistent. In the present paper, we view the entire decision process as a non-cooperative game
with one player at each time point over the whole investment time horizon, who can be viewed as the
future incarnation of the decision-maker, and resort to the (subgame perfect Nash) equilibrium strategy,
which is a time-consistent solution. The equilibrium strategy has been popularly advocated for tackling
time-inconsistent problems in academia recently; see, for example, [5, 6, 24, 27, 41–43] among others.
There are also some references on time-consistent solutions to reinsurance-investment problems; see, for
example, [30, 31, 46]. They, however, address the problem only from the insurer’s perspective without
taking into account any concern from the reinsurer.
The paper proceeds as follows. The model formulation is given in Section 2 along with a verification
theorem for a general problem. The equilibrium solutions as well as some mathematical analysis are
presented in Section 3, and the numerical analysis is demonstrated in Section 4. Section 5 concludes the
paper, and all of the proofs are relegated to Appendices A and B.
2
2.1
Model formulation
Wealth processes
Let (Ω, F , P ) be a complete probability space with a filtration {Ft , t 0} satisfying the usual conditions,
i.e., the filtration contains all P -null sets and is right continuous. All stochastic processes in this paper
are assumed to be well defined and adapted to this filtration. Consider an insurer with a claim process
described by a Brownian motion with a drift as follows:
dC(t) = adt − bdW (t),
(2.1)
where a and b are two positive constants and W is a standard Brownian motion. The diffusion model for
the claim process is a limit of the classical compound Poisson model (see [20]), and such an approximation
has been widely adopted in actuarial literature (see [1, 21, 31]). Further suppose that the insurance
premium charged on the claims by the insurer is computed according to the expected value principle
with a loading factor θ > 0, so that the premium rate for the insurer is c = (1 + θ)a, and this leads to
the following surplus process for the insurer dR1 (t) = θadt + bdW (t).
To proceed, we assume that a proportional reinsurance is applied between the insurer and a reinsurer,
and let p(t) denote the proportion covered by the reinsurer at time t. Further suppose that the reinsurance
premium is also charged according to the expected value principle with a safety loading η > θ. In other
words, non-cheap reinsurance is considered. In this sense, regarding proportion p(t) as a control variable
is of practical meaning. Therefore, in the presence of the proportional reinsurance contract, the wealth
processes of the insurer and the reinsurer are respectively given by
dR1 (t) = (1 + θ)adt − (1 + η)ap(t)dt − (1 − p(t))dC(t)
= (θ − ηp(t))adt + b(1 − p(t))dW (t)
(2.2)
and
dR2 (t) = (1 + η)p(t)adt − p(t)dC(t) = ηp(t)adt + bp(t)dW (t).
(2.3)
Besides the insurance business, both the insurer and the reinsurer are allowed to invest in a risk-free
asset and a risky asset. We assume that the insurer and reinsurer specialize in two distinct risky assets
in their investment. Asset specialization can be justified by that investors tend to trade only in familiar
assets. Since analyzing stock data takes time and effort, each investor is likely to invest in familiar assets
which are in a subset of all available assets. The price process of the risk-free asset is given by
dB(t) = rB(t)dt,
B(0) = B0 > 0,
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where r > 0 is the risk-free interest rate. The price process of the risky asset available for the insurer to
invest is described by
dS1 (t) = S1 (t)[μ1 dt + σ1 (S1 (t))k1 dW1 (t)],
S1 (0) = s1,0 > 0,
(2.4)
S2 (0) = s2,0 > 0.
(2.5)
and for the reinsurer is given by
dS2 (t) = S2 (t)[μ2 dt + σ2 (S2 (t))k2 dW2 (t)],
For convenience, assets S1 (t) and S2 (t) will be respectively referred to as “risky asset 1” and “risky
asset 2”. It is reasonable to assume that the expected return rates μi > r, σi > 0, for i = 1, 2, and
the parameters k1 and k2 in the instantaneous volatilities σ1 (S1 (t))k1 and σ2 (S2 (t))k2 satisfy the general
condition k1 0 and k2 0 as in [14, 19]. Here, W1 and W2 are the randomness sources associated with
the financial assets S1 and S2 . In addition, we assume W , W1 and W2 are independent of each other.
Processes (2.4) and (2.5) are called the CEV models, which are proposed by Cox [11] to model the
dynamics of financial assets and is the first one to relax the constant volatility assumption of the celebrated
as an improvement from the Black-Scholes model [7]. In Models (2.4) and (2.5), the existence of the
exponent ki , i = 1, 2, called the elasticity parameters, allows the instantaneous conditional variance
of asset returns to depend on the asset price level, thus exhibiting an implied volatility smile similar
to volatility curves empirically observed. Due to the various advantages of the CEV model, it has been
widely adopted for modelling financial asset’s price processes in the literature; see, for example, [12,34,39]
among many others. In recent years, the CEV models have also been widely used in actuarial literature,
including [18, 21, 32] among many others.
Denote mi (t) = (Si (t))−2ki , for i = 1, 2. By Itô’s lemma, we can see
dmi (t) = (ki (2ki + 1)σi2 − 2ki μi mi (t))dt − 2ki σi m(t)dWi (t).
(2.6)
This is a mean-reverting square-root model. From [25, Theorem 1.5.5.1], we know that (2.6) has a unique
strong solution. When ki > 0, the Feller condition is satisfied, i.e., 2ki (2ki + 1)σi2 > 4ki2 σi2 , so mi (t) > 0,
a.s.. When ki = 0, the CEV model reduces to the geometric Brownion motion model. Therefore, for
ki 0, (2.4) and (2.5) have unique solutions such that S1 (t) ∈ (0, ∞) and S2 (t) ∈ (0, ∞), a.s., respectively.
In [16], it is found that all the moments of an integrated square-root process are finite. Using this
result, we can further deduce that for any ρ ∈ [1, ∞),
t
T
ρ ρ + E sup
|mi (v)|dv
mi (v)dWi (v)
E sup |mi (v)|ρ K 1 + E
v∈[0,T ]
K 1+E
0
|mi (v)|dv
0
t∈[0,T ]
ρ T
0
T
ρ/2 |mi (v)|dv
+E
< ∞.
(2.7)
0
In (2.7), the second line follows from the Burkholder-Davis-Gundy inequality. Here and throughout this
paper, K denotes a positive generic constant, which may vary from line to line.
The primary objective of the present paper is to investigate the optimal reinsurance between the insurer
and the reinsurer as well as their optimal investment strategies in an integrated way. To proceed, let
π1 (t) and π2 (t), respectively denote the money amounts invested in risky asset 1 by the insurer and in
risky asset 2 by the reinsurer at time t, t 0. Then, the triplet u(t) = (π1 (t), π2 (t), p(t)) encompasses the
reinsurance strategy as well as the investment strategies for both the insurer and the reinsurer. Hereafter
the triplet u(t) is referred to as a trading strategy and the primary objective of the present paper is to
explore the optimal trading strategies from certain perspective.
Given a trading strategy u(t), it is easy to obtain the following dynamics for the wealth process X u of
the insurer and Y u of the reinsurer:
dX u (t) = [rX u (t) + a(θ − ηp(t)) + π1 (t)(μ1 − r)]dt
+ b(1 − p(t))dW (t) + π1 (t)σ1 (S1 (t))k1 dW1 (t),
(2.8)
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and
dY u (t) = [rY u (t) + aηp(t) + π2 (t)(μ2 − r)]dt + bp(t)dW (t) + π2 (t)σ2 (S2 (t))k2 dW2 (t),
(2.9)
respectively. In the present paper, we consider the following weighted sum process for decision making:
Z u (t) = αX u (t) + βY u (t),
(2.10)
where α and β are two constants from interval [0, 1]. Combining (2.8)–(2.10) yields the following dynamics
for the weighted sum process:
dZ u (t) = [rZ u (t) + απ1 (t)(μ1 − r) + βπ2 (t)(μ2 − r) + aαθ − aηp(t)(α − β)]dt
+ απ1 (t)σ1 (S1 (t))k1 dW1 (t) + βπ2 (t)σ2 (S2 (t))k2 dW2 (t)
+ [αb − bp(t)(α − β)]dW (t).
(2.11)
Remark 2.1. Z u (t) in (2.10) can be interpreted in at least two distinct ways, which indeed constitute
our motivation for the present paper. First, it can be observed that an insurer and a reinsurer may
belong to the same corporations. In such case, Z u in (2.10) can be interpreted as the total surplus of
the corporations, if the corporations own 100α% shares of the insurer and 100β% shares of the reinsurer.
When the shares on both the insurance and reinsurance companies the corporations hold are enough to
dominate the management boards of both companies, the decision-making on both companies is up to
the corporations, and the analysis of optimal investment-reinsurance strategies should be based on the
weighted sum process Z u (t) from the interests of the corporations.
Second, if we set β = 1 − α in the weighted sum process Z u (t), then the parameter α takes a role
of balancing the interests between the insurer and the reinsurer in deciding the optimal investmentreinsurance strategies. In particular, Z u is simply the half of total surplus of the insurer’s and reinsurer’s
for α = 1/2, β = 1/2, whereas it reduces to the wealth process of the insurer with α = 1, β = 0. For the
above reasons, α and β may be referred to as decision weights. Throughout this paper, α and β are given
constants, which measure the relative decision-making power of the insurer and the reinsurer. They are
predetermined by the nature of the corporations, thereby not considered as control variables.
Definition 2.2.
A trading strategy {u(v) := (π1 (v), π2 (v), p(v)), v ∈ [t, T ]} is said to be admissible
with respect to an initial condition (t, z, s1 , s2 ) ∈ [0, T ] × R × R+ × R+ , if for any t ∈ [0, T ], it satisfies
the following conditions:
(a) ∀ v ∈ [t, T ], p(v) ∈ [0, 1];
T
T
(b) Et,z,s1 ,s2 [ t |π1 (v)|2 (1+|S1 (v)|2k1 )dv] < ∞ and Et,z,s1 ,s2 [ t |π2 (v)|2 (1+|S2 (v)|2k2 )dv] < ∞, where
Et,z,s1 ,s2 [·] = E[· | Z(t) = z, S1 (t) = s1 , S2 (t) = s2 ].
Hereafter, U (t, z, s1 , s2 ) denotes the set of all admissible strategies with respect to the initial condition
(t, z, s1 , s2 ) ∈ [0, T ] × R × R+ × R+ .
From Conditions (a) and (b) and the boundedness of parameters, we know that
Et,z,s1 ,s2
t
Et,z,s1 ,s2
T
t
T
|απ1 (v)(μ1 − r) + βπ2 (v)(μ2 − r) + aαθ − aηp(v)(α − β)|2 dv < ∞,
(|απ1 (t)σ1 (S1 (t))k1 |2 + |απ1 (v)σ1 (S1 (v))k1 |2 + |αb − bp(v)(α − β)|2 )dv < ∞.
Thus by [44, Theorem 6.3], the wealth equation admits a unique strong solution such that
Et,z,s1 ,s2
sup |Z u (v)|ρ < ∞,
v∈[t,T ]
for any ρ ∈ [1, ∞) and (t, z, s1 , s2 ) ∈ [0, T ] × R × R+ × R+ .
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Problem formulation and verification theorem
Most existing literature (see [3, 22]) on mean-variance problems only consider the optimality of solutions
at the initial time, with a formulation corresponding to the weighted sum process Z u as follows:
γ
(2.12)
sup E0,Z u (0),S1 (0),S2 (0) [Z u (T )] − Var0,Z u (0),S1 (0),S2 (0) [Z u (T )] ,
2
u∈U
where U denotes the corresponding admissible set and γ is the risk aversion coefficient of the decisionmaker. A formulation like (2.12) is a static optimization problem in the sense that the objective are
purely based on information at the initial time 0, and thus the resulting solutions are precommitment
in that they are not updated with the information accumulated over time. As explained in [27, 31], the
target in a pragmatic decision is often varying over time, making the solutions obtained from (2.12)
time-inconsistent in that they are optimal only at time 0 and no longer as time moves forwards into the
future. Time-consistency is a basic requirement for rational investors on a decision, and thus, from a
practical point of view, a solution based on (2.12) is not usable.
The present paper considers the following formulation with a time-varying objective instead: For any
(t, z, s1 , s2 ) ∈ [0, T ] × R × R+ × R+ , the insurer and the reinsurer aims to derive
γ
Et,z,s1 ,s2 [Z u (T )] − Vart,z,s1 ,s2 [Z u (T )] .
sup
(2.13)
2
u∈U (t,z,s1 ,s2 )
Because the variance term in the objective lacks the iterated expectation property, the value function
for (2.13) does not satisfy Bellman principle of optimality and this makes it a time-inconsistent problem
in that the solutions obtained at time t ∈ [0, T ] is not optimal for a future time v > t. To develop a timeconsistent solution to (2.13), we resort to the so-called equilibrium strategy as defined in Definition 2.3
below. Roughly speaking, the decision in the equilibrium strategy established at time t agrees with the
one derived at time t + Δt for an infinitesimal Δt > 0, which is time-consistent.
For the equilibrium strategy to the mean-variance problem (2.13), a more general problem, which
includes (2.13) as a special case, will be studied using a game theoretic perspective. To proceed, let
Q = R × R+ × R+ and denote
C 1,2 ([0, T ] × Q) = {φ(t, z, s1 , s2 ) | φ(t, z, s1 , s2 ) is once continuously differentiable in the first argument
on [0, T ] and is twice continuously differentiable in the remaining arguments on Q},
D
1,2
([0, T ] × Q) = {φ(t, z, s1 , s2 ) | φ(t, z, s1 , s2 ) ∈ C 1,2 ([0, T ] × Q) and φ(t, z, s1 , s2 ) satisfies the
polynomial growth (PG) condition}.
Here we say φ(t, z, s1 , s2 ) satisfies the polynomial growth (PG) condition if
1 q
2 q
|φ(t, z, s1 , s2 )| C(1 + |z|q + |s−2k
| + |s−2k
| ),
1
2
1 q
2 q
|φz (t, z, s1 , s2 )| C(1 + |z|q + |s−2k
| + |s−2k
| ),
1
2
1 q
2 q
|s1 φs1 (t, z, s1 , s2 )| C(1 + |z|q + |s−2k
| + |s−2k
| ),
1
2
1 q
2 q
|s2 φs2 (t, z, s1 , s2 )| C(1 + |z|q + |s−2k
| + |s−2k
| ),
1
2
for some q 1 and any (t, z, s1 , s2 ) ∈ [0, T ] × Q.
The general problem is defined for any function f ∈ D1,2 ([0, T ] × Q) as follows:
sup
u∈U (t,z,s1 ,s2 )
f (t, z, s1 , s2 , g u (t, z, s1 , s2 ), hu (t, z, s1 , s2 )),
(2.14)
where (t, z, s1 , s2 ) ∈ [0, T ] × Q,
g u (t, z, s1 , s2 ) = Et,z,s1 ,s2 [Z u (T )],
u
u
(2.15)
2
h (t, z, s1 , s2 ) = Et,z,s1 ,s2 [(Z (T )) ],
(2.16)
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and U (t, z, s1 , s2 ) is the set of admissible strategies at state (t, z, s1 , s2 ) with the precise definition given
in the preceding subsection. In particular, with
f (t, z, s1 , s2 , g, h) = g −
γ
(h − g 2 ),
2
(2.17)
(2.14) reduces to (2.13).
For a time-consistent solution to a dynamic problem such as (2.14), Ekeland and Lazrak [17] proposed
the following definition of equilibrium strategy.
Definition 2.3.
strategy
Given an admissible strategy u∗ (t) := (π1∗ (t), π2∗ (t), p∗ (t)) ∈ U (t, z, s1 , s2 ), construct
uτ (v) =
2 , p),
(
π1 , π
∗
t v < t + τ,
t + τ v < T,
u (v),
2 ∈ R , p ∈ [0, 1] and v > 0. If
where π
1 , π
∗
lim inf
τ →0
∗
f (t, z, s1 , s2 , g u , hu ) − f (t, z, s1 , s2 , g uτ , huτ )
0
τ
(2.18)
for all (
π1 , π
2 , p) ∈ R× R× [0, 1] and (t, z, s1 , s2 ) ∈ [0, T ]× Q, then u∗ is said to be an equilibrium strategy.
Correspondingly, the equilibrium value function is defined by
∗
∗
V (t, z, s1 , s2 ) = f (t, z, s1 , s2 , g u (t, z, s1 , s2 ), hu (t, z, s1 , s2 )).
To develop an equilibrium strategy for the mean-variance problem (2.13), a verification theorem, which
gives the extended Hamilton-Jacobi-Bellman (HJB) equations for the general problem (2.14), seems
necessary. To proceed, define a variational operator
Au φ(t, z, s1 , s2 ) := φt + [rz + απ1 (μ1 − r) + βπ2 (μ2 − r) + aαθ − aηp(α − β)]φz
+ μ1 s1 φs1 + μ2 s2 φs2
1
1
2
+ {α2 π12 σ12 s2k
+ β 2 π22 σ22 s2k
+ [αb − bp(α − β)]2 }φzz
1
2
2
1
1
+ σ12 s12k1 +2 φs1 s1 + σ22 s22k2 +2 φs2 s2 + απ1 σ12 s12k1 +1 φzs1
2
2
+ βπ2 σ22 s22k2 +1 φzs2 ,
(2.19)
for any φ(t, z, s1 , s2 ) ∈ C 1,2 ([0, T ] × Q).
Theorem 2.4 (Verification theorem).
Consider (2.14). If there exist three real value functions
F (t, z, s1 , s2 ), G(t, z, s1 , s2 ), H(t, z, s1 , s2 ) ∈ D1,2 ([0, T ] × Q)
satisfying
sup
u∈U (t,z,s1 ,s2 )
{Au F (t, z, s1 , s2 ) − ξ u (t, z, s1 , s2 )} = 0,
∗
Au G(t, z, s1 , s2 ) = 0,
u∗
A H(t, z, s1 , s2 ) = 0,
F (T, z, s1 , s2 ) = f (T, z, s1 , s2 , z, z 2),
G(T, z, s1 , s2 ) = z,
2
H(T, z, s1 , s2 ) = z ,
(2.20)
(2.21)
(2.22)
with
u∗ := arg
sup
u∈U (t,z,s1 ,s2 )
{Au F (t, z, s1 , s2 ) − ξ u (t, z, s1 , s2 )},
then
V (t, z, s1 , s2 ) = F (t, z, s1 , s2 ),
∗
g u (t, z, s1 , s2 ) = G(t, z, s1 , s2 ),
∗
hu (t, z, s1 , s2 ) = H(t, z, s1 , s2 ),
(2.23)
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and u∗ = (π1∗ , π2∗ , p∗ ) is an equilibrium strategy to (2.14), where
ξ u (t, z, s1 , s2 ) = ft + [rz + απ1 (μ1 − r) + βπ2 (μ2 − r) + aαθ − aηp(α − β)]fz
1
1
2
+ μ1 s1 fs1 + μ2 s2 fs2 + [α2 π12 σ12 s2k
+ β 2 π22 σ22 s2k
1
2
2
1
1
+ (αb − bp(α − β))2 ]Πu1 + σ12 s12k1 +2 Πu2 + σ22 s22k2 +2 Πu3
2
2
+ απ1 σ12 s12k1 +1 Πu4 + βπ2 σ22 s22k2 +1 Πu5 ,
⎧
⎪
Πu1 = fzz + 2fzg gzu + 2fzh huz + fhh (huz )2 + fgg (gzu )2 + 2fgh gzu huz ,
⎪
⎪
⎪
⎪
⎪
Πu2 = fs1 s1 + 2fgs1 gsu1 + 2fhs1 hus1 + fgg (gsu1 )2 + fhh (hus1 )2 + 2fgh gsu1 hus1 ,
⎪
⎪
⎪
⎪
⎪
⎪Πu3 = fs2 s2 + 2fgs2 gsu2 + 2fhs2 hus2 + fgg (gsu2 )2 + fhh (hus2 )2 + 2fgh gsu2 hus2 ,
⎨
Πu4 = fzs1 + fzg gsu1 + fzh hus1 + fgs1 gzu + fhs1 huz + fgg gzu gsu1 + fhh huz hus1
⎪
⎪
⎪
⎪
+fgh gsu1 huz + fgh gzu hus1 ,
⎪
⎪
⎪
⎪
⎪
Πu5 = fzs2 + fzg gsu2 + fzh hus2 + fgs2 gzu + fhs2 huz + fgg gzu gsu2 + fhh huz hus2
⎪
⎪
⎪
⎩
+fgh gsu2 huz + fgh gzu hus2
(2.24)
(2.25)
and
f = f (t, z, s1 , s2 , g, h),
Proof.
3
g u = g u (t, z, s1 , s2 ),
hu = hu (t, z, s1 , s2 ).
See Appendix A.
Optimal time-consistent strategy for the mean-variance problem
By using Theorem 2.4, the equilibrium strategy {u∗ (t) = (π1∗ (t), π2∗ (t), p∗ (t)), t ∈ [0, T ]} for (2.13) can be
obtained analytically as summarized in Theorems 3.1 and 3.2 for α = β and α = β, respectively.
Theorem 3.1.
Denote
t1 = T +
1
min
r
ln
αγb2
βγb2
, ln
aη
aη
(3.1)
and
1
t2 = T + max
r
αγb2
βγb2
ln
, ln
.
aη
aη
(3.2)
For (2.13) with α = β, an equilibrium investment strategy for the insurer and the reinsurer is
π1∗ (t) =
(μ1 − r)e−r(T −t)
[1 + (μ1 − r)(1 − e−2k1 r(T −t) )],
γασ12 (S1 (t))2k1
0tT
(3.3)
π2∗ (t) =
(μ2 − r)e−r(T −t)
[1 + (μ2 − r)(1 − e−2k2 r(T −t) )],
γβσ22 (S2 (t))2k2
0 t T,
(3.4)
and
and an equilibrium reinsurance strategy and value function are given depending on the model parameters
as in the following cases:
(1) When the parameters satisfy the conditions of Case III in Table 1,
p∗ (t) =
aη
α
− 2
e−r(T −t) ,
α−β
γb (α − β)
0 t T,
and the equilibrium value function is given by (B.11) with A1 (t), B1 (t), C1 (t) and D1 (t) in (B.23).
(3.5)
Zhao H et al.
Table 1
Classification for optimal reinsurance strategy
α, β
Parameters
max (α, β) min (α, β) aη −rT
e
γb2
aη −rT
e
γb2
1<
<
aη
γb2
1<
aη −rT
e
γb2
<
9
Sci China Math
1
aη
γb2
aη
γb2
min (α, β) aη −rT
e
γb2
aη −rT
e
γb2
Case
aη −rT
e
γb2
II
max (α, β)
III
< max (α, β) <
<
aη
γb2
I
aη
γb2
aη −rT
aη
e
< min (α, β) < max (α, β) < γb
2
γb2
aη −rT
aη
e
<
min
(α,
β)
<
max
(α,
β)
γb2
γb2
aη
min
(α,
β)
and
α = β
2
γb
aη −rT
max α, β γb2 e
and α = β
aη −rT
min (α, β) γb
e
<
max (α, β)
2
aη −rT
e
< min (α, β) and α = β
γb2
α = β
IV
V
VI
VII
VIII
IX
X
(2) When the parameters satisfy the conditions of Case V in Table 1,
⎧
⎪
0 t < t1 ,
⎨0,
∗
p (t) =
aη
α
⎪
⎩
− 2
e−r(T −t) , t1 t T,
α−β
γb (α − β)
and the equilibrium value function is given by (B.24).
(3) When the parameters satisfy the conditions of Case VI in Table 1, p∗ (t) = 0, 0 t T , and the
equilibrium value function is given by the function F in (B.27).
(4) When the parameters satisfy the conditions in any of Cases IV and IX in Table 1,
⎧
⎪
0,
0 t < t1 ,
⎪
⎪
⎪
⎨ α
aη
− 2
e−r(T −t) , t1 t < t2 ,
p∗ (t) =
α
−
β
γb
(α
−
β)
⎪
⎪
⎪
⎪
⎩
1,
t2 t T,
and the equilibrium value function is given by (B.29).
(5) When the parameters satisfy the conditions in any of Cases II and VIII in Table 1,
⎧ α
aη
⎪
− 2
e−r(T −t) , 0 t < t2 ,
⎨
α−β
γb (α − β)
∗
p (t) =
⎪
⎩1,
t2 t T,
and the equilibrium value function is expressed in (B.33).
(6) When the parameters satisfy the conditions in any of Cases I, VII and X in Table 1, p∗ (t) = 1,
0 t T , and the equilibrium value function is given by the function F in (B.35).
Proof.
The proof is relegated to Appendix B.
Theorem 3.2.
For (2.13) with α = β, (π1∗ , π2∗ ) given in (3.3) and (3.4) is also an equilibrium investment strategies for the insurer and the reinsurer, and moreover, any measurable function p∗ (t) : [0, T ]
→ [0, 1] is one equilibrium reinsurance treaty. The equilibrium value function is
F (t, z, s1 , s2 ) = A1 (t)z +
where
L1 (t) =
B1 (t) −2k1 C1 (t) −2k2 L1 (t)
s1
s2
,
+
+
γ
γ
γ
T
γaαθ r(T −t)
γ 2 α2 b2
[e
[1 − e2r(T −t) ] + σ12 k1 (2k1 + 1)
− 1] +
B1 (s)ds
r
4r
t
and A1 (t), B1 (t) and C1 (t) are given in (B.23).
10
Proof.
Zhao H et al.
Sci China Math
The proof is similar to that of Theorem 3.1 and thus we omit it.
Indeed, when α = β, the joint mean-variance utility between the insurer and the reinsurer is irrelevant
to the reinsurance proportion p(t) and this implies that the joint utility cannot be enhanced via any
proportional reinsurance treaty between the two parties. This explains why any measurable function
p∗ (t) : [0, T ] → [0, 1] is one solution of the equilibrium reinsurance treaty.
Remark 3.3. An immediate observation from Theorems 3.1 and 3.2 is that the equilibrium investment
strategies for both the insurer and reinsurer are related to the proportional reinsurance treaty only via
the decision weights α and β in the model. If α = 1, β = 0, the optimal reinsurance strategy reduces to
that in [45, 46] without jumps.
Remark 3.4 (Special case under Black-Scholes model).
If one sets k1 = 0 in equation (2.4), the
insurer’s investment asset follows a geometric Brownian motion, and in this case, it follows from The1 −r −r(T −t)
. If one further
orem 3.1 that the optimal investment strategy for the insurer is given by μγασ
2 e
1
assumes α = 1, the insurer’s optimal investment strategy reduces to that obtained in [45] where a
mean-variance analysis was conducted only from the perspective of the insurer under a Black-Scholes
framework.
Moreover, (3.3) and (3.4) show that the equilibrium investment strategies for both the insurer and the
−r(T −t)
reinsurer can be decomposed into two parts (below lists only the insurer’s strategy), γασ2μ(S1 −r
,
2k e
1 (t)) 1
1
which is similar to the optimal strategy under the GBM model except for the stochastic volatility, and
(μ1 − r)2 e−r(T −t)
(1 − e−2k1 r(T −t) ),
γαrσ12 (S1 (t))2k1
which is a supplementary term resulted from the changes of the volatility from GBM model to the CEV
model and reflects the insurer’s decision to hedge the volatility risk.
Remark 3.5. Theorem 3.1 shows an obvious dependence of the equilibrium strategy on the decision
weights α and β as a consequence of its presence in the weighted sum process (2.10). For p∗ (t) = 0, 1,
the solution developed in Theorem 3.1 is given by
α
aη
p∗ (t) =
− 2
e−r(T −t)
α−β
γb (α − β)
in all the cases. Therefore, the relationship between the reinsurance strategy and α is attributed to two
−r(T −t)
∗
factors. On one hand, the term − aηe
γb2 (α−β) increases the value of p (t) as α increases, and on the other
α
decreases its value as α increases. Moreover, (3.3) and (3.4), respectively show that
hand, the term α−β
α exerts a negative effect on the insurer’s investment strategy and a positive effect on the reinsurer’s
investment strategy. Due to the insurer’s attitude of risk aversion, with a larger α assigned, more weight
is given to the preference of the insurer in the decision formulation, and this leads to a reduction in the
risky investment by the insurer and an increase in the reinsurer’s risky investment to potentially create
more wealth for the whole insurance group.
Remark 3.6.
Based on the results established in Theorem 3.1, the sensitivity of the equilibrium
reinsurance proportion p∗ (·) to each model parameter can be analyzed as follows:
(a) From (3.5), we get
∂p∗
1
1
aη −r(T −t)
aη −r(T −t)
∂p∗
=
=
α
−
.
(3.6)
e
−
β
,
e
∂α
(α − β)2 γb2
∂β
(α − β)2
γb2
aη −r(T −t)
Then the optimal reinsurance strategy increases with α when β < γb
and increases with β when
2e
aη −r(T −t)
α > γb2 e
.
(b) It is easy to derive
arηe−r(T −t)
∂p∗
=− 2
for p∗ (t) = 0, 1.
∂t
γb (α − β)
This means that the equilibrium reinsurance proportion decreases with time t for α > β and increases
for α < β. This may be explained as follows. As time goes by, the surpluses of both the insurer and
Zhao H et al.
Sci China Math
11
the reinsurer are expected to increase on average due to the potential gains from investments in financial
assets and this enhances the insurance risk absorbing capacity of both over time. With α > β, more
weight is given to the insurer’s preference in decision, and thus, according to the insurer’s willingness,
more insurance business is retained by the insurer itself at a later period than an earlier time, leading the
reinsurance proportion decreasing as a function of time t. For α < β, the reinsurer’s preference is given
with more priority, and thus the decision allows the reinsurer to gradually undertake more insurance
business from the insurer over time, making the reinsurance proportion p∗ (t) increasing over time.
(c) The effects of market parameters on the reinsurance strategy depend on the decision weights α
and β too. (3.5) shows that
ηe−r(T −t)
∂p∗
=− 2
for p∗ (t) = 0, 1,
∂a
γb (α − β)
which implies that the equilibrium reinsurance strategy increases with a for α < β and decreases with
a for α > β. The parameter a reflects the expectation of the claim size and thus, with the other model
parameters (particularly the volatility b in the claim process (2.1)) fixed, an increase in the expected
claim a reduces the risk per dollar of the insurance liability and makes the insurance business more
attractive to the writer. Therefore, when more weight is given to the insurer in decision, the reinsurance
proportion is reduced and more insurance policies are retained by the insurer in response of an increase of
a. In contrast, if α < β, the reinsurer is given more weight in decision, and thus, more insurance policies
are transferred to the reinsurer with an increase in a.
(d) Since
ae−r(T −t)
∂p∗
=− 2
for p∗ (t) = 0, 1,
∂η
γb (α − β)
the reinsurer’s safety loading η exerts a positive effect on p∗ (·) for α < β and a negative effect on p∗ (·) for
α > β. This is consistent with our intuition. A larger η means more expensive reinsurance. Therefore,
with α < β the decision-maker places more consideration on the reinsurer, whereby the reinsurance
proportion is increased so that the reinsurer can make more profits in response of an increase in η. With
α > β, the insurer’s preference is laid with more weights so that more risk is retained by the insurer itself
when the reinsurance becomes more expensive, i.e., η increases.
(e) (3.5) shows that
aηe−r(T −t)
∂p∗
= 2 2
for p∗ (t) = 0, 1,
∂γ
γ b (α − β)
which indicates that the equilibrium reinsurance proportion p∗ (·) decreases with the risk aversion coefficient γ of the decision-maker for α < β and increases with γ for α > β. With a decision weight α < β,
the choice of the decision-maker relies on the preference of the reinsurer more than the insurer. Thus, the
more risk averse the reinsurer is, the less reinsurance is taken by the reinsurer, leading p∗ (·) decreasing
in γ. Similarly, when α > β, the preference of the insurer is reflected more in decision, and thus the more
risk averse the insurer is, the more risk is ceded to the reinsurer.
(f) From (3.5), we obtain
2aηe−r(T −t)
∂p∗
=
for p∗ (t) = 0, 1.
∂b
γb3 (α − β)
If α < β, there is a negative relationship between p∗ (·) and the volatility b of the claim process C, whereas
the relationship becomes positive for α > β. This can be interpreted by the change of the risk per dollar
of insurance as similarly commented in Part (b). With a larger b, the risk per dollar of insurance is larger
and thus, the insurance business becomes less attractive to an insurance writer, with more is expected to
be ceded to the reinsurer in case a weight α > β is assigned in decision. Similar explanation applies for
the case α < β.
(g) (3.5) also shows that
aη(T − t)e−r(T −t)
∂p∗
=
∂r
γb2 (α − β)
for p∗ (t) = 0, 1.
12
Zhao H et al.
Sci China Math
This indicates that p∗ (·) decreases with respect to the risk-free interest rate r for α < β and increases
in r for α > β. As r increases, both the insurer and the reinsurer have the expectation to gain more
in the financial market, driving the capitals out of the insurance market. When α < β, the interests
of the reinsurer dominate the decision on the trading strategy of the insurance group, and the reinsurer
prefers to reduce its investment in insurance market and move more investments to the financial market.
Similarly, when α > β the insurer’s interests dominate the decision and it is preferred by the insurer that
more insurance risk is taken by the reinsurer so that it has more wealth to invest in the financial market.
4
Numerical analysis
In this section, a numerical example will be presented to illustrate the effects of parameters on the equilibrium investment-reinsurance strategy derived in Theorem 3.1. Unless otherwise stated, the parameter
values are given by a = 0.5, b = 0.6, η = 0.2, r = 0.05, μ1 = 0.12, σ1 = 0.2, k1 = 0.9, S1 = 0.5, μ2 = 0.15,
σ2 = 0.3, k2 = 1.1, S2 = 0.6, α = 0.3, β = 0.7, γ = 0.5, and T = 10.
Figure 1 shows that the insurance proportion p∗ (t) is increasing along with time t for α < β and
decreasing for α > β. Such an observation can be explained as follows. The desire of both the insurer
and the reinsurer to bear insurance risk becomes gradually strengthen as a result of their wealth accumulation over time. Therefore, both prefer to absorb less insurance risk at the first stage (before time 8)
and become more interested in taking insurance business at the second stage with more capital available.
Consequently, with α < β, more voices are heard from the reinsurer in the decision, and according to the
desire of the reinsurer, more insurance risk is transferred to the reinsurer at a later period than an earlier
The effect of α on the optimal reinsurance strategy when α<β
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
p*(t)
p*(t)
(a)
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
α=0.15,β=0.6
α=0.2, β=0.7
α=0.25, β=0.8
α=0.3, β=0.9
0.1
0
0
1
2
3
Figure 1
4
5
t
6
7
8
9
0
10
0
2
3
4
5
t
6
7
8
9
10
The effect of a on the optimal reinsurance strategy when α>β
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
p*(t)
p*(t)
1
Effect of decision weights α and β on equilibrium reinsurance strategy p∗ (t)
(a)
0.5
(b)
0.5
0.4
0.4
0.3
0.3
0.2
0.2
a=0.45
a=0.5
a=0.55
a=0.6
0.1
0
α=0.6, β=0.15
α=0.7, β=0.2
α=0.8, β=0.25
α=0.9, β=0.3
0.1
The effect of a on the optimal reinsurance strategy when α<β
1.0
The effect of α on the optimal reinsurance strategy when α>β
(b)
0
1
2
3
4
5
t
6
Figure 2
7
8
9
a=0.45
a=0.5
a=0.55
a=0.6
0.1
10
0
0
1
2
3
4
5
t
Effect of a on equilibrium reinsurance strategy p∗ (t)
6
7
8
9
10
Zhao H et al.
The effect of b on the optimal reinsurance strategy when α<β
1.0
0.9
0.8
0.8
0.7
0.7
0.6
0.6
p*(t)
p*(t)
(b)
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
b=0.55
b=0.58
b=0.6
b=0.62
0.1
0
1
2
3
4
5
t
6
Figure 3
7
8
9
0
10
0
0.8
0.8
0.7
0.7
0.6
0.6
p*(t)
p*(t)
0.9
0.5
3
4
5
t
6
7
8
9
10
(b)
0.5
0.4
0.4
0.3
0.3
0.2
0.2
η=0.18
η=0.2
η=0.23
η=0.25
0.1
0
1
2
3
4
5
t
6
Figure 4
7
8
9
η=0.18
η=0.2
η=0.23
η=0.25
0.1
0
10
0
1
2
3
4
5
t
6
7
8
9
10
Effect of η on equilibrium reinsurance strategy p∗ (t)
The effect of γ on the optimal reinsurance strategy when α< β
1.0
1.0
(a)
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
p*(t)
p*(t)
2
The effect of η on the optimal reinsurance strategy when α >β
1.0
0.9
0.5
The effect of γ on the optimal reinsurance strategy when α >β
(b)
0.5
0.4
0.4
0.3
0.3
0.2
0.2
γ=0.4
γ =0.45
γ=0.5
γ=0.55
0.1
0
1
Effect of b on equilibrium reinsurance strategy p∗ (t)
(a)
0
b=0.55
b=0.58
b=0.6
b=0.62
0.1
The effect of η on the optimal reinsurance strategy when α <β
1.0
0
The effect of b on the optimal reinsurance strategy when α > β
1.0
(a)
0
13
Sci China Math
1
2
3
4
5
t
6
Figure 5
7
8
9
γ =0.4
γ=0.45
γ=0.5
γ=0.55
0.1
10
0
0
1
2
3
4
5
t
6
7
8
9
10
Effect of γ on equilibrium reinsurance strategy p∗ (t)
period. In contrast, when α > β, the insurer has more voices over the decision, and therefore, according
to the desire of the insurer, more insurance risk is transferred to the reinsurer at an earlier period than
a later period. Furthermore, Figure 1 also illustrates that the response of reinsurance strategy p∗ (t) to
the weights α and β also depends on the relative magnitudes of α and β. For α < β, p∗ (t) is decreasing
in both α and β. In contrast, p∗ (t) is increasing in both α and β for α > β.
Figures 2–5 show the impacts of market parameters on the equilibrium reinsurance strategy, and the
results consistently confirm our comments presented in Remark 3.6. The effects depend on the value
of the decision weight parameters α and β as well. For α > β, the equilibrium reinsurance proportion
14
Zhao H et al.
Sci China Math
The effect of γ on the optimal investment strategy (π *1(t), π2* (t))
α=0.3, β=0.7,
(a)
π*1(t)
α=0.3, β=0.7, π* (t)
1
(b)
α=0.7, β=0.3, π*1(t)
70
80
α =0.7, β =0.3, π* (t)
70
α=0.7, β=0.3, π* (t)
2
1
α=0.3, β=0.7, π* (t)
2
α=0.3, β=0.7, π*2(t)
α=0.7, β=0.3, π*2(t)
Money amount
60
Money amount
The effect of r on the optimal investment strategy (π 1*(t), π2* (t))
90
80
50
40
30
60
50
40
30
20
20
10
10
0
0.30
0.35
0.40
0.45
0.50
0.55
γ
0.60
0.65
0.70
0.75
0
0.03
0.80
0.04
0.05
0.06
0.07
0.08
0.09
0.10
r
Figure 6 (a) Effect of γ on equilibrium investment strategy (π1∗ (t), π2∗ (t)). (b) Effect of r on equilibrium investment
strategy (π1∗ (t), π2∗ (t))
The effect of μ1 on the optimal investment strategy π1* (t)
80
The effect of μ 2 on the optimal investment strategy π2* (t)
35
(a)
(b)
70
30
60
25
50
π*2(t)
π*1(t)
20
40
15
30
10
20
5
10
α=0.3
α=0.7
0
0.06
0.07
0.08
0.09
0.10
μ1
0.11
0.12
0.13
0.14
0
0.06
0.15
β=0.7
β=0.3
0.07
0.08
0.09
0.10
μ2
0.11
0.12
0.13
0.14
0.15
Figure 7 (a) Effect of μ1 on insurer’s equilibrium investment strategy π1∗ (t). (b) Effect of μ2 on reinsurer’s equilibrium
investment strategy π2∗ (t)
The effect of σ1 on the optimal investment strategy π1* (t)
The effect of σ2 on the optimal investment strategy π2* (t)
300
200
α=0.3
α=0.7
(a)
180
β=0.7
β=0.3
(b)
250
160
140
200
π*2(t)
π*1(t)
120
100
150
80
100
60
40
50
20
0
0.10
0.15
0.20
σ1
0.25
0.30
0.35
0
0.10
0.15
0.20
σ2
0.25
0.30
0.35
Figure 8 (a) Effect of σ1 on insurer’s equilibrium investment strategy π1∗ (t). (b) Effect of σ2 on reinsurer’s equilibrium
investment strategy π2∗ (t)
increases with γ, b, r and decreases with t, a, η. In contrast, for α < β, the reinsurance proportion shows
an opposite response to these parameter. It increases with t, a, η and decreases with γ, b, r in this case.
Figures 6–9 depict the sensitivity of the equilibrium investment strategy (π1∗ (t), π2∗ (t)) to various model
parameters, where, for simplicity but without loss of generality, the investment strategy is computed only
for time 0. Figure 6(a) shows that the risk aversion coefficient γ exerts a negative effect on both π1∗ (t)
Zhao H et al.
The effect of k 1 on the optimal investment strategy π1* (t)
The effect of k 2 on the optimal investment strategy π2* (t)
80
120
β =0.7
β =0.3
α =0.3
α =0.7
100
15
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70
(a)
(b)
60
80
40
60
π*2(t)
π*1(t)
50
30
40
20
20
0
0.5
10
0.6
0.7
0.8
0.9
1.0
k1
1.1
1.2
1.3
1.4
1.5
0
0.5
0.6
0.7
0.8
0.9
1.0
k2
1.1
1.2
1.3
1.4
1.5
Figure 9 (a) Effect of k1 on insurer’s equilibrium investment strategy π1∗ (t). (b) Effect of k2 on reinsurer’s equilibrium
investment strategy π2∗ (t)
and π2∗ (t). The larger γ is, the more risk averse the insurer and the reinsurer are. Thus, as γ increases,
the insurer and the reinsurer choose to reduce their investments in the risky assets to control their risk.
Figure 6(b) demonstrates the negative effect from the return rate of the risk-free asset on the equilibrium
investment strategy (π1∗ (t), π2∗ (t)), in accordance to the comments in Part (f) of Remark 3.6. An increase
of the risk-free return rate r changes the interests of both the insurer and the reinsurer in financial market
and drives capitals out of the insurance market.
Figure 7 demonstrates an increasing trend of π1∗ (t) and π2∗ (t) in response of an increase of μ1 and μ2 ,
respectively. Such a phenomenon is natural, since μ1 and μ2 are respectively the expected return rates of
risky assets. With the volatility fixed, an asset with a higher expected return is naturally more attractive
to the investors. The decreasing trend of π1∗ (t) and π2∗ (t) with respect to the corresponding volatility as
demonstrated in Figure 8 can be explained by the same reason.
Figure 9 shows that π1∗ (t) and π2∗ (t) are increasing functions of k1 and k2 , respectively. Such an
observation is consistent to the economic implications of the parameters k1 and k2 in (2.4) and (2.5).
Larger values for k1 and k2 lead to more probability of larger positive movement in prices of the risky
assets, and hence encourage the insurer and the reinsurer to increase their positions in the risky assets.
5
Conclusion
The investment-reinsurance problem is one of the fascinating topics in actuarial science, and it was
commonly discussed only from the insurer’s perspective. In this paper, we proposed a formulation to
jointly take into account the interests of both the insurer and reinsurer, and developed a time-consistent
solution by resorting to the equilibrium strategies. The theory on the investment-reinsurance problems
by combining the interests of both the insurer and the reinsurer remains scarce in the literature, and we
hope this paper can promote more research in this direction.
Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.
11301376, 71201173 and 71571195), China Scholarship Council, the Natural Sciences and Engineering Research
Council of Canada (NSERC), and Society of Actuaries Centers of Actuarial Excellence Research Grant, Guangdong Natural Science Funds for Distinguished Young Scholar (Grant No. 2015A030306040), Natural Science
Foundation of Guangdong Province of China (Grant No. 2014A030310195) and for Ying Tung Eduction Foundation for Young Teachers in the Higher Education Institutions of China (Grant No. 151081).
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Appendix A: Proof of Theorem 2.4
The proof is adapted from [6, 26], and obtained in the following three steps. Here we just consider the
case of α = β, because the proof is similar for α = β.
Step 1.
Consider an arbitrary strategy u ∈ U (t, z, s1 , s2 ). We first show that if Y (t, z, s1 , s2 ) ∈
D1,2 ([0, T ] × Q), then
t
t
t
t
t
t
and
t
Yz (v, Z u (v), S1 (v), S2 (v))ασ1 π1 (v)[S1 (v)]k1 dW1 (v),
(A.1)
Yz (v, Z u (v), S1 (v), S2 (v))βσ2 π2 (v)[S2 (v)]k2 dW2 (v),
(A.2)
Ys1 (v, Z u (v), S1 (v), S2 (v))σ1 [S1 (v)]k1 +1 dW1 (v)
(A.3)
t
Ys2 (v, Z u (v), S1 (v), S2 (v))σ2 [S2 (v)]k2 +1 dW2 (v)
(A.4)
are all local martingales for ∀ t ∈ [t, T ], ∀ (z, s1 , s2 ) ∈ Q and ∀u ∈ U (t, z, s1 , s2 ). Note that the processes (A.1)–(A.4) we consider are indexed by t rather than t, and the associated filtration contains the
information flow from t to T of {Ft , t 0}.
T
T
For any u ∈ U (t, z, s1 , s2 ), we know t |π1 (v)|2 |S1 (v)|2k1 dv < ∞ and t |Z(v)|ρ dv < ∞, a.s., for any
ρ ∈ [1, ∞). Since Y (t, z, s1 , s2 ) ∈ D1,2 ([0, T ] × Q), there exist two constants K > 0 and q 1 such that
1 q
2 q
|Yz (t, z, s1 , s2 )| K(1 + |z|q + |s−2k
| + |s−2k
| )
1
2
1 q
2 q
and |s1 Ys1 (t, z, s1 , s2 )| K(1 + |z|q + |s−2k
| + |s−2k
| ). We combine all the above analysis to obtain
1
2
t
t
|Yz (v, Z u (v), S1 (v), S2 (v))ασ1 π1 (v)(S1 (v))k1 |2 dv
18
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K sup |Yz (v, Z u (v), S1 (v), S2 (v))|2 ×
v∈[t,T ]
T
t
|π1 (v)|2 |S1 (v)|2k1 dv
K sup (1 + |Z u (v)|q + |m1 (v)|q + |m2 (v)|q )2 ×
v∈[t,T ]
T
t
|π1 (v)|2 |S1 (v)|2k1 dv
K 1 + sup |Z u (v)|2q + sup |m1 (v)|2q + sup |m2 (v)|2q
v∈[t,T ]
×
v∈[t,T ]
v∈[t,T ]
T
|π1 (v)|2 |S1 (v)|2k1 dv
t
<∞
and
t
t
|Ys1 (v, Z u (v), S1 (v), S2 (v))|2 σ12 [S1 (v)]2(k1 +1) dv
K
T
t
[(1 + |Z u (v)|q + |m1 (v)|q + |m2 (v)|q )4 + [S1 (v)]4k1 ]dv
4k1 < ∞.
K 1 + sup |Z u (v)|4q + sup |m1 (v)|4q + sup |m2 (v)|4q + sup |S1 (v)|
v∈[t,T ]
v∈[t,T ]
v∈[t,T ]
v∈[t,T ]
The above two inequalities hold almost surely. Therefore, (A.1) and (A.3) are local martingales. Similarly,
we can show (A.2) and (A.4) are also local martingales.
Next, we show that if a real value function Y (t, z, s1 , s2 ) ∈ D1,2 ([0, T ] × Q) satisfies
Au Y (t, z, s1 , s2 ) = 0,
Y (T, z, s1 , s2 ) = z,
∀ (t, z, s1 , s2 ) ∈ ([0, T ] × R × R+ × R+ ),
(A.5)
then
Y (t, z, s1 , s2 ) = g u (t, z, s1 , s2 ).
(A.6)
Since (A.1)–(A.4) are local martingales, we can take a localizing sequence {τn }n=1,2,..., where t τn
↑ ∞, such that taking the conditional expectation, i.e., Et,z,s1 ,s2 [·], of the following stochastic integrals:
T ∧τn
t
T ∧τn
t
T ∧τn
t
and
Yz (v, Z u (v), S1 (v), S2 (v))ασ1 π1 (v)[S1 (v)]k1 dW1 (v),
Yz (v, Z u (v), S1 (v), S2 (v))βσ2 π2 (v)[S2 (v)]k2 dW2 (v),
Ys1 (v, Z u (v), S1 (v), S2 (v))σ1 [S1 (v)]k1 +1 dW1 (v)
T ∧τn
t
Ys2 (v, Z u (v), S1 (v), S2 (v))σ2 [S2 (v)]k2 +1 dW2 (v)
result in zeros.
Using (2.11) and Itô’s formula, we derive
Y (t, Z u (t), S1 (t), S2 (t))
= Y (T ∧ τn , Z u (T ∧ τn ), S1 (T ∧ τn ), S2 (T ∧ τn )) −
= Y (T ∧ τn , Z u (T ∧ τn ), S1 (T ∧ τn ), S2 (T ∧ τn )) −
−
t
T ∧τn
T ∧τn
t
t
T ∧τn
dY (v, Z u (v), S1 (v), S2 (v))
Au Y (v, Z u (v), S1 (v), S2 (v))dv
{Yz (v, Z u (v), S1 (v), S2 (v))[ασ1 π1 (v)(S1 (v))k1 dW1 (v)
+ βσ2 π2 (v)(S2 (v))k2 dW2 (v) + (αb − b(α − β)p(v))dW (v)]
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+ Ys1 (v, Z u (v), S1 (v), S2 (v))σ1 (S1 (v))k1 +1 dW1 (v)
+ Ys2 (v, Z u (v), S1 (v), S2 (v))σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.7)
Taking the conditional expectation on both sides of (A.7) and applying equations (A.5) and (2.15),
we have
Y (t, z, s1 , s2 ) = Et,z,s1 ,s2 [Y (t, Z u (t), S1 (t), S2 (t))]
= Et,z,s1 ,s2 [Y (T ∧ τn , Z u (T ∧ τn ), S1 (T ∧ τn ), S2 (T ∧ τn ))].
(A.8)
On the other hand, we can derive
E sup |Y (v, Z u (v), S1 (v), S2 (v))|
v∈[t,T ]
< ∞.
K 1 + E sup |Z u (v)|q + E sup |m1 (v)|q + E sup |m2 (v)|q
v∈[t,T ]
v∈[0,T ]
(A.9)
v∈[0,T ]
Thus, {Y (T ∧ τn , Z u (T ∧ τn ), S1 (T ∧ τn ), S2 (T ∧ τn ))}n=1,2,... is dominated by an integrable random
variable, supv∈[t,T ] |Y (v, Z u (v), S1 (v), S2 (v))|. By the dominated convergence theorem, we have
Y (t, z, s1 , s2 ) = lim Et,z,s1 ,s2 [Y (T ∧ τn , Z u (T ∧ τn ), S1 (T ∧ τn ), S2 (T ∧ τn ))]
n→∞
= Et,z,s1 ,s2 lim Y (T ∧ τn , Z u (T ∧ τn ), S1 (T ∧ τn ), S2 (T ∧ τn ))
n→∞
= Et,z,s1 ,s2 [Z u (T )] = g u (t, z, s1 , s2 ).
(A.10)
Similar to the above derivation, we can show that if there exists a real function J(t, z, s1 , s2 ) ∈
D1,2 ([0, T ] × Q) such that
Au J(t, z, s1 , s2 ) = 0
and J(T, z, s1, s2 ) = z 2 ,
∀ (t, z, s1 , s2 ) ∈ ([0, T ] × Q),
(A.11)
then
J(t, z, s1 , s2 ) = hu (t, z, s1 , s2 ).
(A.12)
Thirdly, we use (A.5) and (A.11) to develop an expression for f u (t) := f (t, Z u (t), S1 (t), S2 (t), Y u (t),
J (t)), where Y u (t) = Y (t, Z u (t), S1 (t), S2 (t)) and J u (t) = J(t, Z u (t), S1 (t), S2 (t)). In view of (A.6)
and (A.12), we get f (t, Z u (t), S1 (t), S2 (t), g u (t, Z u (t), S1 (t), S2 (t)), hu (t, Z u (t), S1 (t), S2 (t))) = f u (t). Let
ξ u (t) = ξ u (t, Z u (t), S1 (t), S2 (t)). We apply Itô’s formula and (2.11) to obtain
u
u
u
T
f (T ) = f (t) +
= f u (t) +
+
t
T
t
T
df u (v)
t
[fgu (v)Au Y u (v) + fgu (v)Au J u (v) + ξ u (v)]dv
{(fzu (v) + fgu (v)Yzu (v) + fhu (v)Jzu (v))[(αb − αbp(v) + bβp(v))dW (v)
+ ασ1 π1 (v)(S1 (v))k1 dW1 (v) + βσ2 π2 (v)(S2 (v))k2 dW2 (v)]
+ (fsu1 (v) + fgu (v)Ysu1 (v) + fhu (v)Js1 (v))σ1 (S1 (v))k1 +1 dW1 (v)
+ (fsu2 (v) + fgu (v)Ys2 (v) + fhu (v)Js2 (v))σ2 (S2 (v))k2 +1 dW2 (v)}.
Inputting (A.5) and (A.11) into the above equation implies
u
u
f (T ) = f (t) +
T
u
ξ (v)dv +
t
t
T
{(fzu (v) + fgu (v)Yzu (v) + fhu (v)Jzu (v))
× [(αb − αbp(v) + bβp(v))dW (v) + ασ1 π1 (v)(S1 (v))k1 dW1 (v)
+ βσ2 π2 (v)(S2 (v))k2 dW2 (v)] + (fsu1 (v) + fgu (v)Ysu1 (v)
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Zhao H et al.
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+ fhu (v)Jsu1 (v))σ1 (S1 (v))k1 +1 dW1 (v)
+ (fsu2 (v) + fgu (v)Ysu2 (v) + fhu (v)Jsu2 (v))σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.13)
Finally, we show that ∀ (t, z, s1 , s2 ) ∈ ([0, T ] × Q),
F (t, z, s1 , s2 ) sup
u∈U(t,z,s1 ,s2 )
f (t, z, s1 , s2 , g u (t, z, s1 , s2 ), hu (t, z, s1 , s2 )).
(A.14)
Let F u (t) := F (t, Z u (t), S1 (t), S2 (t)) for short. We apply (2.11) to obtain
T
dF u (v)
F u (t) = F u (T ) −
t
u
= F (T ) −
t
T
u
u
A F (v)dv −
t
T
{Fzu (v)[(αb − αbp(v) + bβp(v))dW (v)
+ ασ1 π1 (v)(S1 (v))k1 dW1 (v) + βσ2 π2 (v)(S2 (v))k2 dW2 (v)]
+ Fsu1 (v)σ1 (S1 (v))k1 +1 dW1 (v) + Fsu2 (v)σ2 (S2 (v))k2 +1 dW2 (v)}.
Besides, (2.20) implies that Au F (t, z, s1 , s2 ) ξ u (t, z, s1 , s2 ) for ∀ (t, z, s1 , s2 ) ∈ ([0, T ] × Q), and hence,
T
T
ξ u (v)dv −
{Fzu (v)[(αb − αbp(v) + bβp(v))dW (v)
F u (t) F u (T ) −
t
t
k1
+ ασ1 π1 (v)(S1 (v)) dW1 (v) + βσ2 π2 (v)(S2 (v))k2 dW2 (v)]
+ Fsu1 (v)σ1 (S1 (v))k1 +1 dW1 (v) + Fsu2 (v)σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.15)
Moreover, (2.20) also implies that F u (T ) = f u (T ). Hence, inputting (A.13) into (A.15) gives
T
F u (t) f u (t) +
{(fzu (v) + fgu (v)Yzu (v) + fhu (v)Jzu (v) − Fzu (v))
t
× [(αb − αbp(v) + bβp(v))dW (v) + ασ1 π1 (v)(S1 (v))k1 dW1 (v)
+ βσ2 π2 (v)(S2 (v))k2 dW2 (v)] + (fsu1 (v) + fgu (v)Ysu1 (v) + fhu (v)Jsu1 (v)
− Fsu1 (v))σ1 (S1 (v))k1 +1 dW1 (v) + (fsu2 (v) + fgu (v)Ysu2 (v) + fhu (v)Jsu2 (v)
− Fsu2 (v))σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.16)
Similarly, it can be shown that all stochastic integrals with respect to Brownian motions are local martingales. By using the localization technique, taking conditional expectation on both sides of (A.16) and
supremum over U (t, z, s1 , s2 ), we get (A.14).
Step 2.
Consider the specific admissible strategy π ∗ . The assumptions of Theorem 2.4 show that
G(t, z, s1 , s2 ) and H(t, z, s1 , s2 ) satisfy (A.5) and (A.11) with strategy u∗ . Thus,
∗
G(t, z, s1 , s2 ) = g u (t, z, s1 , s2 )
∗
and H(t, z, s1 , s2 ) = hu (t, z, s1 , s2 ). Moreover, the inequality (A.16) at u = u∗ becomes an equation, i.e.,
T
∗
∗
∗
∗
∗
∗
u∗
u∗
{(fzu (v) + fgu (v)Yzu (v) + fhu (v)Jzu (v) − Fzu (v))
F (t) = f (t) +
t
× [(αb − αbp(v) + bβp(v))dW (v) + ασ1 π1∗ (v)(S1 (v))k1 dW1 (v)
∗
∗
∗
+ βσ2 π2∗ (v)(S2 (v))k2 dW2 (v)] + (fsu1 (v) + fgu (v)Ysu1 (v)
∗
∗
∗
∗
∗
∗
∗
∗
∗
+ fhu (v)Jsu1 (v) − Fsu1 (v))σ1 (S1 (v))k1 +1 dW1 (v) + (fsu2 (v) + fgu (v)Ysu2 (v)
+ fhu (v)Jsu2 (v) − Fsu2 (v))σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.17)
As in Step 1, by using the localization technique and taking conditional expectation on both sides
of (A.17), we obtain
∗
∗
F (t, z, s1 , s2 ) = f (t, z, s1 , s2 , g u (t), hu (t)) sup
u∈U (t,z,s1 ,s2 )
f (t, z, s1 , s2 , g u (t), hu (t)),
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which together with (A.14) gives
F (t, z, s1 , s2 ) =
sup
u∈U (t,z,s1 ,s2 )
f (t, z, s1 , s2 , g u (t), hu (t)).
Thus, u∗ is optimal and the supremum of f u (t) is F (t, z, s1 , s2 ).
Step 3. Prove u∗ is indeed an equilibrium strategy. For any uτ defined in Definition 2.3 and t + τ , we
rewrite (A.13) as
f uτ (t + τ ) = f uτ (t) +
t+τ
ξ uτ (v)dv +
t
t+τ
t
{(fzuτ (v) + fguτ (v)Yzuτ (v)
+ fhuτ (v)Jzuτ (v))[(αb − αbp(v) + bβp(v))dW (v) + ασ1 π1 (v)(S1 (v))k1 dW1 (v)
+ βσ2 π2 (v)(S2 (v))k2 dW2 (v)] + (fsu1τ (v) + fguτ (v)Ysu1τ (v)
+ fhuτ (v)Jsu1τ (v))σ1 (S1 (v))k1 +1 dW1 (v) + (fsu2τ (v) + fguτ (v)Ysu2τ (v)
+ fhuτ (v)Jsu2τ (v))σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.18)
Replacing T by t + τ and u by u∗ in (A.15), we obtain
∗
∗
F u (t) F u (t + τ ) −
t+τ
t
∗
ξ u (v)dv −
t+τ
t
∗
{Fzu (v)[(αb − αbp(v) + bβp(v))dW (v)
k1
+ ασ1 π1 (v)(S1 (v)) dW1 (v) + βσ2 π2 (v)(S2 (v))k2 dW2 (v)]
∗
∗
+ Fsu1 (v)σ1 (S1 (v))k1 +1 dW1 (v) + Fsu2 (v)σ2 (S2 (v))k2 +1 dW2 (v)}.
(A.19)
∗
Based on the definition of uτ , we have f uτ (t + τ ) F u ((t + τ )). Hence, substituting (A.18) into (A.19)
gives
t+τ
t+τ
t+τ
∗
∗
ξ uτ (v)dv −
ξ u (v)dv +
{(fzuτ (v) + fguτ (v)Yzuτ (v)
F u (t) f uτ (t) +
+
+
+
t
t
t
uτ
uτ
fh (v)Jz (v))[(αb − αbp(v) + bβp(v))dW (v) + ασ1 π1 (v)(S1 (v))k1 dW1 (v)
βσ2 π2 (v)(S2 (v))k2 dW2 (v)] + (fsu1τ (v) + fguτ (v)Ysu1τ (v)
fhuτ (v)Jsu1τ (v))σ1 (S1 (v))k1 +1 dW (v) + (fsu2τ (v) + fguτ (v)Ysu2τ (v) + fhuτ (v)Jsu2τ (v))
k2 +1
× σ2 (S2 (v))
dW (v)} −
(t+τ )
t
∗
{Fzu (v)[(αb − αbp(v) + bβp(v))dW (v)
+ ασ1 π1 (v)(S1 (v))k1 dW1 (v) + βσ2 π2 (v)(S2 (v))k2 dW2 (v)]
∗
∗
+ Fsu1 (v)σ1 (S1 (v))k1 +1 dW1 (v) + Fsu2 (v)σ2 (S2 (v))k2 +1 dW2 (v)}.
∗
(A.20)
∗
Noting that F u (t) = f u (t), using the localization technique and taking conditional expectation on both
sides of (A.20), we obtain
t+τ
t+τ
u∗
uτ
uτ
u∗
ξ (v)dv −
ξ (v)dv ,
f (t) f (t) + Et,z,s1 ,s2
t
which implies
t
∗
lim inf
τ →0
f u (t) − f uτ (t)
0,
τ
and thus the proof is complete.
Appendix B: Proof of Theorem 3.1
Suppose that three real functions F (t, z, s1 , s2 ), G(t, z, s1 , s2 ) and H(t, z, s1 , s2 ) satisfy the conditions
given in Theorem 2.4 and Πu1 > Fzz (t, z, s1 , s2 ) for all (t, z, s1 , s2 ) ∈ [0, T ] × Q and u ∈ U (t, z, s1 , s2 ).
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Equation (2.17) indicates that
ft = fz = fs1 = fs2 = fzz = fzg = fzh = fgh = fzs1 = fgs1 = fhs1 = fzs2 = fgs2 = fhs2
= fhh = fs1 s1 = fs2 s2 = 0,
γ
fg = 1 + γg, fh = − , fgg = γ.
2
(B.1)
From (2.24) and (2.25), we obtain
Πu1 = γ(gzu )2 ,
Πu2 = γ(gsu1 )2 Πu3 = γ(gsu2 )2 , Πu4 = γgzu gsu1 , Πu5 = γgzu gsu2 ,
γ
1
2
ξ u (t, z, s1 , s2 ) = [α2 π12 σ12 s2k
+ β 2 π22 σ22 s2k
+ (αb − bp(α − β))2 ](gzu )2
1
2
2
γ
γ
+ σ12 s12k1 +2 (gsu1 )2 + σ22 s22k2 +2 (gsu2 )2 + γαπ1 σ12 s12k1 +1 gzu gsu1
2
2
+ γβπ2 σ22 s22k2 +1 gzu gsu2 .
(B.2)
Inputting (2.19) and (B.2) into (2.20), we have
sup
Ft + [rz + απ1 (μ1 − r) + βπ2 (μ2 − r) + aαθ − aηp(α − β)]Fz + μ1 s1 Fs1
u∈U (t,z,s1 ,s2 )
1
1
2
+ μ2 s2 Fs2 + [α2 π12 σ12 s2k
+ β 2 π22 σ22 s2k
+ α2 b2 − 2αb2 p(α − β) + b2 p2 (α − β)2 ]Fzz
1
2
2
1
1
+ σ12 s12k1 +2 Fs1 s1 + σ22 s22k2 +2 Fs2 s2 + απ1 σ12 s12k1 +1 Fzs1 + βπ2 σ22 s22k2 +1 Fzs2
2
2
γ 2 2 2 2k1
γ
γ
2 2 2 2k2
− (α π1 σ1 s1 + β π2 σ2 s2 + (αb − bp(α − β)2 )2 )G2z − σ12 s12k1 +2 G2s1 − σ22 s22k2 +2 G2s2
2
2
2
− γαπ1 σ12 s12k1 +1 Gz Gs1 − γβπ2 σ22 s22k2 +1 Gz Gs2
= 0.
Differentiating (2.20) with respect to π1 , π2 and p, respectively and using (2.23), we obtain the following
first-order optimality conditions:
⎧
−(μ1 − r)Fz − σ12 s12k1 +1 Fzs1 + γσ12 s12k1 +1 Gz Gs1
⎪
∗
⎪
,
⎪
⎨ π1 =
1
2
ασ12 s2k
1 [Fzz − γGz ]
(B.3)
⎪
−(μ2 − r)Fz − σ22 s22k2 +1 Fzs2 + γσ22 s22k2 +1 Gz Gs2
⎪
∗
⎪
,
⎩ π2 =
2
2
βσ22 s2k
2 [Fzz − γGz ]
and
p0 =
aηFz
α
+
.
α−β
(α − β)b2 [Fzz − γG2z ]
(B.4)
The expressions for π1∗ and π2∗ in (B.3) depend on function F and G. As it can be seen shortly, however,
they eventually have a uniform expressions as given in (3.3) and (3.4).
To proceed, define sets
A1 = {(t, z, s1 , s2 ) ∈ [0, T ] × Q; 0 p0 1},
A2 = {(t, z, s1 , s2 ) ∈ [0, T ] × Q; p0 < 0},
A3 = {(t, z, s1 , s2 ) ∈ [0, T ] × Q; p0 > 1}.
For (t, z, s1 , s2 ) ∈ A1 , the supremum of (2.20) over p is attained at p0 given by (B.3). Introducing (B.3)
and (B.4) into (2.20) and (2.21) gives
1
Ft + (rz + aα(θ − η))Fz + μ1 s1 Fs1 + μ2 s2 Fs2 + σ12 s12k1 +2 Fs1 s1
2
1 2 2k2 +2
γ 2 2k1 +2 2
γ 2 2k2 +2 2
a2 η 2 Fz2
+ σ2 s2
Fs2 s2 − σ1 s1
Gs1 − σ2 s2
Gs2 − 2
2
2
2
2b (Fzz − γG2z )
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−
[(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1 ]2
1
2
2σ12 s2k
1 (Fzz − γGz )
−
[(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2 ]2
=0
2
2
2σ22 s2k
2 (Fzz − γGz )
(B.5)
and
1
1
Gt + (rz + aα(θ − η))Gz + μ1 s1 Gs1 + μ2 s2 Gs2 + σ12 s12k1 +2 Gs1 s1 + σ22 s22k2 +2 Gs2 s2
2
2
2 2
2 2k1 +1
Fzs1 − γσ12 s12k1 +1 Gz Gs1
(μ1 − r)Fz + σ1 s1
a η Fz
Fz Gzz
Gz −
−
− 2
2
2
b (Fzz − γGz )
2(Fzz − γGz )
(Fzz − γG2z )
[(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1 ]
μ1 − r
Gzz
× 2 2k1 Gz + s1 Gzs1 −
1
2
σ1 s1
2σ12 s2k
1 (Fzz − γGz )
(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2 μ2 − r
Gz + s2 Gzs2
−
2
(Fzz − γG2z )
σ22 s2k
2
(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2
2
(B.6)
−
(F
−
γG
)G
zz
zz = 0.
z
2
2σ22 s2k
2
For (t, z, s1 , s2 ) ∈ A2 , (2.20) reaches its maximum at p∗ = 0. Consequently, (2.20) and (2.21), respectively
become
1
1
Ft + (rz + aαθ)Fz + μ1 s1 Fs1 + μ2 s2 Fs2 + σ12 s12k1 +2 Fs1 s1 + σ22 s22k2 +2 Fs2 s2
2
2
γ 2 2k1 +2 2
γ 2 2k2 +2 2
1 2 2
γ 2 2 2
− σ1 s1
Gs1 − σ2 s2
Gs2 + α b Fzz − α b Gz
2
2
2
2
[(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1 ]2
−
1
2
2σ12 s2k
1 (Fzz − γGz )
−
[(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2 ]2
2
2
2σ22 s2k
2 (Fzz − γGz )
=0
(B.7)
and
1
1
Gt + (rz + aαθ)Gz + μ1 s1 Gs1 + μ2 s2 Gs2 + σ12 s12k1 +2 Gs1 s1 + σ22 s22k2 +2 Gs2 s2
2
2
2 2k1 +1
2 2k1 +1
Fzs1 − γσ1 s1
Gz Gs1
1
(μ1 − r)Fz + σ1 s1
+ α2 b2 Gzz −
2
(Fzz − γG2z )
μ1 − r
[(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1 ]
× 2 2k1 Gz + s1 Gzs1 −
Gzz
1
2
σ1 s1
2σ12 s2k
1 (Fzz − γGz )
(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2
(Fzz − γG2z )
μ2 − r
(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2
× 2 2k2 Gz + s2 Gzs2 −
Gzz = 0.
2
2
σ2 s2
2σ22 s2k
2 (Fzz − γGz )
−
(B.8)
Similarly, the maximum of (2.20) over p on A3 is attained at p∗ = 1. Putting (B.3) and p∗ = 1 into (2.20)
and (2.21) gives
1
1
Ft + (rz + aα(θ − 2η) + aη)Fz + μ1 s1 Fs1 + μ2 s2 Fs2 + σ12 s12k1 +2 Fs1 s1 + σ22 s22k2 +2 Fs2 s2
2
2
γ 2 2k1 +2 2
γ 2 2k2 +2 2
1 2 2
2
2
− σ1 s1
Gs1 − σ2 s2
Gs2 + [α b − 2αb (α − β) + b (α − β)2 ](Fzz − γG2z )
2
2
2
[(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1 ]2
−
1
2
2σ12 s2k
1 (Fzz − γGz )
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[(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2 ]2
=0
2
2
2σ22 s2k
2 (Fzz − γGz )
(B.9)
and
1
Gt + (rz + aα(θ − 2η) + aη)Gz + μ1 s1 Gs1 + μ2 s2 Gs2 + σ12 s12k1 +2 Gs1 s1
2
1 2 2k2 +2
1 2 2
2
2
+ σ2 s2
Gs2 s2 + [α b − 2αb (α − β) + b (α − β)2 ]Gzz
2
2
(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1
−
(Fzz − γG2z )
[(μ1 − r)Fz + σ12 s12k1 +1 Fzs1 − γσ12 s12k1 +1 Gz Gs1 ]
μ1 − r
Gzz
× 2 2k1 Gz + s1 Gzs1 −
1
2
σ1 s1
2σ12 s2k
1 (Fzz − γGz )
(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2
(Fzz − γG2z )
μ2 − r
(μ2 − r)Fz + σ22 s22k2 +1 Fzs2 − γσ22 s22k2 +1 Gz Gs2
× 2 2k2 Gz + s2 Gzs2 −
= 0.
G
zz
2
2
σ2 s2
2σ22 s2k
2 (Fzz − γGz )
−
(B.10)
The above three pairs of equations can be solved by the same procedure and we demonstrate the
procedure with the pair of (B.5) and (B.6) only. To proceed, we conjecture the solutions in the following
form:
B1 (t) −2k1 C1 (t) −2k2 D1 (t)
s1
s2
,
(B.11)
F (t, z, s1 , s2 ) = A1 (t)z +
+
+
γ
γ
γ
G(t, z, s1 , s2 ) = A2 (t)z +
B2 (t) −2k1 C2 (t) −2k2 D2 (t)
s1
s2
+
+
γ
γ
γ
(B.12)
with boundary conditions A1 (T ) = A2 (T ) = 1, B1 (T ) = C1 (T ) = D1 (T ) = B2 (T ) = C2 (T ) = D2 (T ) = 0.
Then
B1 (t) −2k1 C1 (t) −2k2 D1 (t)
−2k1 B1 (t) −2k1 −1
+
+
,
s1
s2
, Fz = A1 (t), Fs1 =
s1
γ
γ
γ
γ
−2k2 C1 (t) −2k2 −1
2k1 (2k1 + 1)B1 (t) −2k1 −2
2k2 (2k2 + 1)C1 (t) −2k2 −2
s2
s1
s2
Fs2 =
, Fs1 s1 =
, Fs2 s2 =
,
γ
γ
γ
B (t)
C (t)
D (t)
1
2
Fzz = Fzs1 = Fzs2 = Fs1 s2 = 0, Gt = A2 (t)z + 2 s−2k
+ 2 s−2k
+ 2 ,
1
2
γ
γ
γ
2k1 (2k1 + 1)B2 (t) −2k1 −2
2k2 (2k2 + 1)C2 (t) −2k2 −2
Gz = A2 (t), Gs1 s1 =
s1
s2
, Gs2 s2 =
,
γ
γ
−2k1 B2 (t) −2k1 −1
−2k2 C2 (t) −2k2 −1
s1
s2
Gs1 =
, Gs2 =
, Gzz = Gzs1 = Gzs2 = Gs1 s2 = 0.
γ
γ
Ft = A1 (t)z +
Substituting the above derivatives into (B.5) and (B.6) yields
(A1 (t) + rA1 (t))z + B1 (t) − 2k1 μ1 B1 (t) − 2σ12 k12 B22 (t)
+
[(μ1 − r)A1 (t) + 2σ12 k1 A2 (t)B2 (t)]2
2σ12 A22 (t)
1
s−2k
1
+ C1 (t) − 2k2 μ2 C1 (t) − 2σ22 k22 C22 (t)
γ
2
D (t)
[(μ2 − r)A1 (t) + 2σ22 k2 A2 (t)C2 (t)]2 s−2k
2
+ 1
+ aα(θ − η)A1 (t)
+
2
2
2σ2 A2 (t)
γ
γ
1
a2 η 2 A21 (t)
1
=0
+ σ12 k1 (2k1 + 1)B1 (t) + σ22 k2 (2k2 + 1)C1 (t) +
γ
γ
2γb2 A22 (t)
and
(A2 (t) + rA2 (t))z + B2 (t) − 2k1 μ1 B2 (t)
(B.13)
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1
(μ1 − r)2 A1 (t) + 2k1 σ12 (μ1 − r)A2 (t)B2 (t) s−2k
1
+
C2 (t) − 2k2 μ2 C2 (t)
σ12 A2 (t)
γ
2
D (t)
(μ2 − r)2 A1 (t) + 2k2 σ22 (μ2 − r)A2 (t)C2 (t) s−2k
2
+ 2
+ aα(θ − η)A2 (t)
+
2
σ2 A2 (t)
γ
γ
1
a2 η 2 A21 (t)
1
+ σ12 k1 (2k1 + 1)B2 (t) + σ22 k2 (2k2 + 1)C2 (t) +
= 0.
γ
γ
γb2 A2 (t)
+
(B.14)
In order to eliminate the dependence on z, s1 and s2 , we can decompose (B.13) and (B.14) into
A1 (t) + rA1 (t) = 0,
(B.15)
B1 (t) − 2k1 μ1 B1 (t) − 2σ12 k12 B22 (t) +
C1 (t) − 2k2 μ2 C1 (t) − 2σ22 k22 C22 (t) +
[(μ1 − r)A1 (t) + 2σ12 k1 A2 (t)B2 (t)]2
2σ12 A22 (t)
[(μ2 − r)A1 (t) + 2σ22 k2 A2 (t)C2 (t)]2
2σ22 A22 (t)
= 0,
(B.16)
= 0,
(B.17)
D1 (t)
1
1
a2 η 2 A21 (t)
+ aα(θ − η)A1 (t) + σ12 k1 (2k1 + 1)B1 (t) + σ22 k2 (2k2 + 1)C1 (t) +
= 0,
γ
γ
γ
2γb2 A22 (t)
(B.18)
A2 (t) + rA2 (t) = 0,
(B.19)
B2 (t) − 2k1 μ1 B2 (t) +
C2 (t) − 2k2 μ2 C2 (t) +
(μ1 − r) A1 (t) + 2k1 σ12 (μ1 − r)A2 (t)B2 (t)
σ12 A2 (t)
2
(μ2 − r) A1 (t) + 2k2 σ22 (μ2 − r)A2 (t)C2 (t)
σ22 A2 (t)
2
= 0,
(B.20)
= 0,
(B.21)
D2 (t)
1
1
a2 η 2 A21 (t)
+ aα(θ − η)A2 (t) + σ12 k1 (2k1 + 1)B2 (t) + σ22 k2 (2k2 + 1)C2 (t) +
= 0.
γ
γ
γ
γb2 A2 (t)
(B.22)
Considering the boundary conditions, we obtain
A1 (t) = A2 (t) = er(T −t) ,
(μ1 − r)2 −2k1 μ1 (T −t)
(μ1 − r)2 (2μ1 − r)
[e
− e−2k1 r(T −t) ] +
[1 − e−2k1 μ1 (T −t) ],
2
2k1 σ1 r
4k1 μ1 rσ12
(μ2 − r)2 −2k2 μ2 (T −t)
(μ2 − r)2 (2μ2 − r)
[e
− e−2k2 r(T −t) ] +
[1 − e−2k2 μ2 (T −t) ],
C1 (t) =
2
2k2 σ2 r
4k2 μ2 rσ22
T
γaα(θ − η) r(T −t)
a2 η 2
2
(e
D1 (t) =
− 1) +
(T − t) + σ1 k1 (2k1 + 1)
B1 (s)ds
r
2b2
t
T
C1 (s)ds,
+ σ22 k2 (2k2 + 1)
B1 (t) =
(B.23)
t
(μ1 − r)2
B2 (t) =
(1 − e−2k1 r(T −t) ),
2k1 σ12 r
(μ2 − r)2
(1 − e−2k2 r(T −t) ),
2k2 σ22 r
T
γaα(θ − η) r(T −t)
a2 η 2
2
(e
D2 (t) =
− 1) +
(T
−
t)
+
σ
k
(2k
+
1)
B2 (s)ds
1
1 1
r
2b2
t
T
2
C2 (s)ds.
+ σ2 k2 (2k2 + 1)
C2 (t) =
t
In the rest of the proof, we shall discuss the equilibrium strategies in each of the cases as listed in Table 1.
We shall rely on the first-order conditions given in (B.3) and (B.4), the procedure as demonstrated in
the above for solving the three pairs of equations, as well as the continuity of the functions F and G.
To proceed, we note that the supreme in (2.20) is attained at p∗ (t) = 0 for p0 (t) < 0 and p∗ (t) = 1 for
p0 (t) > 1; moreover, according to (B.4), 0 p0 1 if and only if t1 t t2 , where t1 and t2 are given
in (3.1) and (3.2), respectively. The solutions are obtained by checking when the condition 0 p0 1 is
satisfied or not over the investment time horizon as given below for each case outlined in Table 1.
Case III. In each of these four cases, t1 and t2 satisfy t1 < 0 < T t2 and therefore the equilibrium
reinsurance strategy is p∗ (t) = p0 (t), 0 t T. The above derivation gives F and G expressed in (B.11)
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and (B.12) with A1 (t), B1 (t), C1 (t), D1 (t), A2 (t), B2 (t), C2 (t) and D2 (t) given in (B.23). Inserting (B.11)
and (B.12) into (B.3), we derive (3.3) and (3.4).
Case V.
In each of the two cases, we have 0 t1 < T t2 and hence,
0,
0 t < t1 ,
∗
p (t) =
0
p (t), t1 t T.
Similar to the preceding case, we obtain the expression of F and G as following:
⎧
1 (t)
⎪
B1 (t) −2k1 C1 (t) −2k2 D
⎪
⎪
⎨ A1 (t)z +
s1
s2
, 0 t < t1 ,
+
+
γ
γ
γ
F (t, z, s1 , s2 ) =
⎪
B (t)
C1 (t) −2k2 D1 (t)
⎪
⎪
1
⎩ A1 (t)z + 1 s−2k
s2
, t1 t T,
+
+
1
γ
γ
γ
(B.24)
⎧
2 (t)
⎪
B2 (t) −2k1 C2 (t) −2k2 D
⎪
⎪
⎨ A2 (t)z +
+
+
s1
s2
, 0 t < t1 ,
γ
γ
γ
G(t, z, s1 , s2 ) =
⎪
⎪
⎪ A (t)z + B2 (t) s−2k1 + C2 (t) s−2k2 + D2 (t) , t t T,
⎩
2
1
1
2
γ
γ
γ
where A1 (t), B1 (t), C1 (t), D1 (t), A2 (t), B2 (t), C2 (t) and D2 (t) are given in (B.23) and
(B.25)
2 2 2
1 (t) = γaαθ [er(T −t) − 1] + γaαη [1 − er(T −t1 ) ] + γ α b [e2r(T −t1 ) − e2r(T −t) ]
D
r
r
4r
T
T
a2 η 2
2
2
+
(T
−
t
)
+
σ
k
(2k
+
1)
B
(s)ds
+
σ
k
(2k
+
1)
C1 (s)ds,
1
1
1
1
2
2
1
2
2b2
t
t
2 2
2 (t) = γaαθ [er(T −t) − 1] + γaαη [1 − er(T −t1 ) ] + a η (T − t1 )
D
r
r
b2
T
T
2
2
+ σ1 k1 (2k1 + 1)
B2 (s)ds + σ2 k2 (2k2 + 1)
C2 (s)ds.
(B.26)
and
t
t
Equations (B.24) and (B.25) give the equilibrium investment strategies expressed as in (3.3) and (3.4).
Case VI. In this case, 0 < T t1 < t2 and thus, p∗ (t) = 0, 0 t T. For 0 t T , F and G
satisfy (B.7) and (B.8). Considering the boundary conditions in (2.20) and (2.21), we obtain
F (t, z, s1 , s2 ) = A1 (t)z +
B1 (t) −2k1 C1 (t) −2k2 D̃1 (t)
s1
s2
,
+
+
γ
γ
γ
B2 (t) −2k1 C2 (t) −2k2 D̃2 (t)
s1
s2
,
G(t, z, s1 , s2 ) = A2 (t)z +
+
+
γ
γ
γ
where
D̃1 (t) =
T
γaαθ r(T −t)
γ 2 α2 b2
[e
[1 − e2r(T −t) ] + σ12 k1 (2k1 + 1)
− 1] +
B1 (s)ds
r
4r
t
T
2
C1 (s)ds,
+ σ2 k2 (2k2 + 1)
(B.27)
(B.28)
t
D̃2 (t) =
T
T
γaαθ r(T −t)
− 1] + σ12 k1 (2k1 + 1)
B2 (s)ds + σ22 k2 (2k2 + 1)
C2 (s)ds
[e
r
t
t
and A1 (t), B1 (t), C1 (t), A2 (t), B2 (t) and C2 (t) are defined in (B.23). Inserting (B.27) into (B.3)
yields (3.3) and (3.4).
Cases IV and IX.
In each of the two cases, 0 t1 < t2 T and thus,
⎧
⎪
0 t < t1 ,
⎪
⎨0,
p∗ (t) =
p0 (t),
⎪
⎪
⎩1,
t 1 t < t2 ,
t2 t T.
Zhao H et al.
27
Sci China Math
Solving (B.7) and (B.8) for 0 t < t1 and solving (B.9) and (B.10) for t2 t T , noting the continuity
of F and G and taking the boundary conditions into account, we can similarly obtain
⎧
B1 (t) −2k1 C1 (t) −2k2 D1 (t)
⎪
⎪
s1
s2
,
+
+
⎪ A1 (t)z +
⎪
⎪
γ
γ
γ
⎪
⎪
⎨
1 (t)
B (t)
C1 (t) −2k2 D
1
F (t, z, s1 , s2 ) = A1 (t)z + 1 s−2k
s2
,
+
+
1
⎪
γ
γ
γ
⎪
⎪
⎪
⎪
⎪
B (t)
C1 (t) −2k2 D̂1 (t)
⎪
1
⎩ A1 (t)z + 1 s−2k
s2
,
+
+
1
γ
γ
γ
and
0 t < t1 ,
t1 t < t2 ,
(B.29)
t2 t T,
⎧
B2 (t) −2k1 C2 (t) −2k2 D2 (t)
⎪
⎪
A2 (t)z +
s1
s2
, 0 t < t1 ,
+
+
⎪
⎪
⎪
γ
γ
γ
⎪
⎪
⎨
2 (t)
B (t)
C2 (t) −2k2 D
1
G(t, z, s1 , s2 ) = A2 (t)z + 2 s−2k
s2
, t1 t < t2 ,
+
+
1
⎪
γ
γ
γ
⎪
⎪
⎪
⎪
⎪
B (t)
C2 (t) −2k2 D̂2 (t)
⎪
1
⎩ A2 (t)z + 2 s−2k
s2
, t2 t T,
+
+
1
γ
γ
γ
(B.30)
where
T
γaαθ r(T −t)
γ 2 α2 b2 2r(T −t1 )
[e
[e
− 1] +
− e2r(T −t) ] + σ12 k1 (2k1 + 1)
B1 (s)ds
r
4r
t
T
γaαη
a2 η 2
[2 − er(T −t1 ) − er(T −t2 ) ] +
C1 (s)ds −
(t2 − t1 )
+ σ22 k2 (2k2 + 1)
r
2b2
t
γ 2 [α2 b2 − 2αb2 (α − β) + b2 (α − β)2 ]
γaη r(T −t2 )
[e
[1 − e2r(T −t2 ) ],
− 1] +
+
r
4r
2 2
1 (t) = γaα(θ − η) [er(T −t) − 1] + a η (t2 − t) + γaβη [er(T −t2 ) − 1]
D
r
2b2
r
γ 2 [α2 b2 − 2αb2 (α − β) + b2 (α − β)2 ]
[1 − e2r(T −t2 ) ]
+
4r
T
T
2
2
+ σ1 k1 (2k1 + 1)
B1 (s)ds + σ2 k2 (2k2 + 1)
C1 (s)ds,
D1 (t) =
t
(B.31)
t
γ[aα(θ − 2η) + aη] r(T −t)
γ 2 [α2 b2 − 2αb2 (α − β) + b2 (α − β)2 ]
[e
− 1] +
D̂1 (t) =
r
4r
T
T
2r(T −t)
2
2
× [1 − e
] + σ1 k1 (2k1 + 1)
B1 (s)ds + σ2 k2 (2k2 + 1)
C1 (s)ds,
t
t
2 2
γaαθ r(T −t)
a η
γaαη
[e
[2 − er(T −t1 ) − er(T −t2 ) ]
− 1] + 2 (t2 − t1 ) +
r
b
r
T
T
γaη r(T −t2 )
[e
+
− 1] + σ12 k1 (2k1 + 1)
B2 (s)ds + σ22 k2 (2k2 + 1)
C2 (s)ds,
r
t
t
2 2
2 (t) = γaα(θ − η) [er(T −t) − 1] + a η (t2 − t) + γa(1 − α)η [er(T −t2 ) − 1]
D
r
b2
r
T
T
+ σ12 k1 (2k1 + 1)
B2 (s)ds + σ22 k2 (2k2 + 1)
C2 (s)ds,
D2 (t) =
t
D̂2 (t) =
(B.32)
t
T
γ[aα(θ − 2η) + aη] r(T −t)
(e
− 1) + σ12 k1 (2k1 + 1)
B2 (s)ds
r
t
T
C2 (s)ds
+ σ22 k2 (2k2 + 1)
t
and A1 (t), B1 (t), C1 (t), A2 (t), B2 (t) and C2 (t) are given in (B.23). Putting (B.29) and (B.30) into (B.3)
yields (3.3) and (3.4).
28
Zhao H et al.
Cases II and VIII.
Sci China Math
In each of the three cases, t1 < 0 t2 < T and therefore,
p0 (t), 0 t < t2 ,
p∗ (t) =
1,
t2 t T.
Similar to the preceding cases, F and G can be solved explicitly as follows:
⎧
1 (t)
B1 (t) −2k1 C1 (t) −2k2 D
⎪
⎪
s1
s2
,
+
+
⎨ A1 (t)z +
γ
γ
γ
F (t, z, s1 , s2 ) =
⎪
⎪
⎩ A1 (t)z + B1 (t) s−2k1 + C1 (t) s−2k2 + D̂1 (t) ,
1
2
γ
γ
γ
and
⎧
2 (t)
B2 (t) −2k1 C2 (t) −2k2 D
⎪
⎪
s1
s2
,
+
+
⎨ A2 (t)z +
γ
γ
γ
G(t, z, s1 , s2 ) =
⎪
⎪
⎩ A2 (t)z + B2 (t) s−2k1 + C2 (t) s−2k2 + D̂2 (t) ,
1
2
γ
γ
γ
0 t < t2 ,
(B.33)
t2 t T
0 t < t2 ,
(B.34)
t2 t T
1 (t), D̂1 (t), D
2 (t), D̂2 (t) are given in (B.23), (B.31)
with A1 (t), B1 (t), C1 (t), A2 (t), B2 (t), C2 (t) and D
and (B.32). Inserting (B.33) and (B.34) into (B.3), we obtain (3.3) and (3.4).
Cases I, VII and X. In each of the three cases, t1 and t2 satisfy t1 < t2 < 0 < T , and therefore,
p∗ (t) = 1, 0 t T . Similarly, we can obtain
F (t, z, s1 , s2 ) = A1 (t)z +
B1 (t) −2k1 C1 (t) −2k2 D̂1 (t)
s1
s2
,
+
+
γ
γ
γ
B2 (t) −2k1 C2 (t) −2k2 D̂2 (t)
s1
s2
G(t, z, s1 , s2 ) = A2 (t)z +
+
+
γ
γ
γ
(B.35)
with A1 (t), B1 (t), C1 (t), A2 (t), B2 (t), C2 (t), D̂1 (t) and D̂2 (t) defined in (B.23), (B.31) and (B.32).
Using (B.35) and (B.3), we can derive (3.3) and (3.4).
Clearly, the forms of F, G, H satisfy the continuity and polynomial conditions we impose in Theorem 2.4. Also, it is evident that the optimal proportion reinsurance p(v) ∈ [0, 1], for any v ∈ [t, T ],
i.e., Condition (a) in Definition 2.2 is satisfied. Finally, as πi∗ (t) ∼ (Si (t))−2ki , for i = 1, 2, we can see
|πi∗ (t)|2 (1 + |(Si (t))|2ki ) ∼ [(Si (t))−2ki + (Si (t))−4ki ]. Then (2.7) immediately confirms that the optimal
investment strategies satisfy Condition (b). Indeed,
E
t
T
|πi (v)|2 (1 + |Si (t)|2ki )dv K E sup |mi (v)|2 + E sup |mi (v)| < ∞.
Therefore, u∗ (t) is admissible.
v∈[0,T ]
v∈[0,T ]