Methodol Comput Appl Probab (2018) 20:1477–1502
https://doi.org/10.1007/s11009-018-9630-7
Optimal Reinsurance-Investment Strategy Under Risks
of Interest Rate, Exchange Rate and Inflation
Chang Guo1 ·Xiaoyang Zhuo2 ·
Corina Constantinescu3 ·Olivier Menoukeu Pamen3,4,5
Received: 29 December 2016 / Revised: 14 March 2018 /
Accepted: 20 March 2018 / Published online: 3 April 2018
© The Author(s) 2018
Abstract In this paper, we pursue the optimal reinsurance-investment strategy of an insurer
who can invest in both domestic and foreign markets. We assume that both the domes-
tic and the foreign nominal interest rates are described by extended Cox-Ingersoll-Ross
(CIR) models. In order to hedge the risk associated to investments, rolling bonds, trea-
sury inflation protected securities and futures are purchased by the insurer. We use the
dynamic programming principles to explicitly derive both the value function and the opti-
mal reinsurance-investment strategy. As a conclusion, we analyze the impact of the model
parameters on both the optimal strategy and the optimal utility.
Keywords Optimal reinsurance-investment strategy · Foreign exchange market ·
Extended CIR · Stochastic inflation · Dynamic programming principle
 Olivier Menoukeu Pamen
menoukeu@liv.ac.uk
Chang Guo
gchang@mail.nankai.edu.cn
Xiaoyang Zhuo
zhuoxy3600@mail.nankai.edu.cn
Corina Constantinescu
C.Constantinescu@liverpool.ac.uk
1 School of Finance, Nankai University, Tianjin 300350, China
2 Business School, Nankai University, Tianjin 300071, China
3 Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences,
University of Liverpool, Liverpool L69 7ZL, UK
4 African Institute for Mathematical Sciences, Accra, Ghana
5 University of Ghana, Accra, Ghana
1478 Methodol Comput Appl Probab (2018) 20:1477–1502
Mathematics Subject Classification (2010) 49L20 · 91G80
1 Introduction
As reinsurance and investment are becoming increasingly important in the recent years,
models that enable both risk reduction and profit maximization have attracted a lot of
interest. The purpose of this paper is two-fold. Firstly, it aims at solving an optimal
investment-and-reinsurance problem of an insurance company which can invest domesti-
cally and abroad. This is done by maximizing the expected constant relative risk aversion
(CRRA) utility of terminal wealth. More precisely, by considering investments in a foreign
market, we allow additional sources of risks: the foreign interest rate risk, the inflation risk
and the exchange rate risk. They make the optimal reinsurance-investment solution more
applicable to nowadays insurance markets, but the problem becomes more complex mathe-
matically. The wealth process of the insurance company would account for these additional
sources of uncertainty. Secondly, it aims at studying the impact of the model parameters on
the optimal strategy and the optimal utility of the insurer. The sensitivity analysis in this
case shows that, unlike in the case of no foreign market investments, the investor has to
short sale cash and futures.
In our model, the nominal interest rates follow extended Cox-Ingersoll-Ross (CIR) mod-
els (see (2)), in contrast with the mean reverting models considered in the existing literature.
This makes the setting more realistic, but less tractable. Both the exchange rate and the
stock price follow geometric Brownian motions (GBM) (see (3) and (12)). Due to the pos-
sibility of long term investments, the risk of inflation is introduced in the model. The risk
related to having stochastic nominal interest rates can be reduced by purchasing rolling
bonds with dynamics described by GBMs (see (6)). In order to hedge the risk of exchange
rates, the insurer needs to buy futures, driven also by GBMs (see (11)). As for the infla-
tion risk, the insurer can purchase Treasury Inflation Protected Securities (TIPS). The TIPS
are also assumed to follow GBMs (see (15)). After all these considerations, the solution of
the Hamilton Jacobi Bellman equation derived can be presented in terms of solutions of a
Riccati equation with constant coefficients.
Browne (1995) focused on minimizing the probability of ruin for an insurance company
facing an uncontrollable stochastic cash flow while investing on a risky stock. Schmidli
(2001) considered dynamic proportional reinsurance strategies and derived optimal strate-
gies in a diffusion setup and in a classical compound Poisson risk model. Promislow and
Young (2005) extended the work of the above authors and found the optimal investment-
and-proportional-reinsurance strategy that minimizes the risk of ruin of an insurer whose
claim process follows a Brownian motion with drift. Bai and Guo (2008) studied the case of
portfolio of one risk-free asset and n risky assets while purchasing proportional reinsurance
under the constraint of no-shorting. They obtained optimal strategies by: either maximizing
the expected exponential utility of terminal wealth or minimizing the probability of ruin. Li
et al. (2012) considered a stochastic volatility model for the insurer’s portfolio. They inves-
tigated the optimal time-consistent investment-and-reinsurance strategy for an insurer in a
financial market consisting of one risk-free asset and one risky asset whose price process
follows the Heston stochastic volatility model. Other papers in this area include Liang et al.
(2011), Gu et al. (2012), and Yi et al. (2015) and the references therein.
Methodol Comput Appl Probab (2018) 20:1477–1502 1479
The inflation risk has also been considered by many authors. Brennan and Xia (2002)
developed a framework for analysing a finite-horizon investor asset allocation problem
under inflation when cash, equity, and a single nominal bond are available, and gave closed
forms of investor’s optimal investment strategy and indirect utility depending on investor’s
horizon and its risk aversion. Jarrow and Yildirim (2003) found a relationship between
the forward nominal interest rate and the inflation index, and use an Heath Jarrow Merton
(HJM) model to price TIPS. HJM models are often used to hedge the risk of inflation. Guan
and Liang (2014b) included the inflation risk in the optimal proportional-reinsurance-and-
investment problem with the objective to maximize the expected utility of terminal wealth.
The current paper can be seen as a generalization of Guan and Liang (2014b) into several
directions, the presence of foreign markets, exchange rates, inflation risk and the use of
extended CIR models.
The reason for including foreign markets that they may be attractive to insurers since
returns are often subject to exchange rates and are not directly influenced by the domes-
tic market. Levy and Sarnat (1970) introduced the method of empirical determination of
the composition of optimal international portfolios and analyzed the implications of the
international risk diversification for the investment decisions. Jorion (1989) summarized
the historical evidence on foreign investments and examined the risk and return character-
istics of the foreign stocks and bonds when these assets are stripped of their currency risk.
Brennan and Cao (1997) developed a model for the international equity portfolio invest-
ment flows based on the differences in informational endowments between the foreign and
domestic investors. Allayannis and Ofek (2001) found evidence that firms using currency
derivatives significantly reduced their exchange-rate exposure. They also found that the
number of derivatives used depends only on the firm’s exposure to foreign sales and trade.
In this paper we describe the insurer’s surplus process by a Brownian motion with drift.
We consider that the insurer invests in both domestic and foreign risky assets. Further, under
the hypothesis that the insurer has a CRRA utility function, the insurer’s objective is the
derivation of the optimal reinsurance-investment strategy which maximizes the expected
utility of the terminal wealth. These additional features (foreign market and extended CIR
models) alter the optimal strategy. Although the Hamilton Jacobi Bellman (HJB) equation
is still a second order ordinary differential equation with constant coefficients, these coeffi-
cients incorporate both the interest rate and the domestic inflation rate. However, the optimal
strategy is still unique and it can be derived explicitly. A simulation of the optimal strategy
and a sensitivity analysis of the model parameters are performed under this scenario.
This paper consists of five sections. Section 2 introduces the framework and describes
the optimization problem. Section 3 reduces the original optimization problem to the self-
financing case, which can be solved via an HJB equation. Explicit expressions of the optimal
value function and the corresponding optimal reinsurance-investment strategy are derived in
this section. Section 4 discusses the significance/effect of the choice of parameters on the opti-
mal strategy, the reinsurance strategy and the optimal utility of the insurer. Section 5 concludes.
2 The Financial Market
In this section, we present the general model of the insurer. Let (,F ,P) be a complete
probability space with filtration {Ft }t∈[0,T ]. The interval [0, T ] describes a specified time
1480 Methodol Comput Appl Probab (2018) 20:1477–1502
horizon, within which the insurer adjusts the reinsurance and investment strategy to max-
imise its expected utility of terminal wealth. We assume that all the processes are adapted
to the filtration {Ft }t∈[0,T ].
The surplus process of the insurer is as in Guan and Liang (2014b). Assuming that the
claims Yi , i = 1, 2, . . . , N are independent and identically distributed with E(Yi) = μ1 and
E(Y 2i ) = μ2, and that the number of claims up to time t follows a homogeneous Poisson
process with intensity λ, let a(t) be the reinsurance proportion, meaning that at time t , the
insurer assumes only a(t) of the risk. Let η and θ be the safety loading of the insurance
and reinsurance business respectively, with η < θ . The surplus process, denoted by X(t),
satisfies the following stochastic differential equation (SDE):
= − + +√dX(t) λμ1(η θ)dt λμ1θa(t)dt λμ2a(t)dW0(t), (1)
where {W0(t)}t∈[0,T ] denotes the standard Brownian motion.
2.1 Domestic and Foreign Nominal Exchange Rate
The dynamics of the nominal interest rate are given by the following SDE:
= )
√
drl (t) al(bl − rln n(t) dt − kl l1rn(t)+ kl l2dWr(t), l = d, f, (2)
where al , bl , kl1, k
l
2 are positive coefficients and {Wlr (t)}t∈[0,T ] is standard Brownian motion.
The superscript l = d, f stands for either the domestic or foreign rate and will be used
throughout this paper. This model has been studied by Duffie and Kan (1996) and used in
Guan and Liang (2014a). Vasicek (1977) and Cox et al. (1985) analyzed the special cases
of the model for kl1 = 0 and kl2 = 0, respectively.
We assume that the exchange rate satisfies the following SDE:
dQ(t) = ( f ) ( )rdn (t)− rn (t) dt + σQ λQdt + dWQ(t) , (3)Q(t)
where λQ is the market price of the risk of WQ(t) and WQ(t) a standard Brownian motion.
The standard Brownian motions {W0(t)}, {Wlr (t)} and {WQ(t)} are independent.
The risk-free asset (i.e. the cash) in domestic financial market, denoted by S0(t), evolves
according to:
dS0(t) = rdn (t)S0(t)dt. (4)
2.2 Domestic and Foreign Bond
In order to hedge the risk of nominal interest rates in the domestic and the foreign market,
respectively, the insurer needs zero-coupon bonds fBdn (t, T ) and Bn (t, T ). These are assets
bought at time t and paying £1 at maturity T . It can be shown (see Guan and Liang 2014a)
that the price of Bln(t, T )(l = d, f ) under extended CIR interest rates evolves according to
the following backward stochastic differential equation:
{ √ ( √ )
dBln(t,T ) = rl (t)dt + σ l (T − t) kl rl (t)+ kl l l ll n B 1 n 2 λr k1rn(t)+ kl2dt + dWlr (t)Bn(t,T ) (5)
Bln(T , T ) = 1, l = d, f,
l √m t
where l = 2(e −1)σ l l lB(t) − + + l + − , m = (a − k1λl )2 + 2k
l , and
ml al kl λl em t r√ 1 r (m
l al kl l1λr )
1
λlr k
l l l l
1rn(t)+ k2 is the market price of the risk of Wr(t). However, no zero-coupon bonds
Methodol Comput Appl Probab (2018) 20:1477–1502 1481
with random maturity T exists, hence the insurer invests in rolling bonds BlK(t) with a fixed
maturity, K = T or T l1 2. From (5), the dynamics of BK(t) satisfy
dBl √ √K(t)
( )
= rl l l l l l l l l l
l n
(t)dt + σB(K) k1rn(t)+ k2 λr k1rn(t)+ k2dt + dWr(t) , (6)
BK(t)
with (l,K) = {(d, T1), (f, T2)}.
From the domestic investor’s point of view, a foreign asset value must be converted
in terms of its own currency (see Lioui and Poncet 2002). Let CT2(t) be the price of the
foreign rolling bond in the domestic currency. Then fCT2(t) and BT (t) have the relationship:2
f
CT2(t) = BT (t) ·Q(t). Using Itô’s formula, CT2(t) satisfies the following SDE:2
d f f fCT2 (t) = Q(t)dBT (t)+ BT (t)dQ(t)+ d〈Q(t), BT (t)〉2 2 √ 2 ( √ )
= d d + f f f + f f f f f fCT2 (t)rn (t) t CT2 (t)σB (T2) k1 rn (t) k2 λr k1 rn (t)+ k2 dt + dWr (t)
+ ( )CT2 (t)σQ λQdt + dWQ(t) . (7)
2.3 Futures
In order to hedge the exchange rate risk, the insurer needs to purchase futures. Let FT3(t) and
QF (t, T3) be the price of the exchange rate futures and, respectively, the price of exchange
rate forward, with maturity T3. Then FT3(t) andQ
F (t, T3) are connected via (see Amin and
Jarrow 1991, (39))
{ ∫ T3 }
F (t) = QF (t, T ) exp σd(T − s)2T3 3 B 3 (kd d d1 rn (s)+ k2 )ds . (8)
t
Using the relationship
QF (t, T3) · Bd fn (t, T3) = Bn (t, T3) ·Q(t), (9)
we obtainQF (t, T3) and then FT3(t) through (8). Substituting (3) and (5) into (9) and using
Itô’s formula, it is straightforward to see that the dynamics of QF (t, T3) are given by the
following SDE:
dQF √ √(t, T ) ( )3 = −σd(T − t) kdrd(t)+ kd λd kdrdB 3 1 n 2 r 1 n (t)+ kd2 dt + dWd(t)QF (t, T r3) √ ( √ )
+ f − f f f f f f f fσB (T3 t) k1 rn (t)+ k2 λr k1 rn (t)+ k2 dt + dWr (t)
+ ( )σQ λQdt + dWQ(t) + σdB(T3 − 2
( )
t) kd d1 rn (t)+ kd2 dt. (10)
Applying Itô’s formula to (8) and substituting (10) into the obtained differential equation, it
is easy to see that FT3(t) satisfies the following SDE:
d √ √FT3 (t)
( )
= ( )σQ λQdt + dWQ(t) − σd(T3) kdrd (t)+ kd λd kdrd (t)+ kddt + dWd(t)
FT3 (t)
B 1 n 2 r 1 n 2 r
√ ( √ )
+ f f f f f f f f fσB (T3) k1 rn (t)+ k2 λr k1 rn (t)+ k2 dt + dWr (t) . (11)
1482 Methodol Comput Appl Probab (2018) 20:1477–1502
2.4 The Risky Asset Price
The insurer invests in domestic and foreign stock markets, in which the dynamics of the
risky asset Sl(t) are given by:
d l √ ( √S (t) )= rl (t)dt + σ l kl rl l l l l l l
Sl(t) n S1 1 n
(t)+ k2 λr k1rn(t)+ k2dt + dWr(t)
+ ( )σ lS λlSdt + dWlS(t) , l = d, f. (12)2
We assume that the standard Brownian motion {WlS(t)} is independent from
{W0(t)}, {Wlr (t)} and {WQ(t)}, and that λl lS is the market price of the risk of WS(t).
Let C(t) be the price of the foreign asset in the domestic currency. Then C(t) and
Sf (t) are connected by the relationship C(t) = Sf (t) ·Q(t). Itô’s formula implies that the
dynamics of C(t) satisfies the following SDE:
dC(t) = Q(t)dSf (t)+ Sf (t)dQ(t)+ d〈Q(t), Sf (t)〉
= + ( ) (d d d + d + f f f )C(t)rn (t) t C(t)σQ λQ t WQ(t) C(t)σS λS dt + dWS (t)
√ ( √
2
)
+ f f f f f f f fC(t)σS k r (t)+ k λ k r (t)+ k dt + d fW (t) . (13)1 1 n 2 r 1 n 2 r
2.5 The Inflation
We also assume there exits the inflation risk I l(t), l = d, f in the financial markets and
rlr (t), l = d, f represent the real interest rates. I l(t) evolves according to the following
SDE:
dI l √ √(t) ( ) ( )= rln(t)− rlr (t) dt + σ l l l l l l lI k1rn(t)+ k2 λr k1rn(t)+ kl lI l(t) 1 2dt + dWr(t)
+ ( + )σ lI λlIdt dWlI (t) , l = d, f, (14)2
We assume that λlI is the market price of the risk of the standard Brownian motion {WlI (t)},
and that {WlI (t)} is independent from {W0(t)}, {Wlr (t)}, {WQ(t)} and {WlS(t)}.
To hedge the inflation risk, TIPS are available in the market. Let P lK(t) be the price of a
TIPS with maturity K . The dynamics of P lK(t) are described by the following SDE:
d √ √P l ( )K(t) = rl ln(t)dt + σI kl rl (t)+ kl l1 n 2 λr kl1rln(t)+ kl2dt + dWl(t)
P l (t) 1
r
K
+ ( )σ l lI λIdt + dWlI (t) , l = d, f, (15)2
with (l,K) = {(d, T4), (f, T5)}.
Similarly, the price of the foreign TIPS needs to be expressed in domestic currency,
denoted by CT5(t). Due to the relationship
f
CT5(t) = PT (t) ·Q(t), the dynamics of CT5(t)5
evolves according to the following SDE:
d f f fCT5(t) = Q(t)dPT (t)+ PT (t)dQ(t)+ d〈PT (t),Q(t)〉5 5 √ 5 ( √ )
= f f f f f f f f fCT5(t)rdn (t)dt + CT5(t)σI k1 rn (t)+ k2 λr k1 rn (t)+ k2 dt + dWr (t)1
+ f ( fCT5(t)σI λI dt + f
) ( )
dWI (t) + C2 T5(t)σQ λQdt + dWQ(t) . (16)
Methodol Comput Appl Probab (2018) 20:1477–1502 1483
2.6 The Wealth Process
Let θ0(t), θ lB(t), θ
l
S(t) and θ
l
P (t)(l = d, f ) be the money invested in cash, bonds, stocks
and TIPS respectively, in our financial market. Let θF (t) be the futures position the insurer
holds at time t . Since there is no cost to enter the futures market and the contract’s gains and
losses are settled on a continuous basis (marking-to-market mechanism), the wealth process
of the insurer is X(t) = f f fθ0(t) + θdB(t) + θ dB (t) + θS (t) + θS (t) + θdP (t) + θP (t) and it
satisfies the SDE:
√ dS
d 0
(t)
X(t) = λμ1(η − θ)dt + λμ1θa(t)dt + λμ2a(t)dW0(t)+ θ0(t)
S0(t)
dBd (t) dC d+ d T1 + f T2(t) + dFT3(t) + d dS (t)θB(t) θB (t) θF (t) θ (t)
Bd (t) CT2(t) F (t)
S Sd(t)
T T1 3
d dPd+ f C(t) T (t) f dCT (t)θS (t) + θdP (t) 4 + θ (t) 5P . (17)C(t) P d (t) CT5(t)T4
3 The Utility Maximization
3.1 The Problem
Consider the previousmarket and let ū(t)=(a(t),θd f d f fB(t), θB(t), θF (t), θS (t), θS (t), θdP (t), θP (t))′
be the reinsurance-investment strategy of the insurer. Our aim is to find a strategy that
maximizes the expected utility of terminal wealth of the insurer. Substituting (4), (6), (7),
(11), (12), (13), (15) and (16) into (17), we obtain
dX(t) = λμ1(η − θ)dt + rd(t)X(t)dt + ū(t)′n σ t (tdt + dW t ), (18)
where
⎛√ ⎞
λμ2 √ 0 0 0 0 0 0 0
⎜⎜ 0 σdB(T1) k
d
1 r
d
n (t)+ kd2 0 0 0 0 0 0 ⎟⎜ √ ⎟⎜ ⎟⎜ 0 0 f f f f ⎟σ (T⎜ √ B 2)√k1 rn (t)+ k2 σQ 0 0 0 0 ⎟⎜ ⎟⎜ 0 − d d d + d f f f + f ⎟⎜ σB(T3) k1 rn (t) k2 σB (T3) k1 rn (t) k2 σQ 0 0 0 0 ⎟
σ = ⎜ √ ⎟t ,⎜ 0 σd kd d d ⎟⎜ S 1 rn (t)+ k2 0 0 σd 0 0 0 ⎟1 √ S2⎜ ⎟⎜ f f f f f ⎟⎜ 0 √ 0 σS k1 rn (t)+ k2 σ ⎟1 Q 0 σS 0 0⎜ 2 ⎟⎜ ⎟0 σd⎝ I kd d1 rn (t)+ kd2 0 0 0 0 σdI 0 ⎟1 √ 2 ⎠
0 0 f f fσI k1 rn (t)+ f fk σ1 2 Q 0 0 0 σI2
( √ √= √λμ1θ d d d + d f f f + f f f )′t , λr k1 rn (t) k2 , λr k1 rn (t) k2 , λQ, λd dS, λS , λI , λ ,
( λμ2
I
)
d f f f ′W t = dW0(t), dWdr (t), dWr (t), dW d d ′Q(t), dWS (t), dWS (t), dWI (t), dWI (t) . Here “ ”
denotes for the transposition of a vector.
We assume that the insurer has a CRRA utility function of the form
x1−γ
U(x) = − , γ > 0, γ = 1. (19)1 γ
1484 Methodol Comput Appl Probab (2018) 20:1477–1502
The insurer’s objective is to adjust the strategy ū(t) within the time horizon [0, T ] in order
to maximize the expected utility of its terminal wealth X(T ). Due to the inflation risk, we
have to maximize the expected utility of the real value of X(T ).
Problem 1 The optimization problem becomes
[ ( )]
max X(T )ū(t) E U Id(T ) (20)
s.t. X(0) = x,
with I d,X,U as in (14), (18) and (19).
3.2 The Solution
Due to the fact that the insurer receives a continuous premium income, the portfolio is not
self-financing and hence the classical optimization techniques are not directly applicable.
In the spirit of Guan and Liang (2014b), we transform our wealth process. An auxiliary
wealth process makes our portfolio self-financing and thus we define a new optimization
problem. The following lemma shows that this “discounted” wealth process is a conditional
expectation.
∫ ∫
Lemma 1 Define t tH(t) := exp{ 0 (rd(s)+ 1‖  ‖2n 2 s )ds + ′0 sdW s}. Then H(t) satisfies
the following SDE:
{
dH(t) = ( )H(t) rd ′ ′n (t)+tt dt +H(t)tdW t (21)
H(0) = 1.
Moreover, X(t) has the following form:
[ ∫ ∣X(T )H(t) T λμ1(η − θ)H(t) ∣ ]
X(t) = E − ds∣Ft , t ∈ [0, T ]. (22)
H(T ) t H(s)
Proof The proof is analogous to the one in Guan and Liang (2014b, Lemma 3.1).
As in a general investment problem, H(t) can be regarded as the pricing kernel of the
financial market. A self-financing case requires = [X(T )H(t)X(t) E |Ft ], but this conditionH(T )
is not satisfied. It follows, however, from (22) that X(t) is a supermartingale. In order to
have a self-financing setup, we define the process G(t) by
∫ T ∫
:= [ λμ1(η − θ)H(t)
T
G(t) E ds| ] = H(t)Ft λμ1(η − θ) E[ |Ft ]ds,
t H(s) t H(s)
with G(0) = g. The following lemma gives the SDE satisfied by G(t).
∫
Lemma 2 TG(t) can be rewritten as G(t) = λμ1(η − θ) Bdn (t, s)ds. Then it satisfies thet
following SDE:
⎧
⎪⎨ dG(t) = −λ√μ1(η − θ)dt + rdn (t)G(t)dt + λdr (kd d1 rn (t)+ kd2 )σG(t, T )G(t)dt
+ kd d⎪ 1 rn (t)+ kd2σG(t, T )G(t)dWdr (t) (23)⎩
G(T ) = 0,
∫
where TσG(t, T ) = 1 λμ (η − θ)Bd(t, s)σ d(s − t)ds.G(t) t 1 n B
Proof The proof is analogous to the one in Guan and Liang (2014b, Lemma 3.2).
Methodol Comput Appl Probab (2018) 20:1477–1502 1485
Let Y (t) be an auxiliary process defined by Y (t) := X(t)+G(t). Then
dY (t) = dX(t)+ dG(t)
= rdn (t)Y (t)dt + ū(t)′σ ( dt + dW )+ λd(kd d dt t t r 1 rn (t)+ k2 )σG(t, T )G(t)dt√
+ kd1 rdn (t)+ kd2σG(t, T )G(t)dWdr (t)
= rdn (t)Y (t)dt + u(t)′σ t (tdt + dW t ), (24)
where u σ (t,T )G(t)(t) = ū(t)+ (0, G d , 0, 0, 0, 0, 0, 0)′. Y (t) is a self-financing process.σB(T1)
Thus, the optimization problem (20) can be re-written as a self-financing optimization
problem as follows:
Problem 2
[ ( Y (T ) )∣ ]f ∣ f f
V (t, rdn , rn , i
d , y) : = maxE U ∣rdn (t) = rdn , rn (t) = rn , I d (t) = id , Y (t) = y
u(t) I d (T )
s.t. Y (0) = x + g. (25)
The following theorem, whose proof is straightforward, reveals the Hamilton Jacobi
Bellman (HJB) equation satisfied by the value function fV (t, rdn , rn , i
d , y).
Theorem 1 Let be described by (24). Then the value function d fY (t) V (t, r , r , idn n , y) of
the optimization problem (25) satisfies the following HJB equation
{
sup V + V [rdy + u(t)′t y n σ tt ] + Vrd [ad(bd − rdn )] + V f [af f(bf − rn )]n rn
u(t)
+V · id [rd − rd + d d d d + d 1σ λ (k r k )+ σd λd ] + V u(t)′σ σ ′id n r I1 r 1 n 2 I I yy t tu(t)2 2
+ 1V σ ′ σ + 1 ′ + 1V σ σ V · (id)2σ ′ σ + V u(t)′d d d f f d d d σ σ
2 rn rn d 2 r r f
f i i I I t d
n n 2 yrn
+ }V f u(t)′σ σ d ′t f + Vyid · i u(t) σ tσ I + Vidrd · idσ ′yr n dσ I = 0, (26)n
√ √
where = 0 − d d + d 0 0 0 0 0 0 ′, = 0 0 − f f fσ d ( , k1 rn k2 , , , , , , ) σ f ( , , k1 rn + k2 , 0, 0, 0, 0, 0)′,√
σ I = (0, σ d kd d d d ′I1 1 rn + k2 , 0, 0, 0, 0, σI , 0) .2
Next we derive the closed-form solution of the value function V (t, rd fn , r
d
n , i , y) in terms
of the solution to a partial differential equation.
Theorem 2 The closed-form of fV (t, rdn , rn , i
d , y) is
f 1 y f
V (t, rd , r , id , y) = ( )1−γ dn n − h(t, r1 γ id n , rn ), (27)
where
f f
h(t, rdn , rn ) = exp{rdn q1(t)+ rn q2(t)+ q3(t)}, (28)
1486 Methodol Comput Appl Probab (2018) 20:1477–1502
with q1(t), q2(t) and q3(t) given by
α β e(αj−βj )(T−t)= − j j · − 1qj (t) , j = 1, 2, (29)
N β e(αj−βj )(T−t)j j − αj
∣
= −Mβ
[
1 − + 1 ln ∣∣β e
(α1−β1)(T−t)− ∣α ∣] Nβ [ 1 ∣∣β e(α2−β2)(T−t)− ∣α ∣]1 1 ∣ − 2 − 2 2q3(t) T t − T t+ ln ∣ ∣N1 β1 β1 α1 N2 β2 β2−α2
α β2kd[ − ∣ (α1−β1)(T−t) ∣+ 1 1 2 T t + α1+β1 ∣β1e −α1∣− α1−β1 1 1
]
ln −
2γN2 2
∣ ∣
α α 2 (α −β )(T−t) 21 1 1β1 β1−α1 β1 β1e 1 1 −α1 β1
α β2
f
k [ ∣T −t α +β ∣β e(α2−β2)(T−t) ∣+ 2 2 2 + 2 2 2 −α2∣− α2−β2 1 − 1
]
ln∣ ∣
2γN2 2 2 (α2−β2)(T−t) 22 α2 α2β2 β2−α2 β2 β2e −α2 β2
∫ T
− (γ − 1) rdr (s)ds + R(T − t). (30)
t
Proof For the proof and the details of αj , βj ,Nj (j = 1, 2),M,N,R, see Appendix A.1.
Remark 1 Theorem 2 can be seen as a generalization of Proposition 3.4 in Guan and Liang
(2014b). In fact, if one sets kd1 = 0 and eliminates the foreign investment, the value function
is reduced to the one given by ?[ ()(3.11)]Guan2014. One simply needs to observe that, in
that situation, M = ad1 , N1 = R1 = 0, which implies that q1(t) = 0, and since there is no
foreign investment, q2(t) = 0 as well.
In the next theorem we derive the optimal control for the auxiliary optimization problem
(25).
Theorem 3 (Solution to Problem 2) Assume Y (t) is given by (24). Then the optimal
reinsurance and investment strategy u∗(t) of Problem 2 is given by
u∗(t) = 1 Y ∗(t)·
⎛ γ ⎞
μ1θ
⎜ μ2⎜ d d f f f f f f ⎟1 − σS λ σ
d λd d f (σ −σ (T ))λ (σ −σ (T ))λ
⎜ (λd 1 S − I1 I − + σ (T3) f − σ (T2)λQ − S1 B 2 S − I1 B 2 I ⎟d r d d q1(t)) Bf − f (λ
B
r f f −q2(t))⎟σ (T ) σ σ d σ
⎜⎜ B 1 S I σB (T2 2 1)(σB (T3) σB (T2))
Q σ σ ⎟S2 I2
⎜ ⎟⎜ f f f f f f f ⎟⎜ 1 − f + σB (T3)λ (σS −σQ 1 B (T3))λS⎜ f − f ( λr + f +
(σI −σB (T3))λ1 I ⎟⎟
σQ f
+ q2(t))
⎜ σB (T3) σ ⎟B (T2) σS σ2 I2⎜ ⎟⎜ f f f f f f f ⎟1 f − σ (T2)λ − (σS −σB (T2))λS − (σI −σB (T2))λ⎜ B Q 1 1 I ⎟⎜ ⎟f − f (λr − q (t))σ (T ) σ (T ) σ f f 2⎜ 3 2 Q σ σ ⎟,B B S⎜ 2 I2 ⎟⎜ 1 ⎟λd
⎜⎜
⎟
σd S ⎟S2
⎜ ⎟1 f
⎜⎜
⎟
f λ
σ S
⎟
⎜ S2 ⎟⎜ 1 λdd I + γ − ⎟1⎜ ⎟σ
⎝ I
⎟
2
1 f ⎠
f λ
σ II2
(31)
where
∗ (
d
= + κ t S0(t)
)κ (B (t) )κ (C (t) )κ ( F (t) ) ( d )2 T 3 T 4 T κ5 S (t) κ6
Y (t) (x g)e 1 1 2 3
S (0) Bd0 (0) CT2(0) FT3(0) S
d(0)
T1
( ) ( Pd× C(t) κ
) ( )
7 T (t) κ κ4 8 CT5(t) 9 Z(t) , (32)
C(0) P dT (0) CT5(0) Z(0)4
Methodol Comput Appl Probab (2018) 20:1477–1502 1487
with
⎛ ⎞ ⎛σdB(T1) 0 −
⎞
d −1⎛σB(T3) σ
d
S 0 σ
d 0 d
κ I λ +(γ−1)σ d −
⎞
q1(t)3
⎜ ⎟ ⎜⎜
1 1 ⎟ r If f f f 1
κ 0 σB(T2) σ ⎜
f
⎜ 4⎟ ⎜ B
(T3) 0 σS 0 σI ⎟1 1⎟ ⎜ λr − q2(t) ⎟⎟⎜⎜κ5⎟⎟ ⎜⎜ 0 σQ σQ 0 σ 0 σ ⎜ ⎟Q Q⎟⎟ ⎜ λ1 Q⎜ ⎟⎜ d d ⎟⎜κ6⎟⎟ = ⎜⎜ 0 0 0 σ ⎟S 0 0 02 ⎟ ⎜ λS ⎟ ,(33)⎜ ⎟ γ ⎜ ⎜ ⎟⎜κ7⎟ ⎜ 0 0 0 0 fσS 0 0 ⎟ ⎜ fλ ⎟⎝ ⎠ ⎜ 2 ⎟ ⎜ S ⎟κ8 ⎝ 0 0 0 0 0 ⎟σd ⎝ λd + (γ − 1)σ d ⎠I 02 ⎠ I I2
κ9 f0 0 0 0 0 0 fσ λI2 I
κ2 = 1− κ3 − κ4 − κ6 − κ7 − κ8 − κ9, (34)
2 2
κ1 = λμ1θ − A4. (35)
γμ2
Proof For a detailed proof of A4, see Appendix A.2.
Remark 2 Theorem 3 generalises the first part of Proposition 3.4 in Guan and Liang
(2014b), in the sense that our optimal strategy depends also on the coefficients pertaining
to the foreign market. It is worth mentioning that, when fλr = = f = fλQ λS λI = kd1 =
q1(t) = q2(t) = 0, our Eq. 32 coincides with equation (3.15) in Guan and Liang (2014b).
Now we are ready to present the optimal reinsurance-investment strategy for the original
optimal control problem (20).
Corollary 1 (Solution to Problem 1) Suppose u∗(t) is given by (31). Then the optimal
reinsurance and investment strategy ū∗(t) of Problem 1 is given by
∗ ( σ (t, T )G(t) )G ′ū (t) = u∗(t)− 0, , 0, 0, 0, 0, 0, 0 . (36)
σdB(T1)
Proof (36) follows from the relationship u σ (t,T )G(t)(t) = ū(t)+(0, G d , 0, 0, 0, 0, 0, 0)′.σB(T1)
Remark 3 Let us mention that when u∗(t) coincides with that of Guan and Liang (2014b),
the solution to the original problem ū∗(t) also coincides.
4 Sensitivity Analysis
In this section we use the Monte Carlo method to analyze the parameters effect on the
optimal strategy and on the optimal utility. Unless otherwise specified, the values of the
model parameters are given by: x = 1, γ = 2, λ = 3, μ1 = 0.08, μ2 = 0.05, η = 0.05,
θ = 0.1, ad = 0.1, bd = 0.03, kd1 = 0.005, kd2 = 0.001, af = 0.12, bf = 0.05,
f = 0 004, fk1 . k2 = 0, λdr = 0, fλr = 0.02, σQ = 0.05, λQ = 0.05, σdS = 0.02, σ d1 S =2
0.15, λdS = 0.11, f f f fσS = 0.2, σS = 0.1, λS = 0.11, σdI = 0.08, σdI = 0.08, λd = 0, σ =1 2 1 2 I I1
0 05 f f f. , σI = 0.05, λI = 0.02, rdr = 0.025, rr = 0.03, T = 20, T1 = 10, T2 = 10, T3 = 1,2
T4 = 10, T = 10, rd f5 n (0) = 0.03, rn (0) = 0.05, S0(0) = 1, Q(0) = 1.2, BdT (0) = 0.8114,1
1488 Methodol Comput Appl Probab (2018) 20:1477–1502
2  
1.5
1
0.5
0
reinsurance
−0.5 cash
domestic bond
foreign bond
−1 futures
domestic stock
−1.5 foreign stock
domestic TIPS
foreign TIPS
−2  
0 5 10 15 20
time(year)
Fig. 1 Domestic and foreign, = 2 f fγ , λQ = 0.05, σS = 0.1, σ = 0.052 I2
C (0) = 0.7382, F (0) = 0.9108, SdT2 T3 (0) = 1, C(0) = 1.2, PdT (0) = 0.7788, CT5(0) =4
0.8890, I d(0) = 1.
0.6  
0.4
0.2
0
reinsurance
cash
−0.2 domestic bond
foreign bond
futures
domestic stock
−0.4 foreign stock
domestic TIPS
foreign TIPS
−0.6  
0 5 10 15 20
time(year)
Fig. 2 Domestic and foreign, γ = 2, λQ = 0 05 f. , σS = 0.1 f, σI = 0.052 2
proportions of optimal strategy optimal reinsurance−investment strategy
Methodol Comput Appl Probab (2018) 20:1477–1502 1489
4.1 Sensitivity Analysis of Optimal Strategy
In Figs. 1 and 2 the value of the parameters are those given in the text above. Figure 1
indicates that the investment in foreign bonds and stocks, domestic and foreign TIPS, as
well as the reinsurance proportion are all positive and grow nonlinearly over time, while the
investment in domestic bonds decreases. In addition, it shows that the insurer shorts futures
and cash at an increasing rate. Since the foreign market is “better” than the domestic one,
in the sense of smaller volatility (σ ) and greater market price of risk (λ) of corresponding
assets, the investments in the foreign stocks are larger than those in domestic stocks. The
same is true for the bonds after the first few years. However, the investments in the domestic
TIPS are higher than those in the foreign TIPS. This is due to the fact that the domestic
inflation directly influences the expected utility. Comparing Figs. 2 to 1, we note that the
proportions of each asset in the optimal strategy evolve in the same direction, except the
foreign ones behave more steadily.
In Figs. 3 and 4 we use the same parameters, but consider only the domestic market.
Comparing Figs. 3 to 1, we note that the reinsurance allocation and the investment in the
assets are smaller, but the trends are the same, except for cash. The main difference is that
the insurer does not short cash as in the case of foreign markets. Comparing Figs. 4 to 3,
except for cash, the proportion of other assets in the optimal strategy has the same, but more
stable tendency. The proportion of cash has a slight decline after a slow growth. The final
wealth of the insurer in the situations captured in Figs. 1 and 3 are X(20) = 3.3049 and
X(20) = 2.7038 respectively, which indicate that investing abroad increases its wealth by
22.23%.
In Figs. 5, 6, 7, 8 and 9 the value of parameters γ , fλQ, σS and σ
2
I are changed in2 2
order to highlight their impact on the optimal reinsurance-investment strategy. Since the
investments and the corresponding proportion of each asset in the optimal strategy have
1.4
reinsurance
cash
1.2 domestic bond
domestic stock
1 domestic TIPS
0.8
0.6
0.4
0.2
0
−0.2
0 5 10 15 20
time(year)
Fig. 3 Only domestic, γ = 2
optimal reinsurance−investment strategy
1490 Methodol Comput Appl Probab (2018) 20:1477–1502
0.5  
reinsurance
cash
0.4 domestic bond
domestic stock
domestic TIPS
0.3
0.2
0.1
0
−0.1  
0 5 10 15 20
time(year)
Fig. 4 Only domestic, γ = 2
the same tendency (see for example Figs. 1 and 2), we only focus on the investments in
the risky assets. Figure 5, γ = 4 reflect the fact that the insurer has a higher aversion
towards uncertainty and that the domestic inflation directly affects the expected utility. The
2.5  
reinsurance
cash
2 domestic bond
foreign bond
futures
1.5 domestic stock
foreign stock
domestic TIPS
1 foreign TIPS
0.5
0
−0.5
−1  
0 5 10 15 20
time(year)
Fig. 5 domestic and foreign, γ = 4, λQ = 0.05 f, σS = 0.1 f, σI = 0.052 2
optimal reinsurance−investment strategy proportions of optimal strategy
Methodol Comput Appl Probab (2018) 20:1477–1502 1491
2  
reinsurance
cash
domestic bond
1.5 domestic stock
domestic TIPS
1
0.5
0
−0.5  
0 5 10 15 20
time(year)
Fig. 6 only domestic, γ = 4
investments in domestic TIPS increase while in other assets decrease compared to Fig. 1.
When the investments in stocks are reduced to the domestic market stocks, the result is the
same (see for example Fig. 6 for γ = 4 and Fig. 2). The final wealth of the insurer under
2  
1.5
1
0.5
0
reinsurance
−0.5 cash
domestic bond
foreign bond
−1 futures
domestic stock
−1.5 foreign stock
domestic TIPS
foreign TIPS
−2  
0 5 10 15 20
time(year)
Fig. 7 domestic and foreign, γ = 2, λQ = 0 1 f. , σS = 0 1 f. , σI = 0.052 2
optimal reinsurance−investment strategy optimal reinsurance−investment strategy
1492 Methodol Comput Appl Probab (2018) 20:1477–1502
2.5  
2
1.5
1
0.5
0
reinsurance
−0.5 cash
domestic bond
−1 foreign bond
futures
−1.5 domestic stock
foreign stock
−2 domestic TIPS
foreign TIPS
−2.5  
0 5 10 15 20
time(year)
Fig. 8 domestic and foreign, γ = 2, λQ = 0 05 f = 0 08 f. , σS . , σI = 0.052 2
situations of Figs. 5 and 6 is X(20) = 2.7003 and X(20) = 2.4196, respectively. This
shows that, even if the insurer has a higher level of risk aversion, foreign market investments
can still improve its wealth by 11.60%.
2  
1.5
1
0.5
0
−0.5 reinsurance
cash
−1 domestic bond
foreign bond
−1.5 futures
domestic stock
foreign stock
−2 domestic TIPS
foreign TIPS
−2.5  
0 5 10 15 20
time(year)
Fig. 9 domestic and foreign, γ = 2 f, λQ = 0.05, σS = 0.1 f, σI = 0.032 2
optimal reinsurance−investment strategy optimal reinsurance−investment strategy
Methodol Comput Appl Probab (2018) 20:1477–1502 1493
0.35  
domestic&foreign, μ =0.05
1
0.3 domestic&foreign, μ =0.081
domestic&foreign, μ =0.11
1
only domestic, μ =0.05
0.25 1
only domestic, μ =0.08
1
only domestic, μ =0.11
0.2 1
0.15
0.1
0.05
0  
0 5 10 15 20
time(year)
Fig. 10 effect of μ1 on optimal reinsurance strategy
Figure 7 illustrates the impact for λQ = 0.1. Comparing it to Fig. 1, one notices that the
futures are in the long position instead of short. This is due to the increase in the market
price of the exchange rate, leading the insurer to more willingly invest in futures. At the
same time, the insurer longs domestic bonds and shorts foreign bonds to balance risks.
0.3  
domestic&foreign, μ =0.05
2
domestic&foreign, μ =0.10
2
0.25 domestic&foreign, μ =0.15
2
only domestic, μ =0.05
2
0.2 only domestic, μ =0.102
only domestic, μ =0.15
2
0.15
0.1
0.05
0  
0 5 10 15 20
time(year)
Fig. 11 effect of μ2 on optimal reinsurance strategy
a(t) a(t)
1494 Methodol Comput Appl Probab (2018) 20:1477–1502
0.35  
domestic&foreign, θ=0.06
domestic&foreign, θ=0.10
0.3 domestic&foreign, θ=0.14
only domestic, θ=0.06
0.25 only domestic, θ=0.10
only domestic, θ=0.14
0.2
0.15
0.1
0.05
0  
0 5 10 15 20
time(year)
Fig. 12 effect of θ on optimal reinsurance strategy
Figures 8 and 9 capture the variations when fσS = 0.08 and fσI = 0.03, respectively.2 2
Comparing these two figures to Fig. 1, we observe that, when the volatility of the foreign
stocks (foreign inflation) decreases, there is an improvement in the investments in foreign
0.3  
domestic&foreign, λ=2
domestic&foreign, λ=3
domestic&foreign, λ=4
0.25 only domestic, λ=2
only domestic, λ=3
only domestic, λ=4
0.2
0.15
0.1
0.05  
0 5 10 15 20
time(year)
Fig. 13 effect of λ on optimal reinsurance strategy
a(t) a(t)
Methodol Comput Appl Probab (2018) 20:1477–1502 1495
stocks (foreign TIPS). In order to hedge the risk of such strategy, the insurer shorts more
futures and cash.
4.2 Sensitivity Analysis of Reinsurance
Reinsurance plays an important role in our model. Hence in this section, we will focus on
how various parameters influence the reinsurance policy. Recall that a(t) represents the
proportion of insurance risk that the insurer keeps.
Figure 10 describes the fact that a(t) increases with the expected value of claims, denoted
by μ1, whether or not there is a foreign investment. With other parameters unchanged, a
bigger μ1 leads to less variance of claims, so the insurer can retain more insurance risk. In
addition, it shows that for a specific μ1, the value of a(t) when there is both domestic and
foreign investment is higher than the value when there is only domestic investment. This is
because the foreign investments increase the insurer’s wealth, hence allowing the insurer to
take on more risk. This feature can be visualized in Figs. 11, 12 and 13. As demonstrated in
Fig. 11, when μ2 decreases, a(t) increases, meaning that the insurer retains more risk when
the insurance risk is low. Figure 12 shows that a(t) increases with reinsurance premium
θ . In fact, the higher the reinsurance premium, the higher the cost of the insurer, so the
insurer has less interest in reinsurance. At last, as exhibited in Fig. 13, a(t) decreases when
λ increases. Recall that λ represents the intensity of claims. Therefore the bigger the claims
(that is increase in λ) in a specific time horizon, the more the insurer would like to purchase
more reinsurance to hedge the insurance risk. Another feature shown in Fig. 13 is that in
the last few years, the insurer who invests domestically and abroad always has a bigger a(t)
than the one who only invests in the domestic market. The reason behind this could be that
the increase in wealth from foreign investments can offset the impact of high intensity of
claims.
−0.4  
domestic&foreign
only domestic
−0.45
−0.5
−0.55
−0.6
−0.65
−0.7  
0.025 0.03 0.035 0.04 0.045
rd(0)
r
Fig. 14 effect of rdr (0) on optimal utility
V(0,rd,rf ,id,y)
n n
1496 Methodol Comput Appl Probab (2018) 20:1477–1502
−0.61  
domestic&foreign
only domestic
−0.62
−0.63
−0.64
−0.65
−0.66
−0.67
−0.68  
0.03 0.035 0.04 0.045 0.05
rd(0)
n
Fig. 15 effect of rdn (0) on optimal utility
4.3 Sensitivity Analysis of Optimal Utility
In this section, we study effects of parameters on the optimal utility.
Although the domestic real interest rate rdr (t) has no impact on the optimal reinsurance-
investment strategy, it does positively influence the optimal utility of the insurer regardless
 
−0.6306 domestic&foreign
−0.6306
−0.6306
−0.6306
−0.6306
−0.6306
−0.6306
−0.6306
−0.6306  
0.03 0.035 0.04 0.045 0.05
rf (0)
n
Fig. 16 effect of frn (0) on optimal utility
V(0,rd,rf ,id,y) V(0,rd,rf ,id,y)
n n n n
Methodol Comput Appl Probab (2018) 20:1477–1502 1497
−0.55  
domestic&foreign
only domestic
−0.6
−0.65
−0.7
−0.75
−0.8
−0.85
−0.9  
2 2.5 3 3.5 4 4.5 5 5.5 6
λ
Fig. 17 effect of λ on optimal utility
whether or not it invests home or/and abroad as demonstrated in Fig. 14. The explanation
may be: bigger real interest rates mean that the money raised by the insurer is worth more.
For the same value of rdr (0), the optimal utility gain from investing domestically and abroad
is larger than that obtained from investing domestically. This could be the effect of inter-
national diversification. This phenomenon also exists in Figs. 15 and 17. From Figs. 15
and 16, we can find that the optimal utility increases with rdn (0), while it decreases with
f
rn (0). Note that the increase in the interest rate means the appreciation of the correspond-
ing currency. Thus, the appreciation of domestic currency will improve the insurer’s utility.
However, the appreciation of the foreign currency will reduce the insurer’s utility because
the insurer needs to denominate assets in domestic currency. Comparing Figs. 15 and 16 in
the situation of existing foreign investments, fV (0, rdn , r , i
d
n , y) increases from −0.6306 to
−0.6154 whenever rdn (0) increases from 0.03 to 0.05, while V (0, rd fn , rn , id , y) decreases
from −0.6305874 to −0 f.6305879 when rn (0) increases from 0.03 to 0.05. This means that
rdn (0) has a stronger effect on the optimal utility than
f
rn (0). Figure 17 shows that the opti-
mal utility decreases with λ, due to the fact that larger λ means more claims in a specified
time horizon.
5 Conclusion
In this paper, we studied the optimal reinsurance-investment strategy of an insurer who
invests in domestic and foreign risky assets. In order to hedge the insurance risk, the risk
of nominal interest rates and inflation rates in both domestic and foreign markets, and the
exchange rate risk, the insurer purchased proportional reinsurance, rolling bonds, TIPS and
futures. We solved the problem via a dynamic programming approach and obtained both
the explicit value function and the optimal reinsurance-investment strategy. We conduct a
V(0,rd,rf ,id,y)
n n
1498 Methodol Comput Appl Probab (2018) 20:1477–1502
sensitivity analyses of the model parameters effect on the optimal reinsurance-investment
strategy, optimal reinsurance strategy and optimal utility.
Acknowledgements This research has been carried out with funding provided by the Alexander von
Humboldt Foundation, under the programme financed by the German Federal Ministry of Education and
Research entitled German Research Chair No 01DG15010 and by the European Union’s Seventh Framework
Programme for research, technological development and demonstration under grant agreement No. 318984-
RARE. The second author gratefully acknowledge the support of Natural Science Foundation of China under
grant agreement No. 71532001.
Appendix
A.1 The proof of Theorem 2
Proof Suppose fV (t, rd , r , idn n , y) has the form of (27). Substituting (27) into (26) yields
the following differential equation for fh(t, rdn , rn ):
h h ft + hrd [1− γ ′ + − ] [ ]n σ  ad(bd rd + γ − 1 ′ + r 1− γn ft ) σ σ d σ ′ t + af (bf − rn )
h h γ d n γ I h γ f
1− γ [( ) (h f )+ hrd 2n ′ + r 2n ′
]
+ 1
[hrd h ]n rd ′ f fn + rn rσ σ σ σ σ σ n σ ′ 1− γ ′d d f f d d σ f +  t2γ h h 2 h h f 2γ t
− (γ − 1)
2 ′ − γ − 1σ It (γ − 1)[rd − σd λd d d d d d ′γ r I1 r (k1 rn + k2 )− σI λ2 I ] − σ2γ Iσ I = 0. (37)
From (37), we may assume that h(t, rd fn , rn ) can be separated by independent variables and
write f fh(t, rdn , rn ) = exp{rdn q1(t) + rn q2(t) + q3(t)} with boundary conditions q1(T ) =
f
q2(T ) = q3(T ) = 0. Substituting h(t, rdn , rn ), t , σ d , σ f , and σ I into (37), we obtain
rd [q ′ fn 1(t)−M 2 ′ 21q1(t)−N1q1(t) − R1] + rn [q2(t)−M2q2(t)−N2q2(t) − R2]
+ ′ 1 1 fq (t)+Mq (t)+ kdq (t)2 +Nq (t)+ k q (t)23 1 2 1 2 2 2 − (γ − 1)rd2γ 2γ r + R = 0, (38)
where
d γ − 1 1 γ − 1 [ ]M d d d1 = a + k1 (σI − λr ), N1 = − kd1 , R1 = kd1 (λd)2 − 2λdσd d 2γ 1 2γ 2γ r r I + (σ ) ,1 I1
f 1− γ f f 1 f γ − 1 f fM 22 = a + k1 λr , N2 = − k1 , R2 = kγ 2γ 2γ 1 (λr ) ,
= d d + γ − 1 d γ − 1 f fM a b k2 (λdr − σdI ), N = af bf + k2 λr ,γ 1 γ
= 1− γ [λμ
2
1θ
2
R + kd d 2 f f 2 22 (λr ) + k2 (λr ) + λQ + (λd )2S + f f
]
(λ 2S ) + (λd)2 + (λ 22γ μ I I )2
+ γ − 1 γ − 1 [ ](kdλdσ d2 r I + λd d d dI σI )− k2 (σI )2 + (σ d )2 .γ 1 2 2γ 1 I2
Since (38) is true for all rd fn and rn , it follows one must have
q ′ 2j (t) = Rj +Mjqj (t)+Njqj (t) , qj (T ) = 0, j = 1, 2, (39)
d f
q ′3(t) = −
k
Mq1(t)− 2 q1(t)2 − kNq2(t)− 2 q2(t)2 + (γ − 1)rdr − R, q3(T ) = 0. (40)2γ 2γ
Methodol Comput Appl Probab (2018) 20:1477–1502 1499
For j = 1, 2, qj (t) is a Riccati equation with constant coefficients. Then using Nawalkha
et al. (2007), qj (t) is explicitly given by
α β e(αj−βj )(T−t)= − j j · − 1qj (t) − − − , j = 1, 2, (41)N β e(αj βj )(T t)j j αj
√ √
= Mj+ M
2
j−4NjRj = M
2
j− Mj−4NjRjwhere αj 2 , βj 2 . Since
[ γ − 1 ]2 γ − 1
M21 − 4N1R1 = ad + kd(σ d − λd) + (kd)2(λd − σd )2, (42)γ 1 I1 r 2 1 rγ I1
2 − (4 = f + 1− γ f f )2 + γ − 1 f fM2 N2R2 a k1 λr (k1 λr )2. (43)γ γ 2
Let us observe that (41) makes sense if γ > 1, so that we haveM2j −4NjRj > 0(j = 1, 2).
Integrating both sides of (40) from t to T and using the fact that q3(T ) = 0, we can obtain
(30).
A.2 The Proof of Theorem 3
Proof Since supremum (26) is attained at u∗(t), applying the first order maximum
condition, we have
V σ  + V σ σ ′u∗y t t yy t t (t)+ Vyrdσ tσ d + V f σ tσ f + V d · idσ tσ I = 0.n yr yin
Then
u∗(t) = − 1 −1σ t (V dyt + Vyrdσ d + V f σ f + Vyid · i σ I ),
V n yryy n
where  = σ ′tσ t . Substitute (27) and (28) into the last equation to obtain
∗ = 1u (t) Y ∗(t)(σ ′ )−1t [t + q1(t)σ d + q2(t)σ f + (γ − 1)σ I ]. (44)γ
Equation (31) follows by substituting σ t , t , σ d , σ f and σ I into (44).
In order to obtain the optimal strategy u∗(t), we need to find the SDE satisfied by Y ∗(t)
corresponding to u∗(t). Now, substituting (44) into (24), we have
∗ = ∗ { 1dY (t) Y (t) rdn(t)dt+ [′t+q (t)σ ′1 d+q2(t)σ ′f +(γ−1)σ ′I]tdtγ
+1[′ +q (t)σ ′ +q (t)σ ′ +(γ−1)σ ′ }t 1 d 2 f I]dW tγ
∗ {= 1 f fY (t) rdn (t)+ (kd d d d d d1 rn (t)+ k2 ) λr [λr + (γ − 1)σI − q1(t)] + (k1 1 rn (t)γ
+ f 1 f fk2 ) λr [λr − q2(t)] +
1 1 f 1 1
λ2Q + (λd 2 2S) + (λS ) + λd [λdγ γ γ γ I γ I
2
+ − d ] + f 2 1 + λμ1θ
2 }
(γ 1)σI (λI ) dt + Y ∗(t)D′ dW t , (45)2 γ γμ 12
where √ √
D = 1(1 √λμ1θ , kd d1 rn (t)+ d d + − 1 d − f f f fk2 (λr (γ )σI q1(t)), k r (t)+kγ λμ2 1 1 n 2 (λr −
f )
q (t)), λ , λd , λ , λd + (γ − 1 f ′)σ d , λ . Y ∗(t) can also be obtained by using Y ∗2 Q S S I I I (t) =2
1500 Methodol Comput Appl Probab (2018) 20:1477–1502
X∗(t) + G(t). In order to get Y ∗(t), let us introduce an auxiliary process {Z(t)}t∈[0,T ]
representing the insurance risk and satisfying:
dZ(t) = λ√μ1θ dW0(t), t ∈ [0, T ].
Z(t) γ λμ2
Observing (17), let us assume that Y ∗(t) is of the following form:
d
∗ (= + κ t S0(t)
)κ ( ) ( ) ( ) ( d )2 BT (t) κ1 3 CT2(t) κ4 FT (t) κ3 5 S (t) κ6Y (t) (x g)e 1
S (0) Bd0 (0) CT2(0) F
d
T T1 3
(0) S (0)
( d
× C(t)
)κ ( P (t) )κ (C (t) )7 T4 8 T κ5 9 Z(t) .
C(0) P d (0) CT5(0) Z(0)T4
Using Itô’s formula, we get
∗ [= ∗ + dS0(t) + dB
d
T (t)1 + dCT2 (t) dF (t) dS
d(t)
d TY (t) Y (t) κ 31dt κ2 κ3 κ + κd 4 5 + κ6S0(t) B (t) CT2 (t) FT3 (t) Sd(t)T1
d d d+ C(t) + dPT (t) + dCT (t) + dZ(t)
]
+ 1 ∗
[ (
− dBT (t)
)2
κ 4 57 κ8 κ9 Y (t) κ3(κ3 1)
1
C(t) P d (t) CT5 (t) Z(t) 2
d
T BT (t)4 1
(dC ) ( ) ( d )+ − 1 T2 (t) 2 + − dFT (t) 2 dS (t) 2κ4(κ4 ) κ5(κ 35 1) + κ6(κ6 − 1)
CT2 (t) FT3 (t) S
d(t)
(dC(t))2 ( d ) ( ) ]+κ7(κ7 − 1) + −
dP (t) 2 dC (t) 2
κ8(κ8 1
T
) 4 + Tκ9(κ9 − 1) 5
C(t) P d (t) CT5 (t)T4
[ d〈Bd , F 〉 d〈Bd , Sd 〉 d〈Bd , P d+ T T3 t T t T T 〉t d〈CT , FT 〉tY ∗(t) κ κ 1 1 1 4 2 33 5 + κd 3κ6 + κ3κd 8 + κ4κd d 5BT (t)FT3 (t) B (t)SdT (t) BT (t)P (t) CT (t)FT (t)1 1 1 T4 2 3
+ d〈CT2 , C〉t + d〈CT2 , CT5 〉t + d〈F , S
d
T 〉t
κ 34κ7 κ4κ9 κ5κ6 + d〈FT ,C〉tκ5κ 3d 7CT2 (t)C(t) CT2 (t)CT5 (t) FT3 (t)S (t) FT3 (t)C(t)
d〈F , P d 〉 d〈 〉 d〈Sd, P d+ T3 T4 t + FT3 , C
]
T5 t T
〉
4 t d〈C,CT5 〉tκ5κ8 κd 5κ9 + κ6κ8 + κd 7κ9FT (t)P (t) FT3 (t)CT5 (t) Sd3 T (t)P (t) C(t)CT (t)4 T4 5
= ∗ { d + + + + + + + d d + d + f fY (t) rn (t)(κ2 κ3 κ4 κ6 κ7 κ8 κ9) (k1 rn (t) k2 )A1 (k1 rn (t)
+ fk2 )A2 +
}
A3 + A4 + κ1 dt + Y ∗(t)D′2dW t ,
where
= 1A κ σd(T )λd − κ σd(T )λd + κ σd λd + κ σd λd + κ (κ − 1)σ d (T )21 3 B 1 r 5 B 3 r 6 S1 r 8 I1 r 3 32 B 1
+1κ5(κ d 25 − 1)σB(T3) +
1 1
κ6(κ6 − 1)(σ d )2S + κ d 2 d d1 8(κ8 − 1)(σI ) − κ3κ5σB(T1)σB(T3)2 2 2 1
+κ d d d d d d d d d d3κ6σB(T1)σS + κ3κ8σB(T1)σI − κ5κ6σB(T3)σS − κ5κ8σB(T3)σI + κ6κ8σS σI ,1 1 1 1 1 1
= f f + f f + f f + f f 1 fA2 κ4σB (T2)λr κ5σB (T3)λr κ7σS λr κ9σI λr + κ4(κ4 − 1)σB (T2)21 1 2
+1 f 1κ5(κ 25 − 1)σB (T3) + κ7(κ7 − 1 f)(σ )2 +
1 f f f
S κ9(κ9 − 1)(σI )2 + κ4κ5σB (T2)σ (T3)2 2 1 2 1 B
+ f f f f f f f f f fκ4κ7σB (T2)σS + κ κ σ (T )σ + κ κ σ (T )σ + κ κ σ (T )σ + κ κ σ σ ,1 4 9 B 2 I1 5 7 B 3 S1 5 9 B 3 I1 7 9 S1 I1
= + + + + d d + f f f fA3 λQσQ(κ4 κ5 κ7 κ9) λSκ6σS λS κ7σS + λd d2 2 I κ8σI + λ2 I κ9σI ,2
Methodol Comput Appl Probab (2018) 20:1477–1502 1501
A4 = 2 1 1 1 1σQ[ κ4(κ4 − 1)+ κ5(κ5 − 1)+ κ7(κ7 − 1)+ κ9(κ9 − 1)+ κ4κ5 + κ4κ7 + κ4κ9 + κ5κ72 2 2 2
+ + 1 1 f 1 1 fκ5κ9 κ7κ9]+ κ6(κ6 − 1)(σ dS )2+ κ7(κ7−1)(σS )2+ κ8(κ8−1)(σ dI )2+ κ9(κ9−1)(σ )2,2 2 2 2 2 2 2 I2
( √ √λμ1θ f f f
D = √ , kd2 1 rdn (t)+ kd d d2 (κ3σB(T1)− κ5σB(T3)+ κ σd6 S + κ8σdI ), k1 rn (t)+ k ·γ λμ 1 1 22
f + f + f f f f )′(κ4σB (T2) κ5σB (T3) κ7σS + κ9σI ), σQ(κ4 + κ5 + κ7 + κ9), κ σ d6 S , κ7σS , κ σd , κ1 1 2 2 8 I2 9σI .2
By the uniqueness of the solution of SDE, we can obtain (33)–(35).
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