optimal investment-reinsurance with dynamic risk constraint and regime switching

This article was downloaded by: [Temple University Libraries]On: 13 November 2014, At: 05:35Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Scandinavian Actuarial JournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/sact20

Optimal investment-reinsurance withdynamic risk constraint and regimeswitchingJingzhen Liu a , Ka-Fai Cedric Yiu a , Tak Kuen Siu b & Wai-KiChing ca Department of Applied Mathematics , The Hong KongPolytechnic University , Hong Kong , PR Chinab Department of Applied Finance and Actuarial Studies, Facultyof Business and Economics , Macquarie University , Sydney ,Australiac Department of Mathematics , University of Hong Kong , HongKong , PR ChinaPublished online: 10 Aug 2011.

To cite this article: Jingzhen Liu , Ka-Fai Cedric Yiu , Tak Kuen Siu & Wai-Ki Ching (2013) Optimalinvestment-reinsurance with dynamic risk constraint and regime switching, Scandinavian ActuarialJournal, 2013:4, 263-285, DOI: 10.1080/03461238.2011.602477

To link to this article: http://dx.doi.org/10.1080/03461238.2011.602477

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,

http://www.tandfonline.com/loi/sact20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/03461238.2011.602477

http://dx.doi.org/10.1080/03461238.2011.602477

systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Original Article

Optimal investment-reinsurance with dynamic risk constraintand regime switching

JINGZHEN LIUa, KA-FAI CEDRIC YIUa*, TAK KUEN SIUb

and WAI-KI CHINGc

aDepartment of Applied Mathematics, The Hong Kong Polytechnic University,

Hong Kong, PR ChinabDepartment of Applied Finance and Actuarial Studies, Faculty of Business and

Economics, Macquarie University, Sydney, AustraliacDepartment of Mathematics, University of Hong Kong, Hong Kong, PR China

(Accepted June 2011)

We study an optimal investment�reinsurance problem for an insurer who faces dynamic risk constraint

in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected

utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional

Value at Risk over different economic states. The rationale is to provide a prudent investment�reinsurance strategy by taking into account the worst case scenario over different economic states.

Using the dynamic programming approach, we obtain an analytical solution of the problem when the

insurance business is modeled by either the classical Cramer�Lundberg model or its diffusion

approximation. We document some important qualitative behaviors of the optimal investment�reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the

optimal strategies.

Keywords: optimal reinsurance and investment; regime-switching; utility maximization; dynamic

programming; maximal conditional Value at Risk (MCVaR); regime-switching Hamilton�Jacobi�Bellman (HJB) equations

1. Introduction

Reinsurance provides insurance companies a mean to manage and control their exposures

to risk. Insurance companies can use reinsurance to transfer parts of their risk exposures

to reinsurance companies so as to protect themselves from undesirable or unexpected

potential large losses. An effective use of reinsurance may reduce the volatility of the

insurers’ earnings and enhance their profitabilities. The proportional reinsurance and the

excess-of-loss reinsurance are two popular types of reinsurance policies. They have been

investigated extensively in the actuarial literature. Schmidli (2001, 2002) considered the

proportional reinsurance and determined an optimal proportional reinsurance strategy by

*Corresponding author. E-mail: [email protected]

Scandinavian Actuarial Journal, 2013

Vol. 2013, No. 4, http://dx.doi.org/10.1080/03461238.2011.602477263–285,

# 2011 Taylor & Francis

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

minimizing the probability of ruin. Taksar & Markussen (2003) extended the analysis

using a diffusion model with investment and proportional reinsurance. Schmidt (2004)

dealt with optimal proportional reinsurance for dependent lines of business. Some other

works on optimal proportional reinsurance include Asmussen & Taksar (1997), Taksar

(2000), Højgaard & Taksar (2004), Promislow & Young (2005), and the references therein.

Traditionally insurance companies invest their assets in fixed interest securities.

Nowadays, due to the rapid convergence of insurance and financial markets, many

insurance companies are actively involving in investment activities in capital markets.

They invest their assets in shares, indices, commodities and also risky derivatives. Some of

them even become major players in trading activities of some sophisticated structured

products, such as credit derivatives. Some investment decisions of insurance companies

become more and more complicated. Some quantitative, or scientific, methods may

provide insurance companies with systematic ways to make justified investment decisions.

In particular, an optimal investment policy obtained from a quantitative asset allocation

model may be helpful for insurance companies to make their investment decisions.

Different from portfolio selection problems studied in the finance literature, an optimal

investment problem of an insurance company needs to take into account of both financial

and insurance risks. More specifically, one needs to build stochastic models for the price

dynamics of the investment assets and the insurance risk process. Optimal investment

problems of insurance companies have been studied in the actuarial literature. Compound

Poisson risk processes and diffusion-based risk processes are two major classes of

insurance risk processes considered in studying optimal investment problems. Some works

include Browne (1995, 1997, 1999), Hipp & Plum (2000), Hipp & Taksar (2000), Liu &

Yang (2004), Yang & Zhang (2005), and Bai & Guo (2008).

Maximizing profits and controlling risks are two equally important goals of insurance

companies. These two goals should be taken into account in determining optimal

investment strategies of insurance companies. To control and manage risks, the first step

is to describe and quantify risk. Various quantitative risk measures have been proposed in

the finance and insurance literature. Among these measures, Value at Risk (VaR) is widely

adopted in both finance and insurance industries. However, it is well-known that VaR is

not subadditive. In other words, allocating assets in two risky positions can increase risk.

Some alternative risk measures have been proposed in the literature. Conditional Value at

Risk (CVaR) is an example. It is defined as the average loss when the loss exceeds the VaR

level. When the loss distribution is continuous, CVaR is sub-additive. When the loss

distribution is discrete, an adjustment term is required to make CVaR sub-additive.

Quantitative risk measures have been adopted to study optimal reinsurance and

investment problems. An optimal reinsurance is investigated by Cai & Tan (2007), Cai

et al. (2008), and Tan et al. (2009). While taking VaR as a risk constraint, Kostadinova

(2007) investigated an optimal investment problem for an insurer. For the VaR

approximation, the constant strategy was used by Kostadinova (2007). Over a long risk

horizon, it seems rather not practical since VaR is invariably computed under the premise

that portfolio composition remains unchanged over the risk horizon. Indeed, in practice,

portfolio composition is adjusted frequently and future portfolio adjustments are not

J. Liu et al. 264

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

anticipated. It seems more appropriate to consider a risk measure dynamically which

allows changes in portfolio composition over time.

In this paper, we study an optimal investment�reinsurance problem of an insurer who

faces a ‘dynamic’ risk constraint in a Markovian regime-switching environment. More

specially, we consider a Markovian regime-switching model for the price dynamics of a

risky investment security so as to incorporate the impact of structural changes in econo-

mic conditions on price dynamics and optimal asset choices. For example, a Markovian

regime-switching model modulated by a two-state Markov chain can distinguish the

impact of a good economy on asset prices from that of an economy experiencing crisis

on asset prices. This distinction seems important given the recent global financial crisis.

The object of the insurer is to maximize the expected utility of terminal wealth at a finite

future time horizon. Using the optimal control theory (see, for example, Teo et al. (1980)

and Ahmed & Teo (1981)), and the dynamic programming principle (Flemming & Soner

(1993)), we derive a regime-switching Hamilton�Jacobi�Bellman (HJB) equation and give

an analytical expression of the optimal strategy for the insurer. In practice, portfolio can

only be adjusted discretely over time due to the presence of transaction costs and other

operating costs. So it is not unreasonable to assume that there is no change in the portfolio

composition over a short time duration when we evaluate the risk constraint in this time

duration, see, for example, Yiu (2004). Here we use the maximal conditional Value at Risk,

(MCVaR), as the risk constraint. This constraint is allowed to changeover time. However,

for any fixed time, it is evaluated under the assumption that there is no change in the

portfolio composition in a short duration. We investigate the impact of the risk constraint

on the optimal portfolio strategy as well as the impact of the changing economic conditions

on the optimal insurance demand.

This paper is structured as follows. In Section 2, we first describe the price dynamics of

the model and formulate the optimal investment�reinsurance problem in the classical

Cramer�Lundberg model. Then we derive the regime-switching HJB equation and the

risk constraint. The similar problem under a diffusion approximation to the classical risk

model is discussed in Section 3. In Section 4, we present numerical experiments and

document the effects of the risk constraint and changing economic environment. The final

section summarizes the paper.

2. The classical case

Firstly, the surplus process of an insurer in the classical Cramer�Lundberg model is given

by

dX (t)�cdt�dC(t);X (0)�x;

8<: (2:1)

where X(0) is the initial surplus, or capital, and

Scandinavian Actuarial Journal265

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

C(t)�XN(t)

i�1

Yi (2:2)

is the aggregate insurance claim process.

We assume N(t) is a Poisson process with intensity parameter l, which represents the

number of claims up to time t; for each i�1,2, . . .,Yi represents the size of the ith-claim.

As usual, we suppose that {Yi} is a sequence or independent and identically distributed

(i.i.d.) random variables with common distribution F as a random variable Y, which has a

finite mean. That is, E[Yi]�EYB�, for i�1, 2 . . ..

We assume that the premium is paid to the insurer continuously over time and that the

premium rate c is proportional to the expected payout. The premium rate c is supposed to

be evaluated using the expected value principle with risk loading. In particular,

c�(1�u)lE[Y ];

where u�0 is the relative safety loading.

We assume here that the company is allowed to invest its surplus in the capital market

and to purchase proportional reinsurance.

2.1. Price dynamics in the capital market

We consider a continuous-time capital market, where trading activities occur in a finite

time horizon [0, T], with TB�. Denote the time horizon [0, T] by T : To describe

randomness, we consider a complete probability measure space (V;F ;P); where P is a

real-world probability. This probability space is assumed to be rich enough to incorporate

random fluctuations in financial prices, insurance risk processes, and economic

conditions.

We model the evolution of the state of an economy over time by a continuous-time,

finite-state, Markov chain Z:�fZ(t)gt �T with state space Z:�(z1; z2; . . . ; zN ): We

suppose that the chain is observable and that the states of the chain represents proxies

for observable macro-economic factors, such as gross domestic product (GDP) and

consumer price index (CPI). Without loss of generality, as in Elliott et al. (1994), we

identify the state space of the chain as a finite set of unit vectors o:�fe1; e2; . . . ; eNg;where ei�(0; . . . ; 1; . . . ; 0)? �RN ; for each i�1; 2; . . . ;N; and ? represents the transpose

of a matrix, or in particular, a vector. Suppose Q is the rate matrix, or the generator,

[qij]i;j�1;2;...;N of the chain Z: The statistical properties of the chain Z are completely

specified by the rate matrix Q: Then, with the canonical representation o of the state space

of the chain, Elliott et al. (1994) obtained the following semi-martingale dynamics for the

chain Z:

Z(t)�Z(0)�gt

0

QZ(s)ds�M(t): (2:3)

J. Liu et al. 266

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

Here fM(t)gt �T is an RN-valued martingale with respect to the filtration generated by Z

under P:For each t � T ; let m(t) and s(t) be the expected rate of return and the volatility of a

risky asset S at time t. We suppose that both m(t) and s(t) are modulated by the chain Z

as:

m(t):�hm;Z(t)i;s(t):�hs;Z(t)i; (2:4)

where m:�(m1;m2; . . . ;mN)? �RN and s:�(s1;s2; . . . ;sN)? �RN

with mI�0 and sI�0, for each i�1,2, . . . ,N the brackets h�; �i represent the scalar

product in RN .

Suppose W :�fW (t)gt �T is a standard Brownian motion on (V;F ;P)with respect to

fF (t)gt �T ; the P-augmentation of the natural filtration generated by fW (t)gt�T : The price

process fS(t)gt �T of the risky asset S is assumed to be

dS(t)�S(t)(m(t)dt�s(t)dW (t)); S(0)�s: (2:5)

For each t � T ; let G(t):�FZ(t)�F(t), an enlarged information set generated by FZ(t) and

F(t): Write G:�fG(t)gt�T :

2.2. The reinsurance strategy

If the risk exposure of the insurance company is fixed, then the reinsurer pays q(t) of each

claim while the rest is paid by the cedent. Here we suppose that the proportional

reinsurance process q:�fq(t)gt �T is a G-predictable process with 05q(t)51; for t � T : To

this end, the cedent diverts part of the premiums to the reinsurer at the rate of (1�b)q(t)lEY with a proportional loading of b�u:

2.3. The problem formulation and the regime-switching HJB equation

Let p:�fp(t)gt �T be a portfolio process, where p(t) is the amount of wealth invested in

the risky asset S at time t, the rest put in the risk free asset with zero interest rate. Suppose

that no short-selling is allowed, (i.e. (p(t)]0; t � T ): The process P is an G-predictable,

R-valued process. Here, the investor makes his/her portfolio and reinsurance decisions

according to information about market prices and observable states of the economy. This

is different from some traditional optimal investment�reinsurance models, where only the

price information is taken into account in determining the optimal investment�reinsurance decisions.

Let fX (t)gt �T be the wealth process of the agent with initial wealth X(0)�x�0 and the

strategy (p,q):�(p(t),q(t)). Then the evolution of the wealth process over time is governed

by the following equation:

dX (t)� [m(t)p(t)�c�(1�b)lq(t)E[Y ]]dt

�s(t)p(t)dW (t)�(1�q(t))dC(t)

X (0)�x: (2:6)


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

A reinsurance-investment strategy (P, q¯) is said to be admissible if

E

�g

T

0

s2(t)p2(t)dt

�B�:

05q51: (2:7)

We denote the set of admissible strategies by U:Suppose U(�):R� 0 R� is a twice continuously differentiable, strictly increasing and

concave utility function. Then we state our main problem as follows.

Problem 2.1 The optimal investment�reinsurance problem of the insurer is to select an

G-predictable control process (p, q¯) so as to maximize the expected utility of terminal

wealth. That is, to solve the following maximization problem:

V (t; x; z)�supu �U

E[U(X (T))½X (t)�x;Z(t)�z]:

Here V (t; x; z) is the value function.

Let V(t; x):� (V (t; x; e1);V (t; x; e2); . . . ;V (t; x; eN ))? �RN : We define the operators

L(p;q)V (t; x; ei)�([m(t)p(t)�c�(1�b)lq(t)E(Y )])@V

@x

�1

2s2(t)p2(t)

@2V

@x2�l(E[V (t; x�(1�q(t))Y ;Z(t))]�V (t; x;Z(t)))

�hV(t; x);QZ(t)i; for i�1; . . . ;N: (2:8)

Then by the dynamic programming principle in stochastic optimal control (see Flemming

& Soner (1993)), it can be shown that the value function V satisfies the following regime-

switching HJB equation:

@V

@t�sup

(p;q)

L(p;q)V (t; x; ei)�0: (2:9)

Then the following verification theorem is standard (see Sotomayor & Cadenillas (2009)).

We state it here without giving the proof.

THEOREM 2.1 Suppose, for each i�1; 2; � � � ;N;V (t; x; ei) � C1;2(T �R) and V (t; x; z) is a

solution of the HJB equation (2.9). Then

1. V (t; x; z) is the value function of the optimization problem of the insurer;

2. Assume that the Markov control (p�; q�) satisfies the following equation:

@V

@t�L(p�;q�)V (t; x; ei)�0: (2:10)

Then (p�; q�) is optimal. I

J. Liu et al. 268

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

In this work we consider the following exponential utility:

U(x)��e�ax

a: (2:11)

We further assume that the chain has two states, (i.e. N�2) and that the rate matrix Q of

the chain is given by:

Q:��p p

p �p

� �: (2:12)

Here p � (0; 1): Note that the results derived here can be easily extended to the general case

where N�2 and a general rate matrix Q: However, the two-state chain is good enough to

distinguish a ‘Good’ economy and one experiencing crisis.

To simplify the notation, we write Vi:�V (t; x; ei) and V:�(V1;V2; � � � ;VN)? �RN : In

this case, the value functions Vi, i�1,2, over different states satisfy the following pair of

coupled HJB equations:

@Vi

@t�sup

(p;q)

[(c�mip(t)�(1�b)lq(t)E[Y ])@Vi

@x�

1

2s2

i p2(t)

@2Vi

@x2

�l(E[Vi(t; x�(1�q(t))Y )]�Vi(t; x))]

�hV;Qeii�0; i�1; 2; (2:13)

with the terminal condition:

Vi(T ; x)��e�ax

a: (2:14)

Assume that Equation (2.13) has a smooth solution with Vi;x�0 and Vi;xxB0; then we

consider the following trial function:

Vi(t; x)��e�ax

ahi(t); i�1; 2: (2:15)

Substituting it into Equation (2.13), then

sup(p;q)

[([c�mip(t)�(1�b)lq(t)E[Y ]])@Vi

@x�

1

2s2

i p2(t)

@2Vi

@x2

�l(E[Vi(t; x�(1�q(t))Y )]�Vi(t; x))]

is reduced to

sup(p;q)

�c�

la�

�mip(t)�

1

2as2

i p2(t)

��

�(1�b)q(t)lE[Y ]�

la

E[ea(1�q(t))Y ]

��e�axhi(t):

Define, for each i�1, 2,

Ci(p(t); q(t)):��mip(t)�

1

2as2

i p2(t)

��

�(1�b)q(t)lE[Y ]�

la

E[ea(1�q(t))Y ]

�; (2:16)

then we have the following result.


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

LEMMA 2.1 1.Ci(p(t); q(t)) is concave in (p(t); q(t)) .

2. Let q0(t) be the solution of

(1�b)E[Y ]�E[ea(1�q(t))Y Y ]�0; (2:17)

and denote

p(i;�)(t)�mi

as2i

;

q(i;�)(t)�

q0(t); if q0(t)�0;

0; if q0(t)50;

8>><>>:

(2:18)

respectively. Then Ci(p(t); q(t)) is increasing, (decreasing), in the interval [0; p(i;�)];

([p(i;�);�)); with respect to p, and increasing, (decreasing),in the interval [0; q(i;�)];

([q(i;�); 1]) with respect to q, respectively.

Since q0(t)B1 from Equation (2.17), obviously, Equation (2.18) gives the optimal

condition. Substituting Equations (2.18) and (2.15) into Equation (2.13), the pair of coupled

HJBs is then reduced to the following pair of coupled ordinary differential equations (ODES).

h?1(t)�A1(t)h1(t)�ph2(t)�0;

h?2(t)�A2(t)h2(t)�ph1(t)�0; (2:19)

where

Ai(t)��a(c�(1�b)lq(i;�)(t)E[Y ])�(mi � ri)

2

2s2i

�l(M(1�q(i;�)(t))Y (a)�1)�p; i�1; 2 (2:20)

with terminal conditions:

h1(T)�h2(T)�1: (2:21)

Here M(1�q(i;�)(t))Y is the moment generating function of (1�q(i;�)(t))Y : The solution of the

optimal reinsurance-investment problem of the insurer in the exponential utility case is

summarized in the following theorem.

THEOREM 2.2 Let (p(i;�); q(i;�)) be given by Equation (2.18), h�i (t); i�1, 2, be the solution

of Equation (2.19). Then (p(i;�); q(i;�)) is an optimal investment�reinsurance strategy and

V �i (x; t)��

e�ax

ah�

i (t); i�1; 2:

is the value function.

Proof: The result follows from THEOREM 2.1. I

2.4. The problem with MCVaR constraint

2.4.1. MCVaR constraint. In this subsection, we consider the optimal reinsurance-

investment problem in the presence of the risk constraint given by the maximal

J. Liu et al. 270

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

conditional VaR. We adopt a modified version of the approach considered in Yiu (2004)

to solve the problem.

Before we discuss the risk constraint for the optimal reinsurance-investment problem,

we first give a brief introduction to the background and definition of the maximal

conditional VaR we adopted here. Indeed, the risk measure we used here and the

technique of its derivation follows from those for the CVaR in Rockafellar & Uryasev

(2000).

Let g(x, y) be the loss from a portfolio x with a loss probability density function p(y).

The probability of g(x, y) not exceeding a threshold level, say h, is given by

M(x;h)�gg(x;y)5h

p(y)dy: (2:22)

This quantity plays a fundamental role in defining VaR and CVaR.

For any given probability k � (0; 1); VaR and CVaR are defined by:

VaR:�minfh½M(x;h)]kg; (2:23)

and

CVaR:�1

1 � k g g(x;y)]VaR

g(x; y)p(y)dy; (2:24)

respectively.

The technique adopted in Rockafellar & Uryasev (2000) evaluates VaR and optimizes

CVaR simultaneously. The key idea of their approach is to characterize VaR and CVaR in

terms of the function defined by

F(x;h):�h�1

1 � k g [g(x; y)�h]�p(y)dy; (2:25)

where [x]��x if x]0 and 0 otherwise.

By THEOREM 1 in Rockafellar & Uryasev (2000), F(x; a)is convex and continuously

differentiable in a, and

CVaR�minh

F(x;h): (2:26)

Let

A:�arg minh

F(x;h): (2:27)

Then VaR is given by the left endpoint of A. For further details about this technique,

interested readers may refer to Rockafellar & Uryasev (2000).

We suppose that there is no change in the portfolio composition but with a worst case

scenario in that the one claim takes place in a short time duration [t; t�Dt): Then

Xi(t�Dt)�X (t)�(p(t)mi�c�(1�b)lq(t)EY )Dt

�p(t)siW (Dt)�(1�q)Y : (2:28)

Denote the loss X (t)�X (t�Dt)by L(t) and denote


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

�f[pmi�c�(1�b)lqE(Y )]Dt�psix�(1�q)yg (2:29)

by gi(p; q;h; x; y): Let f(x) and g(y) denote the probability density of standard normal

distribution and the loss Y, respectively. Denote

Fi(p; q;h): �h�1

1 � k g�

0g

�

��

(gi(p; q;h; x; y)�h)�f (x)dxg(y)dy: (2:30)

DEFINITION 2.1 The maximal conditional VaR (MCVaR) is defined by:

MCVaR(p; q): �maxi�1;2

CVaRi(p; q):�maxi�1;2

infh

Fi(p; q;h): (2:31)

Indeed, MVaR gives a conservative way to evaluate risk. It takes into account the case

when the economy transits to a Bad state. Furthermore, it follows from Equation (2.29) to

Equation (2.31) that MCVaR is coherent in the sense of Artzner et al. (1999). By

exploiting the convexity property, the minimization of the MCVaR can be done by some

standard techniques, see, for example, Rockafellar & Uryasev (2000).

REMARK 2.1 The second expression in Equation (2.31) follows from Theorem 2 in

Rockafellar Uryasev (2000), which is used later for minimizing the risk.1

2.4.2. The problem with risk constraint. Suppose the risk is constrained not to exceed a

given level R, i.e.

MCVaR(p; q)5R; (2:32)

and write C for the set of (p, q) satisfying the above constraint (2.32).

Then the optimal investment-insurance problem of the insurer becomes:

Problem 2.2

V (t; x; z)�supu �U

E[U(X (T))½X (t)�x;Z(t)�z];

subject to the constraint Equation (2.32).

With MCVaR imposed as the risk constraint, we still use the trial function Equation

(2.14), then

sup(p;q) �C

[(c�mip(t)�(1�b)lq(t)E[Y ])@Vi

@x

�1

2s2

i p2(t)

@2Vi

@x2�lE[Vi(t; x�(1�q(t))Y )]] (2:33)

is reduced to:

sup(p;q)�C

�mip(t)�

1

2as2

i p2(t)�

�(1�b)lq(t)E[Y ]�

la

E[ea(1�q)Y ]

��: (2:34)

1 As pointed in Rockafellar & Uryasev (2000), to work directly with the function CVaR(pi, q), may be hard to

do because of the nature of its definition in terms of the k-VaR value and the often troublesome mathematical

properties of that value.

J. Liu et al. 272

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

Obviously, if the constraint is inactive, the optimal strategy follows that without

constraint. Suppose that the constraint set is not empty.

THEOREM 2.3 (a) There exists (p(i;�)� ; qi;�

� ) � C such as

Ci(p(i;�)� ; q(i;�)

� )� sup(p;q) �C

Ci(p; q): (2:35)

Proof. The existence is obvious from the concavity and the fact that Ci(p; q) goes to

infinity if p goes to negative infinity. I

THEOREM 2.4 Let p(i;�)� ; q(i;�)

�

�be given by Equation (2.35), h̆

(�)i (t); i�1; 2 be the solution

of Equation (3.12) with

Ai(t)��[a�(1�b)lq(i;�)� (t)E(Y )]a�

1

2s2

i (p(i;�)� )2a2

�fi(M(Y�q(i;�)

� (Y ))(a)�1)�p; i�1; 2 (2:36)

Then (p(i;�)� ; q(i;�)

� ) is the optimal investment�reinsurance strategy and

V̆(�)i (x; t)��

e�ax

ah̆

(�)i (t); i�1; 2:


Proof. From the optimal condition and the construction of the trial function, it is also

easily verified by THEOREM 2.1 that V̆(�)i (x; t) coincides with the value function and

(p̆(�)i ; q̆(i;�)) is the optimal strategy with constraint. I

With the objective function Equation (2.34) and the constraint Equation (2.32),

(p(i;�)� ; q(i;�)

� ) can be obtained by using the standard method of Lagrangian multiplier, see

(Yiu (2004)) for the details, the procedure of which is standard. Then we adopt the

fmincon function from Matlab optimization tool to solve the problem.

3. The diffusion approximation of the claim process

In this section, we use the surplus model considered in Promislow & Young (2005). The

accumulated claim process C(t) is then modeled as follows:

dC(t)�adt�bdW0(t); (3:1)

where W0(t) is another standard Brownian motion defined on (V;F ;P); which is

independent of W(t).

With u being the safely loading, the constant premium rate is supposed to be

c:�(1�u)a:

Let fXtgt �T be the surplus process of the insurer. Then without investment, the surplus

process of the insurer is governed by


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

X (t)�cdt�dC(t)

�uadt�bdW0(t);X (0)�x;

8<: (3:2)

where x is the initial capital.

We now incorporate both investment and (proportional) reinsurance in our formula-

tion. With the combined reinsurance-investment strategy u(t):�(q(t); p(t)) in Equation

(3.2), the wealth process fX u(t)½t � T g of the insurer is governed by

dX u(t) �(p(t)m(t)dt�(u�bq(t))adt�b(1�q(t))dW0(t)

�p(t)s(t)dW (t)

X (0) �x;

8<: (3:3)

Set p̆(t):�(1�q(t); p(t)) . Then a strategy u(t): �(q(t); p(t)) , (or equivalently, p̆(t); t � T ;

is said to be admissible if

1. it is G-progressively measurable;

2. fT

0ks̃T(t)p̃(t)k2

dtB� P-a:s:;3. q(t) � [0; 1]; �t � T :

Write U for the space of such admissible strategies u.

3.1. HJB and optimal conditions

We still consider Problem 2.1. Again using the dynamic programming principle, the value

functions Vi, i�1, 2, over different states satisfy the following pair of coupled HJB

equations:

@Vi

@t� sup

p(t);q(t)

�(u�bq(t))a

@Vi

@x�mip(t)

@Vi

@x�

1

2[(1�q(t))2b2

�s2i p

2(t)]@2Vi

@x2�hV;Qeii

��0; i�1; 2; (3:4)


Vi(T ; x)�U(x): (3:5)

Consequently, the optimal conditions are then given by

p(i)(t)��miVi;x

s2i Vi;xx

; (3:6)

q(i)(t)�

1�baVi;x

b2Vi;xx

; if 1�baVi;x

b2Vi;xx

]0;

0; if 1�baVi;x

b2Vi;xx

50:

8>>><>>>:

(3:7)

For the exponential utility function defined by Equation (2.11), we consider the following

trial function:

J. Liu et al. 274

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

Vi(x; t)��e�ax

ahi(t); i�1; 2: (3:8)

Substituting the trial function above into Equation (3.4), we have from Equations (3.6)

and (3.7) that

pi: �mi

as2i

; (3:9)

qi: �1�

ba

b2a; if 051�

ba

b2a51;

0; if 1�ba

b2aB0;

8>>><>>>:

�max

�1�

ba

b2a; 0

�: (3:10)

Plugging Equations (3.9) and (3.10) into Equation (3.4), the pair of coupled HJBs is then

reduced to the following pair of coupled ODEs:

h?1(t)�B1(t)h1(t)�ph2(t)�0;h?2(t)�B2(t)h2(t)�ph1(t)�0;

(3:11)

where

Bi(t)��a(u�bq)a�1

2a2

�(1�q)2b2�

m2i

2s2i

a2

��p; i�1; 2 (3:12)


h1(T)�h2(T)�1: (3:13)

Then as the classical case, we have the following results.

THEOREM 3.1 Let p(i;�) and q(i;�) be given by Equations (3.9), (3.10), respectively, and

h�i (t); i�1, 2, be the solution of the ODEs Equation (3.11). Then (p(i;�); q(i;�)) is the optimal

investment�reinsurance strategy and

V �i (t; x)��

e�ax

ah�

i (t); i�1; 2:

are the value functions over the two states.

3.2. The CVaR constraint

For a short time interval [t; t�Dt]; we give an approximation to the loss variable.

DX (t)� [p(t)(m(t))�(u�bq(t))a]Dt�b(1�q(t))W 0(Dt)

�p(t)s(t)W (Dt)

X0�x:

8<: (3:14)

The loss variable is then given by:


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

�f[p(t)mi�(u�bq(t))a]Dt�b(1�q(t))W 0(Dt)�p(t)siW (Dt)g: (3:15)

Now

gi(p(t); q(t);h; x; y): ��f[p(t)mi�(u�bq(t))a]Dt�b(1�q(t))x�p(t)siyg; (3:16)

where g(y) is also the density of standard normal distribution. Now the MCVaR

constraint can be written as

�f[p(t)(mi)�(u�bq(t))a]Dt�c(k)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(b(1�q(t)))2Dt�(p(t)si)

2Dt

q5R; (3:17)

for i�1, 2, where c(k)�ffiffiffiffiffiffi2p

p(exp(er f�1(2k�1))2(1�k))�1, see Rockafellar & Uryasev

(2000) for the details.

When the risk constraint is imposed, we still assume that the value function has the

form �e�gx

gh(t); and let

Ci(p; q):�(u�bq)a�mip�1

2a[(1�q)2b2�s2

i p2]: (3:18)

The following Theorems are just similar to Theorems 2.3 and 2.4, respectively.

THEOREM 3.2 There exists ((p(i;�)� ; q(i;�)

� )) such that

Ci((p(i;�)� ; q(i;�)

� ))� sup(p;I) �C

Ci(p; q): (3:19)

THEOREM 3.3 Let (p(i;�)� ; q(i;�)

� ) be given by Equation (3.19), h̆�i (t); i�1, 2 be the solution

of Equation (3.11) with

Bi(t)��a�

(u�bq(i;�)� )a�mip

(i;�)�

��

1

2a2

�(1�q

(i;�)� )2b2�

1

2a2(p(i;�)

� )2

��p; (3:20)

for i�1, 2.

Then (p(i;�)� ; q(i;�)

� ) is the optimal investment�reinsurance strategy and

V̆(�)i (x; t)��

e�ax

ah̆

(�)i (t); i�1; 2:


4. Numerical examples

In this section, we present some numerical results and we employ the f mincon function in

Matlab. In this case, the objective is given by

s:t:supCi(p; q);MVaR5R

05q51

(4:1)

J. Liu et al. 276

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

Note that we use Fi(p; q;h)5R; i�1; 2; instead of MVaR(p; q)5R for the numerical

experiments. Indeed, they coincide with each other for the problem formulation and the

former can simplify the calculation.

4.1. The classical model

In this subsection, we assume that the loss has the exponential distribution with density

function

g(y)�ge�gy; y]0: (4:2)

Consequently, without the risk constraint,

p(i;�)�mi

s2i a

;

q(i;�)�1�

g�gffiffiffiffiffiffiffiffiffiffiffiffi

1 � ap

g; if 1�


1 � ap

g]0;

0; if 1�


1 � ap

gB0

8>>>>>>>><>>>>>>>>:

�max

�1�


1 � ap

g; 0

�(4:3)

The MCVaR is given by

MCVaR(p; q)�maxi�1;2

infh

�h�

1

1 � k g�

0g

�

��

(gi(p; q;h; x; y)�h)�f (x)dxg(y)dy

�; (4:4)

With gi(p; q;h; x; y) defined by Equation (2.29).

Example 4.1 In this example, we consider the risky investment and proportional

reinsurance problem when MCVaR is imposed.

. The effect of the risk constraint.

Hypothetical parameters: m1�m2�0.2, s1�s2�0.3, l�0.1, a�0.1, b�0.69,

g�0.2, R �[3,8.2459], dt�0.1 In this example, we examine how the risky

investment and proportional reinsurance vary with the constraint level when

there are no regimes. For the set of parameters, the constraint is active when R �

[3,8.2459].

Figures 1 and 2 plot the risky investment and proportional reinsurance against the

risk constraint level, respectively. When R is small (i.e. a small risk tolerance is

allowed), the insurer would decrease the risky investment and increase the

proportional reinsurance. Furthermore, to satisfy a lower risk constraint (i.e. to


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

decrease the risk), p should be reduced more, and the insurer should purchase a

higher proportional reinsurance. These results make intuitive sense.

Suppose that the current state is Z(t)�e1, say State I. To investigate the effect of the

potential regime, we shall vary m2 and s2 in the following items, respectively. Let CVaR0,1

and CVaR0,2 be the CVaR corresponding to State I and State II, respectively. With

MCVaR being imposed, we denote the CVaR associated with State I and State II by

CVaR1,1 and CVaR1,2, respectively.

. The effect of m2.

Hypothetical parameters: m1�0.2, s1�s2�0.3, l�0.1, a�0.1, b�0.69, g�0.2,

m2 �[0.1,0.3], dt�0.1, R�6. In this example, m2 is the expected return in the future

regime. R�6 is active for any m2�[0.1,0.3].

Figure 3 plots the CVaR from State I and State II in the case both with and without

the MCVaR constraint. Figure 4 plots the MCVaR. From the two figures, it can be

3 4 5 6 7 8 912

14

16

18

20

22

24

R

risky

inve

stm

ent

Figure 1. The optimal risky investment against the risk constraint level R.

3 4 5 6 7 8 90.4

0.6

0.8

1

1.2

1.4

R

rein

sura

nce

Figure 2. The proportional reinsurance against the risk constraint level R.

J. Liu et al. 278

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

seen that, for the case without the risk constraint, if m2 is smaller than m1, the

MCVaR is equal to CVaR0,2, which decreases with m2, and MCVaR is equal to

CVaR0,1 if m2 is larger than m1. When the MCVaR is stabilized, CVaR1,2 is equal to

MCVaR when m2Bm1, and CVaR1,1 is equal to MCVaR if m2�m1.

With MCVaR being imposed as the risk constraint, Figures 5 and 6 plot the risky

investment and proportional reinsurance, respectively. If m2 is smaller, Figure 5

shows that the amount invested in the risky security should be less, and Figure 6

shows that the proportional reinsurance should be increased.

. The effect of s2.

Hypothetical parameters: m1�m2�0.2, s1�0.3, l�0.1, a�0.1, b�0.69, g�0.2,

s2 �[0.2, 1], dt�0.1, R�6. R�6 is active for any s2 �[0.2, 1].

From Figures 7�10, we can find that the effect of s2 is opposite to that of m2.

0.1 0.15 0.2 0.25 0.3 0.355.5

6

6.5

7

7.5

8

8.5

µ2

CV

aR

CVaR0,1

CVaR0,2

CVaR1,1

CVaR1,2

Figure 3. The comparison of CVaR from different regime against m2.

0.1 0.15 0.2 0.25 0.3 0.355.5

6

6.5

7

7.5

8

8.5

µ2

MC

VaR

without risk constriantwith constraint

Figure 4. The comparison of MCVaR from different regime against m2.


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

0.1 0.15 0.2 0.25 0.3 0.3521

21.2

21.4

21.6

21.8

22

22.2

22.4

µ2

π 1


Figure 5. The optimal risky investment against m2.

0.1 0.15 0.2 0.25 0.3 0.350.5

0.6

0.7

0.8

0.9

µ2

q


Figure 6. The proportional reinsurance against m2.

0.2 0.4 0.6 0.8 10

5

10

15

20

25

σ2

CV

aR

CVaR0,1

CVaR0,2

CVaR1,1

CVaR1,2

Figure 7. The comparison of CVaR from different regime against s2.

J. Liu et al. 280

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

0.2 0.4 0.6 0.8 15

10

15

20

25

σ2

π 1


Figure 9. The optimal risky investment against s2.

0.2 0.4 0.6 0.8 15

10

15

20

25

σ2

MC

VaR


Figure 8. The comparison of MCVaR from different regime against s2.

0.2 0.4 0.6 0.8 10.4

0.6

0.8

1

1.2

1.4

σ2

q without risk constriantwith constraint

Figure 10. The proportional reinsurance against s2.


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

4.2. The normal approximation

Example 4.2 In this example, we consider the effect of MCVaR and the regime parameters

for the diffusion approximation model in Section 3. Here we assume that only one parameter

in State II is different from the State I, which represent a ‘Good’ economy and a ‘Bad’ one,

respectively.

. The effect of m2.

Hypothetical parameters: m1�0.2, m2 �[0.1, 0.3], s1�0.3, s2�0.3, a�0.5,

a�0.2, b�0.6, k�0.95, dt�0.1, u�0.2, b�0.1, p�2.

Figures 11 and 12 show that the risky investment and proportional

reinsurance should be reduced when the risk constraint is active.

Hypothetical parameters: m1�m2�0.2, s1�0.3, s2 �[0.2, 1] a�0.5, a�0:2;

b�0.6, k�0.95, dt�0.1, u�0.2, b�0.1, p�2. The parameter effect of State

II, together with the risk constraint, is shown in Figures 13 and 14. And the

effect of risk constraint is that the risky investment and proportional

reinsurance should be reduced to meet the risk management requirement.

Also, they are similar to those in the previous cases.

5. Summary

With the risk constraints being imposed in the utility maximization, we considered the

optimal investment�reinsurance problem of an insurer when the price dynamics of

the risky asset are governed by a Markov-modulated Geometric Brownian motion. The

market parameters were assumed to switch over time according to a continuous-time,

finite-state, observed Markov chain. We derived a system of coupled HJB equations for

the optimization problem and obtained explicit solutions to the problem in different cases.

For two different types of surplus models, the results show that the investment and

0.1 0.15 0.2 0.25 0.3 0.359.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

µ2

π 1


Figure 11. The optimal risky investment against m2.

J. Liu et al. 282

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

0.1 0.15 0.2 0.25 0.3 0.350.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

µ2


q

Figure 12. The proportional reinsurance against m2.

0.2 0.4 0.6 0.8 15

6

7

8

9

10

11

12

σ2

π 1


Figure 13. The optimal risky investment against s2.

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

σ2

q


Figure 14. The proportional reinsurance against s2.


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

reinsurance strategies are only related to the regime state, and the investment keeps the

Merton’s structure.

With the risk measures (MCVaR) being imposed as risk constraints, we analysis the

quantitative properties of the investment and reinsurance. Moreover, the numerical

examples illustrated the risky investment and the reinsurance are also affected by the

potential state, especially when the state is worse than the current state.

Acknowledgements

The authors would like to thank the reviewer for helpful comments. The first and second

authors would like to thank the Research Committee of the Hong Kong Polytechnic

University for support. Tak Kuen Siu would like to acknowledge the Discovery Grant

from the Australian Research Council (ARC), (Project No. DP1096243).

References

Ahmed, N. U. & Teo, K. L. (1981). Optimal control of distributed parameter systems. New York: North Holland.

Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9, 203�228.

Asmussen, S. & Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance:

Mathematics and Economics 20, 1�15.

Bai, L. & Guo, J. (2008). Optimal proportional reinsurance and investment with multiple risky assets and no-

shorting constraint. Insurance: Mathematics and Economics 42, 968�975.

Browne, S. (1995). Optimal investment policies for a firm with a random risk process: exponential utility and

minimizing the probability of ruin. Mathematics of Operations Research 20, 937�958.

Browne, S. (1997). Survival and growth with a liability: optimal portfolio strategies in continuous time.

Mathematics of Operations Research 22, 468�493.

Browne, S. (1999). Beating a moving target: optimal portfolio strategies for outperforming a stochastic

benchmark. Finance and Stochastics 3, 275�294.

Cai, J. & Tan, K. S. (2007). Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measure.

Astin Bulletin 37, 93�112.

Cai, J., Tan, K. S., Weng, C. G. & Zhang, Y. (2008). Optimal reinsurance under VaR and CTE risk measures.

Insurance: Mathematics and Economics 43, 185�196.

Elliott, R. J., Aggoun, L. & Moore, J. B. (1994). Hidden Markov models: estimation and control. New York:

Springer-Verlag.

Flemming, W. H. & Soner, H. M. (1993). Controlled Markov processes and viscosity solutions. New York:

Springer-Verlag.

Hipp, C. & Plum, M. (2000). Optimal investment for insurers. Insurance Mathematics and Economics 27, 215�228.

Hipp, C. & Taksar, M. (2000). Stochastic control for optimal new business. Insurance Mathematics and Economics

26 (2�3), 185�192.

Højgaard, B. & Taksar, M. (2004). Optimal dynamic portfolio selection for a corporation with controllable risk

and dividend distribution policy. Quantitative Finance 4, 315�327.

Kostadinova, R. (2007). Optimal investment for insurers when the stock price follows an exponential Levy

process. Insurance: Mathematics and Economics 41, 250�263.

Liu, C. & Yang, H. (2004). Optimal investment for an insurer to minimise its probability of ruin. North American

Actuarial Journal 8, 11�31.

Promislow, D. S. & Young, V. R. (2005). Minimizing the probability of ruin when claims follow Brownian motion

with drift. North American Actuarial Journal 9, 109�128.

Rockafellar, R. T. & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk 2, 21�41.

Schmidli, H. (2001). Optimal proportional reinsurance polices in dynamic setting. Scandinavian Actuarial Journal

1, 55�68.

Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Annals of Applied

Probability 12, 890�907.

Schmidt, K. D. (2004). Optimal quota share reinsurance for dependent lines of business. Schweizerische

Aktuarvereinigung. Mitteilungen 2, 173�194.

J. Liu et al. 284

Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

Sotomayor, L. R. & Cadenillas, A. (2009). Explicit solutions of consumption-investment problems in financial

markets with regime switching. Mathematical Finance 19, 251�279.

Taksar, M. (2000). Dependence of the optimal risk control decisions on the terminal for a financial corporation.

Annals of Operations Research 98, 89�99.

Taksar, M. & Markussen, C. (2003). Optimal dynamic reinsurance policies for large insurance portfolios. Finance

and Stochastics 7, 97�121.

Tan, K. S., Weng, C. G. & Zhang, Y. (2009). VaR and CTE criteria for optimal quota-share and stop-loss

reinsurance. North American Actuarial Journal 9, 459�482.

Teo, K. L., Reid, D. W. & Boyd, I. E. (1980). Stochastic optimal control theory and its computational methods.

International Journal on Systems Science 11, 77�95.

Yang, H. & Zhang, L. (2005). Optimal investment for insurer with jump-diffusion risk process. Insurance

Mathematics and Economics 37, 615�634.

Yiu, K. F. C. (2004). Optimal portfolio under a value-at-risk constraint. Journal of Economic Dynamics and

Control 28, 1317�1334.


Dow

nloa

ded

by [

Tem

ple

Uni

vers

ity L

ibra

ries

] at

05:

35 1

3 N

ovem

ber

2014

optimal investment-reinsurance with dynamic risk constraint and regime switching

Documents