This article was downloaded by: [Memorial University of Newfoundland]On: 03 August 2014, At: 16:54Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Stochastic Analysis and ApplicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsaa20

A BSDE Approach to Optimal Investment of an Insurerwith Hidden Regime SwitchingTak Kuen Siu aa Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics ,Macquarie University , Sydney , New South Wales , AustraliaPublished online: 13 Dec 2012.

To cite this article: Tak Kuen Siu (2013) A BSDE Approach to Optimal Investment of an Insurer with Hidden Regime Switching,Stochastic Analysis and Applications, 31:1, 1-18, DOI: 10.1080/07362994.2012.727144

To link to this article: http://dx.doi.org/10.1080/07362994.2012.727144

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Stochastic Analysis and Applications, 31: 1–18, 2013Copyright © Taylor & Francis Group, LLCISSN 0736-2994 print/1532-9356 onlineDOI: 10.1080/07362994.2012.727144

ABSDEApproach to Optimal Investment of anInsurer with Hidden Regime Switching

TAK KUEN SIU

Department of Applied Finance and Actuarial Studies,Faculty of Business and Economics, Macquarie University,Sydney, New South Wales, Australia

We discuss an optimal investment problem of an insurer in a hidden Markov, regime-switching, modeling environment using a backward stochastic differential equation(BSDE) approach. Filtering theory is used to transform the optimal investmentproblem into one with complete observations. Using BSDEs with jumps, we discussthe problem with complete observations.

Keywords Backward stochastic differential equations; Hidden regime switching;Insurance risk; Non-markovian framework; Optimal investment.

Mathematics Subject Classification 97M30; 91G80; 93E11; 93E20.

1. Introduction

Optimal investment is one of the key topics in financial economics. The pioneeringand foundational work on this topic was done by Markowitz [1], where anelegant mathematical modeling framework for an optimal investment problem wasestablished in a simplified, single-period model. Samuelson [2] and Merton [3, 4]considered the problem in a multi-period model and a continuous-time model,respectively. Recently, there has been a considerable interest in studying optimalinvestment problems in regime-switching modeling frameworks. Some examplesinclude Zhou and Yin [5], Sass and Haussmann [6], Yin and Zhou [7], Baeuerle andRieder [8–10], Elliott et al. [11], and others. The main feature of regime-switchingmodels is that the model parameters are modulated by a Markov chain. The statesof the chain are interpreted as different states of an economy, or different modes ofa market. The incorporation of the regime-switching effect on optimal investmentdecisions is a timely and important topic as highlighted by the recent global financialcrisis since there has been substantial changes in economic conditions pre- andpost-financial crisis. It seems that many existing works on optimal investments

Received April 14, 2012; Accepted April 19, 2012Address correspondence to Tak Kuen Siu, Department of Applied Finance and

Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney,NSW 2109, Australia; E-mail: ken.siu@mq.edu.au or ktksiu2005@gmail.com

1

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

2 Siu

in regime-switching models primarily focus on investment decisions made byhouseholds, banks, and financial institutions. However, relatively little attention hasbeen paid to investigate optimal investment problems of insurance companies underregime shifts which may be attributed to market catastrophes such as financialcrises.

What makes the optimal investment problems of insurance companiesinteresting and challenging is the presence of insurance liabilities attributed mainlyto insurance claims. Elliott and Siu [12] considered the optimal investment problemof an insurer in a Markov regime-switching model modulated by an observableMarkov chain. Elliott and Siu [13] considered an optimal investment problem of aninsurer in a hidden, regime-switching, environment, where the modulating hiddenMarkov chain describes the evolution of the hidden “true” state of the model overtime. Elliott and Siu [12, 13] only considered a Markovian modeling frameworkand adopted the dynamic programming approach to discuss the optimal investmentproblem. The problem remains unresolved in a general non-Markovian situation.

In this article, we discuss an optimal investment problem of an insurerin a hidden Markov, regime-switching, modeling framework using a BackwardStochastic Differential Equation (BSDE) approach. The modeling frameworkconsidered here is a non-Markovian generalization of that in Elliott and Siu [13]in the sense that the market coefficients depend on the past values of financialprice and insurance risk processes. We discuss the optimal investment problemsin two steps. First, as in Elliott and Siu [13], we employ filtering theory to turnthe optimal investment problem with partial observations into one with completeobservations. Second, we adopt BSDEs with jumps to discuss the problem withcomplete observations. This provides a general and theoretically sound approachto discuss control problem in a non-Markovian situation. Three typical utilityfunctions, namely, the exponential utility, the power utility and the logarithmicutility, are considered using the BSDE approach.

We organize this article as follows. In the next section, we present the modeldynamics. Section 3 presents the optimal investment problem and discusses thefiltering approach to turn the problem into one with complete observations. InSection 4, we use BSDEs with jumps to discuss the control problem with completeobservations for the three utility functions. The final section gives concludingremarks.

2. The Dynamics

We consider a continuous-time model, where uncertainty is modelled by a completeprobability space ���� � P� and the time parameter set � is given by a finite timehorizon �0� T�, with T < �. Here P is a real-world probability measure.

Suppose X �= �X�t� � t ∈ � is a continuous-time, finite-state, hidden Markovchain on ���� � P� whose state space is the set of standard unit vectors�e1� e2� � eK in �K , where K is the number of states of the chain and the jthcomponent of ei is the Kronecker delta �ij , for each i� j = 1� 2� � K. As in Elliottand Siu [13], we suppose that the chain X describes the evolution of the unobserved“true” state of the model parameters over time in both the financial price processand the insurance risk process to be defined below.

To describe the probability law of the chain X under P, we specify a family ofintensity matrices A�t� �= �aji�t��, t ∈ � , where aji�t� is the instantaneous transition

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 3

intensity of the chain X from state ei to state ej at time t and it is assumed tobe a deterministic function of time t here. Let �X �= ��X�t� � t ∈ � be the right-continuous, P-complete, natural filtration generated by the chain X. Then Elliott etal. [14] obtained the following semimartingale dynamics for the chain X under P:

X�t� = X�0�+∫ t

0A�u�X�u−�du+M�t�� t ∈ � (1)

Here M �= �M�t� � t ∈ � is an �K-valued, ��X� P�-martingale.For each t ∈ � , let r�t� be the instantaneous interest rate of the bond B at time

t, where r�t� > 0. Then the price process of the bond �B�t� � t ∈ � evolves over timeas follows:

B�t� = exp( ∫ t

0r�u�du

)� t ∈ � �

B�0� = 1

For each t ∈ � , let �X�t� and �t� be the appreciation rate and the volatilityof the share index S at time t, respectively. Here we use the superscript X toemphasize the dependence of �X�t� on X and �t� > 0, for each t ∈ � . Suppose�W�t� � t ∈ � is a standard Brownian motion on ���� � P� with respect to itsP-complete, right-continuous, natural filtration, denoted by �W �= ��W�t� � t ∈ � .We assume that the interest rate process �r�t� � t ∈ � and the volatility process� �t� � t ∈ � are �Y -progressively measurable and uniformly bounded processes on���� � P�, where �Y is the natural filtration generated by observations about returnsfrom the share index which will be defined later in this section. Since �Y isobservable, both �r�t� � t ∈ � and � �t� � t ∈ � are observable processes.

We further suppose that the appreciation rate �X�t� is modulated by the chainX as follows:

�X�t� �= ���t��X�t��� t ∈ �

Here ��t� �= ��1�t�� �2�t�� � �K�t��′ ∈ �K where �i�t� ∈ �, for each i = 1� 2� � K,

and y′ denotes the transpose of a vector (or a matrix), y. We suppose that ���t� � t ∈� is an �K-valued, �W -progressively measurable and uniformly bounded processon ���� � P�. The scalar product �·� ·� selects the component of ��t� in force at timet depending on the state X�t� of the chain at time t.

Then we suppose that the price process of the share index �S�t� � t ∈ � is governed by the following Markov, regime-switching, functional geometricBrownian motion:

dS�t� = �X�t�S�t�dt + �t�S�t�dW�t�� t ∈ � �

S�0� = s > 0

This is a non-Markovian generalization of the model considered in Elliott andSiu [13].

We now present a model for insurance liabilities based on a hidden, regime-switching, random measure, which is a non-Markovian generalization of that inElliott and Siu [13]. Suppose �Z�t� � t ∈ � is a real-valued, pure jump process on

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

4 Siu

���� � P�, where Z�t� is the aggregate amount of claims up to time t. Let �Z�u� �=Z�u�− Z�u−�. Then

Z�t� = ∑0<u≤t

�Z�u�� Z�0� = 0� P-a.s.� t ∈ �

Write � for the state space of random claim sizes �0���. Consider a randommeasure � on the product measurable space �� ×����� �⊗�����, which selectsclaims arrival times and sizes z �= Z�u�− Z�u−�, where ��� � and ���� are theBorel -fields generated by open subsets of � and �, respectively. Then

Z�t� =∫ t

0

∫ �

0z��du� dz�� t ∈ � (2)

Write �Z �= �� Z�t� � t ∈ � , the right-continuous, P-complete, natural filtrationgenerated by the insurance claims process �Z�t� � t ∈ � .

For each t ∈ � , let N�t� be the number of claim arrivals up to time t. Wesuppose that �N�t� � t ∈ � is a “general” doubly stochastic process on ���� � P� withstochastic intensity process being modulated by the chain X as follows:

�X�t� �= ���t��X�t��� t ∈ �

Here ��t� �= ��1�t�� �2�t�� � �K�t��′ ∈ �K where �i�t� > 0 and it represents the

stochastic intensity of claims arrival times at time t when X�t� = ei, for each i =1� 2� � K. Again we suppose that ���t� � t ∈ � is an �K-valued, �Z-progressivelymeasurable, uniformly bounded process on ���� � P�.

For each i = 1� 2� � K, let fi�z� be the probability density function of theclaim size z �= Z�u�− Z�u−� when X�u−� = ei under P. Define, for each t ∈ � ,

��t� �= � Z�t� ∨ �X�t��

the minimal -field containing � Z�t� and �X�t�. Write � �= ���t� � t ∈ � . Thenthe compensator, or the dual predictable projection, of the random measure � withrespect to � under P is:

�X�u−��du� dz� =K∑i=1

�X�u−�� ei��i�u�fi�z�dzdu (3)

For each t ∈ � , let p�t� be the premium rate at time t. We suppose thatthe premium rate process �p�t� � t ∈ � is �Z-progressively measurable, uniformlybounded process on ���� � P� taking values in �0���.

Suppose, for each t ∈ � , ��t� is the amount of money invested in the share indexS at time t. Let �V ��t� � t ∈ � be the surplus process associated with the investmentprocess ���t� � t ∈ � . To simplify the notation, write V�t� for V��t�, for each t ∈ � .Then the surplus process �V�t� � t ∈ � evolves over time as follows:

dV�t� = �p�t�+ r�t�V�t�+ ��t���X�t�− r�t���dt + ��t� �t�dW�t�

−∫ �

0z��dt� dz� �

V�0� = v (4)

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 5

To simplify our analysis, we assume here that the initial endowment v is a constant.In general, one may consider the situation where the initial endowment is a randomvariable.

For each t ∈ � , let

��t� �= � Z�t� ∨ � Y �t��

where �Y �= �� Y �t� � t ∈ � is the right-continuous, P-complete, filtration generatedby the logarithmic return process �Y�t� � t ∈ � of the share index S, (i.e., Y�t� �=ln�S�t��, for each t ∈ � ).

Write � for ���t� � t ∈ � . Then we define the notion of admissible investmentprocesses as follows:

Definition 2.1. An investment process ���t� � t ∈ � is admissible if it satisfies thefollowing conditions:

1. ���t� � t ∈ � is �-progressively measurable;2. for each t ∈ � , ��t� ∈ �, where � is a compact, convex and non-empty subset

of �;3. ∫ T

0���t��2dt < �� P-a.s.�

4.

K∑i=1

[ ∫ T

0

(�p�t�+ r�t�V�t�+ ��t���i�t�− r�t��� + �2�t� 2�t�

)dt

+∫ T

0

∫ �

0z2�i�t�fi�z�dzdt

]� P-a.s.�

5. the stochastic differential equation for the wealth process �V�t� � t ∈ � has aunique strong solution.

We write � for the space of admissible investment processes.

3. Optimal Investment and Filtering

In this section, we consider an optimal investment problem of an insurer whosesurplus process is defined in the last section. The object of the insurer is to select anoptimal investment process ��†�t� � t ∈ � which maximizes the expected exponentialutility of terminal surplus given observations about the price process of the shareindex and the insurance liabilities. That is to select ��†�t� � t ∈ � such that

��v� �= sup�∈�

E�− exp�−�V��T�� �V�0� = v�

= E�− exp�−�V�†�T�� �V�0� = v�� � > 0�

subject to the surplus dynamics (4).

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

6 Siu

Consider the following �-adapted process �W �t� � t ∈ � defined by putting:

W �t� �= W�t�+∫ t

0

(�X�u�− �X�u�

�u�

)du� t ∈ � �

where ��X�t� � t ∈ � is the �-optional projection of ��X�t� � t ∈ � under P.It can be shown that �W �t� � t ∈ � is a ��� P�-Brownian motion, (see,

e.g., [15, 16]).For each t ∈ � , let �t� �= � Y �t� ∨ � Z�t� ∨ �X�t�. Write �= ��t� � t ∈ � , so

is the enlarged filtration representing the full information structure in our model.Consider the following -adapted process �Q�t� � t ∈ � on ���� � P�

defined by:

Q�t� �=∫ t

0

∫ �

0z���du� dz�− �X�u��du� dz��� t ∈ � �

so Q is an �� P�-martingale.Write

��dt� dz� �=K∑i=1

�X�t�� ei��i�t�fi�z�dzdt�

where X�t� is the optimal filtered estimate of X�t� given ��t� and is given by:

X�t� = E�X�t� ���t��� P-a.s.� t ∈ �

Define a random measure q�du� dz� on � � by:

q�du� dz� �= ��du� dz�− ��du� dz�

For each t ∈ � , let

Q�t� �=∫ t

0

∫ �

0zq�du� dz�

Then it can be shown that the process �Q�t� � t ∈ � is a ��� P�-martingale, (see, forexample, Elliott [17]).

Consequently, under P, the surplus process �V�t� � t ∈ � can be written as:

dV�t� =(p�t�+ r�t�V�t�+ ��t���X�t�− r�t��−

K∑i=1

∫ �

0�X�t�� ei�z�i�t�fi�z�dz

)dt

+ �t���t�dW �t�−∫ �

0zq�dt� dz��

V�0� = v (5)

In what follows we present a non-Markovian version of the filtering results inElliott and Siu [13], which are used to turn the optimal investment problem into onewith complete observations. As in Elliott and Siu [13], these results are established

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 7

using a change of measure technique, a version of the Bayes’ rule and a gaugetransformation technique. We only state the main results without giving the proof.

For each u ∈ � , let Z�u� ·� � � → �0��� be a random variable with a strictlypositive density function f under the reference probability measure P. Note that Pis the probability measure on ���� � under which the observation processes do notdepend on the hidden Markov chain X. Write, for each i = 1� 2� � K,

Gi�u��� �=�i�u�fi�Z�u����

f�Z�u����

Here fi�Z�u���� and f�Z�u���� are the values of fi�z� and f�z� evaluated at z =Z�u���, respectively. We suppress “�” and write Gi�u� for Gi�u��� to simplify thenotation.

Consider the -adapted process ���t� � t ∈ � defined by putting:

��t� �= exp( ∫ t

0 −2�u��h�u��X�u��dY�u�− 1

2

∫ t

0 −2�u��h�u��X�u��2du

)

· exp(−

∫ t

0

K∑i=1

�X�u−�� ei�( ∫ �

0��i�u�fi�z�− f�z��dz

)du

+∫ t

0

∫ �

0

( K∑i=1

�X�u−�� ei� ln(�i�u�fi�z�

f�z�

))��du� dz�

)

Here h�t� �= �h1�t�� h2�t�� � hK�t��′ ∈ �K , and hi�t� �= �i�t�− 1

2 2�t�, i =

1� 2� � K.Write, for each t ∈ � ,

q�t� �= E���t�X�t� ���t�� ∈ �K�

where E�·� is an expectation under the reference probability measure P.For each i = 1� 2� � K, we define a scalar-valued process li �= �li�t� � t ∈ � by

li�t� �= exp( ∫ t

0hi�u�

−2�u�dY�u�− 12

∫ t

0h2i �u�

−4�u�du

+∫ t

0�1− �i�u��du+

∫ t

0logGi�u�dN�u�

) (6)

Write, for each t ∈ � , L�t� �= diag�l1�t�� l2�t�� � lK�t��, a diagonal matrix withnon-zero elements �l1�t�� l2�t�� � lK�t��. Define the transformed process �q�t� � t ∈� by:

q�t� �= L−1�t�q�t�

Then it has been shown that �q�t� � t ∈ � satisfies the following first-order, lineardifferential equation:

dq�t�dt

�= L−1�t�A�t�L�t�q�t�� q�0� = q�0� = E�X�0�� (7)

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

8 Siu

4. BSDE Solutions to the Optimal Investment Problem

In this section, we adopt BSDEs with jumps to discuss the optimal investmentproblem for three typical utility functions used in the finance and insuranceliterature, namely, the exponential utility, the power utility, and the logarithmicutility. For the cases of the exponential utility and the power utility, (semi)-analytical solutions for the optimal investment processes are obtained, which dependon the solutions of the BSDEs and the filtering result presented in the last section.For the case of the logarithmic utility, we obtain an analytical solution for theoptimal investment process which depends on the filtering result. In the followingdevelopment, we adapt the techniques in Lim and Quenez [18] and Øksendal andSulem [19] to the situation of a “filtered” market described in the last section.The theory of BSDEs with random jumps were originally studied by Tang andLi [20].

4.1. Exponential Utility

First, for any 0 ≤ t ≤ s ≤ T , let Vt�v�s� be the value of the surplus process at times such that Vt�v�t� = v. We then define the conditional expected exponential utilityprocess �J��t� � t ∈ � associated with the investment process ���t� � t ∈ � ∈ � given��t� by:

J��t� �= −E�exp�−�V�t�v�T�� ���t��

Here J��T� = − exp�−�v� in the filtered market.Note that for each ���t� � t ∈ � ∈ �,

V�t�v�T� = V�

0�v�T�− V�0�0�t�

Write, for each t ∈ � and ���t� � t ∈ � ∈ �,

C��t� �= exp��V �0�0�t���

and

D��t� �= E�− exp�−�V�0�v�T�� ���t��

Then, J��t� has the following multiplicative decomposition:

J��t� = C��t�D��t�

The following lemma gives a representation for �D��t� � t ∈ � .

Lemma 4.1. For each ���t� � t ∈ � ∈ �, there exist �-predictable processes����t� � t ∈ � in L2�m⊗ P� and ����t� z� � t ∈ � � z ∈ �0��� in L2�m⊗ P ⊗ f� suchthat

D��t� = D��0�+∫ t

0���u�dW�u�+

∫ t

0

∫ �

0���u� z�q�du� dz�

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 9

Here m is the Lebesgue measure on � ; L2�m⊗ P� and L2�m⊗ P ⊗ f� are the spacesof square integrable processes on � ×� and � ×�× �0��� with respect to m⊗ P

and m⊗ P ⊗ f , respectively.

Proof. Note that �D��t� � t ∈ � is a ��� P�-martingale, that �W �t� � t ∈ � isa ��� P�-standard Brownian motion and q is a ��� P�-martingale randommeasure, (see [16, Definition 15.39]). Then the result follows from the martingalerepresentation for a double martingale in [21, Theorem 5.1]. �

Then the following theorem gives the BSDE with jumps for �J��t� � t ∈ � .

Theorem 4.1. For each t ∈ � , ���t� � t ∈ � ∈ � and z ∈ �0���, let

K�1 �t� �= �J��t� �t���t�+ C��t����t��

K�2 �t� z� �= J��t��exp�−�z�− 1�+ C��t����t� z� exp�−�z��

and

g�t� J��t�� K�1 �t�� K

�2 �t� z�� ��t��

�= −J��t�

[�

(p�t�+ r�t�V�t�+ ��t���X�t�− r�t��

)− 1

2�2 2�t��2�t�

]

+K∑i=1

∫ �

0�exp��z�− 1��J��t�+ K�

2 �t� z���X�t�� ei��i�t�fi�z�dz

− � �t���t�K�1 �t�

Then the process �J��t� � t ∈ � satisfies the following BSDE with jumps:

dJ��t� = −g�t� J��t�� K�1 �t�� K

�2 �t� z�� ��t��dt + K�

1 �t�dW �t�+∫ �

0K�

2 �t� z�q�dt� dz��

J��T� = − exp�−�v�

Proof. By Lemma 4.1,

D��t� = D��0�+∫ t

0���u�dW�u�+

∫ t

0

∫ �

0���u� z�q�du� dz�

Applying Itô’s differentiation rule to C��t� gives:

dC��t� = C��t�

{[�

(p�t�+ r�t�V�t�+ ��t���X�t�− r�t��

)+ 1

2�2 2�t��2�t�

+K∑i=1

∫ �

0�exp�−�z�− 1��X�t�� ei��i�t�fi�z�dz

]dt

+ � �t���t�dW �t�+∫ �

0�exp�−�z�− 1�q�dt� dz�

}

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

10 Siu

Then applying Itô’s product rule to C��t�D��t� gives:

dJ��t� = C��t�dD��t�+D��t�dC��t�+ d�C��D���t�

= J��t�

{[�

(p�t�+ r�t�V�t�+ ��t���X�t�− r�t��

)+ 1

2�2 2�t��2�t�

+K∑i=1

∫ �

0�exp�−�z�− 1��X�t�� ei��i�t�fi�z�dz

]dt + � �t���t�dW �t�

+∫ �

0�exp�−�z�− 1�q�dt� dz�

}

+ C��t�

(���t�dW �t�+

∫ �

0���t� z�q�dt� dz�

)

+ �C��t���t� �t����t�dt + C��t�∫ �

0�exp�−�z�− 1����t� z���dt� dz�

={J��t�

[�

(p�t�+ r�t�V�t�+ ��t���X�t�− r�t��

)+ 1

2�2 2�t��2�t�

+K∑i=1

∫ �

0�exp�−�z�− 1��X�t�� ei��i�t�fi�z�dz

]+ C��t�

[� �t���t����t�

+K∑i=1

∫ �

0�exp�−�z�− 1����t� z��X�t�� ei��i�t�fi�z�dz

]}dt

+ ��J��t� �t���t�+ C��t����t��dW �t�

+∫ �

0�J��t��exp�−�z�− 1�+ C��t����t� z� exp�−�z�q�dt� dz�

From the definitions of K�1 �t� and K�

2 �t� z�,

���t� = K�1 �t�− �J��t� �t���t�

C��t��

and

���t� z� = K�2 �t� z�− J��t��exp�−�z�− 1�

C��t� exp�−�z�

Hence, the result follows. �

We now define the value process ���t� � t ∈ � of the optimal investmentproblem as:

��t� �= ess sup�∈�

J��t�

The following theorem gives a sufficient condition for an optimal investmentstrategy and a BSDE representation for the value process of the optimal investmentproblem. This result and its proof were given in Øksendal and Sulem [19,Theorem 2.1]. The key result used in the proof is the comparison theorem forBSDEs with jumps. We state the result here without giving the proof.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 11

Theorem 4.2. Suppose, for each �t� �� k1� k2�·�� �� ∈ � ×�×�×�×�, thereexists ��t� �= ��t� �� k1� k2�·�� �� such that

g�t� �� k1� k2�·�� ��t�� = ess sup��t�∈�

g�t� �� k1� k2�·�� ��t��

Assume, further, that ���t� � t ∈ � ∈ � and that for each ���t� � t ∈ � ∈ �, there existsa unique solution ��J��t�� K�

1 �t�� K�2 �t� ·�� � t ∈ � of the following BSDE with jumps:1

dJ��t� = −g�t� J��t�� K�1 �t�� K

�2 �t� ·�� ��t��dt + K�

1 �t�dW �t�+∫ �

0K�

2 �t� z�q�dt� dz��

J��T� = − exp�−�v�

Then ��t� = J ��t�, for each t ∈ � . Furthermore, the control

�†�t� = ��t� J ��t�� K�1 �t�� K

�2 �t� ·�� ���

is an optimal control for the optimal investment problem.

Then the optimal investment strategy �†�t� is given in the following corollary.

Corollary 4.1. The optimal investment strategy ��†�t� � t ∈ � is given by:

�†�t� = �X�t�− r�t�

� 2�t�+ K1�t�

J�t�

Here ��J�t�� K1�t�� K2�t� z� � �t� z� ∈ � × �0��� is the unique solution of the followingBSDE with jumps:

dJ�t� = −g�t� J�t�� K1�t�� K2�t� ·�� �†�t��dt + K1�t�dW �t�+∫ �

0K2�t� z�q�dt� dz��

J�T� = − exp�−�v��

and

�X�t� = ���t��L�t�q�t���L�t�q�t�� 1� �

where �q�t� � t ∈ � satisfies the linear ordinary differential equation (7).

1 �= �1� 1� 1�′ ∈ �K

Proof. By Theorem 4.2 and the first-order condition for maximizing g with respectto ��t�, we have

�†�t� = �X�t�− r�t�

� 2�t�+ K1�t�

J�t�

1For the existence and uniqueness theorem for BSDEs with jumps, please refer to Tangand Li [20] and Situ [22, Chapter 8].

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

12 Siu

By a version of the Bayes’ rule and the definition of q�t�,

X�t� =E���t�X�t� ���t��E���t� ���t�� = q�t�

�q�t�� 1� = L�t�q�t��L�t�q�t�� 1�

Consequently,

�X�t� = ���t��L�t�q�t���L�t�q�t�� 1�

�

The processes “�K1�t� � t ∈ � ” and “�K2�t� z� � t ∈ � � z ∈ �0���” may beidentified by the Malliavin derivatives with respect to the Brownian motion�W �t� � t ∈ � and the random measure �q�t� z� � t ∈ � , respectively. Since thedistribution of random jump sizes depends on time, the process �D��t� � t ∈ �

is a time-inhomogeneous Lèvy process. Consequently, we may need to extendthe Malliavin calculus for time-homogeneous Lévy processes in Di Nunno et al.[23] to one for the time-inhomogeneous case when identifying the processes“�K1�t� � t ∈ � ” and “�K2�t� z� � t ∈ � � z ∈ �0���” by Malliavin derivatives. For theMarkovian case, one may consider the use of a stochastic flows approach to identifythe processes “�K1�t� � t ∈ � ” and “�K2�t� z� � t ∈ � � z ∈ �0���.” The stochasticflows approach may also provide a way to solve backward stochastic differentialequations with jumps in the Markovian case. This may represent a potential topicfor future research.

When the market coefficients �X�t�, r�t� and �t� are deterministic functionsof time t, K1�t� = K2�t� z� = 0, for all �t� z� ∈ � × �0���. In this case, the optimalinvestment strategy is given by:

�†�t� = �X�t�− r�t�

� 2�t�

This is a generalization of the Merton ratio.

4.2. Power Utility

In this case, we consider the situation where for each t ∈ � , Z�t� is the aggregateamount of claims up to time t per unit of the surplus of the insurance companyand p�t� is the premium rate per unit of time and per unit of the surplus of theinsurance company. So we assume that the jump size of Z�t� takes a value in (0,1). For each t ∈ � , let ��t� be the proportion of the surplus invested in the shareindex S at time t. We suppose that ���t� � t ∈ � is �-progressively measurable andsatisfies the standard boundedness and integrability conditions similar to those of���t� � t ∈ � described in Definition 2.1. Write �† for the space of admissibleinvestment processes.

With a slight abuse of the notation, we again use �V�t� � t ∈ � here to denotethe surplus process of the insurance company with investment. Consequently,under P, the surplus process �V�t� � t ∈ � associated with the investment process

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 13

���t� � t ∈ � ∈ �† in the “filtered” market is governed by the following stochasticdifferential equation:

dV�t� =(p�t�+ r�t�+ ��t���X�t�− r�t��−

K∑i=1

∫ �

0�X�t�� ei�z�i�t�fi�z�dz

)V�t�dt

+ �t���t�V�t�dW �t�−∫ 1

0V�t�zq�dt� dz�

By Itô’s differentiation rule,

V�T� = v exp[ ∫ T

0 �t���t�dW �t�

+∫ T

0

(p�t�+ r�t�+ ��t���X�t�− r�t��− 1

2 2�t��2�t�

)dt

+K∑i=1

∫ T

0�X�t�� ei�

( ∫ 1

0ln�1− z��i�t�fi�z�dz

)dt

+∫ T

0

∫ 1

0ln�1− z�q�dt� dz�

]

For any 0 ≤ t ≤ s ≤ T and any ���t� � t ∈ � ∈ �†, write

R�t �s� �=

∫ s

t �u���u�dW�u�

+∫ s

t

(p�u�+ r�u�+ ��u���X�u�− r�u��− 1

2 2�u��2�u�

)du

+K∑i=1

∫ s

t�X�u�� ei�

( ∫ 1

0ln�1− z��i�u�fi�z�dz

)du

+∫ s

t

∫ 1

0ln�1− z�q�du� dz�

We consider the following power utility function:

U�v� = x�

�� v ≥ 0� � ∈ �−�� 1�\�0

Then, in this case, the conditional expected utility process �J�1 �t� � t ∈ � associated

with the investment process ���t� � t ∈ � ∈ �† is given by:

J�1 �t� = E

[v� exp��R�

t �T��

����t�

]

Again we note that for each ���t� � t ∈ � ∈ �† and each t ∈ � ,

R�t �T� = R�

0�T�− R�0�t�

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

14 Siu

For each t ∈ � and ���t� � t ∈ � ∈ �†, we define

C�1�t� �= exp�−�R�

0�t���

and

D�1�t� �=

v�

�E�exp��R�

0�T�� ���t��

By definition, �C�1�t� � t ∈ � is �-adapted and �D�

1�t� � t ∈ � is a ��� P�-martingale.Then by Lemma 4.1, for each ���t� � t ∈ � ∈ �†, there exist �-predictable

processes ����t� � t ∈ � ∈ L2�m⊗ P� and ����t� z� � t ∈ � � z ∈ �0� 1� ∈ L2�m⊗ P ⊗f� such that

D�1�t� = D�

1�0�+∫ t

0��

1�u�dW�u�+∫ t

0

∫ 1

0��

1�u� z�q�du� dz�

Again, we have the following multiplicative decomposition:

J�1 �t� = C�

1�t�D�1�t�� t ∈ �

Then as in Theorem 4.1 we have the following theorem:

Theorem 4.3. For each t ∈ � , ���t� � t ∈ � ∈ �† and z ∈ �0� 1�, let

L�1�t� �= C�

1�t���1�t�− �J�

1 �t� �t���t��

L�2�t� z� �= J�

1 �t���1− z�−� − � ln�1− z�− 1�

− C�1�t���1− z�−� − � ln�1− z����

1�t� z��

and

h1�t� J�1 �t�� L

�1�t�� L

�2�t� z�� ��t��

�= J�1 �t�

[�

(p�t�+ r�t�+ ��t���X�t�− r�t��+ 1

2�� − 1� 2�t��2�t�

)

−k∑

i=1

�X�t�� ei�∫ 1

0

(2�1− z�−� − 2� ln�1− z�− 1− L�

2�t� z�

�1− z�−� − � ln�1− z�

)

× ��1− z�−� − � ln�1− z�− 1��ifi�z�dz]+ � �t���t�L�

1�t�

Then the process �J�1 �t� � t ∈ � satisfies the following BSDE with jumps:

dJ�1 �t� = −h1�t� J

�1 �t�� L

�1�t�� L

�2�t� z�� ��t��dt + L�

1�t�dW �t�+∫ 1

0L�2�t� z�q�dt� dz��

J�1 �T� =

v�

�

The proof of Theorem 4.3 resembles to that of Theorem 4.1, so we omit it.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 15

Similarly to Corollary 4.1, an optimal investment strategy and its value functionin the power utility case are given in the following corollary. Again, we state theresult without giving the proof.

Corollary 4.2. The optimal investment strategy ��†1�t� � t ∈ � is given by:

�†1�t� =�X�t�− r�t�

�1− �� 2�t�+ L1�t�

�t��1− ��J1�t�

Here ��J1�t�� L1�t�� L2�t� z� � �t� z� ∈ � × �0� 1� is the unique solution of the followingBSDE with jumps:

dJ1�t� = −h1�t� J1�t�� L1�t�� L2�t� z�� �†1�t��dt + L1�t�dW �t�+

∫ 1

0L2�t� z�q�dt� dz��

J1�T� =v�

��

and

�X�t� = ���t��L�t�q�t���L�t�q�t�� 1� �

where �q�t� � t ∈ � satisfies the linear ordinary differential equation (7).

Again, when the market coefficients are deterministic, L1�t� = L2�t� z� = 0, forall �t� z� ∈ � × �0� 1�. Consequently, the optimal investment strategy is given by:

�†1�t� =�X�t�− r�t�

�1− �� 2�t�

This is, again, a generalization of the Merton ratio.

4.3. Logarithmic Utility

We now consider the case of a logarithmic utility function. That is,

U�x� = ln x� x ∈ �0���

Again we consider the surplus process and the investment process defined inSection 4.2. Then the conditional expected utility process �J�

2 �t� � t ∈ � associatedwith the investment process ���t� � t ∈ � ∈ �† is given by:

J�2 �t� = ln v+ E�R�

t �T� ���t��Note that

R�t �T� = R�

0�T�− R�0�t�

Write, for each t ∈ � ,

C�2�t� �= R�

0�t��

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

16 Siu

and

D�2�t� �= E�lnV��T� ���t�� = ln v+ E�R�

0�T� ���t��Then for each t ∈ � and ���t� � t ∈ � ∈ �†,

J�2 �t� = D�

2�t�− C�2�t�

By definition, �D�2�t� � t ∈ � is a ��� P�-martingale. Consequently, by Lemma 4.1,

for each ���t� � t ∈ � ∈ �†, there exist �-predictable processes ���2�t� � t ∈ � ∈

L2�m⊗ P� and ���2�t� z� � t ∈ � � z ∈ �0� 1� ∈ L2�m⊗ P ⊗ f� such that

D�2�t� = D�

2�0�+∫ t

0��

2�u�dW�u�+∫ t

0

∫ 1

0��

2�u� z�q�du� dz�

Similarly to Theorem 4.3, the following theorem is obtained. We only state the resultwithout giving the proof.

Theorem 4.4. For each t ∈ � , ���t� � t ∈ � ∈ �† and z ∈ �0� 1�, let

M�1 �t� �= ��

2�t�− �t���t��

M�2 �t� z� �= ��

2�t� z�− ln�1− z��

and

h2�t� J�2 �t��M

�1 �t��M

�2 �t� z�� ��t�� �= p�t�+ r�t�+ ��t���X�t�− r�t��− 1

2 2�t��2�t�

+K∑i=1

�X�t�� ei�∫ 1

0ln�1− z��ifi�z�dz

Then the process �J�2 �t� � t ∈ � satisfies the following BSDE with jumps:

dJ�2 �t� = −h2�t� J

�2 �t��M

�1 �t��M

�2 �t� z�� ��t��dt +M�

1 �t�dW �t�+∫ 1

0M�

2 �t� z�q�dt� dz��

J�2 �T� = ln v

In this case an optimal investment process and its value function are given inthe following corollary.

Corollary 4.3. The optimal investment process ��†2�t� � t ∈ � is given by:

�†2�t� =�X�t�− r�t�

2�t�

Here ��J2�t��M1�t��M2�t� z� � �t� z� ∈ � × �0� 1� is the unique solution of the followingBSDE with jumps:

dJ2�t� = −h2�t� J2�t��M1�t��M2�t� z�� �†2�t��dt +M1�t�dW �t�+

∫ 1

0M2�t� z�q�dt� dz��

J2�T� = ln v�

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

BSDE and Optimal Investment of an Insurer 17

and

�X�t� = ���t��L�t�q�t���L�t�q�t�� 1� �

where �q�t� � t ∈ � satisfies the linear ordinary differential Equation (7).

In this case, an analytical solution to the optimal investment process is obtained.

5. Summary

An optimal investment problem of an insurer was considered in a general non-Markovian, regime-switching, model for financial prices and insurance liabilities.We discussed the problem in two steps. In Step I, as usual, we adopted filteringtheory to turn the optimal investment problem with partial observations intoone with complete observations. Then, in Step II, we worked on the “filtered”market with complete observations and adopted the backward stochastic differentialequation approach to discuss the optimal investment problem. Three typical utilityfunctions, namely, the exponential utility, the power utility and the logarithmicutility, were considered. An analytical solution to the optimal investment processwas obtained for the logarithmic utility.

References

1. Markowitz, H.M. 1952. Portfolio selection. Journal of Finance 7(1):77–91.2. Samuelson, P. 1969. Lifetime portfolio selection by dynamic programming. The Review

of Economics and Statistics 51(3):239–246.3. Merton, R.C. 1969. Lifetime portfolio selection under uncertainty: the continuous-time

model. Review of Economics and Statistics 51:247–257.4. Merton, R.C. 1971. Optimum consumption and portfolio rules in a continuous-time

model. Journal of Economic Theory 3:373–413.5. Zhou, X.Y., and Yin, G. 2003. Markowitz’s mean-variance portfolio selection with

regime switching: a continuous time model. SIAM Journal on Control and Optimization42(4):1466–1482.

6. Sass, J., and Haussmann, U.G. 2004. Optimizing the terminal wealth under partialinformation: the drift process as a continuous time Markov chain. Finance andStochastics 8:553–577.

7. Yin, G., and Zhou, X.Y. 2004. Markowitz’s mean-variance portfolio selection withregime switching: from discrete-time models to their continuous-time limits. IEEETransactions on Automatic Control 49:349–360.

8. Baeuerle, N., and Rieder, U. 2004. Portfolio optimization with Markov-modulatedstock prices and interest rates. IEEE Transactions on Automatic Control 49:442–447.

9. Baeuerle, N., and Rieder, U. 2005. Portfolio optimization with unobservable Markov-modulated drift process. Journal of Applied Probability 42:362–278.

10. Baeuerle, N., and Rieder, U. 2007. Portfolio optimization with jumps and unobservableintensity process. Mathematical Finance 17(2):205–224.

11. Elliott, R.J., Siu, T.K., and Badescu, A. 2010. On mean-variance portfolio selectionunder a hidden Markovian regime-switching model. Economic Modeling 27(3):678–686.

12. Elliott, R.J., and Siu, T.K. 2011. A stochastic differential game for optimal investmentof an insurer with regime switching. Quantitative Finance 11(3):365–380.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014

18 Siu

13. Elliott, R.J., and Siu, T.K. 2011. A hidden Markov model for optimal investment ofan insurer with model uncertainty. International Journal of Robust and Nonlinear Control22:778–807.

14. Elliott, R.J., Aggoun, L., and Moore, J.B. 2008. Hidden Markov Models: Estimation andControl. 2nd ed. Springer-Verlag, Berlin.

15. Kallianpur, G. 1980. Stochastic Filtering Theory. Springer-Verlag, Berlin.16. Elliott, R.J. 1982. Stochastic Calculus and Applications. Springer-Verlag, Berlin.17. Elliott, R.J. 1990. Filtering and control for point process observations. In Progress in

Automation and Information Systems. Baras, J., and Mirelli, V. (eds.), Springer-Verlag,Berlin, pp. 1–27.

18. Lim, T., and Quenez, M.-C. 2010. Exponential Utility Maximization and Indifference Pricein an Incomplete Market with Defaults. Preprint. Laboratoire de Probabilités et ModèlesAléatoires, Universités Paris 6–Paris 7.

19. Øksendal, B., and Sulem, A. 2011. Portfolio optimization under model uncertainty andBSDE games. Quantitative Finance 11:1665–1674.

20. Tang, S.J., and Li, X.J. 1994. Necessary conditions for optimal control of stochasticsystems with random jumps. SIAM Journal on Control and Optimization 32:1447–1475.

21. Elliott, R.J. 1979. Double martingales. Probability Theory and Related Fields 34(1):17–28.

22. Situ, R. 2005. Theory of Stochastic Differential Equations With Jumps and Applications:Mathematical and Analytical Techniques With Applications to Engineering. Springer-Verlag, Berlin.

23. Di Nunno, G., and Øksendal, B., Proske, F. 2009. Malliavan Calculus for Lèvy ProcessesWith Applications to Finance. Springer-Verlag, Berlin.

Dow

nloa

ded

by [

Mem

oria

l Uni

vers

ity o

f N

ewfo

undl

and]

at 1

6:54

03

Aug

ust 2

014