optimal investment-consumption-insurance with random parameters

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This article was downloaded by: [Eindhoven Technical University] On: 10 July 2014, At: 11:14 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Scandinavian Actuarial Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/sact20 Optimal investment-consumption- insurance with random parameters Yang Shen ab & Jiaqin Wei b a School of Risk and Actuarial Studies and CEPAR, Australian School of Business, University of New South Wales, Sydney, Australia. b Faculty of Business and Economics, Department of Applied Finance and Actuarial Studies, Macquarie University, Sydney, Australia. Published online: 27 Mar 2014. To cite this article: Yang Shen & Jiaqin Wei (2014): Optimal investment-consumption-insurance with random parameters, Scandinavian Actuarial Journal, DOI: 10.1080/03461238.2014.900518 To link to this article: http://dx.doi.org/10.1080/03461238.2014.900518 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 to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and 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 Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, 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

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Page 1: Optimal investment-consumption-insurance with random parameters

This article was downloaded by: [Eindhoven Technical University]On: 10 July 2014, At: 11:14Publisher: 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-consumption-insurance with random parametersYang Shenab & Jiaqin Weiba School of Risk and Actuarial Studies and CEPAR, AustralianSchool of Business, University of New South Wales, Sydney,Australia.b Faculty of Business and Economics, Department of AppliedFinance and Actuarial Studies, Macquarie University, Sydney,Australia.Published online: 27 Mar 2014.

To cite this article: Yang Shen & Jiaqin Wei (2014): Optimal investment-consumption-insurancewith random parameters, Scandinavian Actuarial Journal, DOI: 10.1080/03461238.2014.900518

To link to this article: http://dx.doi.org/10.1080/03461238.2014.900518

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,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

Page 2: Optimal investment-consumption-insurance with random parameters

Scandinavian Actuarial Journal, 2014http://dx.doi.org/10.1080/03461238.2014.900518

Optimal investment-consumption-insurance with randomparameters

YANG SHEN†‡∗ and JIAQIN WEI‡

†School of Risk and Actuarial Studies and CEPAR, Australian School of Business, University of

New South Wales, Sydney, Australia

‡Faculty of Business and Economics, Department of Applied Finance and Actuarial Studies, Macquarie

University, Sydney, Australia

(Accepted February 2014)

This paper discusses an optimal investment, consumption, and life insurance purchase problem for a wageearner in a complete market with Brownian information. Specifically, we assume that the parameters governingthe market model and the wage earner, including the interest rate, appreciation rate, volatility, force of mortality,premium-insurance ratio, income and discount rate, are all random processes adapted to the Brownian motionfiltration. Our modeling framework is very general, which allows these random parameters to be unbounded,non-Markovian functionals of the underlying Brownian motion. Suppose that the wage earner’s preference isdescribed by a power utility. The wage earner’s problem is then to choose an optimal investment-consumption-insurance strategy so as to maximize the expected, discounted utilities from intertemporal consumption,legacy and terminal wealth over an uncertain lifetime horizon. We use a novel approach, which combinesthe Hamilton–Jacobi–Bellman equation and backward stochastic differential equation (BSDE) to solve thisproblem. In general, we give explicit expressions for the optimal investment-consumption-insurance strategyand the value function in terms of the solutions to two BSDEs. To illustrate our results, we provide closed-form solutions to the problem with stochastic income, stochastic mortality, and stochastic appreciation rate,respectively.

Keywords: investment-consumption-insurance; random parameters; the HJB equation; backward stochasticdifferential equation; stochastic Lipschitz condition

1. Introduction

The portfolio optimization problem is of critical importance in both theory and practice. Forone thing, it is a stochastic optimal control problem, which can be solved explicitly via differentapproaches, such as the dynamic programming principle, the maximum principle, and the convexduality martingale method. For another, the solution to the problem is a major concern for bothindividual and institutional investors, who need to allocate the wealth among various asset classesover a certain/uncertain time horizon. The history of the portfolio optimization problem can betraced back to the pioneering works of Markowitz (1952), where a mathematically elegant

*Corresponding author. Emails: [email protected]; [email protected]

© 2014 Taylor & Francis

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2 Y. Shen & J. Wei

mean-variance analysis was introduced to discuss the problem. Incorporating the preferencedistinction and the consumption opportunity, Samuelson (1969) and Merton (1969, 1971) studiedoptimal investment-consumption problems in a multi-period setting and a continuous-timesetting, respectively. It is worth mentioning that Merton (1969, 1971) were the first to introducethe stochastic optimal control theory to the portfolio optimization problem.

Besides the possibilities of investment and consumption, inevitably, the investor shouldconsider the uncertainty in investment horizon, which may affect the well being of his family dueto his premature death and is a big issue if the planning is made over a long-time span. Interestedreaders may refer to Blanchet-Scalliet et al. (2008) for detailed discussion about an investment-only problem when time-horizon is uncertain. In practice, life insurance is a valuable tool toaddress the issue of uncertain lifetime. Indeed, there has been a considerable interest in researchon life insurance purchase. Richard (1975) extended Merton’s classical investment-consumptionproblem to the case incorporating life insurance purchase and a bounded random lifetime.Pliska & Ye (2007) and Ye (2008) studied the optimal investment-consumption-insurance prob-lem for an wage earner with an unbounded random lifetime. Huang et al. (2008) consideredthe problem with stochastic income which was correlated with the risky asset. Duarte et al.(2011, 2012) generalized the complete market with a single risky asset to the incomplete marketwith multiple risky assets for the problem. Pirvu & Zhang (2012) investigated the problemwith a stochastic mean-reverting appreciation rate for the risky asset. Other related works onthe investment-consumption-insurance problem include Huang & Milevsky (2008), Kraft &Steffensen (2008), Kwak et al. (2009), Bruhn & Steffensen (2011), Kronborg & Steffensen(2013) and references therein.

To discuss the investment-consumption-insurance problem, the first task for us is to choosea ‘right’ model to describe the financial market and the insurance market. Which model is the‘right’one is an extremely subjective question. In the last several decades, many models, such asthe Black-Scholes model, the stochastic interest rate model, and the stochastic volatility model,have been proved empirically ‘right’ and adopted to study the portfolio optimization problem.However, with different data sources and statistical methods, the empirical study may lead todifferent ‘right’ models. As said by George Box, a British statistician, ‘essentially, all modelsare wrong, but some are useful.’ There must be a trade-off between rightness and usefulnessof the model. Our principles for choosing the appropriate model are that (1) it must result inmathematically appealing and explicit solutions to the problem; (2) it must be flexible enoughto incorporate recognized features of both the financial and insurance markets; (3) it must besufficiently general to incorporate most existing models. As is known, it is very difficult, if notpossible, to derive closed-form solutions to the utility maximization problem with jumps. So wedo not consider jumps in our modeling framework, although they have been evidenced by manyempirical studies. We adopt a random-coefficient model in this paper. Specially, we use generalrandom processes to describe stochastic features of quantities related to the problem, includingthe interest rate, appreciation rate, volatility, force of mortality, premium-insurance ratio, incomeand discount rate, which not necessarily are bounded and have Markovian structures. This allowsus to make most existing models as special cases of our model.

In this paper, we are concerned with an investment-consumption-insurance problem withrandom parameters, where all randomness is described by a Brownian motion filtration. Suppose

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that there exists a financial market, where one risk-free bond and multiple risky shares arecontinuously traded, and an insurance market, where the life insurance with an infinitesimallysmall term is provided. More specifically, we assume that the dynamics of the risky shares aregiven by generalized geometric Brownian motion models with random interest rate, appreciationrate, and volatility. We consider the investment-consumption-insurance problem of a wageearner with random income, discount rate, and force of mortality whose preference followsa power utility function. The objective of the wage earner is to choose an optimal investment-consumption-insurance strategy that maximizes the expected, discounted utilities derived fromintertemporal consumption, legacy and terminal wealth over an uncertain lifetime horizon. Basedon the dynamic programming principle, we use a combination of the Hamilton–Jacobi–Bellman(HJB) equation and the backward stochastic differential equation (BSDE) approach to solve theproblem. We derive general solutions of the optimal investment-consumption-insurance strategyand the value function to the problem in terms of the solutions to two BSDEs with stochasticLipschitz condition. We also provide closed-form solutions to the problem in special examplesof stochastic income, stochastic mortality and stochastic appreciation rate, respectively.

Our paper contributes to the existing literature in at least three aspects. Firstly, we adopt amodel with random parameters to investigate a classical optimal control problem in insurance.Although the optimal control problems with deterministic coefficients in insurance have beenwell studied in the actuarial community, the research on those with random parameters is stillat an early stage. If the market is complete, our paper extends Duarte et al. (2011, 2012) fromMarkovian deterministic opportunity sets to non-Markovian stochastic opportunity sets. Indeed,the assumption of Markovian deterministic opportunity sets in the lifecycle model is unrealisticsince the planning horizon is as long as several decades. The assumption of non-Markovianstochastic opportunity sets provides flexibility to allow all model parameters to evolve over timein a random and path-dependent fashion. Our modeling framework merits extension to otheroptimal control problems in insurance, such as the optimal investment-reinsurance problem, theoptimal asset-liability management problem and the optimal dividend problem. Secondly, we usea novel approach, combining the HJB equation and BSDE, to solve the investment-consumption-insurance problem. This naturally allows us to introduce the theory of BSDE to actuarial science,which has not received deserved attention in the actuarial community. Thirdly, we do not requirethat any random parameters are bounded. This advances over the existing literature. Portfoliooptimization problems with bounded random parameters have attracted a good deal of interestin financial mathematics. There have been many works on portfolio optimization problems withbounded stochastic opportunity sets without taking into account the uncertain lifetime. TheBSDE approach is found to be useful in these works. Please see Lim & Zhou (2002) and Lim(2004) for mean-variance problems, Cheridito & Hu (2011) and Rieder & Wopperer (2012)for investment-consumption problems. If random parameters are bounded, however, most well-known stochastic models are excluded from the modeling framework since stochastic processesdefined by these models indeed are unbounded. This is really the case for Markovian stochasticopportunity sets (e.g. the Vasciek model, the CIR model, the Dothan model, the Heston model,and others).

The rest of this paper is structured as follows. Section 2 introduces the model dynamics tobe used throughout this paper and formulates the optimal investment-consumption-insurance

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problem. In Section 3, we provide a verification theorem for a combination of the HJB equationand BSDE related to the problem. Section 4 derives explicit expressions for the investment-consumption-insurance strategy and the value function in terms of two BSDEs. In Section 5,we provide closed-form solutions to the problem in three special examples. Section 6 concludesthe paper. We relegate a brief introduction of BSDEs with stochastic Lipschiz condition inAppendix 1.

2. The model dynamics

In this section, we first introduce the model dynamics of our paper. Then, we formulate theoptimal investment, consumption, and life-insurance purchase problem. Throughout this paper,we denote by C� the transpose of a vector or a matrix C , by C−1 the inverse of a nonsingular(square) matrix C , by ||C || the Euclidean norm of a vector C .

We consider a complete probability space (�,F,P), indexed with a finite time horizonT := [0, T ]. Let {W (t)|t ∈ T } = {(W1(t),W2(t), . . . ,Wn(t))�|t ∈ T } be an n-dimensional,(F,P)-standard Brownian motion, where F := {F(t)|t ∈ T } is the right continuous, P-completefiltration generated by W . Moreover, let P be the predictable σ -field on�×T , and B(E) be theBorel σ -algebra of any topological space E . We assume that a wage earner is alive at time t = 0,whose lifetime is a non-negative random variable τ defined on (�,F,P). Let {λ(t)|t ∈ T } bean �+-valued, F-adapted process, where λ(t) represents the force of mortality or the hazardfunction of the wage earner at time t . Then given that F(t), we denote by

F(t) := P(τ > t |F(t)) = exp

(−∫ t

0λ(s)ds

), (1)

the conditional survival probability of the wage earner alive at time t , and by

f (t) := λ(t) exp

(−∫ t

0λ(s)ds

), (2)

the conditional probability density for the death of the wage earner at time t .In what follows, we define the financial market and the insurance market in which the wage

earner has to make his decision regarding investment, consumption, and life insurance purchase.We assume that the financial market consists of n + 1 primitive assets, including one risk-freebond and n risky assets, which are available investment assets for the wage earner. The evolutionof the price process of the risk-free bond {S0(t)|t ∈ T } follows

d S0(t) = r(t)S0(t)dt, t ∈ T , S0(0) = 1, (3)

where r(t) represents the risk-free interest rate at time t . Furthermore, we assume that{r(t)|t ∈ T } is an �+-valued, F-adapted process.

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The price processes of the other n risky assets {Si (t)|t ∈ T }, i = 1, 2, . . . , n, are governedby the following stochastic differential equations (SDEs)

d Si (t) = Si (t)

(μi (t)dt +

n∑j=1

σi j (t)dW j (t)

), t ∈ T , Si (0) = si > 0, (4)

where μi (t) and σi (t) := (σi1(t), σi2(t), . . . , σin(t)) represent the appreciation rate and thevolatility of the i th share at time t , respectively. Furthermore, we assume that {μi (t)|t ∈ T } are�+-valued, F-adapted processes, for each i = 1, 2, . . . , n, and {σi j (t)|t ∈ T } are �+-valued,F-adapted processes, for each i = 1, 2, . . . , n and j = 1, 2, . . . , n.

To simplify our notation, we denote by

μ(t) := (μ1(t), μ2(t), . . . , μn(t))� ∈ �n,

and

σ(t) := [σi j (t)]i=1,2,...,n; j=1,2,...,n ∈ �n×n,

the appreciation rate vector and the volatility matrix of the risky shares at time t , respectively.Throughout this paper, we assume that the volatility matrix σ(t) satisfies the following non-degeneracy condition

�(t) := σ(t)σ (t)� ≥ δ In×n, ∀t ∈ T ,

for some δ > 0. Here In×n is an (n × n)-identity matrix. This condition implies that thevolatility matrix process {σ(t)|t ∈ T } is non-singular. Hence, both the variance matrix process{σ(t)σ (t)�|t ∈ T } and the inversed variance matrix process {(σ (t)σ (t)�)−1|t ∈ T } are positivedefinite. Note that all randomness in our modeling framework comes from the n-dimensionalBrownian motion W . Since there exist exactly n risky assets driven by W with non-degeneratevolatility matrix process, the market in which the optimal investment-consumption-insuranceproblem lives is complete. Essentially, it means that all underlying risks from income, mortality,appreciation rate, volatility, and others are hedgeable. One may consider these risks as someeconomic factors, which are traded directly in the market or correlated with the dynamics of thetraded assets in the market. We shall discuss different cases with hedgeable risks from stochasticincome, mortality, and appreciation rate, respectively, in Section 5.

We assume that there exists an insurance market, where the term life insurance is continuouslytraded and provides coverage for an infinitesimally small period of time. Suppose that thewage earner is willing to pay the premium p(t) at time t for the life insurance contract sothat his beneficiary will be compensated for his death. Let {η(t)|t ∈ T } be an �+-valued,F-adapted process, where η(t) represent the premium-insurance ratio at time t . Unlike previousactuarial literature, we do not restrict the premium-insurance ratio η to a deterministic functionfor generality of the modeling framework. The situation with a stochastic η may arise due tostochastic mortality or stochastic safety loading. The stochastic η implies that the wage earneris facing up to random shocks from insurance premium. Please also refer to Kraft & Steffensen

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(2008) and Bruhn & Steffensen (2011) for the stochastic pricing intensity in the Markovian case.So if the wage earner dies at time t ∈ T , he will leave behind the legacy

l(t) := X (t)+ p(t)

η(t),

to his beneficiary, where {X (t)|t ∈ T } is the wealth process of the wage earner to be definedbelow and p(t)

η(t) is the insurance benefit paid by the insurance company to the beneficiary. In ourmodeling framework, we do not require the premium p(t) is positive, for each t ∈ T . The casethat the premium p(t) is negative can be interpreted as that the wage earner purchases a specialterm pension annuity, where he receives the premium p(t) from the insurance company at timet . However, the amount p(t)

η(t) should be paid by the wage earner’s beneficiary to the insurancecompany if he dies at time t . This situation is related to the reverse mortgage. Interested readersmay refer to Pirvu & Zhang (2012) for more discussions.

Denote by πi (t), i = 1, 2, . . . , n, the amount of the wage earner’s wealth invested in the i thshare at time t , and by c(t) the amount of the wage earner’s wealth consumed at time t . We call{π(t)|t ∈ T } = {(π1(t), π2(t), . . . , πn(t))�|t ∈ T }, {c(t)|t ∈ T } and {p(t)|t ∈ T } a portfolioprocess, a consumption process, and a premium process of the wage earner, respectively. LetX (t) := Xπ,c,p(t) denote the total wealth of the wage earner at time t associated with the triple ofthe investment-consumption-insurance strategy (π, c, p). Suppose that the wage earner receivesan income flow i(t) continuously during the period [0, τ ∧ T ], where {i(t)|t ∈ T } is assumedto be an �+-valued, F-adapted process. As widely accepted in the literature, we assume that (1)the shares are divisible and can be traded continuously over time; (2) there are no transactioncosts, taxes, and short-selling constraints in trading. Then given that the wage earner is endowedwith the initial wealth x0, the wealth process {X (t)|t ∈ T } is governed by the following SDE

{d X (t) = [

r(t)X (t)+ π(t)� B(t)+ i(t)− c(t)− p(t)]dt + π(t)�σ(t)dW (t), t ∈ [0, τ ∧ T ],

X (0) = x0 > 0,(5)

where

B(t) := (μ1(t)− r(t), μ2(t)− r(t), . . . , μn(t)− r(t))�.

We now consider the optimal investment-consumption-insurance problem in which the wageearner maximizes the expected, discounted utilities derived from the intertemporal consumptionduring [0, τ ∧T ], from the legacy if he dies before T , and from the terminal wealth if he survivesuntil time T . We assume that the discount rate process {ρ(t)|t ∈ T } is an �+-valued, F-adaptedprocess. Given the utility function U for intertemporal consumption, legacy and terminal wealth,the wage earner’s problem is then to choose the investment-consumption-insurance strategy soas to maximize the following performance functional:

E

[∫ τ∧T

0e− ∫ s

0 ρ(u)duU (c(s))ds + e− ∫ τ0 ρ(u)duU (l(τ ))1{τ≤T } + e− ∫ T0 ρ(u)duU (X (T ))1{τ>T }

].

(6)

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Scandinavian Actuarial Journal 7

From the definition of the conditional survival probability (1) and the conditional probabilitydensity (2), it is clear that (6) is equivalent to the following performance functional

J (0, x0;π, c, p)

:= E

[ ∫ T

0e− ∫ s

0 ρ(u)du[F(s)U (c(s))+ f (s)U (l(s))]ds + e− ∫ T

0 ρ(u)du F(T )U (X (T ))

]= E

[ ∫ T

0e− ∫ s

0 (ρ(u)+λ(u))du[U (c(s))+ λ(s)U (l(s))]ds + e− ∫ T

0 (ρ(u)+λ(u))duU (X (T ))

]. (7)

In this paper, we only consider the utility function is of the constant relative risk aversion(CRRA) type. Specifically, we assume that the wage earner’s preference towards risk is givenby the power utility

U (x) =⎧⎨⎩

γ, if x ≥ 0,

−∞, if x < 0,(8)

for γ ∈ (−∞, 1) \ {0}.Although we do not require that the random parameters in our modeling framework are

bounded, the following assumptions are needed to make our mathematics rigorous:

(A1) the appreciation rate of at least one share is different from the interest rate, i.e. B(t) �=0n×1, for a.e. t ∈ T , P-a.s.;

(A2) the interest rate, force of mortality, premium-insurance ratio, discount rate, and incomeare bounded away from zero, i.e. ∃ε > 0, | (t)| ≥ ε, for a.e. t ∈ T , P-a.s., where := r, λ, η, ρ, i ;

(A3) for a sufficient large θ , the random parameters satisfy the exponential integrabilityconditions

E

[exp

∫ T

0| (t)|dt

)]< ∞,

where := r, λ, η, ρ, i and B�(σσ�)−1 B.

In the later part of this paper, it can be seen that the above assumptions are essentially importantto guarantee the uniqueness and existence of solutions to the BSDEs related to the problem.

Definition 2.1 A triple of the investment-consumption-insurance strategy (π, c, p) is said tobe admissible if the following conditions hold

(1) the investment-consumption-insurance strategy (π, c, p) is an �n × �+ × �-valued,F-adapted process such that∫ T

0||π(t)||2dt < ∞,

∫ T

0c(t)dt < ∞,

∫ T

0|p(t)|dt < ∞, P-a.s.,

(2) the SDE (5) governing the wealth process X admits a unique strong solution associatedwith (π, c, p) such that

X (t)+ Y1(t) ≥ 0, P-a.s.,

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8 Y. Shen & J. Wei

where Y1(t) is defined by (31), which can be interpreted as the actuarial value of futureincome at time t ;

(3)

E

[ ∫ T

0e− ∫ s

0 (ρ(u)+λ(u))du[U−(c(s))+ λ(s)U−(l(s))]ds + e− ∫ T

0 (ρ(u)+λ(u))duU−(X (T ))]< ∞,

where U− denotes the negative part of U , i.e. U− := max{−U, 0}.

Write A for the space of all admissible triples (π, c, p).

To pave the way for using the dynamic programming principle, we consider the dynamicversion of performance functional

J (t, x;π, c, p)

= Et,x

[ ∫ T

te− ∫ s

t (ρ(u)+λ(u))du[U (c(s))+ λ(s)U (l(s))]ds + e− ∫ T

t (ρ(u)+λ(u))duU (X (T ))

].

(9)

Here Et,x [·] denotes the conditional expectation E[·|X (t) = x,F(t)]. The objective of the wageearner is to maximize the dynamic performance functional (9) over the admissible set A subjectto the wealth Equation (5). Then the value function of the problem can be written as

V (t, x) = sup(π,c,p)∈A

J (t, x;π, c, p) = J (t, x;π∗, c∗, p∗), (10)

where (π∗, c∗, p∗) ∈ A is the optimal investment-consumption-insurance strategy to be deter-mined. Note that V (t, x) is an F(t)-measurable random variable, for each (t, x) ∈ T ×�, sinceall model parameters are random in our modeling framework. So the value function V (t, x) isindeed a functional of the path {W (s)|0 ≤ s ≤ t}. This is different from the case of constant ordeterministic model parameters, where the value function is a deterministic function of t and xand can be determined from a partial differential equation.

3. A combination of the HJB equation and BSDE

In this section, we employ the dynamic programming principle to discuss the optimal investment-consumption-insurance problem. Since the model parameters are all random processes, theordinary HJB equation cannot be applied to solve the problem. Peng (1992) proposed the theoryof the stochastic HJB equation to solve the optimal control problem with random parameters.However, it is very difficult, if not possible, to solve the stochastic HJB equation explicitly, whichis a class of backward stochastic partial differential equations. Instead, we use a combination ofthe HJB equation and BSDE for the problem in our paper, from which the optimal strategy andthe value function of the problem can be derived. Note that throughout this paper, we consider

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the BSDEs with stochastic Lipschitz condition (see Appendix 1) so as to accommodate ourmodeling framework with unbounded model parameters.

First of all, we introduce the following BSDE

dY (t) = − f (t, Y (t), Z(t))dt + Z(t)dW (t), Y (T ) = ξ. (11)

where ξ is an �m-valued, F(T )-measurable random variable and f is a P ⊗ B(�m) ⊗B(�m×n)/B(�m) -measurable function. Here m ≥ 0 is a generic integer. Moreover, we assumethat the pair (ξ, f ) is a standard data for the BSDE (11) given in Definition A2. Therefore usingLemma A1, there exists a unique solution (Y, Z) ∈ M2

F (θ, a, T ; �m × �m×n) to the BSDE(11). Here Definition A2, Lemma A1 and the solution space M2

F (θ, a, T ; �m × �m×n) aregiven in Appendix 1.

Let O := T ×�×�m be the solvency region. We consider a function (t, x, y) → G(t, x, y) :O �→ � such that G(·, ·, ·) ∈ C1,2,2(O). Then we write Gt , Gx , G y , Gxx and G yy for the partialderivatives of G with respect to t , x and y and the second order partial derivatives of G withrespect to x and y, respectively. Define the following partial differential generator Lπ,c,p actingon a function G(·, ·, ·) ∈ C1,2,2(O) as:

Lπ,c,p[G(t, x, Y (t))] = −[ρ(t)+ λ(t)]G(t, x, Y (t))+ Gt (t, x, Y (t))

+ [r(t)x + π� B(t)+ i(t)− c − p

]Gx (t, x, Y (t))

− G y(t, x,Y (t))� f (t, Y (t), Z(t))+ 1

2π�σ(t)σ (t)�πGxx (t, x,Y (t))

+ π�σ(t)Z(t)�Gxy(t, x, Y (t))+ 1

2tr[Z(t)Z(t)�G yy(t, x, Y (t))

].

The following theorem is a verification theorem for the combination of the HJB equation andBSDE associated with the investment-consumption-insurance problem.

Theorem 3.1 Suppose that the pair (ξ, f ) is a standard data for the BSDE (11) given inDefinition A2. Let O be the closure of the solvency region O. Suppose further that there existsa function G such that G(·, ·, ·) ∈ C2(O) ∩ C(O) and an admissible control (π∗, c∗, p∗) ∈ Asuch that

(1) Lπ,c,p[G(t, x, Y (t))] + U (c)+ λ(t)U (x + p/η(t)) ≤ 0, P-a.s., for all (π, c, p) ∈ A and(t, x) ∈ T × �;

(2) Lπ∗,c∗,p∗ [G(t, x, Y (t))] + U (c∗) + λ(t)U (x + p∗/η(t)) = 0, P-a.s., for all (t, x) ∈T × �;

(3) for all (π, c, p) ∈ A,

limt→T − G(t, x, Y (t)) = U (x), P-a.s.;

(4) let K denote the set of stopping times κ ≤ T . The family {G(κ, X (κ), Y (κ))}κ∈K isuniformly integrable.

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10 Y. Shen & J. Wei

Then,

G(t, x, Y (t)) = V (t, x)

= sup(π,c,p)∈A

J (t, x;π, c, p)

= J (t, x;π∗, c∗, p∗),

and (π∗, c∗, p∗) is an optimal control.

Proof First of all, we define a localizing sequence of stopping times as follows:

κ(N )O := T ∧ inf {t ≥ 0|(t, X (t), Y (t)) /∈ O}

∧ inf {t > 0| t ≥ N or |X (t)| ≥ N or ||Y (t)|| ≥ N }.

By the Dynkin formula,

G(t, x, Y (t)) = Et,x

[e− ∫ κ(N )O

t [ρ(u)+λ(u)]du G(κ(N )O , X

(κ(N )O), Y(κ(N )O))

−∫ κ

(N )O

te− ∫ s

t [ρ(u)+λ(u)]duLπ,c,p[G(s, X (s), Y (s))]ds

]. (12)

For all (π, c, p) ∈ A, using Condition 1 to (12) gives

G(t, x, Y (t)) ≥ Et,x

[ ∫ κ(N )O

te− ∫ s

t [ρ(u)+λ(u)]du[U (c(s))+ λ(s)U (X (s)+ p(s)/η(s))]ds

+ e− ∫ κ(N )Ot [ρ(s)+λ(s)]ds G

(κ(N )O , X

(κ(N )O), Y(κ(N )O)) ]

. (13)

Using Conditions 3–4, Fatou’s Lemma and the Dominated Convergence Theorem yield

G(t, x, Y (t))≥ lim infN→∞ Et,x

[ ∫ κ(N )O

te− ∫ s

t [ρ(u)+λ(u)]du[U (c(s))+ λ(s)U (X (s)+ p(s)/η(s))]ds

+ e− ∫ κ(N )Ot [ρ(s)+λ(s)]ds G

(κ(N )O , X

(κ(N )O), Y(κ(N )O)) ]

≥ Et,x

[lim infN→∞

{∫ κ(N )O

te− ∫ s

t [ρ(u)+λ(u)]du[U (c(s))+ λ(s)U (X (s)+ p(s)/η(s))]ds

+ e− ∫ κ(N )Ot [ρ(s)+λ(s)]ds G

(κ(N )O , X

(κ(N )O), Y(κ(N )O))}]

= J (t, x;π, c, p). (14)

Similarly, we can use Conditions 2–4 to derive that

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G(t, x, Y (t)) = J (t, x;π∗, c∗, p∗). (15)

Consequently, combining (14) and (15) completes the proof. �

Therefore, rearranging conditions in Theorem 3.1 implies that the value function is the solutionof the following coupled system of the HJB equation and BSDE⎧⎨⎩ sup

(π,c,p)∈A{Lπ,c,p[G(t, x, Y (t))] + U (c)+ λ(t)U (x + p/η(t))} = 0, G(T, x, ξ) = U (x),

dY (t) = − f (t, Y (t), Z(t))dt + Z(t)dW (t), Y (T ) = ξ.

(16)In the next section, we shall derive explicit solutions to the problem from (16), including theoptimal strategy and the value function. To conclude this section, we provide some explanationsto BSDE techniques for the optimal investment-consumption-insurance problem. As discussedin the previous section, the value function V (t, x) is a functional of the path of the initial data, i.e.{W (s)|s ∈ [0, t]}, rather than only a function of W (t), since all model parameters are F-adaptedprocesses. Finding a solution for V (t, x) is indeed an infinite-dimensional problem, which isdifficult to solve. Using BSDE techniques, we prove that V (t, x) = G(t, x, Y (t)) depends on anF-adapted process Y (t) in a deterministic way. This reduces the infinite-dimensional problem toa finite-dimensional and solvable problem. One may consider Y and Z as the state process and itsvolatility process, respectively, related to various economic, physical, and demographic factors,both of which are not necessarily Markovian with respect to F. In the case of the Markovianstochastic opportunity sets, the processes Y and Z are nothing but some functions of income,force of mortality, appreciation rate and other parameters (please refer to specific examples inSection 5).

4. General solutions

In this section, we provide general solutions to the investment-consumption-insurance problemfor a wage earner with power utility. We derive explicit expressions for the optimal investment-consumption-insurance strategy (π∗, c∗, p∗) and the value function V (t, x) or G(t, x, Y (t))through solving (16).

Denote by

�(t, x, Y (t);π, c, p) := Lπ,c,p[G(t, x, Y (t))] + U (c)+ λ(t)U (x + p/η(t)).

Following the standard procedure to solve the HJB equation, we require that (1) �ππ is anegative-definite matrix; and (2) �cc < 0 and �pp < 0, which lead to a regular interior maxi-mum. Since the power utility function is strictly concave, Condition (2) is satisfied automatically.To guarantee that Condition (1) holds, we only need to assume that Gxx < 0. This conditionwill be verified in the later part of this section.

Applying the first-order conditions to � with respect to (π, c, p), we have

Gx (t, x, Y (t))B(t)+ Gxx (t, x, Y (t))σ (t)σ (t)�π + Gxy(t, x, Y (t))σ (t)Z(t)� = 0,

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12 Y. Shen & J. Wei

and

−Gx (t, x, Y (t))+ ∂U (c)

∂c= 0,

−Gx (t, x, Y (t))+ λ(t)∂U (x + p/η(t))

∂p= 0.

Solving the above three equations gives

π∗(t) = −(σ (t)σ (t)�)−1 B(t)Gx (t, x, Y (t))

Gxx (t, x, Y (t))− (σ(t)�

)−1Z(t)�

Gxy(t, x, Y (t))

Gxx (t, x, Y (t)), (17)

and

c∗(t) = [Gx (t, x, Y (t))] 1γ−1 , (18)

p∗(t) = η(t)

{[η(t)

λ(t)Gx (t, x, Y (t))

] 1γ−1 − x

}. (19)

From the terminal condition of the value function, we try the following parametric form forthe value function

V (t, x) = G(t, x, Y1(t), Y2(t)) = [x + Y1(t)]γγ

[Y2(t)]1−γ , (20)

where the BSDE (11) can be separated into two decoupled, one-dimensional BSDEs as{dY1(t) = − f1(t, Y1(t), Z1(t))dt + Z1(t)dW (t),Y1(T ) = 0,

(21)

and {dY2(t) = − f2(t, Y2(t), Z2(t))dt + Z2(t)dW (t),Y2(T ) = 1.

(22)

Here the drivers f1 and f2 are to be determined below, which is appropriate enough to guaranteethe uniqueness and existence of solutions to (21) and (22), respectively. Note that the genericinteger m = 2.

Substituting (20) into (17)–(19) gives the optimal investment-consumption-insurance strategy

π∗(t) = 1

1 − γ(σ (t)σ (t)�)−1 B(t)[x + Y1(t)] − (

σ(t)�)−1

Z1(t)�

+ (σ(t)�

)−1Z2(t)

� x + Y1(t)

Y2(t), (23)

and

c∗(t) = x + Y1(t)

Y2(t), (24)

p∗(t) = η(t)

[(η(t)

λ(t)

) 1γ−1

x + Y1(t)

Y2(t)− x

]. (25)

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In addition, the optimal legacy is given by

l∗(t) =(η(t)

λ(t)

) 1γ−1

x + Y1(t)

Y2(t). (26)

To unburden our notation, we write := (t), where := B, σ, i, λ, η, Y1, Z1, Y2, Z2, andf j := f j (t, Y j (t), Z j (t)), where j = 1, 2, whenever no confusion arises. Then substituting(23)–(26) into the HJB Equation (16) gives

[x + Y1]γ−1Y 1−γ2

{− f1 − [

r + η]Y1 + i − B�(σ�)−1

Z�1

}+ 1 − γ

γ[x + Y1]γ Y −γ

2

{− f2 +

[1 + η

γγ−1

λ1

γ−1

]+ γ

1 − γB�(σ�)−1

Z�2

+[

− λ+ ρ

1 − γ+ γ

2(1 − γ )2B�(σσ�)−1 B + γ (r + η)

1 − γ

]Y2

}= 0.

Then setting the coefficients of [x + Y1]γ−1Y 1−γ2 and [x + Y1]γ Y −γ

2 equal to zeros leads to

f1(t, Y1(t), Z1(t)) = −[r(t)+ η(t)]Y1(t)+ i(t)− B(t)�

(σ(t)�

)−1Z1(t)

�, (27)

andf2(t, Y2(t), Z2(t)) = K (t)+ L(t)Y2(t)+ γ

1 − γB(t)�

(σ(t)�

)−1Z2(t)

�, (28)

where

K (t) := 1 + ηγγ−1 (t)

λ1

γ−1 (t),

and

L(t) := 1

1 − γ

{− [λ(t)+ ρ(t)

]+ γ[r(t)+ η(t)

]+ γ

2(1 − γ )B(t)�(σ (t)σ (t)�)−1 B(t)

}.

Since we do not assume that any model parameters are bounded, the drivers (27)–(28) are notLipschitz continuous. So, the classical theory of BSDEs in El Karoui et al. (1997) does notwork here. Indeed, under Assumptions (A1)–(A3), the BSDEs (21)–(22) satisfy the stochasticLipschitz condition, which is discussed in detail in Appendix 1.

We now consider the first BSDE (21). From Lemma A2, we can see that Assumptions(A1)–(A3) guarantee that the BSDE (21) has a unique solution (Y1, Z1) ∈ M2

F (θ, a, T ; �×�n).Define a probability measure P equivalent to P on F(T ) as follows:

dPdP

∣∣∣∣F(T ) = �1(T ), (29)

where the Radon–Nikodym derivative is given by

�1(t) := exp

{− 1

2

∫ t

0B(s)�

(σ(s)σ (s)�

)−1B(s)ds −

∫ t

0B(s)�

(σ(s)�

)−1dW (s)

}, t ∈ T .

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14 Y. Shen & J. Wei

Under Assumption (A3), it is clear that the Novikov condition is satisfied. Hence by Girsanov’stheorem,

W (t) := W (t)+∫ t

0(σ (s))−1 B(s)ds, t ∈ T ,

is an n-dimensional, (F, P)-standard Brownian motion. So under P , the BSDE (21) becomes⎧⎨⎩ dY1(t) = −{

− [r(t)+ η(t)

]Y1(t)+ i(t)

}dt + Z1(t)dW (t),

Y1(T ) = 0.(30)

Using Lemma A2 leads to the following expectation representation for the first component ofthe solution to the BSDE (30)

Y1(t) = E

[ ∫ T

te− ∫ s

t [r(u)+η(u)]dui(s)ds

∣∣∣∣F(t)], (31)

where E[·] is the expectation under P . Here Y1 can be interpreted as an actuarial value processof future income and Z1 is its volatility process.

Although we could represent the second component of the solution Z1 via the theory ofthe Malliavin calculus, too much effort would be devoted to discussing the existence of theMalliavin derivatives. We do not pursue this in our paper. Particularly, if all random parametersare Markovian with respect to the Brownian filtration F, we can find explicit expressions forboth the first and second components of the solution (Y1, Z1). For more discussions, interestedreaders may refer to Lemma A3, which will be used in solving special examples presented inthe next section.

Next we consider the second BSDE (22). Similarly, Assumptions (A1)–(A3) guarantee thatthe BSDE (22) has a unique solution (Y2, Z2) ∈ M2

F (θ, a, T ; � × �n). Define a probabilitymeasure P equivalent to P on F(T ) as follows

dPdP

∣∣∣∣F(T ) = �2(T ),

where the Radon–Nikodym derivative is given by

�2(t) = exp

{− 1

2

∫ t

0

γ 2

(1 − γ )2B(s)�

(σ(s)σ (s)�

)−1B(s)ds

+∫ t

0

γ

1 − γB(s)�

(σ(s)�

)−1dW (s)

}.

Given that Assumption (A3) holds, it is clear that the Novikov condition is satisfied. So usingGirsanov’s theorem leads to that

W (t) := W (t)− γ

1 − γ

∫ t

0(σ (s))−1 B(s)ds, t ∈ T ,

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Scandinavian Actuarial Journal 15

is an n-dimensional, (F,P)-standard Brownian motion. Under P , the BSDE (28) then becomes⎧⎨⎩ dY2(t) = −{

K (t)+ L(t)Y2(t)

}dt + Z2(t)dW (t),

Y2(T ) = 1.(32)

In a similar way, we derive the following expectation representation for the first component ofthe solution to the BSDE (32)

Y2(t) = E

[e∫ T

t L(u)du +∫ T

te∫ s

t L(u)du K (s)ds

∣∣∣∣F(t)]. (33)

Here Y2 can be interpreted as a subjective actuarial value process of one unit instantaneousconsumption rate from current to T ∧ τ , ( η(τ)

λ(τ))

γγ−1 unit insurance benefit paying at death time τ

upon death before T , and one unit legacy at terminal time T upon survival until T . In addition,Z2 can be considered as the volatility process of Y2. This interpretation is similar as the casewith deterministic parameters (see, e.g. Kronborg & Steffensen 2013).

Evidently, it can be verified that

Gxx (t, x, Y1(t), Y2(t)) = (γ − 1)[x + Y1(t)]γ−2[Y2(t)]1−γ < 0.

Consequently, substituting the unique solutions (Y1, Z1) and (Y2, Z2) to the BSDE (21)–(22)into (23)–(25) gives the explicit expressions for the value function and the optimal investment-consumption-insurance strategy.

The following theorem summarizes the above analysis, giving the optimal investment-consumption-insurance strategy and the value function for an investor with power utility.

Theorem 4.1 Given that Assumptions (A1)–(A3) are satisfied, the optimal investment-consumption-insurance strategy of the problem is

π∗(t)

= 1

1 − γ(σ (t)σ (t)�)−1 B(t)[x + Y1(t)] − (

σ(t)�)−1

Z1(t)� + (

σ(t)�)−1

Z2(t)� x + Y1(t)

Y2(t),

and

c∗(t) = x + Y1(t)

Y2(t),

p∗(t) = η(t)

[(η(t)

λ(t)

) 1γ−1

x + Y1(t)

Y2(t)− x

].

Moreover, the value function of the problem is

V (t, x) = [x + Y1(t)]γγ

[Y2(t)]1−γ . (34)

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16 Y. Shen & J. Wei

Here (Y1, Z1), (Y2, Z2) ∈ M2F (θ, a, T ; � × �n) are the unique solutions to the BSDEs

(21)–(22), respectively.

5. Special examples

In this section, we derive closed-form solutions of the optimal investment-consumption-insuranceproblem in three special examples. In each of these examples, we assume that only one modelparameter is random and has a Markovian structure with respect to the Brownian motion filtrationF while other model parameters are all deterministic functions of t . From Theorem 4.1, we cansee once explicit expressions for the solutions (Y1, Z1) and (Y2, Z2) are obtained, the problem issolved completely. In what follows, we focus on deriving these expressions for special examplesof stochastic income, stochastic mortality, and stochastic appreciation rate, respectively.

As introduced in Section 2, we are considering the optimal investment-consumption-reinsurance problem in the complete market, where all underlying risks are hedgeable. In theexample of stochastic income, the completeness assumption states that salary-linked securitiesare traded in the market. In practice, the wage earner’s income may be affected by both systematicand idiosyncratic risks, such as inflation and his company’s financial performance. Thus to hedgeincome risk, one may consider trading the inflation-linked bond (Siu 2012) and his company’sstock. Similarly, in the example of stochastic mortality, the completeness assumption statesthat mortality-linked securities are traded in the market. Recent innovations in securitizationof mortality and longevity risks make it possible to hedge such risks. These mortality-linkedsecurities include longevity bonds and survivor swaps (Blake et al. 2006). The asset dynamicswith stochastic appreciation rate can be indeed reformulated as the cointegrated model (Chiu &Wong 2011), where the vector of log-prices depends linearly on the appreciation rate and hencefollows a multivariate OU process. Cointegration has been empirically evidenced in variousmarkets, such as stock, foreign exchange, and commodity. So the completeness in the exampleof stochastic appreciation rate means that any contingent claim in these markets can be replicatedby a combination of primitive assets.1

5.1. Stochastic income

Example 5.1 Let α ∈ � and β ∈ �n be two constant coefficients. We assume that theinstantaneous income process {i(t)|t ∈ T } is given by the following geometric Brownian motionmodel

di(s) = i(s)[αds + βdW (s)], i(t) = i, 0 ≤ t ≤ s ≤ T . (35)

1 The referee raised one interesting point that both the market price of risk and the market preferences (from anequilibrium pricing point of veiw) are stochastic in the example of stochastic appreciation rate. This is the case. Ifthe market price of risk and the market preferences are considered as special contingent claims, one may replicate thoseby using primitive assets. Alternatively, one may consider investing in some kind of market growth index to hedge therisk. If exists, the log contract of primitive assets seems an ideal candidate of market growth index. Otherwise, fromthe perspective of financial engineering, one may construct such log contract using a futures contract and a collection ofcalls and puts (Carr & Madan 1998).

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Scandinavian Actuarial Journal 17

Since r ,μ, σ , ρ, λ, and η are deterministic functions of t , there is no randomness in the secondBSDE (22). Therefore, the closed-form expressions for (Y2, Z2) are given by

Y2(t) = e∫ T

t L(u)du +∫ T

te∫ s

t L(u)du K (s)ds, (36)

and

Z2(t) = 01×n . (37)

Moreover, (30) can be written as

Y1(t) =∫ T

te− ∫ s

t [r(u)+η(u)]du Et [i(s)]ds,

where Et [·] denotes the conditional expectation under P given that F(t), i.e. E[·|F(t)].Clearly, the dynamics of the stochastic income under P is given by

di(s) = i(s){[α − β(σ(s))−1 B(s)]ds + βdW (s)}, i(t) = i, 0 ≤ t ≤ s ≤ T .

Then the solution of Y1 is

Y1(t) = i∫ T

texp

{∫ s

t

[α − β(σ(u))−1 B(u)− r(u)− η(u)

]du

}ds. (38)

Using Lemma A3 gives

Z1(t) = iβ∫ T

texp

{∫ s

t

[α − β(σ(u))−1 B(u)− r(u)− η(u)

]du

}ds. (39)

Therefore, substituting (36)–(37) and (38)–(39) into Theorem 4.1 gives the optimal investment-consumption-insurance strategy and the value function.

5.2. Stochastic mortality

Example 5.2 Letμ ∈ � and ξ ∈ �n be two constant coefficients. We assume that the processof the force of mortality {λ(t)|t ∈ T } is given by the following stochastic mortality model

dλ(s) = λ(s)[μds + ξdW (s)], λ(t) = λ, 0 ≤ t ≤ s ≤ T . (40)

Indeed, (40) is a version of continuous-time Lee-Carter model, since the force of mortality λ islognormally distributed.

As in Example 5.1, since r , η, and i are all deterministic, the solution to the first BSDE (21)is given by

Y1(t) =∫ T

te− ∫ s

t [r(u)+η(u)]dui(s)ds, (41)

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18 Y. Shen & J. Wei

and

Z1(t) = 01×n . (42)

To simplify our notation, we denote by

�(t, s)

= exp

{1

1 − γ

∫ s

t

(− ρ(u)+ γ

[r(u)+ η(u)

]+ γ

2(1 − γ )B(u)�(σ (u)σ (u)�)−1 B(u)

)du

}.

Then (33) becomes

Y2(t) = Et

[exp

{− 1

1 − γ

∫ T

tλ(u)du

}]�(t, T )

+∫ T

tEt

[exp

{− 1

1 − γ

∫ s

tλ(u)du

}K (s)

]�(t, s)ds, (43)

where Et [·] denotes the conditional expectation under P given that F(t), i.e. E[·|F(t)].Clearly, the dynamics of the stochastic mortality under P is given by

dλ(s) = λ(s)

{[μ+ γ

1 − γξ(σ (s))−1 B(s)

]ds + ξdW (s)

}, λ(t) = λ, 0 ≤ t ≤ s ≤ T .

(44)Similarly, since the stochastic mortality process {λ(t)|t ∈ T } is Markovian with respect to F,(Y2, Z2) can be represented as follows

Y2(t) = φ(t, T, λ)�(t, T )+∫ T

tψ(t, s, λ)�(t, s)ds, (45)

and

Z2(t) ={∂φ

∂λ(t, T, λ)�(t, T )+

∫ T

t

∂ψ

∂λ(t, s, λ)�(t, s)ds

}λξ, (46)

where

φ(t, s, λ) = Et

[exp

{− 1

1 − γ

∫ s

tλ(u)du

}], (47)

and

ψ(t, s, λ) = Et

[exp

{− 1

1 − γ

∫ s

tλ(u)du

}K (s)

]. (48)

If we further assume that σ(s) := σ and B(s) := B, then (44) can be reformulated as

d

[1

1 − γλ(s)

]= λ(s)

{[1

1 − γμ+ γ

(1 − γ )2ξσ−1 B

]ds + 1

1 − γ||ξ ||dW 0(s)

},

where {W 0|t ∈ T } is a one-dimensional, (F,P) -standard Brownian motion. So (47) can beconsidered as a zero-coupon bond pricing formula of the Dothan model, which is a well-knownproblem in the interest rate modeling.

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For notational simplicity, we denote by

q1 = 1 − 2[(1 − γ )μ+ γ ξσ−1 B]||ξ ||2 .

Applying Proposition 2.1 in Pintoux & Privault (2011) immediately leads to

φ(t, s, λ) = e− q2

1 ||ξ ||2(s−t)

8(1−γ )2∫ ∞

0

∫ ∞

0e−λv

× exp

(− 2(1 − γ )2(1 + ζ 2)

||ξ ||2v)

g

(4(1 − γ )2ζ

||ξ ||2v ,||ξ ||2(s − t)

4(1 − γ )2

)dv

v

ζ q1+1, (49)

where

g(v, ζ ) = evπ22ζ√

2π2ζ

∫ ∞

0e− u2

2ζ e−v cosh ζ sinh u sin

(πu

ζ

)du.

Under P , applying Itô’s formula gives

dλ1

1−γ (s) = λ1

1−γ (s)

{[1

1 − γμ+ γ

(1 − γ )2ξσ−1 B + γ

2(1 − γ )2||ξ ||2

]ds

+ 1

1 − γ||ξ ||dW 0(s)

}.

Evidently, we have

Et [λ1

1−γ (s)] = λ1

1−γ exp

{[1

1 − γμ+ γ

(1 − γ )2ξσ−1 B + γ

2(1 − γ )2||ξ ||2

](s − t)

}. (50)

By change of measure, (48) can be rewritten as

ψ(t, s, λ) = φ(t, s, λ)+ 1

ηγ

1−γ (s)Et [λ

11−γ (s)]Et

[exp

{− 1

1 − γ

∫ s

tλ(u)du

}], (51)

where Et [·] is the conditional expectation under P given F(t), i.e. E[·|F(t)], and P is aprobability measure equivalent to P on F(s) defined by

dPdP

∣∣∣∣F(s) = λ1

1−γ (s)

E[λ 11−γ (s)]

= exp

{− ||ξ ||2

2(1 − γ )2s + ||ξ ||

1 − γW 0(s)

}.

By Girsanov’s theorem, we can see the dynamics of the stochastic mortality under P is given by

dλ(t) = λ(t)

{[μ+ γ

1 − γξ(σ (t))−1 B(t)+ 1

1 − γ||ξ ||2

]dt + ||ξ ||dW0(t)

},

whereW0(t) = W 0(t)− ||ξ ||

1 − γt,

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20 Y. Shen & J. Wei

is a one-dimensional, (F, P)-standard Brownian motion. To simplify our notation, we denote by

q2 = −1 − 2[(1 − γ )μ+ γ ξσ−1 B]||ξ ||2 .

Again applying Proposition 2.1 in Pintoux & Privault (2011) immediately results in

Et

[exp

{− 1

1 − γ

∫ s

tλ(u)du

}]= e

− q22 ||ξ ||2(s−t)

8(1−γ )2∫ ∞

0

∫ ∞

0e−λv

× exp

(− 2(1 − γ )2(1 + ζ 2)

||ξ ||2v)

g

(4(1 − γ )2ζ

||ξ ||2v ,||ξ ||2(s − t)

4(1 − γ )2

)dv

v

ζ q2+1. (52)

Combining (47)–(52) gives expressions for (Y2, Z2). Therefore, substituting (41)–(42) and(45)–(46) into Theorem 4.1 gives the optimal investment-consumption-insurance strategy andthe value function.

5.3. Stochastic appreciation rate

Example 5.3 Let �(t) ∈ �n be a vector-valued, uniformly bounded, deterministic function,and A ∈ �n×n be a constant matrix, respectively. We assume that the excessive return vectorprocess {B(t)|t ∈ T } satisfies the following stochastic appreciation rate model

d B(s) = [�(s)− AB(s)]ds − Aσ(s)dW (s), B(t) = b, 0 ≤ t ≤ s ≤ T . (53)

Indeed, this is related to the continuous-time, cointegrated asset price model considered inChiu & Wong (2011).

Since r , η, and i are deterministic functions, the closed-form expressions for (Y1, Z1) aregiven by

Y1(t) =∫ T

te− ∫ s

t [r(u)+η(u)]dui(s)ds, (54)

and

Z1(t) = 01×n . (55)

Denote by

�(t, s) := exp

{1

1 − γ

∫ s

t

(− [λ(u)+ ρ(u)

]+ γ[r(u)+ η(u)

])du

}.

Then (33) becomes

Y2(t) = Et

[exp

2(1 − γ )2

∫ T

tB(u)�(σ (u)σ (u)�)−1 B(u)du

}]�(t, T )

+∫ T

tEt

[exp

2(1 − γ )2

∫ s

tB(u)�(σ (u)σ (u)�)−1 B(u)du

}]�(t, s)K (s)ds. (56)

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Scandinavian Actuarial Journal 21

Clearly, the dynamics of the stochastic appreciation rate under P is given by

d B(s) =[�(s)− 1

1 − γAB(s)

]ds − Aσ(s)dW (s). (57)

Since the stochastic appreciation rate process {B(t)|t ∈ T } is Markovian with respect to F,applying Lemma A3 gives that

Y2(t) = ϕ(t, T, b)�(t, T )+∫ T

tϕ(t, s, b)�(t, s)K (s)ds, (58)

and

Z2(t) = −{[∂ϕ

∂b(t, T, b)

]��(t, T )+

∫ T

t

[∂ϕ

∂b(t, s, b)

]��(t, s)K (s)ds

}Aσ, (59)

where

ϕ(t, s, b) = Et

[exp

2(1 − γ )2

∫ s

tB(u)�(σ (u)σ (u)�)−1 B(u)du

}], 0 ≤ t ≤ s ≤ T,

(60)satisfies the following partial differential equation⎧⎨⎩∂ϕ

∂t+[

γ

2(1 − γ )2b��(t)−1b

]ϕ +

(∂ϕ

∂b

)�[�(t)− 1

1 − γAb

]+ 1

2tr

[∂2ϕ

∂b2A�(t)A�

]= 0,

ϕ(s, s, b) = 1.(61)

Here �(t) = σ(t)σ (t)� is defined in Section 2. Since Equation (60) is indeed the expectationof the exponential quadratic of {B(t)|t ∈ T }, which has a Gaussian structure (57), we try thefollowing ansatz

ϕ(t, s, b) = exp

{1

2b�M1(t, s)b + M2(t, s)�b + M3(t, s)

}. (62)

Then substituting (62) into (61) and setting the coefficients of b to be zeros yield that the following(matrix-valued, vector-valued and real-valued) ODEs⎧⎨⎩

∂M1

∂t− 1

1 − γM1 A − 1

1 − γA�M1 + M1 A�(t)A�M1 + γ

(1 − γ )2�(t)−1 = 0n×n,

M1(s, s) = 0n×n,(63)

⎧⎨⎩(∂M2

∂t

)�+ M�

2

[A�(t)A�M1 − 1

1 − γA

]+�(t)�M1 = 0n×1,

M2(s, s) = 0n×1,

(64)

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Page 23: Optimal investment-consumption-insurance with random parameters

22 Y. Shen & J. Wei

and ⎧⎨⎩∂M3

∂t+ M�

2 �(t)+ 1

2tr(M1 A�(t)A�)+ 1

2M�

2 A�(t)A�M2 = 0,

M3(u, u) = 0.(65)

Note that the matrix-valued Riccati equation as Equation (63) may not have closed-form solutionin general. By Radon’s lemma (see Abou-Kandil et al. 2003 or Chiu & Wong 2011), however,the solution of M1 can be represented explicitly in terms of a solvable linear matrix-valued ODEas follows:

M1(�) = M1(t, s) = R2(�)R1(�)−1, � := s − t, (66)

where R1(�), R2(�) ∈ C(T ; �n×n) and R(�) := (R1(�)�, R2(�)

�)� is the fundamentalsolution of the following linear system of ODEs:

d R

d�=(

11−γ A −A�(�)A�

γ

(1−γ )2�(�)−1 − 1

1−γ A�

)R,

with the initial conditions R1(0) = In×n and R2(0) = 0n×n . Particularly, if the volatility matrixσ(t) is constant, i.e. σ(t) := σ , for each t ∈ T . The solution of R(�) is given by

R(�) = exp

[(1

1−γ A −A�A�γ

(1−γ )2�−1 − 1

1−γ A�

)�

](In×n

0n×n

).

Given that M1, we can see that the solution of M2 is

M2(�)� = M2(t, s)� =

∫ �

0�(u)�M1(u) (u, 0)du (�, 0)−1, (67)

where (t, s) is the fundamental solution of the following matrix-valued ODE

∂ (t, s)

∂t=[

1

1 − γA − A�(t)A�M1

]� (t, s), (s, s) = In×n .

Given that M1 and M2, it is easy to see that the solution of M3 is

M3(t, s) =∫ �

0

[M�

2 (u)�(u)+1

2tr(M1(u)A�(u)A

�)+1

2M�

2 (u)A�(u)A�M2(u)

]du. (68)

Then combining (62) and (66)–(68) gives expressions for (Y2, Z2). Therefore, substituting(54)–(55) and (58)–(59) into Theorem 4.1 gives the optimal investment-consumption-insurancestrategy and the value function.

6. Conclusion

We investigated an optimal investment-consumption-insurance problem with random parame-ters. Using the dynamic programming principle, we derived a combination of the HJB equation

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Scandinavian Actuarial Journal 23

and BSDE related to the problem. We obtained general expressions for the optimal investment-consumption-insurance strategy and the value function in terms of the unique solutions to twoBSDEs. We also provided closed-form solutions to the problem in special examples of stochasticincome, stochastic mortality, and stochastic appreciation rate.

References

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Bender, C. & Kohlmann, M. (2000). BSDEs with stochastic Lipschitz condition, Preprint, http://cofe.uni-konstanz.de/Papers/dp00_08.pdf .

Blake, D., Cairns,A. J. G. & Dowd, K. (2006). Living with mortality: longevity bonds and other mortality-linked securities.British Actuarial Journal 12, 153–197.

Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M. & Martellini, L. (2008). Optimal investment decisions when time-horizon is uncertain. Journal of Mathematical Economics 44, 1100–1113.

Bruhn, K. & Steffensen, M. (2011). Household consumption, investment and life insurance. Insurance: Mathematics andEconomics 48, 315–325.

Carr, P. & Madan, D. (1998). Towards a theory of volatility trading. In R. Jarrow (Ed.), Risk book on volatility(pp. 458–476). New York: Risk.

Cheridito, P. & Hu, Y., (2011). Optimal consumption and investment in incomplete markets with general constraints.Stochastics and Dynamics 2011(11), 283–299.

Chiu, M. C. & Wong, H. Y. (2011). Mean-variance portfolio selection of cointegrated assets. Journal of EconomicDynamics and Control 35, 1369–1385.

Duarte, I., Pinheiro, A., Pinto, A. & Pliska, S. (2011). An overview of optimal life insurance purchase, consumption andinvestment problems. Dynamics, Games and Science 1, 271–286.

Duarte, I., Pinheiro, A., Pinto, A. & Pliska, S., (2012). Optimal life insurance purchase, consumption and investment on afinancial market with multi-dimensional diffusive terms. Optimization: A Journal of Mathematical Programming andOperations Research, 1–20. doi:10.1080/02331934.2012.665054.

El Karoui, N., Peng, S. & Quenez, M. C. (1997). Backward stochastic differential equations in finance. MathematicalFinance 7, 1–71.

Huang, H. & Milevsky, M. A. (2008). Portfolio choice and mortality-contingent claims: the general HARA case. Journalof Banking and Finance 32, 2444–2452.

Huang, H., Milevsky, M. A. & Wang, J. (2008). Portfolio choice and life insurance: the CRRA case. The Journal of Riskand Insurance 75, 847–872.

Lim, A. E. B. (2004). Quadratic hedging and mean-variance portfolio selection in an incomplete market. Mathematics ofOperations Research 29, 132–161.

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Kraft, H. & Steffensen, M. (2008). Optimal consumption and insurance: a continuous-time Markov chain approach. ASTINBulletin 28, 231–257.

Kronborg, M. T. & Steffensen, M. (2013). Optimal consumption, investment and life insurance with surrender optionguarantee. Scandinavian Actuarial Journal. In press. doi:10.1080/03461238.2013.775964.

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Appendix 1. BSDEs with stochastic Lipschitz condition

This appendix is mainly based on the theory of BSDEs with stochastic Lipschitz condition inBender & Kohlmann (2000). We accommodate the uniqueness and existence results there to ourmodeling framework in this appendix.

For any �+-valued, F-adapted process {a(t)|t ∈ T }, we define an increasing continuousprocess {A(t)|t ∈ T } by A(t) = ∫ t

0 a2(s)ds, for each t ∈ T . Let θ ≥ 0. On the filteredprobability space (�,F,F,P), we define the following spaces of processes:

L2,aF (θ, a, T ; �m): the space of all �m-valued, F-progressively measurable processes

{φ(t)|t ∈ T } such that ||φ||2θ,a := ||aφ||2θ = E[∫ T0 a2(t)eθ A(t)||φ(t)||2dt] < ∞;

L2,cF (θ, a, T ; �m): the space of all �m-valued, F-adapted, càdlàg processes {φ(t)|t ∈ T }

such that ||φ||2θ,c := E[sup0≤t≤T eθ A(t)||φ(t)||2] < ∞;

L2F (θ, a, T ; H): the space of all H -valued, F-progressively measurable processes

{φ(t)|t ∈ T } such that ||φ||2θ := E[∫ T0 eθ A(t)||φ(t)||2dt] < ∞.

Then

M2F (θ, a, T ; �m ×�m×n) := (L2,a

F (θ, a, T ; �m)∩L2,cF (θ, a, T ; �m))×L2

F (θ, a, T ; �m×n),

is a Banach space with the norm

||(Y, Z)||2θ := ||Y ||2θ,a + ||Y ||2θ,c + ||Z ||2θ .

Given the data (ξ, f ), where ξ : � → �m is an F(T )-measurable random variable and f isa P ⊗ B(�m)⊗ B(�m×n)/B(�m)-measurable function, we consider the following BSDE:

dY (t) = − f (t, Y (t), Z(t))dt + Z(t)dW (t), Y (T ) = ξ. (A1)

Definition A1 A pair (Y, Z) ∈ M2F (θ, a, T ; �m × �m×n) is said to be a solution to BSDE

(A1), if

Y (t) = ξ +∫ T

tf (s, Y (s), Z(s))ds −

∫ T

tZ(s)dW (s). (A2)

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Scandinavian Actuarial Journal 25

Definition A2 A pair (ξ, f ) is said to be a standard data for BSDE (A1), if the followingconditions hold

(H1) the �m-valued, F(T )-measurable variable ξ satisfies

E[eθ A(T )||ξ ||2] < ∞;

(H2) there exist two �+-valued, F-adapted processes {a1(t)|t ∈ T } and {a2(t)|t ∈ T } suchthat ∀(y, z), (y′, z′) ∈ �m × �m×n

|| f (t, y, z)− f (t, y′, z′)|| ≤ a1(ω, t)||y − y′|| + a2(ω, t)||z − z′||;

(H3) ∃ ε > 0, a2(t) := a1(t)+ a22(t) ≥ ε, for each t ∈ T ;

(H4) f (·,0m ,0m×n)a(·) ∈ L2

F (θ, a, T ; �m).

We refer to Assumption (H2) as the stochastic Lipschitz condition. In what follows, wesuppress ‘ω’ in a1(ω, t) and a2(ω, t), and write a1(t) and a2(t) for a1(ω, t) and a2(t, ω),respectively.

Lemma A1 Let (ξ, f ) be a standard data. Then the BSDE (A1) has a unique solution (Y, Z) ∈M2

F (θ, a, T ; �m × �m×n).

Proof This lemma is a special case of Theorem 3 in Bender & Kohlmann (2000). We omitthe proof here. �

In the following, we consider a particular case of the BSDE satisfying the stochastic Lipschtizcondition., i.e. a one-dimensional linear BSDE with random coefficients.

Lemma A2 Let {(α0(t), α1(t), α2(t))|t ∈ T } be (� × � × �n)-valued, F-adapted process.If the linear BSDE

dY (t) = −[α0(t)+ Y (t)α1(t)+ Z(t)α2(t)]dt + Z(t)dW (t), Y (T ) = ξ, (A3)

satisfying

(H1’) the �-valued, F(T )-measurable random variable ξ is bounded;(H2’) ∃ ε > 0, a2(t) := |α0(t)| + |α1(t)| + ||α2(t)||2 ≥ ε, for each t ∈ T ;(H3’) for a sufficiently large θ , E[eθ A(T )] < ∞,

then it has a unique solution (Y, Z) ∈ M2F (θ, a, T ; �×�n). Furthermore, Y (t) is given by the

following expectation formula

Y (t) = E

[ξ�(t, T )+

∫ T

tα0(s)�(t, s)ds

∣∣∣∣F(t)], (A4)

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26 Y. Shen & J. Wei

where {�(t, s)|t, s ∈ T , s ≥ t} is the adjoint process defined by the forward linear SDE

d�(t, s) = �(t, s)[α1(s)ds + α2(s)dW (s)], �(t, t) = 1, (A5)

satisfying the semi-group property ∀t ≤ s ≤ u, �(t, s)�(s, u) = �(t, u), P-a.s.

Proof Setting a1(t) := |α0(t)| + |α1(t)| and a2(t) := ||α2(t)||. If Conditions (H1’)–(H3’)hold, it is easy to see that Conditions (H1)–(H4) in Lemma A1 are satisfied. Therefore, the linearBSDE (A3) has a unique solution (Y, Z) ∈ M2

F (θ, a, T ; � × �n). The desired result (A4) canbe proved similarly as Proposition 2.2 in El Karoui et al. (1997). So we do not repeat it here. �

Lemma A3 Under conditions given in LemmaA2, we suppose that the process {(α0(t), α1(t),α2(t))|t ∈ T } is Markovian with respect to F in the way that αi (t) := αi (t, P(t)), i = 0, 1, 2,with

d P(s) = β1(s, P(s))ds + β2(s, P(s))dW (s), P(t) = p, 0 ≤ t ≤ s ≤ T, (A6)

where α0 : T × �m → �, α1 : T × �m → �, α2 : T × �m → �n, β1 : T × �m → �m

and β2 : T × �m → �m×n are all deterministic functions of their first and second arguments.Furthermore, ifα0,α1,α2,β1 andβ2 are all continuously differentiable in their second argumentswith uniformly bounded derivatives, then there exists a smooth function u : T × �m → � suchthat

Y (t) = u(t, p), Z(t) =[∂u

∂p(t, p)

]�β2(t, p). (A7)

Proof The proof can be adapted to those of Theorem 4.1 and Corollary 4.1 in El Karoui et al.(1997). So we omit it here. �

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