optimal investment and consumption with default risk: hara utility

Asia-Pacific Finan Markets (2013) 20:261–281DOI 10.1007/s10690-013-9167-2

Optimal Investment and Consumption with DefaultRisk: HARA Utility

Lijun Bo · Xindan Li · Yongjin Wang ·Xuewei Yang

Published online: 10 March 2013© Springer Science+Business Media New York 2013

Abstract In this paper, we consider a portfolio optimization problem in a defaultablemarket. The representative investor dynamically allocates his or her wealth among thefollowing securities: a perpetual defaultable bond, a money market account and adefault-free risky asset. The optimal investment and consumption policies that max-imize the infinite horizon expected discounted HARA utility of the consumption areexplicitly derived. Moreover, numerical illustrations are also presented.

Keywords Optimal control · Portfolio optimization · Perpetual bond ·Defaultable market · HJB equation

1 Introduction

The portfolio optimization problem without default risk is due to the seminal works byMerton (1969, 1971, 1992), in which the author proposed the policy that maximizes thetotal expected discounted utility of the consumption in a market investment problem.So far, the default-free portfolio optimization problems have been extensively studiedin the literature (see, e.g., Bielecki et al. 2005; Callegaro et al. 2010; Fleming and Pang2004; Pang 2006; Pham 2002 and references therein). Among others, a general caseof maximization problem with consumption in a discontinuous default-free market is

L. BoDepartment of Mathematics, Xidian University, Xi’an 710071, China

X. Li · X. Yang (B)School of Management and Engineering, Nanjing University, Nanjing 210093, Chinae-mail: [email protected]

Y. WangSchool of Business, Nankai University, Tianjin 300071, China

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discussed in Callegaro et al. (2010). Therein the authors used the martingale approachto find an explicit solution for the corresponding HJB equation.

The study of credit risk has become increasingly attractive in both industry and aca-demic (see, e.g., Moody’s Investor Service 2004–2005, Bélanger et al. 2004; Bieleckiand Rutkowski 2002 and the references therein). In particular, portfolio optimiza-tion problems with defaultable securities have been investigated by many scholars. InBielecki and Jang (2006), Jang (2005), Bielecki and Jang discussed an optimal alloca-tion problem associated with a defaultable risky asset. The goal was to maximize theexpected HARA utility of terminal wealth. Lakner and Liang (2008) analyzed the opti-mal investment strategy in a defaultable (corporate) bond and a money market accountin a continuous time model. In Shouda (2006), by considering an optimal investmentproblem for bond holders, the author decomposed credit risk into default-timing risk,recovery risk, and spread risk. Given the tractability of recovery of market value, Houand Jin (2002) considered an optimal allocation problem including a defaultable debt.In Bo et al. (2010), the authors explored an optimal portfolio problem in a default-able market under Log utility. An optimal investment problem under multiple defaultrisk using a BSDE-decomposition approach is presented in Jiao et al. (2012). In Jiaoand Pham (2011), combined duality and dynamic programming principle to optimizeCRRA utility in a defaultable market including a riskless bond and a stock subject tocounterparty risk. These literature share one common characteristic, i.e., the intensity-based reduced-form was adopted to model default risk. However, in Korn and Kraft(2003), the authors used a firm value approach (we also call it structural approach) toexplore optimal portfolios with defaultable securities.

In this paper, we consider a portfolio optimization problem with a defaultableperpetual bond. Herein we assume that an investor dynamically chooses a consump-tion ratio and allocates the wealth into the following securities: a perpetual default-able bond, a money market account, and a default-free risky asset. The goal is tomaximize the infinite horizon expected discounted HARA utility of consumption.A dynamic programming principle is used to derive the HJB equation. Explicit for-mulas for optimal control strategies and the corresponding value function would bederived

The outline of this paper is as follows: In the coming section, we describe our modeland derive the price dynamics for defaultable perpetual bond. In Sect. 3, we formulatethe portfolio optimization problem with default risk, and present the optimal controlstrategy and the corresponding value function. Section 4 is devoted to providing theverification theorems. Finally, in Sect. 5, we carry out a sensitivity analysis for theoptimal control strategies and the corresponding value function in non-Log utilitycase. All proofs are arranged in Appendix.

2 Model Description

In this section, we present a model with specifications of a reduced-form frameworkfor an intensity-based defaultable market and of the dynamics of financial securi-ties: a defaultable perpetual bond, a money market account and a default-free riskyasset.

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Optimal Investment and Consumption 263

2.1 Reduced-Form Framework for Default

This subsection describes a reduced-form framework for an intensity-based defaultablemarket.

Let (�,F , P) be a complete real-world probability space and τ be the (random)first jump time of a Poisson process with constant intensity λ > 0. For t ≥ 0, definea default indicator process Z = (Zt ; t ≥ 0) by

Zt = 1{τ≤t}. (1)

Suppose that W = (Wt ; t ≥ 0) is a standard Brownian motion (independent of theaforementioned Poisson process) on the above probability space and F = (Ft ; t ≥ 0)

is the augmented natural filtration of the Brownian motion. Let Dt =σ(Zu; 0≤u ≤ t)and Gt = Ft ∨ Dt where t ≥ 0. Then G = (Gt ; t ≥ 0) is the smallest filtrationsuch that the random time τ is a stopping time. Moreover, the conditional survivalprobability is given by

St :=P(τ > t |Ft ) = e−λt , (2)

and the following process related to default

Mt :=Zt − λ

t∫

0

(1 − Zs)ds, t ≥ 0 (3)

is a (P, G)-martingale.

2.2 Price Dynamics of Financial Securities

In this subsection, we are concerned with the price dynamics of several financialsecurities mentioned above under the real-world probability measure P.

As in Bielecki and Jang (2006), Jang (2005), denote by 1/η ≥ 1 the default riskpremium and let ρ ∈ (0, 1) be the constant loss rate when a default occurs. We makethe assumption that under the chosen risk-neutral measure Q (which is equivalentto the real-world probability measure P) the arrival intensity of default is given byλ = λ/η. Now we use Y = (Yt ; t ≥ 0) to represent the cum-dividend price of adefaultable perpetual bond that pays constant coupon C > 0 per unit time under theconvention of recovery of market value.1

Recall the default indicator process Z defined by 1 and the martingale M definedby 3. We have the following proposition.

1 In this paper, we deal with the cum-dividend price process for defaultable bond: the recovery of the bondreceived at default time τ will be re-invested in the saving account (see 22 below).

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264 L. Bo et al.

Proposition 1 The P-dynamics of the cum-dividend price process Y = (Yt ; t ≥ 0)

is given by

dYt = rYt dt + λρηYt Zt dt − C Zt dt − ρYt−dMt , (4)

where the constant r ≥ 0 is the spot risk-free interest rate, η:=1η − 1 ≥ 0 andZt :=1 − Zt = 1τ>t for all positive t .

Recalling that the spot interest rate is r ≥ 0, the corresponding money marketaccount ζ = (ζt ; t ≥ 0) is given by

ζt = ert , t ≥ 0. (5)

Finally the P-price dynamics of a default-free risky asset is assumed to obey a geo-metric Brownian motion:

dξt = bξt dt + aξt dWt , ξ0 > 0, (6)

where b ∈ R is the return rate and a > 0 denotes the volatility.Now let Xt denote the total wealth at time t , and κt and �t are the proportions at

time t in the wealth of Y = (Yt ; t ≥ 0) and ξ = (ξt ; t ≥ 0) respectively. Accordingly1 − κt − �t is the time-t proportion in the wealth of ζ = (ζt ; t ≥ 0). We assume thatthe consumption ratio at time t is ct (ct Xt is the consumption rate at time t). Usingthe self-financing strategy, the P-dynamics of the wealth process is given by

dXt = [1 − κt − �t ]Xt

ζtdζt + κt Xt−

Yt−dYt + �t Xt

ξtdξt + Cκt Xt Zt

Ytdt − ct Xt dt

= Xt

[r − ct + (b − r)�t + λρηκt Zt

]dt + aXt�t dWt − ρκt Xt−dMt , (7)

X0 = x > 0,

where x > 0 denotes the initial wealth held by the investor. Moreover, by Itô’s formulawith jump (see Protter 1990, Theorem 37), the total wealth Xt at time t admits thefollowing unique explicit form:

Xt = x exp

⎡⎣

t∫

0

r − cs + (b − r)�s + λρ

ηκs Zsds

⎤⎦

× exp

⎡⎣

t∫

0

a�sdWs − 1

2

t∫

0

a2�2s ds

⎤⎦∏

s≤t

[1 − ρκs�Zs] . (8)

Since �Zt ∈ {0, 1} for all t ≥ 0, we conclude that the solvency constraint Xt > 0holds as long as κt < 1/ρ for all t ≥ 0.

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3 Optimal Investment and Consumption Under HARA Utility

The purpose of this section is to find the optimal allocation pair (κt , �t ; t ≥ 0) andthe optimal consumption ratio (ct ; t ≥ 0) to maximize the infinite horizon expecteddiscounted HARA utility of consumption. We will provide the analytic expressions foroptimal control policy (κt , �t , ct ; t ≥ 0) and the corresponding value function underHARA utility: Log utility and non-Log utility.

3.1 Pre-Default and Post-Default Value Functions and HJB Equations

In this subsection, we introduce the definitions of pre-default and post-default valuefunctions and the corresponding HJB equations.

Next, we restrict the allocation pair (κt , �t ; t ≥ 0) and the consumption ratio(ct ; t ≥ 0) to be in an admissible control space A(G).

Definition 2 A r.c.l.l. G-adapted stochastic control policy (κt , �t , ct ; t ≥ 0) is in theadmissible control space A(G), if there is a finite constant K such that:

κt ∈ [0, 1/ρ), |�t | ≤ K , ct > 0, ∀t ≥ 0.

Remark 3 The condition κ· ∈ [0, 1/ρ) in Definition 2 refers to avoiding bankruptcy(see also Jang (2005) Remark 5.2.4). We call each element belonging to A(G) abankruptcy avoiding portfolio. On the other hand, if there exists a control policy(κt , �t , ct ; t ≥ 0) ∈ A(G), then the wealth process X = (Xt ; t ≥ 0) admits thesolvency constraint condition, i.e., Xt > 0 for all positive t .

Toward this end, we introduce value functions in term of HARA utility. Let U (x)

be a HARA type utility function defined on R+. For an admissible control policy(κt , �t , ct ; t ≥ 0) ∈ A(G) and an initial pair (x, z) ∈ R+ × {0, 1}, define thefollowing objective functional by

J(x, z; κ., �., c.) = E(x,z)

⎡⎣

∞∫

0

e−αtU (ct Xt )dt

⎤⎦

:= E

⎡⎣

∞∫

0

e−αtU (ct Xt )dt∣∣∣X0 = x, Z0 = z

⎤⎦ ,

where α > 0 is the discount factor. The corresponding value function under consid-eration is then

V(x, z) = max(κ.,�.,c.)∈A(G)

J(x, z; κ., �., c.). (9)

Hence the pre-default and post-default value functions are respectively given by

V0(x) = V(x, 0), (pre-default value function);

and

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266 L. Bo et al.

V1(x) = V(x, 1), (post-default value function).

Using the dynamic programming principle, we have the following pre-default andpost-default HJB equations respectively:

αV0 = r xV

0x + λ · max

κ∈[0, 1ρ)

[κ

ρ

ηxV

0x + V

1(x − xκρ) − V0(x)

]

+ max�∈R

[(b − r)�xV

0x + 1

2a2�2x2

V0xx

](10)

+ maxc∈R+

[U (cx) − cxV

0x

],

and

αV1 = r xV

1x + max

�∈R

[(b − r)�xV

1x + 1

2a2�2x2

V1xx

]

+ maxc∈R+

[U (cx) − cxV

1x

]. (11)

Remark 4 Observe the forms of HJB equations 10 and 11. We find that the pre-defaultHJB equation (10) associated to the pre-default value function V

0 also depends on thepost-default value function V

1. This is different from the HJB equation correspondingto the value function without default risk (see Merton 1969, 1971, 1992).

3.2 Log Utility Case

In this subsection, we concentrate on the following Log HARA utility. In other words,the utility function U is given by

U (x) = log x, where x > 0.

Under the Log-utility case, it is not hard to verify that the value function V admitsthe property: for all (x, z) ∈ R+ × {0, 1},

V(x, z) = 1

αlog x + V(1, z),

where α > 0 is the discount rate. Appealing to this property, we can first solve the post-default HJB equation 11 and then solve the pre-default HJB equation 10 explicitly.The corresponding analytic value functions are given by the following lemma.

Lemma 5 Recall that a > 0 is the volatility rate, b ∈ R is the return rate of thedefault-free risky asset, r ≥ 0 denotes the risk-free spot interest rate, λ > 0 is the

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arrival intensity of intensity and 1/η ≥ 1 denotes the default risk premium. Then thepost-default value function is

V1(x) = 1

αlog x + W

1, x > 0,

where the constant W1 is given by

W1 = 1

α2

[(b − r)2

2a2 + r

]+ log α − 1

α. (12)

Moreover, the pre-default value function has the form:

V0(x) = 1

αlog x + W

0, x > 0,

where the constant W0 is given by

W0 = W

1 + λ

α(α + λ)

[1

η+ log η − 1

]. (13)

Remark 6 By virtue of the relationship between W0 and W

1 given by 13, we find thatif there is no default, i.e., λ = 0 or the jump risk premium 1/η = 1, then it holds thatW

0 = W1 and hence V

0(x) = V1(x). This implies that the value function

V(x, z) = (1 − z)V0(x) + zV1(x)

= V1(x),

which reduces to the case without default risk. However in a general case where thedefault risk exists, we have the pre-default value function V

0(x) is always greater thanor equal to the post-default value function V

1(x) for each fixed initial wealth x > 0.Moreover, the function

V(x, z):= 1

αlog x +

[zW1 + (1 − z)W0

], where x > 0, and z = 0, 1, (14)

is the nontrivial solution to the HJB equation associated with the value function V(x, z).

Finally we give a G-adapted stochastic control policy (κLt , �Lt , cLt ; t ≥ 0) expressedas

κLt ={

1−ηρ

, 0 ≤ t ≤ τ,

0, t > τ.(15)

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where ρ ∈ (0, 1) denotes the loss rate given default. The other two controls are givenby

�Lt = b − r

a2 , and cLt = α, t ≥ 0. (16)

We shall show that the above stochastic control policy is admissible and optimal, andthat the value function is nothing but V(x, z) given by 14.

3.3 Non-Log Utility Case

In contrast to the Log utility case, this subsection concerns the following non-LogHARA utility

U (x) = 1

γxγ ,

where x > 0 and γ ∈ (0, 1).The value function V corresponding to the above utility admits the following homo-

theticity property (see Shreve and Soner 1994, Proposition 3.3 or Fleming and Pang2004, Lemma 2.3):

V(x, z) = xγV(1, z), where x > 0, z = 0, 1.

Accordingly, we have the following forms concerning the post-default value functionand pre-default value function respectively

Lemma 7 The post-default value function is given by

V1(x) = 1

γxγ

W1γ , x > 0,

where the constant W1γ is

W1γ =[α − rγ

1 − γ− γ (b − r)2

2a2(1 − γ )2

]γ−1

. (17)

Moreover, the pre-default value function admits the form

V0(x) = 1

γxγ

W0γ , x > 0,

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where the constant W0γ is given by

W0γ =

⎡⎢⎢⎣[W

1γ

] 1γ−1 + λ(η−γ )

η(1−γ )

1 + ληγ

1−γ

[W1

γ

] 11−γ

⎤⎥⎥⎦

γ−1

. (18)

By virtue of Lemma 7,

V(x, z) = 1

γxγ[zW1

γ + (1 − z)W0γ

], x > 0, z = 0, 1 (19)

is the nontrivial solution to the HJB equation associated with the value function V inthe case of non-Log utility.

Finally we give a G-adapted stochastic control policy (κγt , �

γt , cγ

t ; t ≥ 0) expressedas

κγt =

⎧⎪⎨⎪⎩

1ρ

[1 −[

W0γ

ηW1γ

] 1γ−1]

, 0 ≤ t ≤ τ

0, t > τ.

(20)

where ρ ∈ (0, 1) denotes the loss rate given default. The other two controls are givenby

�γt = b − r

a2[1 − γ ] , and cγt =

⎧⎪⎨⎪⎩

[W

0γ

] 1γ−1

, 0 ≤ t < τ[W

1γ

] 1γ−1

, t ≥ τ.

(21)

We shall show that the above stochastic control policy is admissible and optimal, andthat the final value function is nothing but V(x, z) given by 19.

Remark 8 (1) Remark 6 also holds in the non-Log utility case. (2) It is known that theLog utility function corresponds to the HARA utility function in the limiting case ofrelative risk aversion 1 − γ equal to 1, that is in the limiting case of γ = 0. We canverify that the stochastic control strategy (κ

γt , �

γt , cγ

t ; t ≥ 0) in the case of non-Logutility will approach the policy (κLt , �Lt , cLt ; t ≥ 0) in the case of the Log utility asγ → 0. In fact, as γ → 0, we have W

1γ → 1/α, W

0γ → 1/α, and thus

κγt → κLt , �

γt → �Lt , cγ

t → cLt .

4 Verification Theorems

In this section, we present two verification theorems corresponding to the respectiveLog-utility and non-Log utility cases. The first verification theorem shows that the

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270 L. Bo et al.

stochastic control policy (κLt , �Lt , cLt ; t ≥ 0) given by 15–16 is optimal and thefunction V given by 14 is just the value function V defined by 9 in the Log-utility case.The second one proves that the stochastic control policy (κ

γt , �

γt , cγ

t ; t ≥ 0) givenby 20–21 is optimal and the function V given by 19 is just the value function 9 in thenon-Log utility case.

Theorem 9 (Log utility case) We have

(i) For all admissible control policies (κt , �t , ct ; t ≥ 0) ∈ A(G), it holds that

V(x, z) ≥ E(x,z)

⎡⎣

∞∫

0

e−αtU (ct Xt )dt

⎤⎦ , x > 0, z = 0, 1, and

(ii) The stochastic control policy (κLt , �Lt , cLt ; t ≥ 0) ∈ A(G) and it holds that

V(x, z):=E(x,z)

⎡⎣

∞∫

0

e−αtU(cLt · XL

t

)dt

⎤⎦ = V(x, z), x > 0, z = 0, 1,

where XL = (XLt ; t ≥ 0) denotes the wealth process satisfying (7) with

(κt , �t , ct ; t ≥ 0) replaced by (κLt , �Lt , cLt ; t ≥ 0).

In what follows we assume the following conditions hold:

(A1)γ (b − r)2

2a2(1 − γ )+ γ r < α,

(A2)[W

1γ

] 1γ−1 + λ(η − γ )

η(1 − γ )> 0,

(A3)(b − r)2

a2(1 − γ )+ λ

(1

η− 1

)+ r < α.

A sufficient condition for the condition (A2) is γ ≤ η. The conditions (A1) and (A3)can be satisfied if the discounted factor α > 0 is taken relatively large (see the finalnumerical section).

Theorem 10 (Non-Log utility case) Let the conditions (A1) and (A2) be satisfied. Wehave

(i) For all admissible control policies (κt , �t , ct ; t ≥ 0) ∈ A(G), it holds that

V(x, z) ≥ E(x,z)

⎡⎣

∞∫

0

e−αtU (ct Xt )dt

⎤⎦ , x > 0, z = 0, 1, and

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Table 1 Parameter definitionsand preferred values

Symbol Definition Value

r Risk free interest rate 0

b Return rate of default-free risky asset 0.06765

a Volatility of default-free risky asset 0.15

α Discount factor 1

γ Risk aversion parameter 0.15

η Inverse of default risk premium 1/2.53

ρ Loss rate 1

λ Default intensity 0.013

(ii) Moreover, if the condition (A3) holds, we have

V(x, z):=E(x,z)

⎡⎣

∞∫

0

e−αtU(cγ

t Xγt)

dt

⎤⎦ = V(x, z), x > 0, z = 0, 1,

where Xγ = (Xγt ; t ≥ 0) denotes the wealth process (7) with (κt , �t , ct ; t ≥ 0)t≥0

replaced by (κγt , �

γt , cγ

t ; t ≥ 0).

The proof of Theorem 10 will be given in Appendix. Since the proof of Theorem 9is very similar and much simpler than that of Theorem 10 (see Bo et al. 2010), weomit the details.

5 Sensitivity Analysis

In this section, we carry out a sensitivity analysis on the optimal control policy(κ

γt , �

γt , cγ

t ; t ≥ 0) and the corresponding value function V in Theorem 10. Through-out this section, we adopt the following preferred values (see, e.g., Bielecki and Jang2006):

Recall (21). The optimal control �γt = b−r

a2(1−γ )has an obvious representation con-

cerning γ ∈ (0, 1). Hence we here discuss the parameter sensitivity of the control(κ

γt , cγ

t ; t ≥ 0) only (the preferred parameter values are given in Table 1).

5.1 Optimal Control (κγt , cγ

t ; t ≥ 0)

We now discuss the parameter sensitivity of the optimal control (κγt , cγ

t ; t ≥ 0) inthe pre-default case. Recall that

κγt = 1

ρ

⎡⎣1 −[

W0γ

ηW1γ

] 1γ−1⎤⎦ , 0 ≤ t < τ.

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272 L. Bo et al.

LR=0.5

LR=0.6

LR=0.7

0 1 2 3 4

DI

0.9

1.0

1.1

1.2

1.3

1.4

1.5O

IPD

B

DRP=2.53

DRP=1.30

DRP=1.00

0.2 0.4 0.6 0.8 1.0

LR

0

1

2

3

4

OIP

DB

Fig. 1 Left Optimal investment proportion of the defaultable bond (OIPDB) versus Default Intensity (DI)as the Loss Rate (LR) is 0.50, 0.60, 0.70. Right OIPDB versus LR as DRP is 1.00, 1.30, 2.53

0.20.4

0.60.8

1.0IDRP

0.2 0.4 0.6 0.8 1.0

LR

0

1

2

3

4

OIP

DB

Fig. 2 OIPDB versus IDRP and LR

At first, we analyze the relationship between default intensity λ and κγ . Since ahigher default intensity leads to a high yield before the default occurs, so we guessthat there is a positive relationship between default intensity and κγ . The left panelof Fig. 1 shows that κγ increases as the default intensity increases. This verifies ourconjecture. We also note that the slope of these curves decreases as the default intensityincreasing.

Second, we analyze the relationship between the loss rate ρ and κγ . Since a higherloss rate induces a higher potential loss, the investors will reduce their investmentproportion of the defaultable bond. The right panel of Fig. 1 depicts a negative rela-tionship between the loss rate and κγ , which accords with our conjecture. Figure 2provides a full description for κγ versus the Inverse of Default Risk Premium (IDRP)η and the loss rate ρ.

Third, we investigate the relationship between the risk aversion parameter γ andκγ . Since the utility function has a constant Pratt’s measure of relative risk aversion1 − γ (see, e.g., Jang 2005). This implies that the investors with less risk aversionparameter detest risk much more and thus will reduce their investment proportion ofthe defaultable bond. Figure 3 depicts a positive relationship between the risk aversionparameter and κγ , which accords with our conjecture.

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0.1 0.2 0.3 0.4

RAP

1.2

1.3

1.4

1.5

OIP

DB

Fig. 3 OIPDB versus Risk Aversion Parameter (RAP)

0.0050.010

0.0150.020

0.025

DI

0.2

0.4

0.60.8

1.0

IDRP

1.140

1.145

1.150

1.155

1.160

OC

R

Fig. 4 Optimal consumption ratio (OCR) versus IDRP and DI

Finally, we consider the optimal consumption ratio cγ in the pre-default case. Recallthat

cγt =[W

0γ

] 1γ−1

, 0 ≤ t < τ.

Since a smaller IDRP (a larger DRP) leads to a higher yield by investing in the default-able bond, the investors will reduce their consumption. The case for DI is similar.Figure 4 depicts cγ versus IDRP and DI, which consistents with intuition.

5.2 Value Function

In this subsection, we discuss the value function V given by 19. Recall that

V(x, z) = 1

γxγ[zW1

γ + (1 − z)W0γ

], x > 0, z = 0, 1.

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274 L. Bo et al.

0.005

0.010

0.015

0.020

0.025

DI

0.2

0.4

0.6

0.8

1.0

IDRP

5.90

5.92

5.94

VF

Fig. 5 The value function (VF) versus IDRP and DI

Figure 5 depicts V versus IDRP and DI. Since a higher DI indicates a (global) higherrisk, a risk aversion investor will tend to consume rather than invest in the defaultablebond. That in turn results in a higher value for the value function. Similarly, a smallerIDRP (a higher DRP) also indicates a higher risk, so the value function decreases asIDRP increasing.

6 Concluding Remarks

In this paper, we consider an optimal investment and consumption problem in a default-able market. The investor dynamically allocates his or her wealth among a perpetualdefaultable bond, a money market account and a default-free risky asset. The goal is tomaximize the infinite horizon expected discounted HARA utility of the consumption.Optimal investment and consumption policies as well as the value function are explic-itly derived. Numerical results are consistent with intuition. An interesting extensionmay be introducing the dependence between the default time and the risky asset.

Acknowledgments We are very grateful to two anonymous referees for providing us valuable commentsand suggestions. This work was partially supported by the NSF of China (Nos. 11001213, 71201074,70932003). The research of Bo was also supported by NCET-12-0914 and the Fundamental ResearchFunds for the Central Universities (No. K5051370001). The research of Yang was also supported by theFundamental Research Funds for the Central Universities (No. 1127011812).

Appendix: Proofs

Proof of Proposition 1 The cum-dividend price of the defaultable perpetual bond thatpays constant coupon C > 0 per unit time under the recovery scheme of market valuehas the following risk-neutral representation:

Yt = Zt EQ

⎡⎣

∞∫

t

Ce−r(s−t) Zsds + (1 − ρ)e−r(τ−t)Yτ−

∣∣∣∣∣∣Gt

⎤⎦+ Zt (1 − ρ)er(t−τ)Yτ−,

(22)

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where EQ is the risk-neutral expectation corresponding to risk-neutral probabilitymeasure Q. Note that it holds that

ZtEQ

[(1 − ρ)e−r(τ−t)Yτ−|Gt

]= Zt

EQ

[(1 − ρ)e−r(τ−t)Yτ−|Ft

]St

= Zt λ(1 − ρ)

∞∫

t

e−(r+λ)(s−t)Ysds,

where we used (2) in the second equality. Moreover, since

EQ

⎡⎣

∞∫

t

Ce−r(s−t) Zsds

∣∣∣∣∣∣Gt

⎤⎦ = Zt

∞∫

t

Ce−(r+λ)(s−t)ds,

it is easy to show that

dYt = rYt dt − Cdt, if Zt = 1 (Zt = 0), (23)

where the adjusted interest rate r = r + ρλ. So that

Yt = Zt

[C

r+(

Y0 − C

r

)er t]

+ Zt

[C

r+(

Y0 − C

r

)erτ

](1 − ρ)er(t−τ). (24)

Noting that

Mt :=Zt − λ

t∫

0

(1 − Zs)ds, t ≥ 0,

is a (Q, G)-martingale. Then P-dynamics of the cum-dividend price follows fromapplying Itô’s formula to 24.

Prior to presenting the proof of Theorem 10, we first state the following lemma.

Lemma 11 Let the condition (A3) be satisfied. Then it holds that

limt→+∞ e−αt

Ex [Xγ

t] = 0.

Proof of Lemma 11 Recall from (7) that

dXγt = Xγ

t

[r − cγ

t + (b − r)�γt + λρηκ

γt Zt

]dt

+ a�γt Xγ

t dWt − ρXγt−κ

γt Zt−dMt .

123

276 L. Bo et al.

For each positive t , define

Mγt =

t∫

0

a�γs Xγ

s dWs −t∫

0

ρXγs−κ

γs Zs−dMs,

and a random function

f (t; x) =[r − cγ

t + (b − r)�γt + λρηκ

γt Zt

]· x, x > 0.

Then

Xγt = X0 + Mγ

t +t∫

0

f (s; Xγs )ds.

A direct calculation shows that f (t; ·) is continuous on R+ for each fixed x > 0, and

C2 ≤ fx (t; x):=∂ f

∂x(t; x) ≤ C1, f (t; 0) = 0,

for all (t, x) ∈ R2+, where the constants

C1 = r + (b−r)2

a2(1−γ )+ λη, and C2 = r + (b−r)2

a2(1−γ )−(W

1γ ∧ W

0γ

) 1γ−1

.

Note that Xγt > 0 for all t ≥ 0. For n ∈ N, define

τn = inf{t ≥ 0; Xγ

t ≥ n}.

By virtue of (3), Hölder’s inequality and Burkhölder–Davis–Gundy (BDG) inequality,it follows that for every finite time T > 0,

Ex

[sup

s≤T ∧τn

∣∣Xγs∣∣2]

≤ 2C1T Ex

⎡⎣

T ∧τn∫

0

∣∣Xγt

∣∣2 dt

⎤⎦+ 2E

x

[sup

t≤T ∧τn

∣∣X0 + Mγt

∣∣2]

≤ 4x2 + 2C1T Ex

⎡⎣

T ∧τn∫

0

∣∣Xγt

∣∣2 dt

⎤⎦

+C2Ex

⎡⎣ ∑

t≤T ∧τn

ρ2∣∣Xγ

t−∣∣2 ∣∣κγ

t

∣∣2 �Zt

⎤⎦

= 4x2 + 2C1T Ex

⎡⎣

T ∧τn∫

0

∣∣Xγt

∣∣2 dt

⎤⎦

123


+C2Ex

⎡⎣

T ∧τn∫

0

∣∣Xγt

∣∣2 (a2|�γt |2 + λρ2|κγ

t |2)

dt

⎤⎦

≤ 4x2 + �1 ·T ∧τn∫

0

Ex[

sups≤t

∣∣Xγs∣∣2]

dt,

where C1, C2, �1 are positive constants depending only on T . Note that τn → ∞, asn → ∞. Applying Fatou’s lemma and Monotone Convergence Theorem for n → ∞,Gronwall’s lemma yields that

Ex[

sups≤t

∣∣Xγs∣∣2]

≤ 4x2e�1T , ∀ 0 ≤ t ≤ T . (25)

We derive thanks to (25) that, Mγ = (Mγt ; t ≥ 0) is a (P, G)-adapted r.c.l.l martingale.

By using a similar procedure of Pang[2006, Lemma 3.1], we deduce

Ex [Xγ

T

] ≤ �2eC1T ,

where �2 is a positive constant which is independent of T . As a consequence

limT →∞ e−αT

Ex [Xγ

T

] = 0,

for all α > C1. Thus the proof is concluded by recalling the definition of C1 aboveand the condition (A3). Proof of Theorem 10 For all (κt , �t , ct ; t ≥ 0) ∈ A(G), Itô formula with jumps (seeProtter 1990) yields that

dV(Xt , Zt ) = Xt Vx (Xt , Zt )[r − ct + �t (b − r) + λ(ρ/η)κt Zt

]dt

+1

2a2 X2

t �2t Vxx (Xt , Zt )dt

+λ[V(Xt − ρXtκt Zt , Zt + 1) − V(Xt , Zt )

]Zt dt + dNt , (26)

where X = (Xt ; t ≥ 0) is the wealth process defined by 7 and N = (Nt ; t ≥ 0) is ar.c.l.l (P, G)-local martingale defined by

Nt = a

t∫

0

XsVx (Xs, Zs)�sdWs

+t∫

0

[V(Xs− − ρXs−κs Zs−, Zs− + 1) − V(Xs−, Zs−)

]dMs .

123

278 L. Bo et al.

Recall the definition of V given by 19. Then for x > 0 and z = 0, 1,

xVx (x, z) = xγ[zW1

γ + (1 − z)W0γ

],

x2Vxx (x, z) = (γ − 1)xγ

[zW1

γ + (1 − z)W0γ

],

and

[1 − ρκt Zt

]γZt = [1 − ρκt ]γ Zt .

Consequently under the conditions (A1) and (A2),

e−αTV(XT , ZT ) − V(x, z) −

T∫

0

e−αt dNt

=T∫

0

e−αt dV(Xt , Zt ) −T∫

0

αe−αtV(Xt , Zt )dt −

T∫

0

e−αt dNt

=T∫

0

e−αt{

Zt Xγt

[r − ct + �t (b − r) + 1

2a2(γ − 1)�2

t

]W

1γ

+Zt Xγt

[(r − ct + �t (b − r) + ρλ

ηκt + 1

2a2(γ − 1)�2

t

)W

0γ − λ

γW

0γ

]

+Zt Xγt

λ

γ(1 − ρκt )

γW

1γ − αV(Xt , Zt )

}dt

≤T∫

0

e−αt[

Zt Xγt

(α

γW

1γ − 1

γcγ

t

)+ Zt Xγ

t

(α

γW

0γ − 1

γcγ

t

)− αV(Xt , Zt )

]dt

=T∫

0

e−αt[α

(Zt

1

γXγ

t W1γ + Zt

1

γXγ

t W0γ

)− 1

γXγ

t

(Zt cγ

t + Zt cγt

)− αV(Xt , Zt )

]dt

= −T∫

0

e−αt U (ct Xt )dt. (27)

Let τR be the first exit time of the wealth process X from BR = {x ∈ R : |x | ≤ R}.Then for each T > 0,2

2 Noting that the positive process X can only jump downward, we have supt≤τR|Xt | ≤ R. It follows that

(Nt∧τR ; t ≥ 0) is a martingale.

123


V(x, z) ≥ E(x,z)

⎡⎣

T ∧τR∫

0

e−αtU (ct Xt )dt

⎤⎦+ E

(x,z)[e−αT ∧τR V(XT ∧τR , ZT ∧τR )

].

(28)

Let R → ∞ and so τR → ∞. Then

V(x, z) ≥ E(x,z)

⎡⎣

T∫

0

e−αtU (ct Xt )dt

⎤⎦ ,

which holds for all admissible control (κt , �t , ct ) ∈ A(G). Hence the conclusion (i)follows by letting T → ∞.

Note that (κγt , �

γt , cγ

t ; t ≥ 0) ∈ A(G), instead of (28), we have

V(x, z) = E(x,z)

⎡⎣

T ∧τR∫

0

e−αtU(cγ

t Xγt)

dt

⎤⎦+ E

(x,z)[e−αT ∧τR V

(Xγ

T ∧τR, ZT ∧τR

)].

(29)

Using Monotone Convergence Theorem, for each fixed T > 0, it follows that

limR→∞ E

(x,z)

⎡⎣

T ∧τR∫

0

e−αtU(cγ

t Xγt)

dt

⎤⎦ = E

(x,z)

⎡⎣

T∫

0

e−αtU(cγ

t Xγt)

dt

⎤⎦ . (30)

Moreover, by the definition (19) of V, we arrive at

e−αT ∧τR V

(Xγ

T ∧τR, ZT ∧τR

)

= 1

γe−αT ∧τR

∣∣∣Xγ

T ∧τR

∣∣∣γ[

ZT ∧τR W1γ + ZT ∧τR W

0γ

]

≤ �3 · sup0≤t≤T

∣∣Xγt

∣∣γ ,

where �3 is a positive constant which is independent of T and R. Noting the estimation(25) and applying Lebesgue’s Dominated Convergence Theorem to (29), we have

V(x, z) = E(x,z)

⎡⎣

T∫

0

e−αtU(cγ

t Xγt)

dt

⎤⎦+ E

(x,z)[e−αT

V(Xγ

T , ZT)]

. (31)

To prove (ii) under the additional condition (A3), it is sufficient to show

lim supT →∞

E(x,z)[e−αT

V(Xγ

T , ZT)] ≤ 0. (32)

123

280 L. Bo et al.

Using the definition of V again, we have

E(x,z)[e−αT

V(Xγ

T , ZT)] = 1

γe−αT

E(x,z)[∣∣Xγ

T

∣∣γ (ZT W1γ + ZT W

0γ

)]

≤ �4 · e−αTE

(x,z) [∣∣Xγ

T

∣∣γ ] ,where �4 > 0 is a constant independent of T . Thus apply Lemma 11 with the condition(A3) to conclude that (32) holds. Recall (31). Let T → ∞, and then

V(x, z) ≤ E(x,z)

⎡⎣

∞∫

0

e−αtU(cγ

t Xγt)

dt

⎤⎦ = V(x, z).

Since V(x, z) ≥ V(x, z) by employing (i), it follows that (ii) holds. Thus we completethe proof of the verification theorem.

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