Hamilton-Jacobi-Bellman Equation of anOptimal Consumption Problem

Shuenn-Jyi Sheu

Institute of Mathematics, Academia Sinica

WSAF, CityU HK

June 29-July 3, 2009

1. Introduction

Xc,πt is the wealth with the consumption policy (c, π).

πt is a trading strategy, ctXc,πt is the rate of consumption.

We consider the optimal consumption problem

(1.1) sup(c,π)∈A

E[∫ ∞

0e−ρt

1γ(ctX

c,πt )γdt],

A is a family of admissible strategies.

ρ is the discount factor.

U(x) = 1γ(x)

γ is the HARA utility with parameter γ,

γ < 1, γ 6= 0.

Purpose: find an optimal strategy.

• We consider a model for the market (factor model).

Some economic factors affect the returns and volatilities

of the stocks.

• Dynamic programming approach is used.

The Hamilton-Jacobi-Bellman (HJB) equation is

derived.

• We study the solution of HJB equation.

A general existence result for the solutions of HJB

equation will be proved from the existence of a pair of

sub/super-solution (of HJB equation).

• We construct a suitable pair of sub/supersolution.

• We give the verification theorem.

The policy constructed from the solution is shown to be

optimal.

• We show that the solutions have different behaviors for

γ > 0 and γ < 0.

References

H.Hata and S.J.Sheu (2009), Hamilton-Jacobi-Bellman

equation for an optimal consumption problem, preprint.

H.Hata and S.J.Sheu (2009), An optimal consumption

and investment problem with linear Gaussian model,

preprint.

2. A Brief History

There are two kinds of investment problem that we can

find many discussions in the literature.

• Optimal consumption problem discussed in this paper.

• Optimization problem of expected utility of final wealth,

(2.1) supπ∈A

E[1γ(Xπ

T)γ],

We show an interesting relation of these two problems

following a recent result of Hata-Sheu.

Fleming-Hernandez (2005) is one of few examples

discussing such relation.

We start a brief review of the studies of these two problems

in the literature.

In Merton(1969), the following problem is discussed,

supE[∫ T

0e−ρtU(C(t))dt+B(X(T ), T )].

C(t) is the consumption rate, X(t) is the wealth process.

U,B are utility functions.

R.C. Merton (1969), Lifetime portfolio selection under uncertainty:

the continuous time case, The Review of Economics and Statistics,

Vol 51, 246-257 ( with 1475 citations).

In Cox-Huang (1986), a similar problem to Merton (1969)

is considered for general complete markets.

J. C. Cox and C.F. Huang (1989), Optimal consumption and

portfolio policies when the asset prices follow a diffusion process, J.

Economic Theory, Vol 49, 33-83(with 690 citations).

In Pliska(1986), the following problem is considered for

the complete market with general diffusion model,

supπE[U(Xπ(T ))].

S. Pliska(1986), A stochastic calculus model of continuous trading:

optimal portfolios, Math. Operation Research Vol.11, 371-382 (with

218 citations)

The developments in Pliska(1986) and Cox-Huang

(1989) start the use of stochastic calculus, martingale

representation theorem and duality argument.

Using their approach, the optimal solution can be explicitly

calculated without solving PDE.

This is very interesting and is also a reason for receiving

so many citations.

Their method is different from the approach in Merton

(1969).

In Merton (1969), the solution is obtained by solving the

HJB (Hamilton-Jacobi-Bellman) equation. HJB is a PDE

(partial differential equation). For a simple model, the

equation can be solved explicitly. In general, it is difficult

to calculate the solution.

The idea in Pliska, Cox-Huang can not be applied to

incomplete markets.

Previous studies have suggested some possible directions

for research:

• Obtain an explicit solution by generalizing the idea of

Pliska and Cox-Huang.

• Study HJB equation.

In practice, it is also important to solve the equation

numerically, if an analytical solution is not possible.

In addition to these, there are also a huge number

of possible applications can suggest many interesting

problems.

The following show some models which have analytical

solution, Wachter(2002),Chacko-Viceira(2005),Jun Liu

(2007)

An initial attempt to use HJB equation in more

complicated models is first proposed in Fleming (1995).

The idea is to reformulate the investment problem

as a stochastic control problem. Then the dynamic

programming approach can be used to derive the HJB

equation. Solving HJB equation will provide a candidate

of optimal investment strategy.

This suggests several interesting questions for the solutions

of such HJB equations.

• Study the regularity, growth conditions of the solutions.

• Obtain suitable estimates for the solutions.

This is needed when we want to prove the candidate

of optimal portfolio derived from a solution is indeed

optimal (verification theorem).

To show how this works, Fleming-Sheu(1999) study a

simple model different from that of Merton(1969) and

provides a detailed analysis.

The model can be briefly described as follows

There is one stock and one banking account that an

investor can trade.

The interest rate for the banking account is constant

r > 0.

The price of stock is given by

Pt = exp(Lt),

dLt = c(µt+ α0 − Lt)dt+ σdWt.

It turns out that the model in Fleming-Sheu (1999), after

reformulation, becomes a special case of factor model.

y(t) = Lt − µt− (α0 −µ

c)

plays the role of factor.

Using this approach, the risk-sensitive portfolio

optimization problem for more general factor models have

been considered in a list of papers:

Fleming-Sheu (1999a, b), Fleming-Sheu(2002),

Kuroda-Nagai(2002), Nagai-Peng(2002), Nagai(2003),

Kaise-Sheu(2004), Hata-Sekine(2005), Bilecki-Pliska-

Sheu(2005)

An useful theorem about the structure of the solutions of

HJB equation is given in Kaise-Sheu(2006).

The studies mentioned above have interesting applications

to the minimization of down-side risk probabilities,

(2.2) minP (logXπ

T

T≤ k).

There is a duality relation between (2.2) and the risk

sensitive portfolio optimization problem (2.1) with a

particular risk sensitive parameter γ = γ(k) < 0.

The following is a list of papers,

Hata-Nagai-Sheu(2009), Hata-Sheu(2008), Hata(2008),

Nagai(2008, 2009)

The result is interesting because of the following reasons.

• The problem (2.2) is not a conventional optimization

problem and a direct solution is not available.

The result shows that we can solve (2.2) using a solution

of (2.1).

• HARA utility appears naturally from (2.2). Although

(2.2) seems to not have relation with utilty function at

the first look.

For this connection, see also works of Pham (2003) on

the maximization problem of up-side chance probabilities,

maxP (logXπ

T

T≥ c).

We remark that a recent work of Follmer-Schachermayer

(2008) seems to also relate to our study,

In this talk, we will discuss another application of the

study of the risk sensitive portfolio optimization problem

(2.1) to the consumption problem in (1.1).

This application seems to not be expected from the results

in the literature.

We are motivated by the three recent papers, Fleming-

Hernandez(2003, 2005), Fleming-Pang(2004).

Fleming-Pang (2004) considers a model that the stock

price is geometric Brownian motion and interest rate of

the banking account is random and is an ergodic 1-d

diffusion process.

We also use an approach similar to Fleming-Pang (2004).

We show that a solution of HJB for (2.1) can be used to

construct a supersolution for the HJB of (1.1). Then a

nice solution of the HJB for (1.1) can be obtained. From

this, (1.1) can be solved.

We develop some useful ideas for general factor models

with multiple stocks.

An attempt to use the duality argument similar to

that in Pliska(1986)and Cox-Huang(1989) is proposed

in Castaneda-Hernandez (2005)for general factor models.

A solution is given in the case of HARA utility. However,

the result for general utilities is not satisfactory.

References

Analytical Solution

• J.A. Wachter(2002), Portfolio and consumption decisions under

mean-reverting ruturns: an exact solution for complete markets, J.

Financ. Quant. Anal. 37, 63-91.

• Chacko, G. and Viceira, L.M.(2005), Dynamic consumption and

portfolio choice with stochastic volatility in incomplete markets,

Rev. Financ. Stud. 18, 1369-1402.

• Jun Liu (2007), Portfolio selection in stochastic enviroments, Rev.

Financ. Stud. 20, 1-39.

Risk Sensitive Portfolio Optimization

• Fleming, W.H.(1995),Optimal investment models and risk-sensitive

stochastic control. Mathematical Finance (Davis M, et al,ed),

Spring-Verlag, Berlin.

• W.H. Fleming and S. J. Sheu (1999), optimal long term growth

rate of expected utility of wealth, Ann. Appl. Probab., Vol 9,

871-903

• W.H. Fleming and S.J. Sheu (1999), Risk sensitive control and an

optimal investment model, Math. Finance 10, 197-213.

• W.H. Fleming and S.J. Sheu (2002), Risk sensitive control and an

optimal investment model II, Ann. Appl. Probab. 12, 730-767.

• K. Kuroda and H. Nagai (2002), risk sensitive portfolio optimization

on infinite time horizon, Stoch. Stoch. Report, 73, 309-331

• H. Nagai and S. Peng (2002), Risk sensitive portfolio optimization

with partial information on infinite time horizon, Ann. Appl.

Probab. 12, 173-195.

• H. Nagai (2003), Optimal strategies for risk-sensitive portfolio

optimization problems for general models, SIAM J. Cont. Optim.

41, 1779-1800.

• H. Kaise and S.J. Sheu (2004), Risk sensitive optimal investment:

solutions of the dynamical programming equation. In Mathematics

of Finance, Contemp. Math. 351, 217-230.

• H. Hata and J. Sekine (2005), Solving long term optimal

investment problems with Cox-Ingersol-Ross interest rates, Advance

in Mathematical Economics 8, 231-255.

• T.R. Bielecki, S. Pliska and S.J. Sheu (2005), Risk-sensitive

portfolio management with Cox-Ingersol-Ross interest rates:HJB

equation, SIAM J. Cont. Optim. 44, 1811-1843.

• H. Kaise and S. J. Sheu (2006), On the structure of solutions of

ergodic type Bellman equations related to risk-sensitive control,

Ann. Probab. 34, 284-320.

Down-side Risk probability

• H. Hata, H. Nagai and S.J. Sheu (2009), Asymptotics of probability

minimizing a down-side risk, to appear in Ann. Appl. Probab.

• H. Hata and S.J. Sheu (2008), Down-side risk probability

minimization problem for a multidimensional model with stochastic

volatility, preprint.

• H. Hata (2008), Down-side risk large deviations control problem

with Cox-Ingersoll-Ross interest rates, preprint.

• H. Nagai (2008), Asymptotics of the probability minimizing a

”down-side” risk under partial information, preprint.

• H. Nagai (2009), Down-side risk minimization as large deviation

control, preprint.

• H. Follmer and W. Schachermayer (2008), Asymptotic arbitrage

and large deviations, preprint.

Optimal Consumption

• W.H. Fleming and D. Hernandez-Hernandez (2003), An optimal

consumption model with stochastic volatility, Finance Stochastics,

7, 245-262.

• W.H. Fleming and T. Pang (2004), An application of stochastic

control theory to financial economics, SIAM J. Control Optim., 43,

502-531

• W.H. Fleming and D. Hernandez-Hernandez (2005), The tradeoff

between consumption and investment in incomplete markets, Appl.

Math. Optim., 52, 219-235.

• N. Castaneda, D. Hernandez-Hernandez (2005), Optimal

consumption-investment problems in incomplete markets with

stochastic coefficients, SIAM J. Cont. Optim., 44, 1322-1344.

3. Factor Model

There are N risky assets and a banking account.

Si(t) is the price of i-th asset, i = 1, 2, · · · , N .

dSi(t) = Si(t)(µi(y(t))dt+ σ(i)P (y(t)) · dB(t)),

The banking account has interest rate r(y(t)).

y(t) = (y1(t), · · · , ym(t)) is the factor process

dy(t) = b(y(t))dt+ σF(y(t))dB(t).

The investment strategy is given by

π(t) = (π1(t), π2(t), · · · , πN(t)),

πi(t) is the proportion of wealth in i-th asset.

1−N∑i=1

πi(t)

is the proportion of wealth in banking account.

The dynamics of the wealth is given by

dXπ(t) = Xπ(t)(N∑

i=1

πi(t)dSi(t)Si(t)

+ (1−N∑

i=1

πi(t))dS0(t)S0(t)

).

dXπ(t) = Xπ(t)((∑

i πi(t)(µi(y(t))dt+ σ(i)P (y(t)) · dB(t))

+(1−∑N

i=1 πi(t))r(y(t))dt).

dXπ(t) = Xπ(t)(∑

i πi(t)µi(y(t)) + r(y(t)))dt+πi(t)σ

(i)P (y(t)) · dB(t)).

Here

µi(y) = µi(y)− r(y).This can be solved,(3.1)Xπ(t) = x exp(

∫ t

0πi(s)σ

(i)P (y(s)) · dB(s)

−12

∫ t

0|σP (y(s))∗π(s)|2ds+

∫ t

0(∑

i πi(s)µi(y(s)) + r(y(s)))ds),

σP(y) is the matrix with columns σ(i)P .

When there is a consumption, the dynamics for the wealthprocess becomes

dXπ,c(t) = Xπ,c(t){(∑

i πi(t)µi(y(t)) + r(y(t))− c(t))dt+πi(t)σ

(i)P · dB(t)}.

Here c(t) is the consumption rate at time t.(3.2)Xπ,c(t) = x exp(

∫ t

0πi(s)σ

(i)P (y(s)) · dB(s)− 1

2

∫ t

0|σP (y(s))∗π(s)|2ds

+∫ t

0(∑

i πi(s)µi(y(s)) + r(y(s)))− c(s))ds),

We assume the following conditions.

(A1) µi, r, σP , σF are bounded smooth with bounded

derivatives of any order.

(A2) There are constants c1, c2 > 0 such

c1 ≤ σPσ∗P(y) ≤ c2, c1 ≤ σFσ

∗F(y) ≤ c2, y ∈ Rn.

(A3) b has bounded derivatives. There are k1, k2 > 0 such

that

y∗b(y) ≤ −k1|y|2 + k2, y ∈ Rn.

4. Optimization of Expected Utility ofFinal Wealth

Let T be fixed and 0 ≤ t ≤ T . Define

(4.1) φ(t, y, x) = supπE[(Xπ(T ))γ], 0 < γ < 1,

φ(t, y, x) = infπE[(Xπ(T ))γ], γ < 0,

when Xπ(t) = x, y(t) = y. Then

φ(t, y, x) = xγ exp(W (t, y)).

To derive an equation for W (t, y), let Xπt = x, we write

(Xπ(T ))γ = ζπt,Txγ exp(

∫ T

t

`(y(t), π(t))dt),

`(y, π) =12γ(1− γ)|σP(y)∗π|2 + γπ∗µ(y) + γr(y),

ζπt,T = exp(γ∫ T

t πi(t)σ(i)P (y(s)) · dB(s)

−12γ

2∫ T

0 |σP(y(s))∗π(t)|2ds).Under some conditions, ζπt,T is a probability density. Then

E[(XπT)

γ] = xγEPπ[exp(

∫ T

t

`(y(t), π(t))dt)],

dP π

dP|FT

t= ζπt,T .

Under P π, y(t) satisfies the equation,(4.2)dy(s) = (b(y(s))+γσF (y(s))σP (y(s))∗π(s))ds+

∑i

σF (y(s))dBπ(s),

Bπ(s) = B(s)−B(t)− γ

∫ s

t

σP (y(u))∗π(u))du,

is a Brownian motion under P π.

Therefore,

(4.3) exp(W (t, y)) = supπEP π

t,y [exp(∫ T

t

`(y(s), π(s))ds)]

By a standard argument, the stochastic control problemwith criterion (4.3) and the dynamics (4.2) for thecontrolled process has HJB equation given in the following.

supπ{dW (t,y)

dt + 12a

ij(y)DijW (t, y) + 12a

ij(y)DiW (t, y)DjW (t, y)+(b(y) + γσF (y)σP (y)∗π)DW (t, y) + `(y, π)} = 0.

After simplification,(4.4)dW (t, y)

dt+

12aij(y)DijW (t, y) + b

(γ)E (y,W (t, y), DW (t, y)) = 0,

where

aij(y) = aijF (y) = σFσ∗F(y),

b(γ)E (y, w, p) =

m∑i=1

b(γ)i (y)pi +

12

m∑ij=1

aijγ (y)pipj + V (γ)(y),

b(γ)(y) = b(y) +γ

1− γσFσ

∗Pa

−1P µ(y),

aγ = aF +γ

1− γσFσ

∗Pa

−1P σPσ

∗F ,

V (γ)(y) =γ

2(1− γ)µ(y) ∗ a−1

P µ(y) + γr(y)

aF = σFσ∗F , aP = σPσ

∗P .

By assuming the separation of variables,

W (t, y) ∼ Λ(T − t) +W (y), T →∞,

we arrive at the equation,

(4.5)12aij(y)DijW (y) + b

(γ)E (y,W (y), DW (y)) = Λ

This is the HJB equation to optimize the long term growth

rate,

(4.6) Λ = supπ

lim supT→∞

1T

logE[(Xπ(T ))γ], 0 < γ < 1,

(4.7) Λ = infπ

lim infT→∞

1T

logE[(Xπ(T ))γ], γ < 0,

A solution of (4.5) (given by a pair (Λ,W )) gives a

candidate of the optimal strategy for the problem (4.6) or

(4.7).

(4.8) u∗(y) =1

1− γa−1P (y)(µ(y) + σP(y)σ∗F(y)∇W (y)),

(4.9) π∗(t) = u∗(y(t)).

Theorem 4.1. (Theorem 7 in Kaise-Sheu(2004)).

Assume (A1)∼(A3). Then for any γ < 1, γ 6= 0, there

is Λ∗(γ) such that for Λ ≥ Λ∗(γ), (4.5) has a smooth

solution.

For Λ = Λ∗(γ), (4.3) has a unique solution W ∗γ up to

the addition of constants. The following properties hold.

(a) For any δ, β > 0, there is a constant cβ,δ such that

|W ∗γ (y)| ≤ δ|y|β + cβ,δ,

|DW ∗γ (y)| ≤ δ|y|β + cβ,δ.

Λ∗(γ) is the value for the problem (4.6) or (4.7).

(c) π∗t defined by (4.9) is an optimal portfolio.

5. Optimal Consumption Problem

We consider the consumption problem,

VC(x, y) = supπ,c

E[∫ ∞

0exp(−ρt)1

γ(c(t)Xπ,c(t))γdt].

For a consumption policy (π, c),

Xπ,c(t) = x exp(∫ t

0πi(s)σ

(i)P (y(s)) · dB(s)− 1

2

∫ t

0|σP (y(s))∗π(s)|2ds

+∫ t

0(∑

i πi(s)µi(y(s)) + r(y(s)))− c(s))ds),

As above, we have

VC(x, y) =1γxγ exp(WC(y)),

and W (y) = WC(y) satisfies the equation,

(5.1)12aij(y)DijW (y) + b

(γ)C (y,W (y), DW (y)) = 0,

b(γ)C (y, w, p) = (1− γ) exp(− w

1−γ)− ρ+∑m

i=1 b(γ)i (y)pi

+12

∑mij=1 a

ijγ (y)pipj + V (γ)(y).

To study (5.1), we introduce an useful concept.

Smooth functions W,W are called a pair of

sub/supersolution of (5.1) if

(a) For all y, we have

12aij(y)DijW (y) + b

(γ)C (y,W (y) +DW (y)) ≥ 0,

12aij(y)DijW (y) + b

(γ)C (y,W (y) +DW (y)) ≤ 0.

(b) W ≤W .

We can state our main results.

Theorem 5.1. Assume W,W is a pair of

sub/supersolution of (5.1). Then there is a solution

W of (5.1) such that W ≤W ≤W .

Theorem 5.2. Assume (A1)∼(A3). Then for any γ <

1, γ 6= 0, there is a solution W(γ)C of (5.1) satisfying the

following properties..

(a) Let 0 < γ < 1. For any δ, β > 0, there is a constant

cβ,δ such that

|W (γ)C (y)| ≤ δ|y|β + cβ,δ,

|DW (γ)C (y)| ≤ δ|y|β + cβ,δ.

(b) Let γ < 0. There are c1 < c2 depending on γ such that

for all y we have

c1 ≤WC(y) ≤ c2,

c1 ≤ |DWC(y)| ≤ c2.

(c) c∗t , π∗t defined by the following is an optimal portfolio.

π∗(t) = u∗(y(t)),

u∗(y) =1

1− γa−1P (y)(µ(y) + σP(y)σ∗F(y)∇W (γ)

C (y)),

c∗(t) = exp(− 11− γ

W(γ)C (y(t))).

µ(y) = µ(y)− r(y)1

6. Proof of Theorem 5.1

There are two steps to prove Theorem 5.1.

The first step is to consider the following boundary value

problem.

For each R > 0 and a smooth function ψ,

(6.1){12a

ijDijW (y) + b(γ)C (y,W (y), DW (y)) = 0, |y| < R,

W (y) = ψ(y), |y| = R.

Smooth functions W,W is called a pair of

sub/supersolution of (5.1) if

(a) For all |y| < R, we have

12aij(y)DijW (y) + b

(γ)C (y,W (y) +DW (y)) ≥ 0,

12aij(y)DijW (y) + b

(γ)C (y,W (y) +DW (y)) ≤ 0.

(b) W (y) ≤W (y), |y| ≤ R.

(c) W (y) ≤ ψ(y) ≤W (y), |y| = R

Theorem 6.1. Assume W,W is a pair of

sub/supersolution of (6.1). Then there is a solution

W of (6.1) such that W ≤W ≤W .

In addition to Theorem 6.1, we also need a comparison

theorem.

Lemma 6.2. Assume smooth functions W1,W2 satisfy

the following properties

(a) For any |y| < R,

12aij(y)DijW1(y) + b

(γ)C (y,W1(y) +DW1(y)) ≥ 0,

12aij(y)DijW2(y) + b

(γ)C (y,W2(y) +DW2(y)) ≤ 0.

(b) For any y with |y| = R, W1(y) ≤W2(y).

Then W1(y) ≤W2(y), |y| ≤ R.

Proof of Theorem 5.1.

We shall use Theorem 6.1 and Lemma 6.2 to prove

Theorem 5.2.

(W,W ) is a pair of sub/supersolution of (5.1).

For each R, consider (6.1) with ψ = W .

By Theorem 6.1, there is a solution of (6.1), denoted by

WR.

By Lemma 6.2, WR is unique.

We now fix R0 > 0. R0 < R1 < R2. Compare

WR1(y),WR2(y), |y| ≤ R0.

Take R = R2 in (6.1).

WR2 satisfies

12aij(y)DijWR2(y) + b

(γ)C (y,WR2(y) +DWR2(y)) = 0, |y| ≤ R2

12aij(y)DijW (y) + b

(γ)C (y,W (y) +DW (y)) ≥ 0, |y| ≤ R2

and

W (y) = WR2(y), |y| = R2.

By Lemma 6.2, we have

W (y) ≤WR2(y), |y| ≤ R2.

In particular,

WR1(y) = W (y) ≤WR2(y), |y| = R1.

We now take R = R1 in (6.1).

Apply Lemma 6.2 to W1 = WR1,W2 = WR2. We have

WR1(y) ≤WR2(y), |y| ≤ R1.

In particular, for R0 < R1 < R2 we have

WR1(y) ≤WR2(y), |y| ≤ R0.

|y| ≤ R0, WR(y) is non decreasing in R > R0.

WR(y), R > R0 is bounded by W (y).

Then WR(y) has limit W (y) as R→∞.

W (·) is a solution of (5.1) satisfying W ≤W ≤W .

Lemma 6.2 follows by a simple comparison argument.

For the proof of Theorem 6.1, we need the following result

giving apriori estimates for the solution of (6.1).

Using these apriori estimates and following a continuity

argument in the theory of PDE, we can prove the existence

of the solution of (6.1) claimed in Theorem 6.1.

Theorem 6.3. R > 0, 0 < τ < 1 and ψ is a smoothfunction . Let Wτ be the solution of the equation(6.2){

12a

ij(y)DijWτ(y) + b(τγ)C (y,Wτ(y), DWτ(y)) = 0, |y| < R,

Wτ,γ(y) = τψ(y), |y| = R.

W is a supersolution of (6.1) (or (6.2) with τ = 1). Then

exp(Wτ(y)) ≤ τ exp(W (y)) + (1− τ)f0(y), |y| ≤ R.

f0 satisfies the equation{ 12a

ij(y)Dijf0(y) + b(y)∗Df0(y)− ρf0(y) + 1 = 0, |y| ≤ R,

f0(y) = 1, |y| = R

f0(y) has the expression,

f0(y) =1ρ

+ (1− 1ρ)Ey[exp(−ρθ)],

θ = inf{t > 0; |Y (t)| = R}.

Moreover,

Wτ(y) ≥ − log(max{ ρ

1− γ, 1})− sup

|y|=R{|ψ(y)|}.

For the proof of Theorem 6.3, we consider

Vτ(x, y) =1τγxτγ exp(Wτ(y)).

We have the equation,

12tr(aDyyVτ) + b∗DyVτ + supc>0,π

[cτγxτγ

τγ − ρVτ

+xπ∗σPσ∗FDxyV + 12x

2π∗σPσ∗PDxxVτ

+xDxVτ{r + π∗(µ− r1− c}] = 0.

We consider

Vτ(x, y) =1τγ

(xτγ exp(Wτ(y))− f0(y)).

We have the equation,

(6.3)12tr(aDyyVτ) + b∗DyVτ + supc>0,π

[1τγ(c

τγxτγ − 1)− ρVτ

+xπ∗σPσ∗FDxyVτ + 12x

2π∗σPσ∗PDxxVτ

+xDxVτ{r + π∗(µ− r1− c}]

= 0.

Theorem 6.3 will follow by applying comparison argument

to (6.3).

The Proof of Theorem 5.2

From Theorem 5.1, to get a solution of (5.1) we need to

construct a pair of sub/supersolution of (5.1).

In the following, we consider 0 < γ < 1.

For a subsolution W , we take a constant function

W = −K

where K is a large positive constant.

For a supersolution W , we take

W (y) = W ∗γ (y) + δ(1 + |y|2)β + C.

Here we take small positive δ, β and large C.

For such choice, by Theorem 4.1, W is a nonnegative

function.

By Theorem 5.1, we get a solution W of (5.1) satisfying

W ≤W ≤W .

We easily deduce a upper bound ofW (y) given in Theorem

5.2(a).

Using an idea in Kaise-Sheu (2004) and the upper estimate

of W , we can get an upper estimate of DW (y) as shown

in Theorem 5.2 (a).

Using the estimate in Theorem 5.2(a), we can prove the

verification theorem stated in Theorem 5.2(c).

The analysis for the case γ < 0 will be similar.