Modeling continuous-time financial markets

with capital gains taxes ∗

Imen Ben Tahar

CEREMADE, Universite Paris Dauphine

and Universite Leonard De Vinci

imen.ben tahar@devinci.fr

Nizar Touzi

Centre de Recherche

en Economie et Statistique

and CEREMADE, Universite Paris Dauphine

touzi@ensae.fr

January 2003

Abstract

We formulate a model of continuous-time financial market with risky asset subjectto capital gains taxes. We study the problem of maximizing expected utility of futureconsumption within this model both in the finite and infinite horizon. Our main resultis that the maximal utility does not depend on the taxation rule. This is shown byexhibiting maximizing strategies which tracks the classical Merton optimal strategy intax-free financial markets. Hence, optimal investors can avoid the payment of taxes bysuitable strategies, and there is no way to benefit from tax credits.

Key Words and phrases: Optimal consumption and investment in continuous-time, capital gains taxes.

∗We are grateful to Stathis Tompaidis for numerous discussions on the modelization issue of this paper.

We have also benefited from interesting comments and discussions with Jose Scheinkman and Mete Soner.

In particular, Jose suggested the introduction of the fixed delay in the tax basis, and Mete gave a clear

motivation to push forward the approximating strategies introduced in this paper.

1

1 Introduction

Since the seminal papers of Merton [6, 7], there has been a continuous interest in the theoryof optimal consumption and investment decision in financial markets. A large literature hasfocused particularly on the effect of market imperfections on the optimal consumption andinvestment decision, see e.g. Cox and Huang [1] and Karatzas, Lehoczky and Shreve [5]for incomplete markets, Cvitanic and Karatzas [2] for markets with portfolio constraints,Davis and Norman [3] for markets with proportional transaction costs.

However, there is a very limited literature on the capital gains taxes which apply tofinancial securities and represent a much higher percentage than transaction costs. Com-pared to ordinary income, capital gains are taxed only when the investor sells the security,allowing for a deferral option. One may think that the taxes on capital gains have anappreciable impact on individuals consumption and investment decisions. Indeed, undertaxation of capital gains, an investor supports supplementary charges when he rebalanceshis portfolio, which alters the available wealth for future consumption, possibly depreciat-ing consumption opportunities compared to a tax-free market. On the other hand, sincetaxes are paid only when embedded capital gains are actually realized, the investor maychoose to defer the realization of capital gains and liquidate his position in case of capitallosses, particularly when the tax code allows for tax credits. Previous works attemptedto characterize intertemporal consumption and investment decisions of investors who havepermanently to choose between two conflicting issues : realize the transfers needs for anoptimally diversified portfolio, or use the ability to defer capital gains taxes.

The taxation code specifies the basis to which the price of a security has to be comparedin order to evaluate the capital gains (or losses). The tax basis is either defined as (i)the specific share purchase price, or (ii) the weighted average of past purchase prices. Insome countries, investors can chose either one of the above definitions of the tax basis. Adeterministic model with the above definition (i) of the tax basis, together with the first infirst out rule of priority for the stock to be sold, has been introduced and studied by Jouini,Koehl and Touzi [9, 10].

The case where the tax basis is defined as the weighted average of past purchase pricesis easier to analyze, as the tax basis can be described by a controlled Markov dynamics.Therefore, it can be treated as an additional state variable in a classical stochastic controlproblem. A discrete-time formulation of this model with short sales constraints and lin-ear taxation rule has been studied by Dammon, Spatt and Zhang [4]. They considered theproblem of maximizing the expected discounted utility of future consumption, and provideda numerical analysis of this model based on the dynamic programming principle. In partic-ular, they showed that investors may optimally sell assets with embedded capital gains, andthat the Merton tax-free optimal strategy is approximately optimal for ”young investors”.We refer to Gallmeyer, Kaniel and Tompaidis [11] for an extension of this analysis to themulti-asset framework.

The first contribution of this paper is to provide a continuous-time formulation of theutility maximization problem under capital gains taxes, see Section 2. The financial marketconsists of a tax exempt riskless asset and a risky one. Transfers are not subject to any

2

transaction costs. The holdings in risky assets are subject to the no-short sales constraints,and the total wealth is restricted by the no-bankruptcy condition. The risky asset is subjectto taxes on capital gains. The tax basis is defined as the weighted average of past purchaseprices. We also introduce a possible fixed delay in the tax basis. In contrast with [4], weconsider a general nonlinear taxation rule. Our results hold both for finite and infinitehorizon models. Section 3 shows that the reduction of our model to the tax-free caseproduces the same indirect utility than the classical Merton model.

The main result of our paper states that the value function of the continuous-time utilitymaximization problem with capital gains taxes coincides with the Merton tax-free valuefunction. In other words, investors can optimally avoid taxes and realize the same indirectutility as in the tax-free market. We also provide a maximizing strategy which shows howtaxes can be avoided. This result is first proved in Section 4 in the case where no taxcredits are allowed by the taxation rule. It is then extended to general taxation rules inSection 5 by reducing the problem to the linear taxation rule. The particular tractabilityof the linear taxation rule case allows to prove that it is optimal to take advantage of thetax credits by realizing immediately capital losses.

From an economic viewpoint, our result shows that capital gains taxes do not induce anytax payment by optimal investors. This suggests that the incorporation of capital gainstaxes in financial market models should be accompanied by another market imperfection,as transaction costs, which prevents optimal investors from implementing the maximizingstrategies exhibited in this paper. This aspect is left for future research.

2 The Model

2.1 The financial Market

We consider a financial market consisting of one bank account with constant interest rater > 0, and one risky asset with price process evolving according to the Black and Scholesmodel:

dSt = µStdt + σStdWt, (2.1)

where µ is a constant instantaneous mean rate of return, σ > 0 is a constant volatilityparameter, and the process W = Wt, 0 ≤ t is a standard Brownian motion defined onthe underlying probability space (Ω,F , P).

Let F be the P-completion of the natural filtration of the Brownian motion. In order forpositive investment in the risky asset to be interesting, we shall assume throughout that

µ > r . (2.2)

We also assume that the financial market is not subject to any transaction costs, and theshares of the stock are infinitely divisible.

2.2 Relative tax basis

The sales of the stock are subject to taxes on capital gains. The amount of tax to be paidfor each sale of risky asset is computed by comparison of the current price to the weighted

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average of price of the assets in the investor portfolio. We therefore introduce the relativetax basis process Bt which records the ratio of the weighted average price of the assets inthe investor portfolio to the current price. When Bt is less than 1, the current price of therisky asset is greater than the weighted-average purchase price of the investor so if she sellsthe risky asset, she would realize a capital gain. Similarly, when Bt is larger then 1, thesale of the risky asset corresponds to the realization of a capital loss.

Example 2.1 Let 0 ≤ t0 < t1 < t2 < t3 be some given trading dates, and consider thefollowing discrete portfolio strategy. :

- buy 5 units of risky asset at time t0,- sell 1 unit of risky asset at time t1,- buy 2 units of risky assets at time t2,- sell 4 units of risky asset at time t3,- buy 2 units of risky assets at time t3.

The relative tax basis is not defined strictly before the first purchase date t0, and is equalto one exactly at t0. We set by convention

Bt = 1 for t ≤ t0.

Sales do not alter the basis. Therefore, we only care about purchases in order to determinethe basis at each time. At times t2 and t4, the relative tax basis is given by

Bt2 =5St0 + 2St2

7St2

and Bt3 =5St0 + 2St2 + 2St3

9St3

.

Although no purchases occur in the time intervals (t0, t2), (t2, t3), the relative tax basismoves because of the change of the current price :

(BS)t =

(BS)t0 for t0 ≤ t < t2 ,

(BS)t2 for t2 ≤ t < t3(BS)t3 for t ≥ t3 .

2.3 Taxation rule

Each monetary unit of stock sold at some time t is subject to the payment of an amountof tax computed according to the relative tax basis observed at the prior time

tδ := (t− δ)+ = max0, t− δ . (2.3)

Here δ ≥ 0 is a fixed characteristic of the taxation rule. Another characteristic of thetaxation rule is the amount of tax to be paid per unit of sale. This is defined by

f (Btδ−) , (2.4)

where f is a map from R+ into R satisfying

f non-increasing, f(1) = 0, and lim infb↑1

f(b)b− 1

> −∞ . (2.5)

4

Example 2.2 (Proportional tax on non-negative gains) Let

f(b) := α(1− b)+ for some constant 0 < α < 1 .

When the relative tax basis is less than unity, the investor realizes a capital gain, and paysthe amount of tax α(1−Bt) per unit amount of sales.

Example 2.3 (Proportional tax with tax credits) Let

f(b) := α(1− b) for some constant 0 < α < 1 .

When the relative tax basis is less than unity, the investor realizes a capital gain, and paysthe amount of tax α(1−Bt) per unit amount of sales. When the relative tax basis is largerthen unity, the investor receives the tax credit α(Bt − 1) per unit amount of sales.

Remark 2.1 When there are no tax credits, i.e. f ≥ 0, it is clear that the total taxpaid by the investor is non-negative, and the investor can not do better than in a tax-freemarket. However, when f is not non-negative, it is not obvious that the investor can nottake advantage of the tax credits, and do better than in a tax-free model. Of course, thiswould not be acceptable from the economic viewpoint. Our analysis of this situation inSection 5 shows that the presence of tax credits does not produce such a non-desirableeffect.

2.4 Investment-consumption strategies

An investor start trading at time t = 0 with an initial capital x in cash and y monetaryunits in the risky asset. At each time t ≥ 0, trading occurs by means of transfers betweenthe two investment opportunities.

We denote by L := (Lt, t ≥ 0) the process of cumulative transfers form the bank accountto the risky assets one, and M := (Mt, t ≥ 0) the process of cumulative transfers from therisky assets account to the bank. Here, L and M are two F−adapted, right-continuous,non-decreasing processes with L0− = M0− = 0.

In addition to the trading activity, the investor consumes in continuous-time at the rateC = Ct, t ≥ 0. The process C is F−adapted and nonnegative.

Given a consumption-investment strategy (C, L, M), we denote by Xt the position on thebank, Yt the position on the risky assets account, and Bt the relative tax basis at time t.We also introduce the total wealth

Zt := Xt + Yt, t ≥ 0 .

2.5 Portfolio constraints

We first restrict the strategies to satisfy the no-bankruptcy condition

Zt ≥ 0 P− a.s. for all t ≥ 0 , (2.6)

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i.e. the total wealth of the portfolio at each time has to be non-negative.We also impose the no-short sales constraint

Yt ≥ 0 P− a.s. for all t ≥ 0 , (2.7)

together with the absorption condition

Yt0(ω) = 0 for some t0 =⇒ Yt(ω) = 0 for a.e. ω ∈ Ω. (2.8)

The latter is a technical condition which is needed for a rigorous continuous-time formula-tion of our problem.

The consumption-investment strategy (C, L, M) is said to be admissible if the resultingstate variables (X, Y,B) satisfy the above conditions (2.6)-(2.7)-(2.8). In particular, theprocess (Z, Y ) is valued in the closure S of the subset of R2:

S := (0,∞)× (0,∞). (2.9)

Finally, given an admissible strategy (C, L, M), we introduce the stopping time :

τ := inft ≥ 0 : Yt 6∈ (0,∞) = inft ≥ 0 : Yt = 0 ,

where the last equality follows from (2.7). In view of (2.8), it is clear that the trading strat-egy can be described by means of the non-decreasing right-continuous processes (Lt,Mt)t≥0

which are related to (Lt, Mt)t≥0 by

Lt :=∫ t

0Y −1

t dLt and Mt :=∫ t

0Y −1

t dMt, t < τ .

Here, dLt and dMt represent the proportion of transfers of risky assets.

2.6 Controlled dynamics

Let (C, L, M) be an admissible strategy, and define (L,M) as in the previous paragraph.We shall denote ν := (C,L,M), and (Xν , Y ν , Bν) := (X, Y,B) the corresponding statevariables

Given an initial capital x on the bank account, the evolution of the wealth on this accountis described by the dynamics :

dXνt = (rXν

t − Ct)dt− Y νt−dLt + Y ν

t−(1− f(Bν

tδ−))dMt and X0− = x , (2.10)

recall from (2.3) that tδ := (t − δ)+. Given an initial endowment y on the risky assetsaccount, the evolution of the wealth on this account is also clearly given by

dY νt = Y ν

t−

(dSt

St+ dLt − dMt

)and Y0− = y . (2.11)

This implies that the total wealth evolves according to

dZνt = r(Zν

t − Ct)dt + Y νt−[(µ− r)dt + σdWt]− Y ν

t−f(Bνtδ−)dMt

and Z0− = y + x .(2.12)

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In order to specify the dynamics of the relative tax-basis, we introduce the auxiliary processKν := BνY ν . By definition of Bν , we have :

dKνt = Y ν

t−dLt −Kνt−dMt and Kν

0− = y , (2.13)

since Bν0− = 1. Observe that the contribution of the sales in the dynamics of Kt is evaluated

at the basis price. We then define the relative basis process Bν by

Bνt = 1Y ν

t =0 +Kν

t

Y νt

1Y νt 6=0 . (2.14)

Hence, the position of the investor resulting from the strategy ν is described by the triple(Zν , Y ν , Bν). We call (Zν , Y ν , Bν) the state process associated with the control ν.

Proposition 2.1 Let ν = (C,L,M) be a triple of adapted process such thatA1 L,M are right-continuous, non-decreasing and L0− = M0− = 0A2 the jumps of M satisfy ∆M ≤ 1A3 C ≥ 0 and

∫ t0 Csds < ∞ a.s. for all t ≥ 0

Then, there exists a unique solution (Zν , Y ν , Bν) to (2.12)-(2.11)-(2.13)-(2.14).Moreover, (Zν , Y ν , Bν) satisfies conditions (2.7)-(2.8).

Proof. Equation (2.11) clearly defines a unique solution Y ν . Given Y ν , it is also clear that(2.13) has a unique solution Y νBν , and (2.14) defines Bν uniquely. Finally, given (Y ν , Bν),it is an obvious fact that equation (2.10) has a unique solution Xν . 2

Remark 2.2 The statement of the above proposition is still valid when A2 is replaced bythe following weaker conditionA2’ the jumps of the pair process (L,M) satisfy ∆L−∆M ≥ −1.However in the case where tax credits are allowed by the taxation rule, see Section 5, it iseasy to construct consumption-investment strategies, satisfying A1-A2’-A3, which increasewithout bound the value function of the problem (2.16) defined below, starting from somefixed positive initial holding in stock. Hence such a model allows for a weak notion ofarbitrage opportunities. Indeed, for each ε > 0 and λ > 0, let ∆Lt = ∆Mt := 0 for t 6= τ ,∆Lτ = ∆Mτ := Λ where τ := inft : Bt > 1 + ε. By sending Λ to infinity, the valuefunction of the problem (2.16) converges to +∞.

Definition 2.1 Let ν = (C,L,M) be a triple of F−adapted processes, and (z, y) ∈ S. Wesay that ν is a (z, y)−admissible consumption-investment strategy if it satisfies ConditionsA1-A2-A3 together with the no-bankruptcy condition (2.6). We shall denote by Af (z, y) thecollection of all (z, y)−admissible consumption-investment strategies.

Remark 2.3 We used the absorption at zero condition in order to express an investmentstrategy by means of the proportions dLt = dLt

Yt−and dMt = dMt

Yt−, instead of the volume of

transfers, dLt and dMt. This modification was needed for the specification of the tax basis

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by means of the process K defined above. Indeed, in terms of (L, M), the dynamics of thestate variables X and Y are given by :

dYt = Yt−dSt

St+ dLt − dMt and dXt = r(Xt − ct)dt− dLt + (1− f(Btδ−)) dMt ,

but the relative basis is defined by means of the process K whose dynamics are given by

dKt = dLt −Kt−Yt−

dMt .

Since the event Y = 0 has positive probability, this may cause trouble for the definitionof the model.

2.7 The consumption-investment problem

Throughout this paper, we consider a power utility function :

U(c) :=cp

pfor all c ≥ 0,

where 0 < p < 1 is a given parameter. We next consider the investment-consumptioncriterion

Jft (z, y; ν) := E

[∫ t

0e−βtU(Ct)dt + U(Xν

t )1t<∞

](2.15)

for t ∈ R+ ∪ +∞, (z, y) ∈ S and ν ∈ Af (z, y). Let

T ∈ R+ ∪ +∞

be a given time horizon, so that our analysis holds for both finite and infinite horizon. Theconsumption investment problem is defined by

V fT (z, y) := sup

ν∈Af (z,y)

JfT (z, y; ν) , (z, y) ∈ S . (2.16)

In the context of financial markets without taxes, i.e. f ≡ 0, a slight modification of thisproblem has been solved by Merton [7, 6] by means of a verification argument. In the finitehorizon case (T < ∞), the tax-free problem can be solved directly by passing to a dualformulation, [8, 1, 5]. In the infinite horizon tax-free problem, Merton [6] singled out thecondition

γ :=1

1− p

[β − rp− 1

2p

1− p

(µ− r

σ

)2]

> 0 whenever T = +∞ , (2.17)

in order to ensure that the value function is finite. The (explicit) solution in this context issimply obtained by sending the time horizon to infinity in the solution of the finite horizonproblem.

We conclude this section by the following easy result which states that, the value functionV f is non-increasing in f .

8

Proposition 2.2 Let (z, y) be some initial holdings pair in S, and let f ≥ f be two mapsfrom R+ into R. Then V f

T (z, y)T ≤ V gT (z).

Proof. Consider some admissible consumption-investment strategy (C,L,M) ∈ Af (z, y).We denote by (Zf , Y f , Bf ) and (Zg, Y g, Bg) the corresponding state process respectivelyunder the taxation rule implied by f and g. Observing that (Y f , Bf ) = (Y g, Bg), wedirectly compute that :

Zgt − Zf

t =∫ t

0r(Zg

s − Zfs )ds +

∫ t

0[f(Bf

sδ−)Y fs− − g(Bg

sδ−)Y gs−]dMs

≥∫ t

0r(Z0

s − Zfs )ds +

∫ t

0(f − g)(Bf

sδ−)Y fs−dMs

≥∫ t

0r(Z0

s − Zfs )ds ,

since f ≥ g. This shows that Zg ≥ Zf so that (C,L,M) is also an admissible strategyin Ag(z, y). Hence Af (z, y) ⊂ Ag(z, y) by arbitrariness of (C,L,M) ∈ Af (z, y), and therequired result follows. 2

3 Financial market without taxes

In this section, we review the solution of the consumption-investment problem in a financialmarket without capital gain taxes, i.e. when f ≡ 0. Since the reduction of our problem tothis case is slightly different from the classical Merton model, we shall study both problems.We will show that they have essentially the same value functions, and discuss the issue ofoptimal strategies.

3.1 The classical Merton model

In the classical formulation of the tax-free consumption-investment problem, the investmentcontrol variable is described by means of unique process π which represents the proportionof wealth invested in risky assets at each time. Given a consumption plan C, the totalwealth process is then defined by the dynamics :

dZ(C,π)t =

(rZ

(C,π)t − Ct

)dt + πtZ

(C,π)t [(µ− r)dt + σdWt] , and Z

(C,π)0 = z.(3.1)

In this context, a consumption-investment strategy is a pair of F−adapted processes (C, π),where C is non-negative and∫ T

0Csds +

∫ T

0|πs|2ds < ∞ P− a.s.

We shall denote by A(z) the collection of all such consumption-investment strategies whichsatisfy the additional no-bankruptcy condition

Z(C,π)t ≥ 0 P− a.s. for all 0 ≤ t ≤ T .

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The relaxed tax-free consumption-investment problem is then defined by :

VT (z) := sup(C,π)∈A(z)

E[∫ T

0e−βtU(Ct)dt + U

(Z

(π,C)T

)1T<∞

]. (3.2)

Set ∂zS := (z, y) ∈ S : z = 0, and let (z, y) be an arbitrary initial data in S ∪ ∂zS= (z, y) ∈ S : y > 0. Clearly, for any admissible consumption-investment strategyν = (C,L,M) ∈ A0(z, y), one can define a pair (C, π) ∈ A(z) such that Zν = Z(C,π). Thisshows that

V 0T (z, y) ≤ VT (z) for all (z, y) ∈ S ∪ ∂zS , (3.3)

and justifies the name of the problem VT . The value function V 0T on the boundary ∂yS :=

∂S \ ∂zS will be studied separately.We shall prove later on (Proposition 3.2) that equality holds in (3.3) by exhibiting a

maximizing strategy for the problem V 0T . In preparation to this, let us first recall the

explicit solution of the relaxed tax-free consumption-investment problem.

Theorem 3.1 Let Condition (2.17) hold. Then, for all z > 0 :

VT (z) =zp

p

[1γ

+(

1− 1γ

)e−γT

]1−p

.

Moreover, existence holds for the problem VT (z) with optimal consumption-investment strat-egy given by :

πt = π :=µ− r

(1− p)σ2, Ct := c(t)Zt ,

where c(.) is the deterministic function

c(t) :=[

1γ

+(

1− 1γ

)e−γ(T−t)

]−1

,

and Z := Z(C,π) is the wealth process defined by the strategy (C, π) :

Z0 = z, dZt = Zt

[(r − c(t)) dt− µ− r

(1− p)σ

(µ− r

σdt + dWt

)].

Observe that- the optimal investment strategy π is constant both in the finite and infinite horizon

cases,- the optimal consumption process is a linear deterministic function of the wealth process,

with slope defined by the function c(t); in the infinite horizon case, the function c reducesto the constant γ,

Remark 3.1 Consider the infinite horizon case T = +∞. Then c(t) = γ. By directcomputation, we see that

e−βtE[U(Zt)

]= zpe−γt for all t ≥ 0 .

Hence Condition (2.17) guarantees that e−βtE[U(Zt)

]−→ 0 as t →∞, and therefore :

V∞(z) = limt→∞

E[∫ t

0e−βsU(γZs)ds + e−βtU(Zt)

].

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3.2 Connection with our tax-free model

We now focus on the reduction of the model of Section 2 to the tax-free case, i.e. f ≡ 0. Inthis context, the state variable B is not relevant any more. Given an initial data (z, y) ∈ Sand an admissible control ν = (C,L,M) ∈ A0(z, y), the controlled state process reduces tothe pair (Zν , Y ν) which evolves according to the dynamics :

dZνt = (rZν

t − Ct) dt + Y νt− [(µ− r)dt + σdWt]

dY νt = Y ν

t− [(µ− r)dt + σdWt + dLt − dMt]

together with the initial condition (Z, Y )0− = (z, y).This model presents some minor differences with the classical Merton model of Section

3.1. First, the investment strategies are constrained to have bounded variation. We shallsee that this induces a non-existence of an optimal control for the problem V 0(z, y), butdoes not entail any difference between V 0 and V . Second, the above dynamics imply thatzero is an absorbing boundary for the Y variable which describes the holdings in stock.From the solution of the classical Merton model reported in Theorem 3.1, notice that theinvestment in stock is always positive, except the case of zero initial capital z = 0. Wetherefore expect that the value functions V and V 0 do coincide except on the boundary

∂yS := (z, y) ∈ S : z > 0 and y = 0.

Observe that the analysis of both problems V and V 0 is trivial on the boundary

∂zS := (z, y) ∈ S : z = 0,

since there is no possibility neither for consumption nor for investment. The following resultcharacterizes the value function V 0 on the boundary of S which, according to the previousnotations, is partitioned into

∂S = ∂zS ∪ ∂yS .

Proposition 3.1 The solution of the problem V 0 on the boundary ∂S is given by :(i) For (z, y) ∈ ∂zS,

V 0(z, y) = V (z) = 0 with optimal controls (C, L, M)t = (0, 0, 1), t ≥ 0 ,

and optimal state process (Z, Y )t = 0 for t ≥ 0.(ii) Set γ0 := β−rp

1−p and assume β > r whenever T = +∞. Then, for (z, y) ∈ ∂yS,

V 0T (z, y) =

zp

p

[1γ0

+(

1− 1γ0

)e−γ0T

]1−p

, (3.4)

with optimal controls

(C, L, M)t = (c0(t)Zt, 0, 0), c0(t) :=[

1γ0

+(

1− 1γ0

)e−γ0(T−t)

]−1

(3.5)

for 0 ≤ t ≤ T ; the optimal state processes are Y = 0 and

Zt := z exp[rt−

∫ t

0c0(s)ds

], 0 ≤ t ≤ T .

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Proof. For item (i), it is sufficient to observe that the investment strategy (C, L, M)t =(0, 0, 1), t ≥ 0 is the only admissible strategy. We now concentrate on item (ii). Since ∂ySis an absorbing boundary, we are reduced to the (deterministic) control problem :

V 0T (t, z, 0) = sup

∫ T

te−βtU (Ct) dt + e−β(T−t)U (ZT )1T<∞ ,

where the state dynamics are given by

dZt = (rZt − Ct)dt .

1. We first solve the finite horizon problem T < ∞. We shall use a verification argumentby guessing a solution to the Hamilton-Jacobi equation of this problem :

0 = −βV 0T (t, z)

∂V 0T

∂t(t, z) + rz

∂V 0T

∂z(t, z) + sup

ξ≥0

U(ξ)− ξ

∂V 0T

∂z(t, z)

= −βV 0

T (t, z)∂V 0

T

∂t(t, z) + rz

∂V 0T

∂z(t, z) +

p

1− p

(∂V 0

T

∂z(t, z)

)p/p−1

;

the argument y = 0 has been omitted for notational simplicity. We guess a solution to theabove first order partial differential equation in the separable form zph(t), and determineh so that the terminal condition

V 0T (T, z) = U(z) or, equivalently, h(T ) =

1p

is satisfied. This leads to the candidate solution defined in (3.4). By usual verificationarguments, this candidate is then shown to be the solution of the problem, and the optimalcontrols are identified.

2. We now concentrate on the infinite horizon case T = +∞. It is clear that the optimalstate process Z should be set to zero at infinity. In terms of optimal control, this is a naturaltransversality condition for the problem. In order to take advantage of this information, wesolve the problem by the calculus of variation approach. Direct calculation from the localEuler equation of the problem leads to the following characterization of the optimal state :

(p− 1)(rZ − Z

)= (β − r)(rZ − Z) ,

where Z = dZ/dt denotes the time derivative of the state Z.. this ordinary differentialequation can be solved explicitly by the technique of variation of the constant. In view ofthe boundary conditions Z0 = z and Z∞ = 0, this provides the unique solution to the localEuler equation :

Zt = z exp−(

β − r

1− p

)t with optimal consumption Ct = rZt − Zt = γ0Zt .

Notice that the condition β > r is here necessary in order to ensure that Z∞ = 0. 2

To conclude our analysis of the tax-free model, we now focus on the value function in theinterior of the domain S. in contrast with the situation on the boundary, trading in the

12

stock is now possible. The following result shows that the value function V 0 coincides withV , the maximal utility in the classical Merton model. The price for the control restrictionto the class of bounded variation processes is that existence does not hold any more for theproblem V 0.

Proposition 3.2 For all (z, y) ∈ S, we have V 0T (z, y) = VT (z).

In view of (3.3), the only non-trivial inequality in the above result is that V 0T (z, y) ≥

VT (z). This follows directly from our main Theorem 4.1, by considering the case f ≡ 0.

4 Optimal consumption-investment under capital gain taxes

In this section, we consider the case of a financial market with no tax credits, i.e.

f(b) ≥ 0 for all b ≥ 0 . (4.1)

The general case will be studied in the subsequent section.In the context of (4.1), we shall prove the first main result of the paper which states that

the maximal utility in the financial market is not altered by the capital gains taxation rule,i.e. V f = V 0.

This result is of course trivial when the initial holding in stock is zero, Y0− = 0, since∂yS is an absorbing boundary. For a non-zero initial holding y in stock, we shall prove thisresult by forcing the relative tax basis Bt to be as close as desired to unity, and trackingMerton’s optimal strategy, i.e. keep the proportion of wealth invested in the risky asset

πt :=Yt

Zt1Zt 6=0, 0 ≤ t ≤ T,

and the proportion of wealth dedicated for consumption

ct :=Ct

Zt1Zt 6=0, 0 ≤ t ≤ T,

close to the pair (π, c(t)) defined in Theorem 3.1.To do this, we first fix some t > 0, and define a convenient sequence (νt,n)n≥1 :=

(Ct,n, Lt,n,M t,n)n≥1 for all (z, y) ∈ S ∪ ∂zS. We shall denote by(Zt,n, Y t,n, Bt,n

)=(

Zνt,n, Y νt,n

, Bνt,n)

the corresponding state processes. For each integer n ≥ 1, the consumption-

investment strategy νt,n is defined as follows.

1. At time 0 choose the transfers (∆Lt,n0 ,∆M t,n

0 ) so as to adjust the proportion of wealthto π :

∆Lt,n0 :=

(π

z

y− 1

)1πz≥y and ∆M t,n

0 :=(

1− πz

y

)1πz<y ,

so that

πt,n0 :=

Y t,n0

Zt,n0

= π ;

13

recall that B0δ− = B0− = 1.

2. At the final time t, fix the jumps(∆Lt,n,∆M (t,n)

)tso that all the wealth is transferred

to the bank :

∆Lt,nt := 0 and ∆M

(t,n)t = 1 .

This implies that

Y t,nt = 0 and Zt,n

t = X(t,n)t .

3. In Step 3 below, we shall construct a sequence of stopping times (τ t,nk )k≥1. Our con-

sumption strategy is defined by

Ct,ns := c(t)Zt,n

s for 0 ≤ s ≤ T .

The investments strategy is piecewise constant :

dLt,ns = dM t,n

s = 0 for all s ∈ [0, T ] \ τ t,nk , k ≥ 1 .

4. We now introduce the sequence of stopping times τ t,nk as the hitting times of the pair

process (π,B) of some barrier close to (π, 1). Set τ t,n0 := 0, and define the sequence of

stopping times

τ t,nk := T ∧ τπ

k ∧ τBk ,

where

τπk := inf

s ≥ τ t,n

k−1 : |πt,ns − π| > n−1

,

τBk := inf

s ≥ τk−1 :

(1−Bt,n

sδ

)> n−1λk

,

where λ is a parameter in (0, 1) to be fixed later on.

5. To conclude the definition of νt,n, it remains to specify the jumps(∆Lt,n,∆M t,n

)at

each time τ t,nk . The idea here is to re-set the proportion πt,n to the constant π, and to

push-back the relative tax basis B to unity. To do this, we first consider some parameterλ ∈ (0, 1) such that :

1 + λ(1− π) > 0 .

We then define for all s ∈ τ t,nk , k ≥ 1 :

∆Lt,ns :=

π

πt,ns−

1− πs−f(Bt,n

sδ−

)1 + λ

[1− πf

(Bt,n

sδ−

)] and ∆Mns := 1− λ∆Ln

s .

Using the dynamics of (Z, Y, B), we have :

πt,ns =

Y t,ns

Zt,ns

=πt,n

s−(1 + ∆Lt,ns −∆M t,n

s )

1− πt,ns−∆M t,n

s f(Bt,n

sδ−

)14

and

Bt,ns Y t,n

s −Bt,ns−Y t,n

s− = Y t,ns−

(∆Lt,n

s −Bt,ns−∆M t,n

s

),

so that with the above definition of the jumps (∆Lt,n,∆M t,n), we have

πt,ns = π and Bt,n

s =1 + λBt,n

s−1 + λ

for s ∈ τ t,nk , k ≥ 0 .

Remark 4.1 Since f(1) = 0 and f is continuous, it is immediately checked from the abovedefinitions that, for sufficiently large n :

0 < ∆Lt,ns < 1 and 0 < ∆M t,n

s < 1 for s ∈ τ t,nk , k ≥ 0 .

This guarantees that the process of holdings in risky assets Y t,n is positive P−a.s. 2

Remark 4.2 The sequence(τ t,nk

)k≥0

is strictly increasing, and converges to T . To see

this, we first make the trivial observation that τ t,nk < τπ

k+1 P−a.s. On the other hand, sinceL and M are constant in the stochastic interval [τ t,n

k−1, τt,nk ), we have 1 − Bt,n

τ t,nk −

≤ λk/n.

Then :

1−Bt,n

τ t,nk

=λ

1 + λ

(1−Bt,n

τ t,nk −

)≤ λk+1

n(1 + λ)<

λk+1

n.

This guarantees that τ t,nk < τB

k+1 P−a.s. In particular τ t,nk −→ τ t,n

∞ ≤ T . The proof of ourclaim is completed by observing that the limit τ t,n

∞ is necessarily equal to T . 2

The main property of the sequence(νt,n

)n

is the following.

Lemma 4.1 Let t > 0 be some fixed time horizon. Then for any map f satisfying (2.5),we have

E[

sup0≤s≤t

∣∣Zt,ns − Zs

∣∣2] ≤ n−2 αeαt ,

for some constant α depending on t.

Proof. By definition of the sequence of stopping times(τ t,nk

)k, we have

sup0≤s≤t

∣∣πt,ns − π

∣∣ ≤ 1n

and supτk−1≤s<τk

∣∣1−Bt,ns

∣∣ ≤ λk

nfor all k ≥ 1 . (4.2)

Set D := Zt,n − Z. Since D0 = 0, we decompose D into :

Ds = Fs + Gs + Hs ,

where

Fs :=∫ s

0Du

[(r − c(u))du + πt,n

u ((µ− r)du + σdWu)]

,

Gs :=∫ s

0Zu

(πt,n

s − π)((µ− r)du + σdWu)

Hs := −∫ s

0πt,n

u−f(Bt,n

uδ−

) (Du− + Zu−

)dM t,n

u =∑u≤s

πt,nu−f

(Bt,n

uδ−

) (Du− + Zu−

)∆M t,n

u .

15

In the subsequent calculation, A will denote a generic (t−dependent) constant whose valuemay change from line to line. We shall also denote by V ∗

s := sup0≤u≤s |Vu| for all process(Vs)s.

We first start by estimating the first component F . Observe that c(.) is bounded and theprocess πt,n is bounded by 2π for large n. Then

|Fs|2 ≤ 2(∫ s

0Du(r − c(u) + πt,n

u (µ− r))du

)2

+ 2(∫ s

0Duπt,n

u σdWu

)2

≤ A

∫ s

0|D∗

u|2du + 2(∫ s

0Duπt,n

u σdWu

)2

.

By the Buckholder-Davis-Gundy inequality, this provides

E|F ∗s |2 ≤ A

(∫ s

0E|D∗

u|2du + E∫ s

0|Du|2|πt,n

u |2σ2du

)≤ A

∫ s

0E|D∗

u|2du . (4.3)

Similarly, it follows from (4.2) that :

|Gs|2 ≤ 2n2

[(µ− r)2

(∫ s

0Zudu

)2

+ σ2

(∫ s

0ZudWu

)2]

.

Using again the Buckholder-Davis-Gundy inequality, this provides

E|G∗s|2 ≤ A

n2. (4.4)

Finally, since the jumps of M t,n are bounded by 1, we estimate the component H by :

|Hs|2 ≤ 2

∑u≤s

|πt,nu−|f

(Bt,n

uδ−

)|Du−|

2

+ 2

∑u≤s

f(Bt,n

uδ−

)Zu−

2

≤ A(|D∗

s |2 + |Z∗t |2

) ∑u≤s

f(Bt,n

uδ−

)2

≤ A(|D∗

s |2 + |Z∗t |2

) ∑u≤T

f(Bt,n

uδ−

)2

≤ A(|D∗

s |2 + |Z∗t |2

) ∑k≥0

λk

n

2

=A

n2

(1 + |D∗

s |2)

, (4.5)

where the last inequality follows from (4.2) and (2.5) which implies that f is locally Lipschitzat b = 1. We now collect the estimates from (4.3), (4.4) and (4.5) to see that :(

1− A

n2

)E|D∗

s |2 ≤ A

n2+ K

∫ s

0E|D∗

u|2du for all s ≤ t .

16

The required result follows from the Gronwall inequality. 2

We are now ready for the first main result of this paper which states that the value functionof the consumption investment problem is not altered by the capital gain taxes rule. Noticethat the proof produces a precise description of optimal consumption-investment behavior :in the finite horizon case T < ∞, (νT,n)n is a maximizing consumption-investment strategy,in the infinite horizon case, a maximizing consumption-investment strategy is obtained bymeans of a diagonal extraction argument from the sequence (νt,n).

Theorem 4.1 Let T ∈ R+ ∪ +∞ be some given maturity. Assume the the function f

defining the taxation rule satisfies (2.5) and (4.1). Then V fT = V 0

T on S. In particular, inS ∪ ∂zS, the value function of the consumption investment problem under taxes coincideswith the value function of the classical Merton problem.

Proof. 1. We first show that

limn→∞

Jft (z, y; νt,n) = Jt(z) := E

∫ t

0e−βsU(CsZs)ds + e−βT U(Zt)

for all (z, y) be in S and t ∈ R+. Indeed, since the utility function U is p-holder continuous

|U(c(s)Zt,ns ) − U(c(s)Zs)| ≤ c(s)|Zt,n

s − Zs|p

Then using the Jensen inequality with the concave function x :7→ xp2 :

E|Zt,ns − Zs|p = E

(|Zt,n

s − Zs|2) p

2 ≤(E|Zt,n

s − Zs|2) p

2

Now, using the estimate provided by lemma (4.1)(E|Zt,n

s − Zs|2) p

2 ≤(n−2 αeαt

) p2 = n−p αe

p2αt

It follows that :∣∣∣Jt(z)− Jft (z, y; νt,n)

∣∣∣ ≤ n−p

[(2pc(0)p + α

)e

p2αt − 2

p

] ∫ t

0e

p2αsds .

2. Combining Proposition 2.2 together with (3.3), we see that V f ≤ V 0 ≤ V on S ∪ ∂zS.In order to prove that equality holds, it suffices to show that V f ≥ V . In the finite horizoncase, the proof is completed by taking t = T in Step 1. We next concentrate on the infinitehorizon case T = +∞. Fix some positive integer k. By Remark 3.1, we have :

VT (z) = limt→∞

Jt(z) .

Then

Jtk(z) ≥ VT (z) − 1k

,

for some tk > 0. By the first step of this proof :

limn→∞

Jftk

(z, y; νtk,n) = Jtk(z) .

17

Then, there exists some integer nk

Jftk

(z, y; νtk,nk) ≥ Jtk(z) − 12k

≥ VT (z)− 1k

.

3. Finally, we define the consumption-investment strategies νk consisting in following νtk,nk

up to tk, then liquidating at tk the risky asset position and making a null consumption onthe time interval (tk, T ). Then:

JfT (z, y; νk) ≥ Jf

tk(z, y; νtk,nk) ≥ VT (z)− 1

kfor all k ≥ 0 .

This proves that V 0T (z, y) ≥ lim supk→∞ Jf

T (z, y; νk) ≥ VT (z). 2

5 Extension to Taxation rules with possible tax credits

In this section we consider the case where the financial market allows for tax credits, andwe restrict our analysis to the case

δ = 0 .

Our main purpose is to extend Theorem 4.1 to this context.

Theorem 5.1 Consider a taxation rule defined by the map f satisfying (2.5), and let δ = 0.Assume further that

infb≥0

f(b)1− b

> −∞ . (5.1)

Then, for all T ∈ R ∪ +∞ and (z, y) ∈ S, we have V fT (z, y) = V 0

T (z, y).

Proof. 1. Since f ≤ f+, V fT ≥ V f+

T by Proposition 2.2. Now f+ defines a taxation rulewithout tax credits as required by Condition (4.1). It then follows from Theorem 4.1 thatV f

T ≥ V 0T .

2. From (5.1) it follows that V fT ≤ V `

T , where ` is the linear map

`(b) := α (1− b) with α := infb≥0

f(b)1− b

.

We shall prove in Proposition 5.2 that V `T = V 0

T , hence V fT ≤ V 0

T . 2

In the above proof, we reduced the problem to the linear taxation rule of Example 2.3.The rest of this section is specialized to this context.

5.1 Immediate realization of capital losses for the linear taxation rule

Consider the taxation rule defined by

f(b) := α(1− b) for all b ≥ 0 . (5.2)

18

We first intend to prove that it is always worth realizing capital losses whenever the taxbasis exceeds unity. In other words, for each (z, y) in S, any admissible consumption-investment strategy for which the relative tax basis exceeds 1 at some stopping time τ canbe improved strictly by increasing the sales of the risky asset at τ . We shall refer to thisproperty as the optimality of the immediate realization of capital losses. In a discrete-timeframework, this property was stated without proof by [4].

Proposition 5.1 Let f be the linear taxation rule of (5.2), δ = 0, and consider someinitial holdings (z, y) in S. Consider some consumption-investment strategy ν := (C,L,M)∈ Af (z, y), and suppose that there is a finite stopping time τ ≤ T with P [τ < T ] > 0 andBν

τ > 1 a.s. on τ < T.Then there exists an admissible strategy ν = V(ν, τ) such that

Y ν = Y ν , Z ν ≥ Zν , Bν ≤ Bν , C ≥ C,

and

J fT (z, y; ν) > J f

T (z, y; ν) .

We start by proving the following lemma which shows how to take advantage of the taxcredit at time τ .

Lemma 5.1 Let f be as in (5.2), δ = 0, and consider some initial holdings (z, y) be in S.Consider some consumption-investment strategy ν := (C,L,M) ∈ Af (z, y), and supposethat there is a finite stopping time τ ≤ T with P [τ < T ] > 0 and Bν

τ > 1 a.s. on τ < T.Define ν = (C, L, M) by

C := C and (L, M) := (L,M) + (1, 1)(1−∆Mτ )1t≥τ . (5.3)

Then ν ∈ AfT (z, y) and the resulting state processes are such that

Y ν = Y ν , Z ν ≥ Zν , Bν ≤ Bν ,

and, almost surely,

Bντ = 1, Z ν

τ > Zντ on τ < T .

Proof. 1. Since ν and ν differ only by the jump at the stopping time τ , and ∆Lτ = ∆Mτ ,we have

Y ν = Y ν ,

and (Z ν

t , Bνt

)= (Zν

t , Bνt ) for all t < τ .

By the definition of ν, it is easily seen that

Bντ = 1

19

and

Z ντ − Zν

τ = −f(Bν

τ−)Y ν

τ− (1−∆Mτ ) = αY ντ−

(Bν

τ− − 1)(1−∆Mτ ) . (5.4)

Since

0 > (1−Bντ ) = (1−Bν

τ−)1−∆Mτ

1 + ∆Lτ −∆Mτon τ < T , (5.5)

it follows that Bτ− > 1 and 1−∆Mτ > 0 a.s. on τ < T. We therefore deduce from (5.4)that

Z ντ − Zν

τ > 0 on τ < T .

2. We next examine the state variable Bνt for t > τ . Since Y ν = Y ν , we have K ν −Kν =

Y ν (Bν −Bν), and

d(K νt −Kν

t ) = −(K νt− −Kν

t−) dMt, with K ντ −Kν

τ = Y ντ (1−Bν

τ ) . (5.6)

This linear stochastic differential equation can be solve explicitly :

K νt −Kν

t = (K ντ −Kν

τ ) e−Mct +Mc

τ

∏τ<u≤t

(1−∆Mu), for all t ≥ τ ,

where M c denotes the continuous part of M . Since K ντ −Kν

τ = Y ντ (1−Bν

τ ) < 0 on τ < T,this shows that

K νt ≤ Kν

t and therefore Bνt ≤ Bν

t for t ≥ τ . (5.7)

3. In this step, we intend to prove that Z νt ≥ Zν

t for t ≥ τ . From the dynamics of theprocesses Z ν and Zν we have, for t > τ :

e−r(t−τ)(Z ν

t − Zνt

)= Z ν

τ − Zντ −

∫ t

τe−r(u−τ)Y ν

u−(f(Bν

u−)− f(Bνu−)

)dMu

= Z ντ − Zν

τ + α

∫ t

τe−r(u−τ)Y ν

u−(Bν

u− −Bνu−)

)dMu

= Z ντ − Zν

τ + α

∫ t

τe−r(u−τ)

(K ν

u− −Kνu−)

)dMu ,

where the last equality follows from the fact that Y ν = Y ν . We next use (5.7) and (5.6) tosee that, for t ≥ τ :

e−r(t−τ)(Z ν

t − Zνt

)≥ Z ν

τ − Zντ + α

∫ t

τ

(K ν

u− −Kνu−)

)dMu

= Z ντ − Zν

τ + α(K ν

τ −Kντ −K ν

t + Kνt

).

Now, since Yτ = Yτ− (1 + ∆Lτ −∆Mτ ), it follows from (5.4) and (5.5) that Z ντ − Zν

τ =−α (K ν

τ −Kντ ). Hence :

e−r(t−τ)(Z ν

t − Zνt

)≥ −α

(K ν

t −Kνt

)≥ 0 for t ≥ τ ,

20

by (5.7).4. Clearly, ν satisfies Conditions A1-A2-A3, and Z ν ≥ Zν by the previous steps of thisproof. Hence ν ∈ Af (z, y). 2

Proof of Proposition 5.1. Consider the slight modification ν = (C, L, M) of the consumption-investment strategy introduced in Lemma 5.1 :

Ct := Ct + ξ(Z ν

t − Zνt

)1t≥τ and

(L, M

):=

(L, M

), (5.8)

where ξ is an arbitrary positive constant. Observe that (Y ν , Bν) = (Y ν , Bν), and Z νt = Z ν

t

for t ≤ τ . In order to check the admissibility of the triple (C, L, M), we repeat Step 3 ofthe proof of Lemma 5.1 :

e−r(t−τ)(Z ν

t − Zνt

)= Z ν

τ − Zντ −

∫ t

τe−r(u−τ)Y ν

u−(f(Bν

u−)− f(Bνu−)

)dMu

+ξ

∫ t

τe−r(u−τ)

(Z ν

u − Zνu

)du

= Z ντ − Zν

τ −∫ t

τe−r(u−τ)Y ν

u−(f(Bν

u−)− f(Bνu−)

)dMu

+ξ

∫ t

τe−r(u−τ)

(Z ν

u − Zνu

)du

· · · ≥ ξ

∫ t

τe−r(u−τ)

(Z ν

u − Zνu

)du .

Since Z ντ −Zν

τ = Z ντ −Zν

τ ≥ 0 by Lemma 5.1, it follows from the Gronwall lemma that Z νt

≥ Zνt a.s. Hence ν ∈ Af

T (z, y).Recall from Lemma 5.1 that Z ν

τ > Zντ on τ < T. Since the process (Z ν − Zν) is right-

continuous, the strict inequality holds on some nontrivial time interval almost surely onτ < T. Hence C > C with positive Lebesgue⊗P measure, and

J fT (z, y; ν) > J f

T (z, y; ν) .

2

5.2 Reduction to the tax-free financial market

In view of the optimality of the immediate realization of capital losses stated in Proposition5.1, we expect that, given (z, y) ∈ S, and for all constant ε > 0, the problem of maximizingJ f

T (z, y; ν) can be restricted to those admissible control processes ν inducing a cumulatedtax credit bounded by ε. This result, stated in Lemma 5.2, will allow to prove that V f =V 0 in the context of a linear taxation rule, hence completing the proof of Theorem 5.1.

Lemma 5.2 Let f be the linear taxation rule defined in (5.2), δ = 0, and let t > 0 be somefinite maturity, ε > 0, and ν in Af (z, y). Then, there exists νε = (Cε, Lε,M ε) in Af (z, y)such that

J ft (z, y; νε) ≥ J f

t (z, y; ν) and∫ t

0

(Bνε

u− − 1)Y νε

u−dMνε

u ≤ ε a.s.

21

Proof. Let θ0 := 0, ν0 := ν,

θn+1 := t ∧ infs > θn : (Bνn

s − 1)(1 ∨Hνn

s

)> ε ,

with

Hsνn

:= Y νn

s− eMνns

c ∏u≤s

(1−∆Mνn

u )−1 ,

and

νn+1 := νn1θn+1=t + V(νn, θn+1)1θn+1<t .

where Mνnc denotes the continuous part of Mνn, and V is defined in Proposition 5.1 so as

to take advantage of the tax credit while decreasing the relative tax basis. We shall simplydenote (Zn, Y n, Bn) := (Zνn

, Y νn, Bνn

). Then :

Cn+1 ≥ Cn, Y n = Y 0, Zn+1 ≥ Zn, Bn+1 ≤ Bn, (5.9)

and

(Bns − 1)

(1 ∨Hνn

s

)≤ ε for t ≤ θn+1. (5.10)

Clearly, for a.e. ω ∈ Ω, θn(ω) = t for n ≥ N(ω), where N(ω) is some sufficiently largeinteger. Therefore the sequence νn(ω) is constant for n ≥ N(ω) and

νn −→ νε a.s.

for some νε ∈ Af (z, y). Also, by construction of the sequence νn, we have :

νεs = νn

s for s ≤ θn+1 . (5.11)

By (5.9), it is immediately checked that J ft (z, y; νε) ≥ J f

t (z, y; ν). We finally use (5.10)and (5.11), together with Ito’s lemma, to compute that∫ t

0

(Bνε

u− − 1)Y νε

u−dMνε

u ≤ ε

∫ t

0Y νε

u−(Hνε

s )−1dMνε

u

= ε

∫ t

0e−Mνε

sc ∏

u≤s

(1−∆Mνε

u )dMνε

u

= ε

1− e−Mνε

t

∏u≤t

(1−∆Mνε

u )

≤ ε ,

by the fact that 1−∆Mνε ≤ 1. 2

We are now ready to state the extension of Theorem 4.1 to the linear taxation rule casewith tax credits.

22

Proposition 5.2 Consider the linear taxation rule of (5.2), and let δ = 0. Then, for allT ∈ R ∪ +∞ and (z, y) ∈ S, we have V f

T (z, y) = V 0T (z, y).

Proof. The result is trivial for (z, y) ∈ ∂yS. We then assume in the sequel that (z, y) ∈S ∪ ∂zS.1. Since f ≤ f+ := maxf, 0, we have V f

T ≥ V f+

T . Now f+ defines a taxation rule withouttax credits as required by Condition (4.1). It then follows from Theorem 4.1 that V f+

T =V 0

T and therefore V fT ≥ V 0

T .2. We next concentrate on the reverse inequality. Notice that

J f∞(z, y, ν) ≤ lim inf

t→∞J f

t (z, y, ν).

This follows by the monotone convergence theorem and the fact that the utility function isnon-negative. Therefore, in order to prove the required inequality, it is sufficient to showthat, for any fixed finite maturity 0 < t ≤ T :

J ft (z, y; ν) ≤ V 0

T (z, y) for all ν ∈ Af (z, y) . (5.12)

Let ε > 0, and consider the consumption-investment strategy νε ∈ Af (z, y) defined inLemma 5.2. Let Zε be the wealth process induced by the strategy νε in a tax-free market,i.e. with f ≡ 0. Since the taxation rule f allows for tax credits, there is no reason for theprocess Zε to be non-negative. Using the dynamics of Zν , it follows from Lemma 5.2 that

e−ru(Zε

u − Zνε

u

)= α

∫ u

0(1−Bνε

s−)Y νε

s−dMνε

s ≥ −αε . (5.13)

Now, from Lemma 5.2 and the increase of the utility function U and its concavity, we have

Jft (z, y; ν) ≤ Jf

t (z, y; νε) ≤ Jft (z + 2αε, y; νε)

= J0t (z + 2αε, y; νε)

+e−βtE[U(2αεert + Zνε

t )− U(2αεert + Zεt )

].

Using (5.13) together with the increase and the concavity of U , this provides :

Jft (z, y; ν) ≤ J0

t (z + 2αε, y; νε) + e−βtE[U(2αεert + Zνε

t )− U(αεert + Zνε

t )]

≤ J0t (z + 2αε, y; νε) + αεe(r−β)tE

[U ′ (αεert + Zνε

t

)]≤ J0

t (z + 2αε, y; νε) + αεe(r−β)tU ′ (αεert)

= J0t (z + 2αε, y; νε) + pe−βtU

(αεert

).

Now, observe from (5.13) that νε ∈ Af (z + 2αε, y). Then :

Jft (z, y; ν) ≤ V 0

t (z + 2αε, y) + e−βtU(αεert

)= Vt(z + 2αε) + e−βtU

(αεert

),

where the last equality follows from Propositions 3.1 and 3.2. The required inequality (5.12)is obtained by sending ε to zero and using the continuity of the Merton value function V .

2

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References

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