Hereditary Portfolio Optimization with Taxes and Fixed Plus Proportional Transaction Costs—Part I

This is the first of the two companion papers which treat an infinite time horizon hereditary portfolio optimization problem in a market that consists of one savings account and one stock account. Within the solvency region, the investor is allowed to consume from the savings account and can make transactions between the two assets subject to paying capital gain taxes as well as a fixed plus proportional transaction cost. The investor is to seek an optimal consumption-trading strategy in order to maximize the expected utility from the total discounted consumption. The portfolio optimization problem is formulated as an infinite dimensional stochastic classical-impulse control problem. The quasi-variational HJB inequality (QVHJBI) for the value function is derived in this paper. The second paper contains the verification theorem for the optimal strategy. It is also shown there that the value function is a viscosity solution of the QVHJBI.


Introduction
This is the first of the two companion papers (see [1] for the second paper) which treat an infinite time horizon hereditary portfolio optimization problem in a financial market that consists of one savings account and one stock account. It is assumed that the savings account compounds continuously with a constant interest rate r > 0 and the unit price process, {S(t), t ≥ 0}, of the underlying stock follows a nonlinear stochastic hereditary differential equation (see (2.5)) with an infinite but fading memory. The main purpose of the stock account is to keep track of the inventories, (i.e., the time instants and the base prices at which shares were purchased or short-sold) of the underlying stock for purpose of calculating the capital gain taxes, and so forth. In the stock price dynamics, we assume that both f (S t ) (the mean rate of return) and g(S t ) (the volatility coefficient) depend on the entire history of stock prices S t over the time interval (−∞,t] instead of just the current stock price S(t) at time t ≥ 0 alone. Within the solvency region κ (to be defined in (2.29)) and under the requirements of paying fixed plus proportional transaction costs and capital gain taxes, the investor is allowed to consume from his savings account in accordance with a consumption rate process C = {C(t), t ≥ 0} and can make transactions between his savings and stock accounts according to a trading strategy -= {(τ(i), ζ(i)), i = 1,2,...}, where τ(i), i = 0,1,2,... denotes the sequence of transaction times and ζ(i) stands for quantities of the transaction at time τ(i) (see Definitions 2.4 and 2.5).
The investor will follow the following set of consumption, transaction, and taxation rules (Rules 1-6). Note that an action of the investor in the market is called a transaction if it involves trading of shares of the stock such as buying and selling. Rule 1. At the time of each transaction, the investor has to pay a transaction cost that consists of a fixed cost κ > 0 and a proportional transaction cost with the cost rate of μ ≥ 0 for both selling and buying shares of the stock. All the purchases and sales of any number of stock shares will be considered one transaction if they are executed at the same time instant and therefore incur only one fixed fee κ > 0 (in addition to a proportional transaction cost).

Rule 2.
Within the solvency region κ , the investor is allowed to consume and to borrow money from his savings account for stock purchases. He can also sell and/or buy back at the current price shares of the stock he bought and/or short-sold at a previous time.
Rule 3. The proceeds for the sales of the stock minus the transaction costs and capital gain taxes will be deposited in his savings account and the purchases of stock shares together with the associated transaction costs and capital gain taxes (if short shares of the stock are bought back at a profit) will be financed from his savings account.
Rule 4. Without loss of generality, it is assumed that the interest income in the savings account is tax-free by using the effective interest rate r > 0, where the effective interest rate equals the interest rate paid by the bank minus the tax rate for the interest income.
Rule 5. At the time of a transaction (say t ≥ 0), the investor is required to pay a capital gain tax (resp., be paid as a capital-loss credit) in the amount that is proportional to the amount of profit (resp., loss). A sale of stock shares is said to result in a profit if the current stock price S(t) is higher than the base price B(t) of the stock and it is a loss otherwise. The base price B(t) is defined to be the price at which the stock shares were previously bought or short-sold, that is, B(t) = S(t − τ(t)) where τ(t) > 0 is the time duration for which those shares (long or short) have been held at time t. The investor will also pay capital gain taxes (resp., be paid as capital-loss credits) for the amount of profit (resp., loss) by short-selling shares of the stock and then buying back the shares at a lower (resp., higher) price at a later time. The tax will be paid (or the credit will be given) at the buying back time. Throughout the end, a negative amount of tax will be interpreted as a capital loss credit. The capital gain tax and capital loss credit rates are assumed to be the same as β > 0 for simplicity. Therefore, if |m| (m > 0 stands for buying and m < 0 stands for selling) shares of the stock are traded at the current price S(t) at the base B(t) = S(t − τ(t)), then Mou-Hsiung Chang 3 the amount of tax due at the transaction time is given by |m|β S(t) − S t − τ(t) .
(1.1) Rule 6. The tax and/or credit will not exceed all other gross proceeds and/or total costs of the stock shares, that is, where m ∈ denotes the number of shares of the stock traded with m ≥ 0 being the number of shares purchased and m < 0 being the number of shares sold.
Under the above assumptions and Rules 1-6, the investor's objective is to seek an optimal consumption-trading strategy (C * ,- * ) in order to maximize the expected utility from the total discounted consumption over the infinite time horizon, where δ > 0 represents the discount rate and 0 < γ < 1 represents the investor's risk aversion factor. Due to the fixed plus proportional transaction costs and the hereditary nature of the stock dynamics and inventories, the problem will be formulated as a combination of a classical control (for consumptions) problem and an impulse control (for the transactions) problem in infinite dimensions. A classical-impulse control problem in finite dimensions is treated in [2]. In this paper a quasi-variational Hamilton-Jocobi-Bellman inequality (QVHJBI) for the value function together with its boundary conditions is derived. The second paper (see [1]) establishes the verification theorem for the optimal investment trading strategy. In there, it is also shown that the value function is a viscosity solution of the QVHJBI (see (QVHJBI ( * )) in Section 4.3.4). Due to the complexity of the analysis involved, the uniqueness result and finite-dimensional approximations for the viscosity solution of (QVHJBI ( * )) will be treated separately in a future paper.
In recent years, there has been extensive amount of research on the optimal consumption-trading problems with proportional transaction costs (see, e.g., [3][4][5][6], and references contained therein) and fixed plus proportional transaction costs (see, e.g., [7]) within the geometric Brownian motion financial market. In all these papers, the objective has been to maximize the expected utility from the total discounted or averaged consumption over the infinite time horizon without considering the issues of capital gain taxes (resp., capital loss credits) when stock shares are sold at a profit (resp., loss). In different contents, the issues of capital gain taxes have been studied in [8][9][10][11][12][13][14][15], and references contained therein. In particular, [9,10] considered the effect of capital gain taxes and capital loss credits on capital market equilibrium without consumption and transaction costs. These two papers illustrated that under some conditions, it may be more profitable to cut one's losses short and never to realize a gain because of capital loss credits and capital gain taxes as some conventional wisdom will suggest. In [8] the optimal transaction time problem with proportional transaction costs and capital-gain taxes was considered in order to maximize the long-run growth rate of the investment (or the so-called Kelley criterion), that is, where V (t) is the value of the investment measured at time t > 0. This paper is quite different from ours in that the unit price of the stock is described by a geometric Brownian motion, and all shares of the stock owned by the investor are to be sold at a chosen transaction time and all of its proceeds from the sale are to be used to purchase new shares of the stock immediately after the sale without consumption. Fortunately due to the nature of the geometric Brownian motion market, the authors of that paper were able to obtain some explicit results.
In recent years, the interest in stock price dynamics described by stochastic delay equations has increased tremendously (see, e.g., [16,17]). To the best of the author's knowledge, this is the first paper that treats the optimal consumption-trading problem in which the hereditary nature of the stock price dynamics and the issue of capital gain taxes are taken into consideration. Due to drastically different nature of the problem and the techniques involved, the hereditary portfolio optimization problem with taxes and proportional transaction costs (i.e., κ = 0 and μ,ν > 0) remains to be solved. This paper is organized as follows. The description of the stock price dynamics, the admissible consumption-trading strategies, and the formulation of the hereditary portfolio optimization problem are given in Section 2. In Section 3, the properties of the controlled state process are further explored and corresponding infinite-dimensional Markovian solution of the price dynamics is investigated. Section 4 contains the derivations of the QVHJBI together with its boundary conditions (QVHJBI ( * )) using a Bellmantype dynamic programming principle.
The verification theorem for the optimal consumption-trading strategy and the proof that the value function is a viscosity solution of the (QVHJBI ( * )) are contained in the second paper [1].

The hereditary portfolio optimization problem
Throughout the end, we use the following convention.

Convention 2.
If t ≥ 0 and φ : → is a measurable function, define Under Conditions 1-2, it can be shown that ρ is essentially bounded and strictly positive on (−∞,0]. Furthermore, 3) The following are two examples of ρ : (−∞,0] → [0,∞) that satisfy Conditions 1 and 2: ρ (−∞,0) (or simply × L 2 ρ for short) be the history space of the stock price dynamics, where L 2 ρ is the class of ρ-weighted Hilbert space of measurable functions φ : For t ∈ (−∞,∞), let S(t) denote the unit price of the stock at time t. It is assumed that the unit stock price process {S(t), t ∈ (−∞,∞)} satisfies the following stochastic hereditary differential equation with an infinite but fading memory: (2.5) In the above equation, the process {W (t), t ≥ 0} is one-dimensional standard Brownian motion defined on a complete filtered probability space (Ω,Ᏺ,P;F), where F = {Ᏺ(t), t ≥ 0} is the P-augmented natural filtration generated by the Brownian motion {W (t), t ≥ 0}. Note that f (S t ) and g(S t ) in (2.5) represent, respectively, the mean growth rate and the volatility rate of the stock price at time t ≥ 0. Note that the stock is said to have a hereditary price structure with infinite but fading memory because both the drift term S(t) f (S t ) and the diffusion term S(t)g(S t ) in the right-hand side of (2.5) explicitly depend on the entire past history prices (S(t),S t ) ∈ × L 2 ρ in a weighted fashion by the function ρ satisfying Conditions 1-2.
Note that we have used the following notation in the above: It is assumed for simplicity and to guarantee the existence and uniqueness of a strong solution S(t), t ≥ 0, that the initial price function (S(0),S 0 ) = (ψ(0),ψ) ∈ + × L 2 ρ,+ is given and the functions f ,g : L 2 ρ → [0,∞) are continuous, and satisfy the following Lipschitz and linear growth conditions (see, e.g., [18][19][20][21][22] for the theory of stochastic functional differential equations with an infinite or a bounded memory).
Assumption 2.2 (Lipschitz condition). There exists a constant c 2 > 0 such that Assumption 2.3. There exist positive constants α and σ such that Note that the lower bound of the mean rate of return f in Assumption 2.3 is imposed to make sure that the stock account has a higher mean growth rate than the interest rate r > 0 for the savings account. Otherwise, it will be more profitable and less risky for the investor to put all his money in the savings account for the purpose of optimizing the expected utility from the total consumption.
Although the modeling of stock prices is still under intensive investigations, it is not the intention of this paper to address the validity of the model stock price dynamics treated in this paper but to illustrate the hereditary optimization problem that is explicitly dependent upon the entire past history of the stock prices for computing capital gain taxes or capital loss credits. The term "hereditary portfolio optimization" is therefore coined in this paper for the first time. We, however, mention here that stochastic hereditary equation similar to (2.5) was first used to model the behavior of elastic material with infinite memory and that stochastic functional differential equations with bounded memory have been used to model stock price dynamics in option pricing problems (see [16,17]).
It can be shown that, for each initial historical price function (ψ(0),ψ) ∈ + × L 2 ρ,+ , the price process {S(t), t ≥ 0} is a positive, continuous, and F-adapted process defined on (Ω,Ᏺ,P;F) but it is not Markovian with respect to any filtration that makes sense. For this reason, we frequently consider the corresponding + × L 2 ρ,+ -valued process {(S(t), S t ), t ≥ 0} instead of the real-valued process {S(t), t ≥ 0}. However, following approaches similar to that of [20,Section 3], it can be shown under Conditions 1-2 and Assumptions 2.
We also note here that, since security exchanges have only existed in a finite past, it is realistic but not technically required to assume that the initial historical price function Mou-Hsiung Chang 7 (ψ(0),ψ) has the property that The stock inventory space. The space of stock inventories, N, will be the space of bounded functions ξ : (−∞,0] → of the following form: where {n(−k), k = 0,1,2,...} is a sequence in with n(−k) = 0 for all but finitely many k, and Note that the function ξ : (−∞,0] → defined above denotes the inventory of the investor's stock account. In particular, when As illustrated in Sections 2.3 and 2.5, N is the space in which the investor's stock inventory lives. The assumption that n(−k) = 0 for all but finitely many k implies that the investor can only have finitely many open positions in his stock account. However, the number of open positions may increase from time to time. Note that the investor is said to have an open long (resp., short) position at time τ if he still owns (resp., owes) all or part of the stock shares that were originally purchased (resp., short-sold) at a previous time τ. The only way to close a position is to sell what he owns and buy back what he owes.
If η : → is a bounded function of the form then for each t ≥ 0, we define, using Convention 2, the function η t : (−∞,0] → by In this case, ρ,+ be the investor's initial portfolio immediately prior to t = 0. That is, the investor starts with x ∈ dollars in his savings account, the initial stock inventory, (2.20) and the initial profile of historical stock prices (ψ(0),ψ) ∈ + × L 2 ρ,+ , where n(−k) > 0 (resp., n(−k) < 0) represents an open long (resp., short) position at τ(−k). Within the solvency region κ (see (2.29)), the investor is allowed to consume from his savings account and can make transactions between his savings and stock accounts under Rules 1-6 and according to a consumption-trading strategy π = (C,-) defined below.
..} is a trading strategy with τ(i), i = 1,2,..., being a sequence of trading times that are G-stopping times such that and for each i = 0,1,..., is an N-valued Ᏻ(τ(i))-measurable random vector (instead of a random variable in ) that represents the trading quantities at the trading time τ(i). In the above, m(i) > 0 (resp., m(i) < 0) is the number of stock shares newly purchased (resp., short-sold) at the current time τ(i) and at the current price of S(τ(i)) and, for k = 1,2,..., m(i − k) > 0 (resp., m(i − k) < 0) is the number of stock shares bought back (resp., sold) at the current time τ(i) and at the current price of S(τ(i)) in his open short (resp., long) position at the previous time τ(i − k) and at the base price of S(τ(i − k)).
For each stock inventory ξ of the form expressed (2.13), Rules 1-6 also dictate that the investor can purchase or short sell new shares and/or buy back (resp., sell) all or part of Mou-Hsiung Chang 9 what he owes (resp., owns). Therefore, the trading quantity {m(−k), k = 0,1,...} must satisfy the constraint set (ξ) ⊂ N defined by (2.24)

Solvency region.
Throughout the end of this paper, the investor's state space S is taken to be S = × N × + × L 2 ρ,+ . An element (x,ξ,ψ(0),ψ) ∈ S is called a portfolio, where x ∈ is investor's holding in his savings account, ξ is the investor's stock inventory, and (ψ(0),ψ) ∈ + × L 2 ρ,+ is the profile of historical stock prices. Define the function H κ : S → as follows: where G κ : S → is the liquidating function defined by (2.26) In the right-hand side of the above expression, x − κ = the amount in his savings account after deducting the fixed transaction cost κ; (2.27) and for each k = 0,1,..., The solvency region κ of the portfolio optimization problem is defined as Note that within the solvency region κ , there are positions that cannot be closed at all, namely, those (x,ξ,ψ(0),ψ) ∈ κ such that This is due to the insufficiency of funds to pay for the transaction costs and/or taxes, and so forth. Observe that the solvency region κ is an unbounded and nonconvex subset of the state space S. The boundary ∂ κ will be described in detail in Section 4.3.

Portfolio dynamics and admissible strategies.
At time t ≥ 0, the investor's portfolio in the financial market will be denoted by the quadruplet (X(t),N t ,S(t),S t ), where X(t) denotes the investor's holdings in his savings account, N t ∈ N is the inventory of his stock account, and (S(t),S t ) describes the profile of the unit prices of the stock over the past history (−∞,t] as described in Section 2.1. Given the initial portfolio and applying a consumption-trading strategy π = (C,-) (see Definition 2.4), the portfolio dynamics of {Z(t) = (X(t),N t ,S(t),S t ), t ≥ 0} can then be described as follows. Firstly, the savings account holding {X(t), t ≥ 0} satisfies the following differential equation between the trading times: (2.32) and the following jumped quantity at the trading time τ(i): As a reminder, m(i) > 0 (resp., m(i) < 0) means buying (resp., selling) new stock shares at τ(i) and m(i − k) > 0 (resp., m(i − k) < 0) means buying back (resp., selling) some or all of what he owed (resp., owned).
Mou-Hsiung Chang 11 Secondly, the inventory of the investor's stock account at time t ≥ 0, N t ∈ N, does not change between the trading times and can be expressed as the following equation: It has the following jumped quantity at the trading time τ(i): Thirdly, since the investor is small, the unit stock price process {S(t), t ≥ 0} will not be in anyway affected by the investor's action in the market and is again described as in (2.5).
Mou-Hsiung Chang 13 Let ( × L 2 ρ ) * be the space of bounded linear functionals (or the topological dual of the space × L 2 ρ ) equipped with the operator norm · * defined by Note that ( × L 2 ρ ) * can be identified with × L 2 ρ by the well-known Riesz representation theorem.
Let ( × L 2 ρ ) † be the space of bounded bilinear functionals Φ : In this case, DΦ(φ(0),φ) ∈ ( × L 2 ρ ) * is called the (first-order) Fréchet derivative of Φ at (φ(0),φ) ∈ × L 2 ρ . The function Φ is said to be continuously Fréchet differentiable if its Fréchet derivative DΦ : and where In this case, the bounded bilinear functional ρ . The second-order Fréchet derivative D 2 Φ is said to be globally Lipschitz on × L 2 ρ if there exists a constant K > 0 such that Assuming all the partial and/or Frechet derivatives of the following exist, the actions of the first-order Fréchet derivative DΦ(φ(0),φ) and the second-order Fréchet D 2 Φ(φ(0),φ) can be expressed as where ∂ φ(0) Φ and ∂ 2 φ(0) Φ are the first-and second-order partial derivatives of Φ with respect to its first variable φ(0) ∈ , D φ Φ and D 2 φ Φ are the first-and second-order Fréchet derivatives with respect to its second variable φ ∈ L 2 ρ , ∂ φ(0) D φ Φ is the second-order derivative first with respect to φ in the Fréchet sense and then with respect to φ(0), and so forth.
Let C 2,2 ( × L 2 ρ ) be the space of functions Φ : × L 2 ρ → that are twice continuously differentiable with respect to both its first and second variables. The space of Φ ∈ C 2,2 ( × L 2 ρ ) with D 2 Φ being globally Lipschitz will be denoted by C 2,2 lip ( × L 2 ρ ).
It will be shown in the proof of Theorem 3.5, however, that any tame function of the above form can be approximated by a sequence of quasi-tame functions that are in Ᏸ(Γ).
We have the following Ito formula in case Φ ∈ × L 2 ρ is a quasi-tame function in the sense defined above.
Proof. The Ito formula for a quasi-tame function Φ : × L 2 ([−h,0]) → for the × L 2 ([−h,0]) solution process {(x(t), x t ), t ≥ 0} of a stochastic function differential equation with a bounded delay h > 0 is obtained in an unpublished dissertation by Arriojas [18] (the same result can also be obtained from [21,22] with some modifications). The same arguments can be easily extended to the infinite memory stochastic hereditary differential equation (2.5) considered in this paper. To avoid further lengthening the paper, we omit the proof here.
Throughout the end of this proof, we define for each φ ∈ L 2 ρ and each k = 1,2,... the function φ (k) (−θ;h) by  (3.26) and c > 0 is the constant chosen so that and by the Lebesque dominating convergence theorem, we have Therefore, for any finite G-stopping time τ, we have from Theorem 3.4 and sample path convergence property of the Ito integrals (see [23,24]) that  This proves the theorem.

Dynkin's formula for the controlled state process.
Combining the above results in this section and results for general jumped processes (see [25,26]), we have the following Dynkin formula for the controlled (by the admissible strategy π) κ -valued state process {Z(t) = (X(t),N t ,S(t),S t ), t ≥ 0}: and A and Γ are as defined in (3.12) and (3.15).

The quasi-variational HJB inequality
The main objective of this section is to derive the dynamic programming equation for the value function in form of an infinite-dimensional quasi variational Hamilton-Jacobi-Bellman (HJB) inequality (or QVHJBI) (see (QVHJBI ( * )) in Section 4.3.4).

4.1.
The dynamic programming principle. The following Bellman-type dynamic programming principle (DPP) was established in [6] and still holds true in our problem by combining it with that obtained in [27][28][29] for optimal classical control of stochastic functional differential equations with a bounded memory. For the sake of saving space, we take the following result as the starting point without proof for deriving our dynamic principle equation.

Derivation of the QVHJBI.
In this section, we will derive the Hamilton-Jacobi-Bellman (HJB) quasi-variational inequality (see (QVHJBI ( * )) in Section 4.3.4) based on the dynamic programming principle described in Proposition 4.1. We emphasize here that it is not our intension to rigorously verify every step involved in the derivations since the rigorous verification is to be done in [1], the continuation of this paper. To derive (QVHJBI ( * )) in Section 4.3.4, we consider the effects on the value function when there is consumption but no transaction and when there is transaction but no consumption.
Mou-Hsiung Chang 21 In this case, V κ (X(t),N t ,S(t),S t ) = V κ (X(t−),N t− ,S(t),S t ) for all t ≥ 0, since there is no jump transaction. Assume that the value function V κ : κ → + is sufficiently smooth. From Proposition 4.1 and (4.4), we have This shows that since the maximum of the above expression is achieved at Note that the Fréchet differential operators A and Γ are defined in (3.12) and (3.15), respectively.

22)
ᏹ κ Φ is as given in (4.8), and the operators A and Γ are given as follows.

Boundary values of the QVHJBI.
4.3.1. The solvency region for κ = 0 and μ > 0. When there is no fixed transaction cost (i.e., κ = 0 and μ > 0), the solvency region can be written as where G 0 is the liquidating function given in (2.26) with κ = 0. This is because In this case, all shares of the stock owned or owed can be liquidated due the absence of a fixed transaction cost κ = 0.   The interface (intersection) between ∂ +,I,1 κ and ∂ +,I,2 κ is denoted by  Mou-Hsiung Chang 25 In this case, ∂ −,ℵ κ = ∅ (the empty set),

Decomposition of
On the other hand, if I = ∅ (the empty set), that is, n(−i) ≥ 0 for all i ∈ ℵ, then (4.31)

Boundary conditions for the value function.
Let us now examine the conditions of the value function V κ : κ → + on the boundary ∂ κ of the solvency region κ defined in (4.25)-(4.26). We make the following observations regarding the behavior of the value function V κ on the boundary ∂ κ .  To guarantee that (X(t),N t ,S(t),S t ) ∈ 0 , we require that Applying Theorem 3.5 to the process {e −rt G 0 (X(t),N t ,S(t),S t ), t ≥ 0} we obtain for every almost surely finite G-stopping time τ, where X(t) and N t are given in (2.32)-(2.36) with κ = 0. Taking into the account of (2.5) and (2.32)-(2.36) and substituting them into the function G 0 , we have G 0 X(t),N t ,S(t),S t = G 0 X(t−),N t− ,S(t),S t . (4.37) Intuitively, this is also because of the invariance of liquidated value of the assets without increase of stock value. Hence (4.36) becomes the following by grouping the terms n(Q(t) − i) according to i ∈ I and i / ∈ I: (4.38) Mou-Hsiung Chang 27 Now fix the first exit time τ ( τ is a G-stopping time) defined by (4.39) We can integrate (4.38) from 0 to τ, keeping in mind that (x,ξ,ψ(0),ψ) ∈ ∂ I,1 0 (or equivalently, G 0 (x,ξ,ψ(0),ψ) = 0), to obtain (4.40) Now use the facts that 0 < μ + β < 1, C(t) ≥ 0, α ≥ f (S t ) > r > 0, n(−i) < 0 for i ∈ I and n(−i) ≥ 0 for i / ∈ I and Rule 6 to obtain the following inequality: Now define the following process: Then by the Girsanov transformation (see [23,24]), { W(t), t ≥ 0} is a Brownian motion defined on a new probability space (Ω,Ᏺ, P;F), where P and P are equivalent probability measures, and hence Since G 0 (X(t),N t ,S(t),S t ) ≥ 0 for all t ≥ 0, this implies that τ = 0 a.s., that is, We need to determine the conditions under which the exit time occurred. Let k be the index of the shares of the stock where the state process violated the condition for the stopping time τ. In other words, if k ∈ I, then We will examine both cases separately.  Mou-Hsiung Chang 29 We have established that X( τ),N τ ,S( τ),S τ ∈ ∂ I,1 0 , (4.50) and this is inconsistent with (4.51) Therefore, we know n(Q( τ − k)) = 0. we see that n(Q( τ) − k) = 0. We conclude from both cases that (X( τ),N τ ) = (0,{0}). This means that the only admissible strategy is to bring the portfolio from (x,ξ,ψ(0),ψ) to (0,0,ψ(0),ψ) by an appropriate amount of the transaction specified in the lemma. This proves the lemma.
We conclude from some simple observations and Theorem 4.5 the following.
Boundary condition (ii). On ∂ I,1 κ for I ⊂ ℵ, then the investor should not consume but buy back n(−i) shares for i ∈ I and sell n(−i) shares for i ∈ I c of the stock in order to bring his portfolio to {0} × {0} × + × L 2 ρ,+ after paying transaction costs and capital gains taxes, and so forth. In other words, bring his portfolio from the position (x,ξ,ψ(0),ψ) ∈ ∂ I,1 κ to (0, 0,ψ(0),ψ) by the quantity that satisfies (4.57)-(4.58). In this case, the value function V κ : ∂ I,1 κ → + satisfies the following equation: (4.61) Note that this is a restatement of Theorem 4.5.

Mou-Hsiung Chang 31
Boundary condition (iii). On ∂ +,I,2 κ for I ⊂ ℵ, the only optimal strategy is to make no transaction but to consume optimally according to the optimal consumption rate function c * (x,ξ,ψ(0),ψ,) = (∂V κ /∂x) 1/(γ−1) (x,ξ,ψ(0),ψ) which is obtained via c * x,ξ,ψ(0),ψ = argmax c≥0 ᏸ c V κ x,ξ,ψ(0),ψ + c γ γ , (4.62) where ᏸ c is the differential operator defined by This is because the cash in his savings account is not sufficient to buy back any shares of the stock but to consume optimally. In this case, the value function V κ : ∂ +,I,2 κ → + satisfies the following equation provided that it is smooth enough: Boundary condition (iv). On ∂ −,I,2 κ , the only admissible consumption-investment strategy is to do no consumption and no transaction but to let the stock price grow as in the boundary condition (i).