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

This paper is the continuation of the paper entitled “Hereditary portfolio optimization with taxes and ﬁxed plus proportional transaction costs I” that treats 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 as-sets 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 due to the hereditary nature of the stock price dynamics and inventories. This paper contains the veriﬁcation theorem for the optimal strategy. It also proves that the value function is a viscosity solution of the QVHJBI.


Introduction and summary of results in [1]
This is the second of the two companion papers (see [1] for the first paper) that 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 (1.23)). 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 sold) for the 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 2 Journal of Applied Mathematics and Stochastic Analysis 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 (1.9)) and under the requirements of paying a 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,... denote the sequence of transaction times and ξ(i) stands for the quantities of transactions at time τ(i) (see Definition 1.10).
The investor will follow the following set of consumption, transaction, and taxation rules (Rules 1.1-1.6). Note that an action of the investor in the market is call a transaction if it involves trading of shares of the stock such as buying and selling. Rule 1.1. 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 1.2.
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 1.3. The proceeds for the sales of the stock minus the transaction costs and capitalgain 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 1.4. 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 1.5. At the time of a transaction (say t ≥ 0), the investor is required to pay a capitalgain tax (resp., be paid 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 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 the Mou-Hsiung Chang 3 amount of tax due at the transaction time is given by (1.1) Rule 1.6. 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 1.1-1.6, 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) and an impulse control (for the transactions) problem in infinite dimensions. In the first paper [1], a quasivariational Hamilton-Jocobi-Bellman inequality (QVHJBI) for the value function together with its boundary conditions are derived. This paper establishes the verification theorem for the optimal investment-trading strategy. It is also shown here that the value function is a viscosity solution of the QVHJBI (see QVHJBI( * ) in Section 2). 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 this and the previous paper, the state space will be S = × N × × L 2 ρ . In the above, (i) the stock inventory space, N, is the space of bounded measurable 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 1 {τ(−k)} is the indicator function at τ(−k).

Journal of Applied Mathematics and Stochastic Analysis
Let · N (the norm of the space N) be defined by Throughout the end of this paper, let ρ : (−∞,0] → [0,∞) be the influence function with relaxation property that satisfies the following conditions.
An element (x,ξ,ψ(0),ψ) ∈ S will be referred to as a portfolio, where x ∈ represents the investor's holding in his savings account, ξ ∈ N represents his stock inventory, and (ψ(0),ψ) ∈ × L 2 ρ stands for a profile of historical stock prices. The solvency region κ of the portfolio optimization problem is defined as where G κ : S → is the liquidating function defined by and S + = + × N + × + × L 2 ρ,+ is the positive cone of the state space S.
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 (1.11) 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 , the investor is allowed to consume from his savings account and can make transactions between his savings and stock accounts under Rules 1.1-1.6 and according to a consumption-trading strategy π = (C,-) defined below.
Mou-Hsiung Chang 5 Definition 1. 10. The pair π = (C,-) is said to be a consumption-trading strategy if (i) the consumption rate process C = {C(t), t ≥ 0} is a nonnegative G-progressively measurable process such that ..} 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 quantity 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 the current price of S(τ(i)) in his open short (resp., long) position at the previous time τ(i − k) and the base price of S(τ(i − k)). Note that G = {Ᏻ(t), t ≥ 0} is the filtration generated by {S(t), t ≥ 0}, that is, (1.15) For each stock inventory ξ of the form expressed (1.4), Rules 1.1-1.6 also dictate that the investor can purchase or short sell new shares and/or buy back (resp., sell) all or part of what he owes (resp., owns). Therefore, the trading quantity {m(−k), k = 0,1, ...} must satisfy the constraint set (ξ) ⊂ N defined by Given the initial portfolio and applying a consumption-trading strategy π = (C,-) (see Definition 1.10), 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 holdings {X(t), t ≥ 0} satisfy the following differential equation between the trading times: (1.18) and the following jumped quantity at the trading time: 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).
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 assumed to satisfy the following nonlinear stochastic hereditary differential equation: Mou-Hsiung Chang 7 with the initial historical price function (S(0),S 0 ) = (ψ(0),ψ) ∈ + × L 2 ρ,+ . Note that f (S t ) and g(S t ) in (1.4) represent, respectively, the mean growth rate and the volatility rate of the stock price at time t ≥ 0 and that they are dependent on the entire history of stock Under the Lipschitz and linear growth conditions (see [ is a strong Markovian with respect to the filtration G. Definition 1.11. If the investor starts with an initial portfolio (1.24) The consumption-trading strategy π = (C,-) defined in Definition 1.10 is said to be ad- The class of consumption-investment strategies admissible at (x,ξ,ψ(0),ψ) ∈ κ will be denoted by ᐁ κ (x,ξ,ψ(0),ψ).

Journal of Applied Mathematics and Stochastic Analysis
The results obtained in [1] include derivations of the infinite dimensional quasivariational Hamilton-Jacobi-Bellman (GVHJB) together with its boundary conditions. The boundary conditions for the value function are given as follows.

The verification theorem
Let Ᏸ(Γ) be the domain of the operator Γ defined in (1.35), that is, Ᏸ(Γ) is the set of (Borel) measurable functions Φ : κ → such that the limit in (1.35) exists for each fixed (x,ξ,ψ(0),ψ) ∈ κ . Let C 1,0,2,2 lip ( κ ) be the collection of functions Φ : κ → that are continuously differentiable with respect to its first variable x and twice continuously differentiable and Fréchet differentiable with respect to its third variable ψ(0) and fourth variable ψ and the second-order Fréchet derivative D 2 Φ(x,ξ,·,·) is said to be globally Lipschitz on × L 2 ρ in operator norm · † , that is, there exists a constant K > 0 such that We have the following verification theorem for the value function V κ : κ → for our hereditary portfolio optimization problem.

The viscosity solution
It is clear that the value function V κ : κ → + has discontinuity on the interfaces Q I,+ and Q I,− and hence it can not be a solution of QVHJBI( * ) in the classical sense. The main purpose of this section is to show that it is a viscosity solution of the QVHJBI( * ). See [6,7] for connection of viscosity solutions of second-order elliptic equations with stochastic classical control and classical-impulse control problems.
To give a definition of a viscosity solution, we first define the upper and lower semicontinuity concept as follows.
Let Ξ be a metric space, and let Φ : Ξ → be a Borel measurable function. Then the upper semicontinuous (USC) envelop Φ : Ξ → and the lower semicontinuous (LSC) envelop Φ : Ξ → of Φ are defined, respectively, by We let USC(Ξ) and LSC(Ξ) denote the set of USC and LSC functions on Ξ, respectively. Note that in general, one has Let ᏸ( × L 2 ρ ) and ( × L 2 ρ ) † be the space of bounded linear and bilinear functionals equipped with the usual operator norms · * and · † , respectively.
The following properties of the intervention operator ᏹ κ can be established similar to [7, Lemmas 3.2, 3.3, and Corollary 3.4] with some modifications to fit our situation.
Lemma 3.2. The following statements hold true regarding ᏹ κ defined by (1.38).
(3.60) Therefore, to complete the proof of Proposition 3.5 we must verify Lemma 3.6.