Model and problem formulation

2.2.1 Assets

Let n ≥ 1 be an integer, W = {Wt, t ≥ 0} be an n-dimensional Brownian motion on the complete probability space (Ω,F,P) and denote by F={F^t, t ≥0} the corresponding com-pleted canonical filtration. We also consider an infinite time horizon T = +∞.

We then consider a (complete) financial market consisting of:

- one non-risky asset (a bank account), evolving with the interest rate r≥0, which will be assumed to be constant, in order to simplify notations; an easy generalization could be done for a deterministic interest rate;

2.2. MODEL AND PROBLEM FORMULATION 41 - n risky assets with dynamics that are given by:

dPt=diag(Pt)[b(Pt)dt+σ(Pt)dWt] (2.1) where Pt : Ω→Rⁿ, b :R₊×Rⁿ→ Rⁿ and σ :R₊×Rⁿ→Mn(R). For a vector x, diag(x) stands for the diagonal matrix with i-th term on the diagonal equal to xⁱ.

We assume that b and σ are Lipschitz continuous and bounded, which guarantees that the previous SDE has a strong solution with continuous paths. We also assume thatσ is regular, σ⁻¹ is bounded and we define the risk premium by:

θ:=σ⁻¹[b−r1n], with 1n=





 1 ...

1





∈Rⁿ.

We focus here on the infinite horizon case, but the results can be extended almost without any change to the finite horizon case. In fact notations are a bit more complex for the finite case (functions depend also on timet), but things work simpler as we will not have to assume integratibility hypotheses unlike in the infinite horizon framework. We will make a few remarks about that along this work. Another easy extension would be to consider the interest rate r to be a deterministic continuous and bounded function.

2.2.2 Taxation rule

As in [3], the aim of this work is to study a problem of optimal consumption and investment, when the capital gains are subject to taxes. We use the same taxation model as theirs, which is the continuous version of the one introduced first by Damon, Spatt and Zhang [13]. We assume that the sales of stocks are subject to taxes on capital gains. The amount of tax for a sale at timet of the i-th risky asset is computed by comparing the current price P_tⁱ to the current i-th index value B_tⁱ, defined as the weighted average price of shares purchased by the investor up to time t. If P_tⁱ < B_tⁱ, the investor realizes a capital gain, while if P_tⁱ > B_tⁱ he realizes a capital loss. When the investor sells a certain number of shares, the proportions of shares are kept constant in order to compute the tax basis, which means that it will not change the tax basis (but it will change the number of shares used to compute it).

For example, if the agent buys x₁ shares of the i-th risky asset at time t₁ at the price P₁ and x2 shares of the same asset at time t2 at the price P2, the tax basis at time t will be B_tⁱ = ^x1P_x^1+x2P2

1+x2 (if nothing else occurs). If he then sells at time t3 z shares, we consider that he still owns x1(1− x1+x^z 2) of thet1−shares and x2(1− x1+x^z 2) of the t2−shares. Thus B_t⁺

3 =B_t⁻

3.

Then the sale of one unit share of the i-th asset at time twill breed the (algebraic) payment of an amount denoted by ℓ(P_tⁱ−Bⁱ_t) which we assume to be linear:

ℓ(P_tⁱ−B_tⁱ) = α(P_tⁱ−B_tⁱ) (2.2) where α∈[0,1) is constant and is called the tax rate coefficient.

Remark 2.1 As claimed before, taxes can be negative. It is almost true in certain markets.

Indeed, in some countries, if P is lower than B, the investor will have a tax deduction for the following year. A linear model is not of course perfectly realistic, but not absolutely inconsistent.

Remark 2.2 As it is pointed out in [3], such a taxation rule might a priori allow better portfolios than the optimal one of the tax-free model, which would not be economically acceptable. Hopefully, in our framework too the upper bound result will show that this will never happen.

2.2.3 Strategies

We assume that the agent allocates his wealth between the bank account (non-risky asset) and the risky assets. The amount of money he owns in the bank account is denoted by (Xt), the amount in the risky assets by (Yt) which takes values in Rⁿ. We also denote by (Ct) the consumption (rate) of the agent. In other words, Ctdt represents the amount consumed within [t, t+dt]. Recall that essentially for notational purposes, we have taken an infinite time horizon T = +∞.

We then introduce the position in the risky assets evaluated at the basis prices:

K_tⁱ =B_tⁱY_tⁱ

P_tⁱ. (2.3)

Comparing P to B is equivalent to comparing Y to K. Notice that B_tⁱ is not defined if Y_tⁱ = 0. We assume that in that case B_tⁱ = Y_tⁱ, but such a choice has no influence on the value of K_tⁱ.

Moreover we make the following assumptions:

− (Xt), (Yt) and (Ct) are progressively measurable with respect toF;

− Ct≥0, P−a.s, ∀t≥0;

− Z S

0

Ctdt <+∞, P−a.s, for all S ∈[0,+∞).

2.2. MODEL AND PROBLEM FORMULATION 43 Remark 2.3 As we will see, X and Y are both c`adl`ag by definition, therefore assuming that they are F-adapted is equivalent to assuming that they are progressively measurable.

We will denote (Zt) the total wealth of the investor after liquidation of the risky asset positions:

Zt=Xt+

n

X

i=1

[(1−α)Y_tⁱ +αK_tⁱ]. (2.4)

Transfers on the financial market are described by means of the transfers between the bank account towards the risky assets and in the opposite direction. Because of taxes, considering only the sum of algebraic transactions does not make sense. Therefore, we denote by (Lt)∈ Rⁿ the absolute transfer fromX to Y, and by (Mt)∈Rⁿ the relativetransfer from Y to X (M_tⁱ is the proportion of Y_tⁱ transfered,Y_t−ⁱ dM_tⁱ is the amount).

We assume that M and L are F-adapted, right-continuous and non-decreasing processes.

We also assume that M₀₋ = L₀₋ = 0. Finally, we assume that short-sales are not allowed, so that we restrict the jumps of M by:

∆M_tⁱ ≤1,∀i,∀t≥0,P-a.s. (2.5)

We assume that the dynamics of Y is given for each component iby:

dY_tⁱ =Y_tⁱdP_tⁱ

P_tⁱ +dLⁱ_t−Y_tⁱ−dM_tⁱ (2.6) as the variation of the wealth in the i-th risky asset is due to three factors: the relative variation of the stock dP_tⁱ

P_tⁱ multiplied by the amount held Y_tⁱ, the absolute transfers from the bank account to the risky asset dLⁱ_t and the relative transfers from the risky asset to the bank account dM_tⁱ multiplied by the amount held just before the transfer (as jumps are allowed) Y_t−ⁱ .

The dynamics of K is assumed to be given, for each iby:

dK_tⁱ =dLⁱ_t−K_tⁱ−dM_tⁱ (2.7) as the variation of Kⁱ is only due to the transfers in one direction or the other.

Then the dynamics of X under the self-financing condition is:

dXt= (rXt−Ct)dt+

n

X

i=1

[−dLⁱ_t+{(1−α)Y_tⁱ⁻ +αK_tⁱ⁻}dM_tⁱ] (2.8)

as the variation of the wealth in the bank account is due to five factors. The non-risky rate brings the rXtdt term. The consumption brings the −Ctdt term. The absolute transfers from the bank account to the risky assets bring the terms −dLⁱ_t, while the tranfers from the risky assets to the bank account bring the term Y_t−ⁱ dM_tⁱ to which we must substract the taxes, explaining the expression −α(Y_t−ⁱ −K_t−ⁱ )dM_tⁱ.

And this implies:

dZt= (rZt−Ct)dt+

n

X

i=1

[(1−α)Y_tⁱ(dP_tⁱ

P_tⁱ −rdt)−rαK_tⁱdt]. (2.9) Notice that L and M do not appear in the dynamics of Z.

Denoting x.y the scalar product of x and y, we will use the following notations:

(i) For any m∈ N^∗, if x, x^′ ∈R^m, we will denote x ≥x^′ (respectively x > x^′) if and only if x−x^′ ∈(R₊)^m (respectively x−x^′ ∈(R^∗₊)^m).

(ii) Let S =

(x, y, k)∈R×Rⁿ×Rⁿ; x+ [(1−α)y+αk].1n >0, ∀i, yⁱ >0, kⁱ >0 . (2.10) Then S is the closure of S, and ∂^zS =

(x, y, k)∈S, z =x+ [(1−α)y+αk].1n= 0 the boundary corresponding to a zero initial wealth after liquidation of the risky positions.

Definition 2.4 (i) A (consumption-investment) strategy is a triple (C, L, M)satisfying the previous hypotheses.

(ii) Given an initial condition s ∈S, and a strategy ν, we note S^s,ν = (X^s,ν, Y^s,ν, K^s,ν) the unique strong solution of the dynamics above such that S₀₋=s.

(iii) Let s∈S, a strategy ν is s-admissible if S^s,ν satisfies the no-bankruptcy condition:

Zt≥0, P-a.s, ∀t≥0. (2.11) We denote byA(s)the set of all s-admissible strategies. A strategy ν is said to be admissible if there exists s∈S such that ν is s-admissible.

2.2.4 The consumption-investment problem

Now we are able to define the agent’s optimization problem. We assume that the investor’s preferences are described by a utility functionU :R₊→R,C¹, (strictly) increasing, strictly concave, and satisfying the Inada conditions:

x→0limU^′(x) = +∞ and lim

x→+∞U^′(x) = 0.

2.3. FIRST PROPERTIES OF THE VALUE FUNCTION 45