In contrast with a frictionless model (Merton’s problem), even in simple cases as in [3] (Black-Scholes dynamics for P, power utility function), we have very few knowledge of the value function as soon asα >0. And in our more general framework, we know even less. But using bounds thanks to the Merton’s problem, we will be able to derive a first order expansion of the value function as the tax rate α and the interest rate r go to zero. Therefore we first make a quick review of Merton’s problem before showing that considering two Merton’s problems with slightly different parameters, we obtain upper and lower bounds.
2.4. BOUNDS USING MERTON’S PROBLEM 47
2.4.1 Review of Merton’s problem
We first recall classical results about Merton’s problem without proofs. We refer the reader to Karatzas and Shreve [49], chapter 3, for the proofs or any further detail.
In this section, we denote by ¯Vα the value function for Merton’s problem with modified parameters (θα, σα). Moreover we call the two problems associated with ¯V and ¯Vα respec-tively standard Merton’s problem and modified Merton’s problem.
Merton’s problem is the same as the previous problem, but without taxes. Consequently, we only consider the dynamics of Z (X andY bring no useful information for the optimization problem), and the strategies take the form (Ct,Πt)∈R+×Rn, where Πit is the position in the i-th risky asset, so that it corresponds to the Yti of the model with taxes. We write ¯Z the portfolio value for this problem, and its dynamics is given by:
dZ¯t= (rZ¯t−Ct)dt+ Πt.σt(θtdt+dWt). (2.14) The set of admissible strategies is bigger than the one defined before. In fact, we do not limit ourselves to transfers with bounded variations, whereas it was the case for L and M previously.
The classical approach consists in considering the optimization dual problem. First, let us define the dual problem using Legendre-Fenchel transform:
V˜(ζ) := sup
z∈R{V¯(z)−zζ}, ∀ζ ∈R As ¯V is concave, we have:
V¯(z) = inf
ζ∈R{V˜(ζ) +zζ}, ∀z ∈R We define in the same way the dual utility function, denoted by ˜U.
We denote by P0 the martingale measure, defined by its Radon-Nikodym density:
dP0 dP
Ft
=e−R0tθ∗udWu−12R0tkθuk2du, (2.15) where we have written θu instead of θ(Pu) for notational purposes. We denote by E0 the expectation under P0, while E stands for the expectation under P. We also define W0 by:
Wt0 = Wt+ Z t
0
θudu, and using Girsanov’s Theorem W0 is a Brownian motion under P0. Finally, we introduce the following process:
Ht:=e−rt−R0tθ∗udWu−12R0tkθuk2du (2.16)
=e−rt dP0 dP
Ft
.
The assumptions on U ensure that U′ has an inverse which we denote by I and we define for y >0:
X(ζ) :=E
Z +∞
0
HtI(ζeβtHt)dt
(2.17)
=E0
Z +∞
0
e−rtI(ζeβtHt)dt
. We assume that:
X(ζ)<∞, ∀ζ >0. (2.18)
Then, X inherits the decrease of I, lim
0+ X = +∞ and lim
+∞X = 0. Moreover we know thatX is C2(0,+∞) and admits an inverse function Y which is decreasing and C2:
X ◦ Y(z) =z.
Then we introduce for ζ ∈(0,∞):
G(ζ) :=E
Z +∞
0
e−βtU ◦I(ζeβtHt)dt
. (2.19)
We also know that Gis C2(0,+∞).
Remark 2.13 If T <+∞, we define similar quantities. See [49] for more details.
We summarize here the important properties for us:
Theorem 2.14 For z >0, V(z) = G◦ Y(z), and in particular V is C2 on (0, T).
Moreover there exists an optimal control ( ˆC,Π), given by:ˆ Cˆt=I(eβtY(z)Ht) Πˆt= (σ∗t)−1
Zˆtθt+ ψt
Ht
,
2.4. BOUNDS USING MERTON’S PROBLEM 49 where Zˆ is the associated optimal wealth satisfying:
Zˆt = 1 Ht
E Z ∞
t
HuCˆudu Ft
,
and ψ in the (unique) adapted and L2 integrand in the martingale representation of:
E Z ∞
0
HuCˆudu|Ft
. And we have:
V′(ζ) =Y(z), V˜(ζ) =G(ζ)−ζX(ζ), V˜′(ζ) =−X(ζ).
Remark 2.15 Notice that assumption (2.18) implies that E Z ∞
0
HuCˆudu
<∞.
Remark 2.16 We also know that, in this Markov framework,V satisfies an HJB equation, but we will not use it.
Remark 2.17 Again, this theorem can be stated for T <∞.
2.4.2 Upper and lower bounds
Now we derive upper and lower bounds from two Merton’s problems, with different param-eters. Again the proofs of the results of this section are very close to the ones given in [3], except for the end of the proof of Proposition 2.20 below. The proofs are given in the appendix, stated in our framework and with the additional arguments for Proposition 2.20.
First we give the upper bound.
Proposition 2.18 Let s= (x, y, k)∈S, we have the following upper bound:
V(s)≤V¯ (x+ [(1−α)y+αk].1n)
Then we introduce a ”modified” Merton’s problem. More precisely, we consider Merton’s problem but with parameters (θα, σα) defined by:
θα=θ− rα
1−ασ−11n (2.20)
σα= (1−α)σ. (2.21)
We write ¯V and ¯Vα respectively the value functions of Merton’s problem with parameters (θ, σ) and (θα, σα) respectively.
Recall that we note J(s, ν) = E Z ∞
0
e−βtU(Ct)dt
where (Ct) is associated to the (admis-sible) strategy ν. For Merton’s problem, we will note as well ¯J(z, γ).
For Merton’s problem, we define the relative consumption and portofolio:
ct:= Ct
Z¯t
πt:= Πt
Z¯t
, if ¯Zt 6= 0.
We denote ˆc and ˆπ (resp. ˆcα and ˆπα) the optimal relative consumption and portfolio asso-ciated to Merton’s problem (resp. Merton’s modified problem).
Together with assumption (2.18), we make the following hypotheses for the optimal control of any Merton’s modified problem (any α≥0):
− for any α≥0 and any T > 0, ˆcα and ˆπα are bounded on [0, T]; (2.22)
− for any α≥0, there exists a version of ˆπα that has continuous paths; (2.23)
− for any α≥0, πˆα is a diffusion with uniformly bounded drift and volatility. (2.24) As H has continuous paths, and I is continuous, it is immediate that for any α, ˆcα has continuous paths.
Remark 2.19 When θ and σ are constant (resp. deterministic) and T = +∞ (resp. T <
+∞), the feedback form given by Corollary 3.9.15 in [49] (resp. 3.8.8) implies that ˆπα has continuous paths, as it is a continuous function of the optimal wealth. Moreover, the optimal terminal wealth is a continuous function of Ht, so that the dynamics of ˆπα is explicit.
Finally we give a lower bound using this modified problem.
Proposition 2.20 Let s= (x, y, k) such that z =x+ [(1−α)y+αk].1n ≥0. Then:
V¯α(z)≤V(s).
Here, the proof is a bit different from the one given in [3]. More precisely, the main idea is the same, but in the end, as we are dealing with a general utility function, it is a little harder to conclude. This additional part corresponds to the part from Lemma 2.36 until the end of the appendix.
Remark 2.21 We could have taken a specific tax rate for each risky asset: α= (αi)∈Rn. Everything would work exactly the same way, we would just need to define σα and θα the following way: σα = (I−diag(α))σ, which means that the i-th line is multiplied by (1−αi), and we will have (σα)−1 =diag
1 1−αi
i
σ−1. Then we would also haveθα =θ−r(σα)−1α.