c xxxx Society for Industrial and Applied Mathematics
⃝
Vol. xx, pp. x
Characterization of Optimal Strategy for Multi-Asset Investment and
Consumption with Transaction Costs∗
Xinfu Chen† and Min Dai‡
Abstract. We consider the optimal consumption and investment with transaction costs on multiple
assets, where the prices of risky assets jointly follow a multi-dimensional geometric Brownian
motion. We characterize the optimal investment strategy and in particular prove by rigorous
mathematical analysis that the trading region has the shape that is very much needed for well
defining the trading strategy, e.g., the no-trading region has distinct corners. In contrast,
the existing literature is restricted to either single risky asset or multiple uncorrelated risky
assets.
Key words. Portfolio selection, Optimal investment and consumption, Transaction costs, Multiple risky
assets, Shape and location of no-trading regions.
AMS subject classifications. 91G10, 93E20
1. Introduction. We consider the optimal investment and consumption decision of a riskaverse investor who has access to multiple risky assets as well as a riskfree asset. Proportional
transaction costs are incurred when the investor buys or sells the risky assets whose prices
are assumed to follow a multi-dimensional geometric Brownian motion. We aim to provide a
theoretical characterization of the optimal strategy.
In the absence of transaction costs, the problem described above has been studied by
Merton (1969, 1971). It turns out that the optimal strategy of a constant relative risk aversion
(CRRA) investor is to keep a constant fraction of total wealth in each assets and consume at
a constant fraction of total wealth. In contrast, the optimal strategy of a constant absolute
risk aversion (CARA) investor is to keep a certain fixed amount in each risky asset and a
consumption that is affine in the total wealth. Merton’s strategy requires continuous trading
in all risky assets and thus must be suboptimal when transaction costs are incurred.
Magill and Constantinides (1976) introduce proportional transaction costs to Merton’s
model with single risky asset and a CRRA investor. They provide a fundamental insight that
there exists an interval, known as the no-trading region, such that the optimal investment
strategy is to keep the fraction of wealth invested in the risky asset within the interval (i.e.,
no-trading region). Hence, as long as the initial fraction falls within the no-trading region,
the future transactions only occurs at the boundary of the region. For a CARA investor, it
can be shown that the optimal investment strategy is to keep the dollar amount in the risky
asset between two levels [cf. Liu (2004) and Chen et al. (2012)].
Since the seminal work of Magill and Constantinides (1976), portfolio selection with trans∗
We thank two anonymous referees for helpful comments and suggestions. Xinfu Chen acknowledges support
from NSF grant DMS-1008905. Min Dai acknowledges support from Singapore AcRF grant R-146-000-138-112 and
NUS Global Asia Institute - LCF Fund R-146-000-160-646.
†
Department of Mathematics, University of Pittsburgh, PA 15260, USA.
‡
Department of Mathematics and Centre for Quantitative Finance, National University of Singapore, Singapore. Email: [email protected]
1
x–x
2
Xinfu Chen and Min Dai
action costs has been extensively studied along different lines, e.g., the effect of transaction
costs on liquidity premium [Constantinides (1986), Jang et al. (1997)], perturbation analysis
for small transaction costs [Shreve and Soner (1994), Atkinson and Wilmott (1995), Janecek
and Shreve (2004), Law et al. (2009), Bichuch and Shreve (2011), Bichuch (2012), Soner and
Touzi (2012) and Possamaı̈ et al. (2013)], utility indifference pricing [Davis et al. (1993),
Constantinides and Zariphopoulou (2001), Bichuch (2011)], martingale approach [Cvitanic
and Karatzas (1996)], numerical solutions [Gennotte and Jung (1994), Akian et al. (1996),
Muthuraman (2006), Muthuraman and Kumar (2006), Dai and Zhong (2010)], risk sensitive
asset management [Bielecki (2000, 2004)], etc. In particular, there is a large body of literature devoted to the characterization of optimal investment strategy. For example, Davis and
Norman (1990) and Shreve and Soner (1994) provide a thorough theoretical characterization
of the optimal strategy for the lifetime optimal investment and consumption. Taksar et al.
(1988) and Dumas and Luciano (1991) present exact solutions for the CRRA utility maximization of terminal wealth as time to maturity goes to infinity. Liu and Loewenstein (2002),
Dai and Yi (2009), Dai et al. (2009), Dai et al. (2010), and Chen et al. (2012) characterize
the finite time horizon investment decisions. Kallsen and Muhle-Karbe (2010) and Gerhold
et al. (2011) study the optimal strategy by determining a shadow price which is the solution
to the dual problem.
Most of existing theoretical characterizations of optimal strategy are for the single riskyasset case.1 In contrast, there is relatively limited literature on the multiple risky-asset case.
Assuming that there are multiple uncorrelated risky assets available for investment, Akian et
al. (1996) obtain some qualitative results on the optimal strategy of a CRRA investor. Liu
(2004) considers a CARA investor who is also restricted to invest in uncorrelated risky assets.
He shows that the problem can be reduced, by virtue of the separability of the CARA utility
function, to the single risky-asset case. This leads to the separability of the optimal investment
strategy which is to keep the dollar amount invested in each asset between two constant levels.
Unfortunately, such a reduction does not work when the risky assets are correlated.
The main contribution of this paper is to provide a thorough characterization of the
optimal investment strategy for a risk-averse investor who can access multiple correlated risky
assets as well as a riskfree asset. We focus on the CARA utility case, and an extension to
the CRRA utility case is explained later. To illustrate our results, we take as an example the
scenario of two risky assets. We will show that the shape of trading and no-trading regions
must be as in Figure 1, where “Si ”, “Bi ”, and “Ni ” represent selling, buying, and no trading
in asset i, respectively. The no-trading region N1 ∩ N2 locates in the center, surrounded
by eight trading regions. Moreover, each intersection ∂S1 ∩ ∂S2 , ∂S1 ∩ ∂B2 , ∂B1 ∩ ∂S2 , and
∂B1 ∩ ∂B2 is a singleton. In addition, we show that the boundary of each of corner regions
S1 ∩ S2 , S1 ∩ B2 , B1 ∩ B2 , and B1 ∩ S2 consists of one vertical and one horizontal half line,
whereas the boundary of each of S1 ∩ N2 , N1 ∩ S2 , B1 ∩ N2 , and N1 ∩ B2 consists of two
parallel either vertical or horizontal half lines and a curve in between connecting the end points
of the two half lines. These characterizations on the shapes of trading regions are extremely
1
There do exist many papers working on perturbation analysis or numerical solutions for the multiple riskyasset case, e.g. Law et al. (2009), Bichuch and Shreve (2011), Muthuraman and Kumar (2006), Dai and Zhong
(2010).
Amount in Asset 2
Multi-Asset Investment and Consumption with Transaction Costs
B1 ∩ S2
N1 ∩ S 2
B1 ∩ N2
N 1 ∩ N2
B1 ∩ B2
N1 ∩ B2
3
S1 ∩ S2
S 1 ∩ N2
S1 ∩ B2
Amount in Asset 1
Figure 1.1. Shape of trading and no-trading regions with CARA utility
important because they are necessary conditions for the trading strategy to be well-defined.2
For example, given an initial portfolio in S1 ∩ B2 , the investor should sell asset 1 and buy
asset 2 to the unique corner ∂S1 ∩ ∂B2 ; similarly, given an initial portfolio in S1 ∩ N2 , the
investor should sell asset 1 and keep asset 2 unchanged to the unique portfolio on ∂S1 ∩ N2 .
Recently, Soner and Touzi (2012) and Possamaı̈ et al. (2013) use homogenization techniques to prove rigorous asymptotic expansion for optimal investment and consumption in
a multidimensional market with proportional transaction costs in a very general setting. In
particular, the model in the papers permits transaction between risky assets (unlike in the
present paper) and the shape of the resulting trading/no-trading regions is different from that
as described above. It should be pointed out that our method and results cannot be extended
to models with such a feature; see Remark 3.1.
We also prove that the no-trading region is contained in a union of uniformly bounded
ellipses. Thus in numerical simulation one need only perform computation on the bounded
union. Furthermore, we provide a precise characterization of the corners of the no-trading
region.
We present the model and collect main theoretical results in Section 2. In section 3, we
explain an extension of our results to the CRRA utility. Section 4 is devoted to the proof of
our main results regarding shape and location of non-trading zone for the “infinite horizon”
problem (Theorems 2.4 and 2.5), with the proof of a technical C 1 regularity of viscosity
solution (Theorem 2.3) being left to Section 5.
Since negative wealth is permitted with the CARA utility, we need to impose some constraint to prevent the investor from unlimited consumptions. This motivates us to start from
a finite horizon problem formulation, where the expected utility is from not only the intermediate consumption but also the terminal wealth (i.e., bequest). We provide some basic
2
It should be pointed out that these results have been conjectured or numerically verified by some researchers
[e.g. Liu (2004), Dai and Zhong (2010), Bichuch and Shreve (2011)]. However no theoretical analysis has been
given so far.
4
Xinfu Chen and Min Dai
properties of the value function associated with the finite horizon problem; in particular, we
let the finite horizon T → ∞ to obtain the “infinite horizon” problem addressed in this paper.
These results are presented as preliminaries in Section 2 and their proofs are placed in the
Appendix.
2. Problem Formulation and Main Results. Consider a portfolio consisting of one riskfree asset (bank account) and n risky assets whose unit share prices are stochastic process
{(St0 , St1 , · · · , Stn )} described by the stochastic differential equations
dSt0
= rdt,
St0
n
∑
dSti
=
α
dt
+
aij dWtj
i
Sti
j=1
for i = 1, · · · , n,
where {Wt1 , · · · , Wtn }t≥0 is a standard n-dimensional Wiener process, r > 0 is the constant
bank rate, αi is the constant expected return rates of the i-th risky asset, and (aij )n×n
is a constant positive definite matrix. We consider optimal strategies of investment and
consumption subject to transaction cost which are proportional to the amount of transactions.
2.1. Investment and Consumption. We introduce a non-negative parameter κ where
κ = 0 corresponds to the no-consumption case. Suppose the terminal time is T and current
time is t < T . For s ∈ [t, T ), we denote by κcs ds the consumption, deducted from the bank
account, during time interval [s, s + ds). Here we assume that κ has the same unit as r, being
1/year, and that cs has the unit of dollars.3 We denote by dLis the transfer of money from the
bank account to the i-th risky assets during [s, s + ds), which incurs purchasing costs λi dLis .
Similarly, we denote by dMsi the money transferred from the i-th risky asset to the bank
account during [s, s + ds), which incurs selling costs µi dMsi . Here λi ≥ 0 and µi ∈ [0, 1) are
the constant proportions of transaction costs for purchasing and selling the i-th risky asset,
respectively.
Let xs and ys = (ys1 , · · · , ysn ) be dollar values at time s ∈ [t, T ] invested in the bank
account and risky assets, respectively. Their evolutions are described by

∑
∑

dxs = (rxs − κcs )ds −
(1 + λi )dLis +
(1 − µi )dMsi ,



i
i
(2.1)
∑


dy i = ysi (αi ds +
aij dWsj ) + dLis − dMsi , i = 1, · · · , n.

 s
j
For simplicity, we define an admissible (investment-consumption) strategy as S = (C, L, M )
where C = {cs }s∈[t,T ] , L = {L1s , · · · , Lns }s∈[t,T ] , and M = {Ms1 , · · · , Msn }s∈[t,T ] are adapted
processes satisfying
(2.2)
dLis ≥ 0,
dMsi ≥ 0,
and {xs , ys }t≤s≤T is the solution of (2.1) subject to constant initial conditions. We denote by
At all the admissible strategies.
3
The parameters κ and K in (2.3) below are both used as the weight between consumption and terminal
wealth. However K is dimensionless while κ has the same unit as r. It should be pointed out that the condition
r < 1 in Chen et al. (2012) indeed means r < κ.
Multi-Asset Investment and Consumption with Transaction Costs
5
2.2. The Merton’s Problem. Given concave utilities U (x, y) and V (c) for the terminal
portfolio and consumption respectively, a constant discount factor β > 0, and positive dimensionless constant weight K, we consider the measure of quality of an investment-consumption
strategy S defined by
(2.3)
J(S, t) := KU (xT , yT )e−β(T −t) +
∫
T
V (cs )e−β(s−t) κds,
t
where {xs , ys }s∈[t,T ] is the solution of (2.1) with given strategy S ∈ At . The Merton’s problem
is to maximize the expected utility:
Φ(x, y, t) = sup Ex,y
t [J(S, t)]
∀ x ∈ R, y ∈ Rn , t ≤ T,
S∈At
where Ex,y
is the expectation under the condition (xt , yt ) = (x, y). In this paper, we mainly
t
consider the exponential utility
V (c) := −e−γc ,
U (x, y) := V (x + ℓ(y)),
where ℓ(y) is the liquidation value of the holdings in the risky assets:
(2.4)
ℓ(y) =
∑
{
ℓi (yi ),
ℓi (yi ) =
i
(1 − µi )yi
(1 + λi )yi
if yi ≥ 0,
if yi < 0.
Notice that ℓi (·) is a concave function and
ℓi (yi ) =
min
1−µi ≤k≤1+λi
{kyi } = min{(1 − µi )yi , (1 + λ)yi } ∀ yi ∈ R.
For later use, we define
σij :=
∑
aik ajk
k
and
(2.5)
mj :=
∑
σ ji (αi − r) ∀ j,
i
A0 :=
1∑
mi σij mj ,
2
i,j
where (σ ij )n×n is the inverse matrix of (σij )n×n . Here m := (m1 , · · · , mn ) is the optimal
investment strategy for the Merton’s problem without transaction costs, being the solution of
the linear system
∑
j
σij mj = αi − r
∀ i = 1, · · · , n.
6
Xinfu Chen and Min Dai
2.3. Preliminary Results. With the exponential utility, we have the following:
Theorem 2.1. There exists a function ψ defined on Rn × [0, ∞) such that
Φ(x, y, t) = −e−γξ(τ )x+(r−β)b(τ )+Z(τ ) ln K−ln ξ(τ )−ψ(z,τ ) ,
(2.6)
where τ = T − t, z = γξ(τ )y, and ξ, Z, b are defined by
(2.7)
ξ(τ ) =
rerτ
,
r + κerτ − κ
Z(τ ) = ξ(τ )e−rτ ,
b(τ ) =
κ(erτ − 1 − rτ ) + r2 τ
,
r(r + κerτ − κ)
i.e., unique solutions of the following
ξ ′ = (r − κξ)ξon [0, ∞),
(2.8)
ξ(0) = 1;
′
Z = −κξZon [0, ∞),
(2.9)
Z(0) = 1;
′
b = −κξb + 1on [0, ∞),
(2.10)
b(0) = 0.
In addition,
(1) for each τ ≥ 0, ψ(·, τ ) is Lipschitz continuous:
ℓ(ẑ − z) ≤ ψ(ẑ, τ ) − ψ(z, τ ) ≤ −ℓ(z − ẑ)
(2.11)
∀ z, ẑ ∈ Rn ;
(2) for A0 defined in (2.5),
ℓ(z) ≤ ψ(z, τ ) ≤ ℓ(z) + A0 b(τ )
(2.12)
∀ z ∈ Rn , τ ≥ 0;
(3) for each τ ≥ 0, ψ(·, τ ) is concave;
(4) ψ is a viscosity solution of

{
}
 min ∂τ ψ − Aτ [ψ], B(∇ψ) = 0 in Rn × (0, ∞),
(2.13)

ψ(·, 0) = ℓ(z) on Rn × {0},
where Aτ and B are defined by
(2.14)
Aτ [ψ] :=
∑
1∑
σij zi zj (∂zi zj ψ−∂zi ψ∂zj ψ) +
(αi −r)zi ∂zi ψ + κξ(τ ) [z · ∇ψ − ψ],
2
i,j
(2.15)
i
B(p) := min min{1 + λi − pi , −1 + µi + pi }
i
∀ p = (p1 , · · · , pn ) ∈ Rn .
We regard the infinite horizon problem as the limit of finite horizon problem under appropriate scales, as the finite horizon T → ∞. For this, we have the following:
Theorem 2.2.Suppose κ > 0. Then there exist a function u and a constant M such that
(2.16)
|∂τ ψ(z, τ )| + |ψ(z, τ ) − u(z)| ≤ M (1 + rτ )e−rτ
∀ τ > 0, z ∈ Rn .
In addition, u is a Lipschitz continuous concave viscosity solution of the following equation
(2.17)
min{−A[u], B(∇u)} = 0,
0≤u−ℓ≤
A0
r
in Rn
Multi-Asset Investment and Consumption with Transaction Costs
7
where ℓ(z), A0 , and B are defined in (2.4), (2.5), and (2.15) respectively, and
∑
1∑
αi zi ∂zi u − ru.
(2.18)
A[u] :=
σij zi zj (∂zi zj u − ∂zi u∂zj u) +
2
i,j
i
The estimate (2.16) implies that ψ and ∂τ ψ converge at an exponential rate governed by
the interest rate r > 0, instead of the discount factor β. If r goes to 0, the analytic infinite
horizon optimal strategy in the absence of transaction costs indicates that the dollar values
invested in risky assets and consumption tend to infinity [see, e.g., Merton (1969) and Chen
et al. (2012)]. 4
The proof of the above two theorems is given in the Appendix.
2.4. Main Results. We are mainly interested in the infinite horizon problem. Recall that
τ = T − t and z = γξ(τ )y. Hence, setting ΦT (x, y) = Φ(x, y, 0) where Φ is the solution of the
finite T -horizon problem, we have, when κ > 0,
Φ∞ (x, y) := lim ΦT (x, y) = − lim e(r−β)b(T )−γξ(T )x+z(T ) ln K−ln ξ(T )−ψ(γξ(T )y,T )
T →∞
T →∞
(r−β)b(∞)−γξ(∞)x+Z(∞) ln K−ln ξ(∞)−u(γξ(∞)y)
= −e
κ
= − e(r−β)/r−γrx/κ−u(γry/κ) .
r
Hence, the solution u to the equation (2.17) can be formally regarded as the value function
associated with the infinite horizon utility maximization problem:
[∫ ∞
]
x,y
∞
−βs
Φ (x, y) = sup E0
V (cs )e ds
S∈A∞
0
where A∞ is certain admissible strategy. Note that without any restriction on A∞ , the
optimal strategy is cs ≡ ∞. Hence, if one considers directly the infinite horizon problem,
some technical conditions should be given [see, e.g., Liu (2004)]. We instead regard the
infinite horizon problem as the limit, as T → ∞, of the finite T -horizon problem with the
addition of a bequest utility which prevents unlimited consumption.
For solution u of the “infinite horizon” problem (2.17), we have the following regularity
result that plays a critical role in the analysis of the shape and location of the trading and
no-trading regions:
Theorem 2.3. Let u be the solution of (2.17). Then zi∏
∂zi u ∈ C(Rn ) for each i = 1, · · · , n;
1
n
consequently, u ∈ C (Ω) where Ω = {(z1 , · · · , zn ) ∈ R | i zi ̸= 0}.
The shape of the trading and no-trading regions is characterized by the following:
Theorem 2.4. Assume that 5 n = 2 and for i = 1, 2 define
Bi := {z | ∂zi u(z) = 1 + λi },
Si := {z | ∂zi u(z) = 1 − µi },
Ni := {z | 1 − µi < ∂zi u(z) < 1 + λi },
4
It should be admitted that the model with the CARA utility discussed here is less plausible than the one
with the CRRA utility.
5
Here we consider only the two dimensional case. We expect that analogous results remain true for the
higher dimensional case.
8
Xinfu Chen and Min Dai
and denote SS = S1 ∩ S2 , SN = S1 ∩ N2 , SB = S1 ∩ B2 , NS = N1 ∩ S2 , NT = N1 ∩
N2 , NB = N1 ∩ B2 , BS = B1 ∩ S2 , BN = B1 ∩ N2 , and BB = B1 ∩ B2 . Then
(1) For i = 1, 2, there are bounded functions li± (·) (defined on Rn−1 = R)such that
(2.19)
Bi = {(z1 , z2 ) | zı̆ ∈ R, zi ≤ li− (zı̆ )},
Si = {(z1 , z2 ) | zı̆ ∈ R, zi ≥ li+ (zı̆ )},
Ni = {(z1 , z2 ) | zı̆ ∈ R, li− (zı̆ ) < zi < li+ (zı̆ )}
where ı̆ = {1, 2} \ {i}, i.e., ı̆ = 2 if i = 1 and ı̆ = 1 if i = 2.
(2) Each intersection ∂S1 ∩ ∂S2 , ∂S1 ∩ ∂B2 , ∂B1 ∩ ∂S2 , and ∂B1 ∩ ∂B2 is a singleton, so
the four boundaries ∂S1 , ∂B1 , ∂S2 , ∂B2 divide the plane into nine regions, with open region
NT in the center surrounded in clockwise order by closed regions SS, SN, SB, NB, BB,
BN, BS, and NS.
(3) The boundary of each of corner regions SS, SB, BB, and BS consists of one vertical
and one horizontal half line, whereas the boundary of each of SN, NS, BN, and NB consists
of two parallel either vertical or horizontal half lines and a curve in between connecting the
end points of the two half lines; c.f. Figure 1.
We remark when n ≥ 3, ı̆ = {1, · · · , n}\{i}, so li (zı̆ ) is indeed a function of n−1 variables.
±
The theorem implies the following: There are intervals [b±
i , si ] such that
+
SS = [s+
1 , ∞) × [s2 , ∞),
+
+
SN = {(z1 , z2 ) | z2 ∈ (b+
2 , s2 ), z1 ≥ l1 (z2 )},
+
SB = [s−
1 , ∞) × (−∞, b2 ],
−
−
NB = {(z1 , z2 ) | z1 ∈ (b−
1 , s1 ), z2 ≤ l2 (z1 )},
−
BB = (−∞, b−
1 ] × (−∞, b2 ],
−
−
BN = {(z1 , z2 ) | z2 ∈ (b−
2 , s2 ), z1 ≤ l1 (z2 )},
−
BS = (−∞, b+
1 ] × [s2 , ∞),
+
+
NS = {(z1 , z2 ) | z1 ∈ (b+
1 , s1 ), z2 ≥ l2 (z1 )}.
The no-trading region NT is bounded by four curves: Γ2+ from the right, Γ2− from the left,
Γ1+ from the top, and Γ1− from the bottom where
±
Γ2± := {(l1± (z2 ), z2 ) | z2 ∈ (b±
2 , s2 )},
±
Γ1± := {(z1 , l2± (z1 )) | z1 ∈ (b±
1 , s1 )}.
These four curves connect each other only at their tips:
 ± ±
li± (zı̆ ),
li± (s±
l (b ) = limzı̆ ↘b± li± (zı̆ ),

ı̆ ) = limzı̆ ↗s±

ı̆
ı̆
 i ı̆
+ +
− −
−
+
+ +
(2.20)
(b+
(l1+ (s+
1 , l2 (b1 )) = (l1 (s2 ), s2 ),
2 ), s2 ) = (s1 , l2 (s1 )),


 − − −
− −
− −
+ +
+
(l1 (b2 ), b2 ) = (b−
(s−
1 , l2 (b1 )),
1 , l2 (s1 )) = (l1 (b2 ), b2 ).
±
Each function li± is constant outside (b±
ı̆ , sı̆ ):
(2.21)
±
li± (zı̆ ) = li± (b±
ı̆ ) ∀ zı̆ ≤ bı̆ ,
±
li± (zı̆ ) = li± (s±
ı̆ ) ∀ zı̆ ≥ sı̆ .
Multi-Asset Investment and Consumption with Transaction Costs
9
As emphasized in the introduction, Theorem 2.4 is very much needed for the trading
strategy to be well-defined. It is well-known that except at the initial time, transactions
occur at the boundary of the no-trading region. When the initial portfolio falls outside the
no-trading region, the shape of the trading regions stated in the theorem implies a unique
trading strategy to move the portfolio to the boundary of the no-trading region. However,
at this stage we cannot prove the smoothness of the curves li± ,6 so we are unable to prove
rigorously the optimal controlled portfolio process as a reflected diffusion as in Davis and
Norman (1990) and Shreve and Soner (1994).
We can now state our main result regarding the location of no trading region:
Theorem 2.5. Assume that n = 2. Let (m1 , m2 ) and A0 be as given in (2.5), and
√
σ1 = σ11 ,
√
σ2 = σ22 ,
√
σ12
ρ= √
,
σ11 σ22
M (h) :=
2(A0 − rh)
.
1 − ρ2
Also, define the ellipse C(k1 , k2 , h) and constant cij by
(2.22)
1∑
{
}
C(k1 , k2 , h) := (z1 , z2 ) σij (ki zi − mi )(kj zj − mj ) = A0 − rh ,
2
i,j
(2.23)
cij =
lim
{u(z) − ℓ(z)},
(−1)i z1 →∞,(−1)j z2 →∞
i, j = 1, 2.
1. The no-trading region NT is contained in the set
∪
(2.24)
D :=
C(k1 , k2 , h).
1−µi <ki <1+λi ,h≥u(0)
+
2. The corner (s+
1 , s2 ) of SS lies on the ellipse C(1 − µ1 , 1 − µ2 , c22 ) and on its top-left
part in the sense that
−ρ ≤
(1 − µ1 )s+
1 − m1
≤ 1,
M (c22 )/σ1
−ρ ≤
(1 − µ2 )s+
2 − m2
≤ 1.
M (c22 )/σ2
+
3. The corner (s−
1 , b2 ) of SB lies on the bottom-right part of C(1 − µ1 , 1 + λ2 , c21 ):
ρ≤
(1 − µ1 )s−
1 − m1
≤ 1,
M (c21 )/σ1
−1 ≤
(1 + λ2 )b+
2 − m2
≤ −ρ.
M (c21 )/σ2
−
4. The corner (b−
1 , b2 ) of BB lies on bottom-left part of C(1 + λ1 , 1 + λ2 , c11 ):
−1 ≤
6
(1 + λ1 )b−
1 − m1
≤ ρ,
M (c11 )/σ1
−1 ≤
(1 + λ1 )b−
2 − m2
≤ ρ.
M (c11 )/σ2
The regularity proof of free boundary in high dimension is always challenging. We leave it for future
research.
10
Xinfu Chen and Min Dai
−
5. The corner (b+
1 , s2 ) of BS lies on top-left of C(1 + λ1 , 1 − µ2 , c12 ):
−1 ≤
(1 + λ1 )b+
1 − m1
≤ −ρ,
M (c12 )/σ1
ρ≤
(1 − µ2 )s−
2 − m2
≤ 1.
M (c12 )/σ2
We remark that
A0
r
since 0 ≤ u(z) − ℓ(z) ≤ A0 /r and u(z) − ℓ(z) is an increasing concave function in each radial
direction.
The theorem shows that the no-trading region is contained in a union of uniformly bounded
ellipses. Moreover, the location of the corners of the no-trading region is estimated. Hence we
can restrict attention to a bounded domain to study the problem. In particular, this allows
us to do computations in a bounded domain.
0 ≤ u(0) ≤ cij ≤
The proof of Theorems 2.4 and 2.5 will be given in Section 4 and the proof of Theorem
2.3 in Section 5.
3. Extension to the CRRA Utility. Now let us examine the case with the CRRA utility,
namely,
1
V (c) = cγ , γ < 1, γ ̸= 0,
γ
for which we require that the liquidated wealth be non-negative:
∑
x+
ℓi (yi ) ≥ 0.
i
Note that we can directly consider the infinite horizon problem for the CRRA utility and the
above solvency constraint. Let Φ (x, y1 , y2 ) be the associated value function which satisfies
(cf. Dai and Zhong (2010))
}
{
(3.1)
min −LΦ, min [(1 + λi ) ∂x Φ − ∂yi Φ] , min [− (1 − µi ) ∂x Φ + ∂yi Φ] = 0,
i
i
∑
in x + i ℓi (yi ) > 0, where L is as given in (A.6).
For illustration, we still consider the case of two risky assets. The homogeneity of the
utility function allows us to make the following transformation:
zi =
yi
, i = 1, 2,
x + y1 + y2
φ (z1 , z2 ) ≡ Φ (1 − z1 − z2 , z1 , z2 ) =
1
(x + y1 + y2 )γ
Φ (x, y1 , y2 ) .
Then (3.1) reduces to
{
[
]
[
]}
∑
∑
b min λi γφ −
min −Aφ,
(δik + λi zk ) ∂zk φ , min µi γφ −
(−δik + µi zk ) ∂zk φ
=0
i
k
i
k
Multi-Asset Investment and Consumption with Transaction Costs
11
{
}
∑
in D = (z1 , z2 ) : 2i=1 [zi − ℓi (zi )] < 1 , where the expression of Ab is omitted.
Now we define, for i = 1, 2,
{
(z1 , z2 ) ∈ D : λi γφ −
Bi =
{
Si =
(z1 , z2 ) ∈ D : µi γφ −
2
∑
k=1
2
∑
}
(δik + λi zk ) ∂zk φ = 0 ,
}
(−δik + µi zk ) ∂zk φ = 0 ,
k=1
Ni = D ∩ Bic ∩ Sic .
We aim to show that D is partitioned into nine regions as shown in Figure 2. In particular,
N1 ∩ N2 has four distinct corners. We point out that the shape of trading/no-trading regions
is the same as that postulated by Bichuch and Shreve (2011) who deal with a slightly different
setting: the prices of risky assets follow arithmetic Brownian motions.
Let us consider, as an example, the region S1 ∩ S2 . It is worthwhile pointing out that
regarding the proof for the CARA utility case, two conditions play critical roles: 1) φ is
concave and is C 1 except on the coordinates planes; 2) ∂zi φ, i = 1, 2 are constants in S1 ∩ S2 .
The former still holds true with the CRRA utility whereas the latter does not. This motivates
us to make a new transformation:
yi
, i = 1, 2,
x + (1 − µ1 ) y1 + (1 − µ2 ) y2
φ (z 1 , z 2 ) ≡ Φ (1 − (1 − µ1 ) z 1 − (1 − µ2 ) z 2 , z 1 , z 2 )
1
=
Φ (x, y1 , y2 ) .
[x + (1 − µ1 ) y1 + (1 − µ2 ) y2 ]γ
zi =
It is easy to verify that
− (1 − µi ) ∂x Φ + ∂yi Φ = [x + (1 − µ1 ) y1 + (1 − µ2 ) y2 ]γ−1 ∂zi φ,
which implies
(3.2)
∂z i φ = 0 in S1 ∩ S2 , i = 1, 2.
Since φ is also concave and is C 1 except on the coordinates planes, we can use the same
analysis as in the CARA utility case to obtain the desired result. We point out that most
results presented in the CARA utility can be extended to the CRRA utility case.
Remark 3.1. If transactions between risky assets are allowed, as modeled in Soner and Touzi
(2012) and Possamaı̈ et al. (2013), we are unable to find a transformation that gives equation
(3.2), and therefore cannot extend our analysis to the case. Indeed, numerical calculation
presented in Possamaı̈ et al. (2013) indicates that there is no analogous conclusion for the
shape of the trading and no trading regions.
Now we return to the CARA case.
12
Xinfu Chen and Min Dai
z
2
1
µ2
2
1
−µ
1
1
z
N ∩S
z
=0
µ
1−
B1∩ S2
2
2
2
−µ
z
1
1+
λ
1
z
2
=0
S1∩ S2
B1∩ N2
N1∩ N2
S1∩ N2
−
1
λ1
1
µ1
N ∩B
B ∩B
1
1
2
2
z
1
S ∩B
1
2
2
=0
1
1−
µ
2
z
1
z
1
+λ
2
+λ
1
z
2
z
=0
λ
1+
− λ12
Figure 3.1. Shape of trading and no-trading regions with CRRA utility
4. Shape and Location of The Trading/No-Trading Zones. In this section, we investigate the shape and location of the trading/no-trading regions in the two dimensional case
for the infinite horizon problem; that is, we prove Theorem 2.4 and Theorem 2.5 using the
properties of u established in Theorem 2.2 and Theorem 2.3 whose proof will be given in the
Appendix and the next section, respectively.
4.1. The Bound of the No-Trading Region.
Let u be the viscosity solution of (2.17). For each z 0 , we define the first order approximation of u by
π(z 0 , z) := u(z 0 ) + ∇u(z 0 ) · (z − z 0 ).
By concavity, we have
u(z) ≤ π(z 0 , z),
which will be often used later on. In particular,
(4.1)
u(0) ≤ π(z 0 , 0) ≤ u(z 0 ) − ∇u(z 0 )z 0 .
Multi-Asset Investment and Consumption with Transaction Costs
13
We now decompose A[u] as A[u] = Lu − f where
(4.2)
(4.3)
1∑
σij zi zj ∂zi zj u(z),
2
i,j
∑
1∑
f (z) =
σij zi zj ∂zi u(z)∂zj u(z) −
αi zi ∂zi u(z) + ru(z)
2
i,j
i
1∑
σij (zi ∂zi u − mi )(zj ∂zj u − mj ) − A0 + r(u − z · ∇u),
=
2
Lu(z) =
i,j
where, mj and A0 are defined in (2.5). Thanks to Theorem 2.3 (to be proven in the next
section), f is continuous. If z ∈ NT, then A[u](z) = 0 so Lu(z) = f (z). Using concavity of u
we derive
1∑
σij (zi ∂zi u − mi )(zj ∂zj u − mj ) − A0 + ru(0),
0 ≥ Lu(z) = f (z) ≥
2
i,j
where we use (4.1) in the last inequality. Thus, z is on or inside the ellipse C(∂z1 u, ∂z2 u, u(0)).
Since ∂zi u(z) ∈ (1 − µi , 1 + λi ) we see that z ∈ D defined in (2.24). The first assertion of
Theorem 2.5 thus follows.
In the sequel, we shall use the following important fact:
Lemma 4.1. If z 0 = (z10 , z20 ) ∈ S1 then [z10 , ∞) × {z20 } ∈ S1 and u(z) = π(z 0 , z) and
∇u(z) = ∇u(z 0 ) for all z ∈ [z10 , ∞) × {z20 }.
Analogous assertions hold also for the cases z 0 ∈ B1 , z 0 ∈ S2 , and z 0 ∈ B2 , respectively.
Proof. Suppose z 0 = (z10 , z20 ) ∈ S1 , i.e., ∂z1 u(z 0 ) = 1 − µ1 . Since ∂z1 z1 u ≤ 0 and
∂z1 u ≥ 1 − µ1 , we have ∂z1 u(z1 , z20 ) = 1 − µ1 for all z1 ≥ z10 ; that is, [z10 , ∞) × {z20 } ∈ S1 .
In addition, for each z = (z1 , z20 ) with z1 ≥ z10 , u(z) = u(z0 ) + (1 − µ1 )(z1 − z10 ) = π(z0 , z).
Since u(z) ≤ π(z0 , z) for all z ∈ R2 , we also have ∇u(z) = ∇π(z 0 , z) = ∇u(z 0 ) for every
z ∈ [z10 , ∞) × {z20 }.
The other cases can be similarly proven. This completes the proof.
4.2. Limit Profile.
With ki+ := 1 − µi and ki− := 1 + λi , we define the limit profile by
(
)
)
(
v2± (z2 ) := lim u(z1 , z2 ) − k1± z1 , v1± (z1 ) := lim u(z1 , z2 ) − k2± z2 .
(4.4)
z1 →±∞
z2 →±∞
Lemma 4.2.
1. With ki+ := 1 − µi and ki− := 1 + λi , vi± defined in (4.4) is concave and
{
}
u(z1 , z2 ) ≤ min v2± (z2 ) + k1± z1 , v1± (z1 ) + k2± z2
∀ z ∈ R2 .
±
±
± ±
2. There are intervals (b±
i , si ) and functions li defined on (bı̆ , sı̆ ) such that

if zi ≥ s±
 = 1 − µi
i ,
±
∈ (1 − µi , 1 + λi ) if zi ∈ (b±
vi± ′ (zi )
i , si ),

±
= 1 + λi
if z ≤ bi ,
14
Xinfu Chen and Min Dai
u(z1 , z2 ) =
 +
v2 (z2 ) + (1 − µ1 )z1





 v − (z2 ) + (1 + λ1 )z1
2

v1+ (z1 ) + (1 − µ2 )z2




 −
v1 (z1 ) + (1 + λ2 )z2
+
if z1 ≥ l1+ (z2 ), z2 ∈ (b+
2 , s2 ),
−
if z1 ≤ l1− (z2 ), z2 ∈ (b−
2 , s2 ),
+
if z2 ≥ l2+ (z1 ), z1 ∈ (b+
1 , s1 ),
−
if z2 ≤ l2− (z1 ), z1 ∈ (b−
1 , s1 ),
∂z1 u(z1 , z2 ) > 1 − µ1
+
if z1 < l1+ (z2 ), z2 ∈ (b+
2 , s2 ),
∂z1 u(z1 , z2 ) < 1 + λ1
−
if z1 > l1− (z2 ), z2 ∈ (b−
2 , s2 ),
∂z2 u(z1 , z2 ) > 1 − µ2
+
if z2 < l2+ (z1 ), z1 ∈ (b+
1 , s1 ),
∂z2 u(z1 , z2 ) < 1 + λ2
−
if z2 > l2− (z1 ), z1 ∈ (b−
1 , s1 ).
3. Define
±
li± (s±
ı̆ ) = lim li (zı̆ ),
± ±
li∗
(sı̆ ) = lim li± (zı̆ ),
zı̆ ↗s±
ı̆
zı̆ ↗s±
ı̆
±
li± (b±
ı̆ ) = lim li (zı̆ ),
zı̆ ↘bı̆
± ±
li∗
(bı̆ ) = lim li± (zı̆ ).
zı̆ ↘bı̆
Then
+ +
+
+
+ +
+ +
[l1+ (s+
2 ), l1∗ (s2 )] × {s2 } ∪ {s1 } × [l2 (s1 ), l2∗ (s1 )] ⊂ ∂NT ∩ SS,
+ +
+
−
− −
− −
[l1+ (b+
2 ), l1∗ (b2 )] × {b2 } ∪ {s1 } × [l2 (s1 ), l2∗ (s1 )]
⊂ ∂NT ∩ SB,
[l1+ (s−
2 ),
+ −
[l1 (b2 ),
⊂ ∂NT ∩ BS,
+ −
l1∗
(s2 )] × {s−
2}
+ −
−
l1∗ (b2 )] × {b2 }
∪
∪
+ +
+ +
{b+
1 } × [l2 (b1 ), l2∗ (b1 )]
+ −
− −
{b−
1 } × [l2 (b1 ), l2∗ (s1 )]
⊂ ∂NT ∩ BB.
Proof. By symmetry, we need only consider the function v := v2+ .
(1) Note that u(z1 , z2 ) − (1 − µ1 )z1 is a concave function, ∂z1 [u(z1 , z2 ) − (1 − µ1 )z1 ] =
∂z1 u−(1−µ1 ) ≥ 0, and 0 ≤ u(z)−ℓ(z) ≤ A0 /r. Hence, v(z2 ) := limz1 →∞ [u(z1 , z2 )−(1−µ1 )z1 ]
exists, v is concave, and v(z2 ) ≥ u(z1 , z2 ) − (1 − µ1 )z1 .
(2) Let z2 ∈ R be a generic point such that 1 − µ2 < v ′ (z2 ) < 1 + λ2 . Since ∂z2 u(z1 , z2 ) →
′
v (z2 ) as z1 → ∞, ∂z2 u(z1 , z2 ) ∈ (1 − µ2 , 1 + λ2 ) for all z1 ≫ 1. As NT is bounded, we must
have ∂z1 u(z1 , z2 ) = 1 − µ1 for all z1 ≫ 1. In addition, since ∂z1 u → 1 + λ1 as z1 → −∞, we
can define
l(z2 ) := min{z1 ∈ R | ∂z1 u(z1 , z2 ) = 1 − µ1 }.
Since u(·, z2 ) is concave, we must have
∂z1 u(z1 , z2 ) < 1 − µ1 ∀ z1 < l(z2 ),
∂z1 u(z1 , z2 ) = 1 − µ1 ∀ z1 ≥ l(z2 ).
Denote z 0 := (l(z2 ), z2 ). Then by Lemma 4.1, u(z) = π(z 0 , z) = v(z20 ) + (1 − µ1 )z1
and ∇u(z) = ∇u(z 0 ) for each z ∈ [l(z2 ), ∞) × {z2 }. Consequently, ∂z2 u(z1 , z2 ) = v ′ (z2 ) ∈
(1 − µ2 , 1 + λ2 ) for all z1 ∈ [l(z2 ), ∞). Thus, for small positive ε, B(∇u(z)) > 0 on [l(z2 ) −
Multi-Asset Investment and Consumption with Transaction Costs
15
ε, l(z2 )) × {z2 }. Hence, [l(z2 ) − ε, l(z2 )) × {z2 } ∈ NT and (l(z2 ), z2 ) ∈ ∂NT. Since NT is
bounded and v is concave, there exist bounded b and s such that
v ′ = 1 + λ2 on (−∞, b],
1 + λ2 > v ′ > 1 − µ2 on (b, s),
v ′ = 1 − µ2 on [s, ∞).
(3) Now we define l(s) = limz2 ↗s l(z2 ), l∗ (s) = limz2 ↗s l(z2 ), and z ∗ = (l(s), s). By
continuity, we have ∂z1 u(l(s), s) = 1 − µ1 . This implies that ∇u(z) = (1 − µ1 , v ′ (s)) =
(1 − µ1 , 1 − µ2 ) for all z ∈ [l(s), ∞) × {s}. Hence, [l(s), ∞) × {s} ∈ SS.
Note that for each z2 < s and z1 ∈ [l(s), ∞), we have
∫ ∞
u(z1 , s) − u(z1 , z2 ) = v(s) − v(z2 ) −
[∂z1 u(ξ, s) − ∂z1 u(ξ, z2 )]dξ
z1
∫ s
∫ ∞
=
v ′ (y)dy +
[∂z1 u(ξ, z2 ) − (1 − µ1 )]dξ > (1 − µ2 )(s − z2 ).
z2
z1
It follows by concavity that ∂z2 u(z1 , z2 ) > 1 − µ2 . Hence,
∂z2 u > 1 − µ2 on [l(s), ∞) × (−∞, s).
(4.5)
It then follows by the definition of l(s) and l∗ (s) that [l(s), l∗ (s)] × {s} ⊂ ∂NT.
Similarly we can work on the other functions vi± to complete the proof of Lemma 4.2.
4.3. The Intersection of ∂NT with B1 ∩ B2 , S1 ∩ S2 and Bi ∩ Sj .
In this subsection we prove the following:
Lemma 4.3. Let cij and C(k1 , k2 , h) be defined as in (2.23) and (2.22).
1. The set ∂NT ∩ SS is a single point on top-right of the ellipse C(1 − µ1 , 1 − µ2 , c22 ).
In addition, if (z10 , z20 ) ∈ SS, then [z10 , ∞) × [z20 , ∞) ⊂ SS.
2. The set ∂NT ∩ SB is a single point on bottom-right of C(1 − µ1 , 1 + λ2 , c21 ). If
(z10 , z20 ) ∈ SB, then [z10 , ∞) × (−∞, z20 ] ⊂ SB.
3. The set ∂NT∩BB is a single point on bottom-left of C(1+λ1 , 1+λ2 , c11 ). If (z10 , z20 ) ∈
BB, then (−∞, z10 ] × (−∞, z20 ] ⊂ BB.
4. The set ∂NT∩BS is a single point on top-left of C(1+λ1 , 1−µ2 , c12 ). If (z10 , z20 ) ∈ BS,
then (−∞, z10 ] × [z20 , ∞) ⊂ BS.
Proof. (i) Suppose z 0 = (z10 , z20 ) ∈ S1 ∩ S2 .
First we show that [z10 , ∞) × [z20 , ∞) ⊂ SS. Indeed by Lemma 4.1, u(z) = π(z 0 , z) on
0
([z1 , ∞) × {z20 }) ∪ ({z01 } × [z20 , ∞)). This implies that u(·) ≥ π(z 0 , ·) on [z10 , ∞) × [z20 , ∞) since
π(z 0 , ·) is linear and u(·) is concave. On the other hand, we have u(z) ≤ π(z 0 , z) on R2 . Thus
we must have u(z) = π(z 0 , z) and ∇u(z) = ∇u(z0 ) = (1−µ1 , 1−µ2 ) so [z10 , ∞)×[z20 , ∞) ⊂ SS.
In addition, on [z10 , ∞) × [z20 , ∞),
u(z) = u(z 0 ) + (1 − µ1 )(z1 − z10 ) + (1 − µ2 )(z2 − z20 ) = c22 + z · ∇u(z0 ).
Next since −A[u] = f (z) − Lu ≥ 0 on R2 , using the linearity of u on [z10 , ∞) × [z20 , ∞) we
obtain
0 = lim Lu(z 0 + se1 + se2 ) ≤ lim f (z 0 + se1 + se2 ) = f (z 0 ).
s↘0
s↘0
16
Xinfu Chen and Min Dai
Using u = c22 + z · ∇u(z0 ) on [z10 , ∞) × [z20 , ∞) we find that f (z 0 ) = f22 (z 0 ) ≥ 0 where
f22 (z) :=
1∑
σij (zi [1 − µi ] − mi )(zj [1 − µj ] − mj ) − A0 + rc22
2
∀ z ∈ R2 .
i,j
This analysis in particular implies that f22 (z) ≥ 0 for every z ∈ SS.
(ii) Next, suppose z 0 ∈ ∂NT ∩ SS. Note that when z ∈ NT, we have 0 = −A[u](z) =
f (z) − Lu[z], i.e. f (z) = L[u](z) ≤ 0 (as u is concave). Hence,
f (z 0 ) =
lim
z∈NT,z→z 0
f (z) =
lim
z∈NT,z→z 0
Lu(z) ≤ 0.
Thus, we must have f (z 0 ) = 0, i.e., z 0 ∈ C := C(1−µ1 , 1−µ2 , c22 ). Moreover, since f22 (z) ≥ 0
for every z ∈ [z10 , ∞) × [z20 , ∞) and f (z) < 0 for each z inside the ellipse C, we see that z 0
lies on the top-right part of the ellipse C. Locating the highest and rightmost points of the
ellipse C, we then derive that
f22 (z 0 ) = 0,
−ρ ≤
(1 − µ1 )z10 − m1
≤ 1,
M (c22 )/σ1
−ρ ≤
(1 − µ2 )z20 − m2
≤ 1.
M (c22 )/σ2
(iii) Now we show that ∂NT ∩ SS is a singleton, by a contradiction argument. Suppose
= (z10 , z20 ) ∈ ∂NT∩SS, ẑ 0 = (ẑ10 , ẑ20 ) ∈ ∂NT∩SS, and ẑ 0 ̸= z 0 . Then both z 0 and ẑ 0 lies on
the upper-right part of the ellipse C(1 − µ1 , 1 − µ2 , c22 ). Exchanging the roles of ẑ 0 and z 0 , we
can assume that z20 < ẑ20 and z10 > ẑ10 . Note that u(z) = π(ẑ 0 , z) for all z ∈ [ẑ10 , ∞) × (ẑ20 , ∞)
and u(z) = π(z 0 , z) for all z ∈ [z10 , ∞) ∩ [z20 , ∞). Hence, π(z 0 , z) = π(ẑ 0 , z) for all z ∈ R2 . On
the other-hand, if u(z) = π(z 0 , z), then ∇u(z) = ∇π(z 0 , z) = ∇u(z0 ) so z ∈ SS. Therefore,
z0
SS := {z ∈ R2 | ∇u(z) = (1 − µ1 , 1 − µ1 )} = {z ∈ R2 | u(z) = π(z 0 , z)}.
Note that u is concave, π(z 0 , ·) is linear, and u(·) ≤ π(z 0 , ·) on Rn . We derive that SS is a
convex set. Consequently, it contains L, the line segment connecting z 0 and ẑ 0 .
Next for each s ∈ R, denote z s = (ẑ10 , z20 ) + (s, s). For s < 0, z s ̸∈ SS since otherwise
it would imply [ẑ10 + s, ∞) × [z20 + s, ∞) ∈ SS, contradicting z 0 ∈ ∂NT. Hence, there exists
s∗ ≥ 0 such that z ∗ = (ẑ10 + s∗ , z20 + s∗ ) ∈ ∂(SS). The point z ∗ lies on or below L so is an
interior point of the ellipse C. There are two cases: (a) s∗ > 0, and (b) s∗ = 0.
Consider case (a) s∗ > 0. Then for each s ∈ [0, s∗ ), z s ̸∈ SS since SS is convex. Also
∂z1 u(z s ) > 1 − µ1 since by Lemma 4.1, ∂z1 u(z s ) = 1 − µ1 would imply that ∇u(z s ) = ∇u(ẑ) =
∇u(z0 ) where ẑ is the intersection of L with the line z2 = z20 + s. Similarly, ∂z2 u(z s ) > 1 − µ2 .
∗
for each s ∈ [0, s∗ ). Thus, z s ∈ NT for all s ∈ [0, s∗ ). This means that z s ∈ ∂NT ∩ SS,
∗
which is impossible since z s ̸∈ C.
Consider case (b) s∗ = 0. First of all ∂z1 u(z1 , z20 ) > 1 − µ1 for all z1 < ẑ10 since otherwise
it would imply ∇u(z1 , z20 ) = ∇u(z 0 ) and thus [z1 , ∞) × [z20 , ∞) ∈ SS contradicting ẑ 0 ∈ ∂NT.
Similarly, ∂z2 u(ẑ10 , z2 ) > 1 − µ2 for every z2 < z20 . Now for every small positive ε, consider the
closed set
Dε := [ẑ10 − ε, ẑ10 ] × [z20 − ε, z20 ] \ (ẑ10 − ε/2, ẑ10 ] × (z20 − ε/2, z20 ].
If z ∗ = (z1∗ , z2∗ ) ∈ SS ∩ Dε we would have z1∗ < ẑ10 and z2∗ < z20 so z 0 is an interior point of
[z1∗ , ∞) × [z2∗ , ∞) ⊂ SS, a contradiction. Thus, SS ∩ Dε = ∅. Consequently, the closed sets
Multi-Asset Investment and Consumption with Transaction Costs
17
Dε ∩ S1 and Dε ∩ S2 are disjoint. Since Dε cannot be written as the union of two disjoint
closed proper subsets, the set Dε \ (S1 ∪ S2 ) is non-empty. This means that Dε ∩ NT ̸= ∅.
Consequently, (ẑ10 , z20 ) ∈ ∂NT ∩ SS, but this is impossible since (ẑ10 , z20 ) does not lie on C.
Therefore ∂NT ∩ SS is a singleton.
The proof for the singleness of ∂NT ∩ BS, ∂NT ∩ SB, and ∂NT ∩ BB is similar. This
completes the proof of the Lemma.
4.4. Completion of the Proof of Theorems 2.4, 2.5.
By Lemmas 4.2 and 4.3, we see that the limits in (2.20) exist, and the limits satisfy the
±
matching condition stated in (2.20). We extend li± from (b±
ı̆ , sı̆ ) to R by (2.21).
+
+ +
+
By the definition of l2 on (b1 , s1 ) we know that when z1 ∈ (b+
1 , s1 ), ∂z2 u(z1 , z2 ) > 1 − µ2
+
if and only if z2 < l2 (z1 ). Also, in view of (4.5) and the matching (2.20), we derive that when
+ +
+
z1 ∈ [s+
1 , ∞), ∂z2 u(z1 , z2 ) > 1 − µ2 if and only if z2 < l2 (s1 ) = l2 (z1 ). Similarly, we can show
+
+
that when z1 ∈ (−∞, b1 ], ∂z2 u(z1 , z2 ) < 1 − µ2 if and only if z2 < l2+ (b+
1 ) = l2 (z1 ). Thus,
S2 := {z | ∂z2 u = 1 − µ2 } = {(z1 , z2 ) | z1 ∈ R, z2 ≥ l2+ (z1 )}.
Similarly, we can show the other equations in (2.19). The rest assertions of Theorems 2.4 and
2.5 thus follow from Lemmas 4.2 and 4.3. This completes the proof of Theorems 2.4 and 2.5.
5. C 1 Regularity. In this section we prove Theorem 2.3; that is, we show that the viscosity
solution u of the “infinite horizon” problem (2.17) is C 1 except on the coordinates planes where
the elliptic operator A is degenerate. The C 1 continuity plays a critical role in the previous
section where a key step is to derive the continuity of f (·) defined in (4.3).
We begin with recalling the definition of a viscosity solution:
Definition 5.1.A function u defined on Rn is called a viscosity solution of (2.17) if u is
continuous, u − ℓ ∈ L∞ (Rn ), and the following holds:
1. If ζ is a C 2 function in Bε (z 0 ) := {z ∈ Rn | |z − z 0 | < ε} for some z 0 ∈ Rn and ε > 0,
and that ζ(z) − u(z) ≥ 0 = ζ(z 0 ) − u(z 0 ) for every z ∈ Bε (z 0 ), then
{
}
min − A[ζ](z 0 ), B(∇ζ(z 0 )) ≤ 0.
2. If ζ is a C 2 function in Bε (z0 ) := {z ∈ Rn | |z − z 0 | < ε} for some z0 ∈ Rn and ε > 0,
and that ζ(z) − u(z) ≤ 0 = ζ(z 0 ) − u(z 0 ) for each z ∈ Bε (z 0 ), then
{
}
min − A[ζ](z 0 ), B(∇ζ(z 0 )) ≥ 0.
One can show that the viscosity solution of (2.17) is unique. Since the solution is the limit
of ψ(·, τ ) that is concave, we see that the viscosity solution of (2.17) is concave. We shall use
this fact to prove the C 1 regularity of u.
For any function f defined on Rn , we define its super-differential by
(5.1)
∂f (z) = {p ∈ Rn : f (ẑ) ≤ f (z) + p · (ẑ − z) ∀ ẑ ∈ Rn }.
We shall use the following fact.
Lemma 5.2.Suppose f is a concave function on Rn . Define its super-differential by (5.1).
Then the following holds:
18
Xinfu Chen and Min Dai
1. The set {(z, p) | z ∈ Rn , p ∈ ∂f (z)} is closed; i.e. if pk ∈ ∂f (z k ) for all k ≥ 1 and
limk→∞ (pk , zk ) = (p, z), then p ∈ ∂f (z).
2. For each z ∈ Rn , ∂f (z) is a non-empty, convex and compact set.
3. If ∂f (z) = {p} is a singleton, then f is differentiable at z and p = ∇f (z).
4. If ∂f (z) is singleton for every z in an open neighborhood of z 0 ∈ Rn , then f is C 1 in
an open neighborhood of z 0 .
5. For each i = 1, · · · , n and fixed z ∈ Rn define
∂i f (z) = {ei · p | p ∈ ∂f (z)},
g(t) = f (z + tei ).
Then
[
∂i f (z) = ∂g(0) =
g(h) − g(0)
g(0) − g(−h) ]
, lim
.
h↘0
h↘0
h
h
lim
The conclusion of the lemma is well-known; see Crandall et al. (1992) and references
therein.
If f is concave, then f is locally Lipschitz continuous and ∂f is non-empty and almost
everywhere singleton, and coincides with the Sobolev gradient. For convenience, we identify
the set ∂f (z) as a generic vector p in ∂f (z).
Proof of Theorem 2.3. For illustration, we consider only the two space dimensional
case.
First we show that ∂1 u(z1 , z2 ) is a singleton if z1 ̸= 0. We use a contradiction argument.
Suppose the assertion is not true. Then there exist z 0 = (z10 , z20 ) with z10 ̸= 0 and p = (p1 , p2 )
and q = (q1 , q2 ) with 1 − µ1 ≤ p1 < q1 ≤ 1 + λ1 such that p, q ∈ ∂u(z 0 ).
By the definition of super-differential, we have
(5.2)
u(z) ≤ min{u(z 0 ) + p · (z − z 0 ),
u(z 0 ) + p · (z − z 0 )} ∀ z ∈ R2 .
Now for any 0 < ε ≪ 1, consider the quadratic concave function
ζ(z) = u(z 0 ) +
[(q − p) · (z − z 0 )]2
p+q
· (z − z 0 ) −
2
4ε
in Qε := {z : |(q−p)·(z−z 0 )| ≤ ε}. By considering separately the cases 0 ≤ (q−p)·(z−z 0 ) ≤ ε
and −ε ≤ (q − p) · (z − z 0 ) < 0, we find that
ζ(z) −
(5.3)
[(q − p) · (z − z 0 )]2
p+q
[(q − p) · (z − z 0 )]2
= u(z 0 ) +
· (z − z 0 ) −
4ε
2
2ε
≥ u(z 0 ) + min{p · (z − z 0 ), q · (z − z 0 )} ≥ u(z)
for every z ∈ Qε . Thus, ζ(z) − u(z) ≥ 0 = ζ(z 0 ) − u(z 0 ) for every z ∈ Qε . Consequently, by
the definition of viscosity solution,
min{−A[ζ](z0 ),
B(∇ζ(z0 ))} ≤ 0.
It is easy to calculate,
1
∇ζ(z0 ) = (p + q),
2
D2 ζ(z0 ) = −
1
(q − p) ⊗ (q − p).
2ε
Multi-Asset Investment and Consumption with Transaction Costs
19
Since z10 ̸= 0 and q1 − p1 > 0, when ε is sufficiently small, we have −A[ζ](z0 ) > 0. Thus, we
mush have B( 12 (p+q)) ≤ 0. Since 1−µ1 ≤ p1 < q1 ≤ 1+λ1 we have 1−µ1 < 12 (p1 +q1 ) < 1+λ1 .
Hence, we must have one of the following:
(i) p2 = q2 = 1 − µ2 ,
(ii) p2 = q2 = 1 + λ2 .
Let’s first consider the case (i) p2 = q2 = 1 − µ2 . Note that u(z10 , ·) is a concave function
with super-differential bounded between 1 − µ2 and 1 + λ2 . Since p2 = q2 = 1 − µ2 , we see
that ∂z2 u(z10 , z2 ) = 1 − µ2 for all z2 > z20 . We define
ẑ20 = inf{z2 ≤ z20 | u(z10 , z2 ) = u(z 0 ) + (1 − µ2 )(z2 − z20 )}.
Since u(z10 , z2 ) ≤ O(1) + (1 + λ2 )z2 for z2 ≤ 0, we see that ẑ20 > −∞. In addition, set
ẑ 0 = (z10 , ẑ20 ) we have
{
u(z10 , z2 ) = u(ẑ 0 ) + (1 − µ2 )(z2 − ẑ20 )
∀ z2 ≥ ẑ20
(5.4)
u(z10 , z2 ) < u(ẑ 0 ) + (1 − µ2 )(z2 − ẑ20 )
∀ z2 < ẑ20 .
From the definition of ζ and the fact that p2 = q2 = 1 − µ2 , we obtain from (5.3) that,
setting β = q1 − p1 ,
[β(z1 − z10 )]2
∀ z ∈ [z10 − εβ −1 , z10 + εβ −1 ] × R,
4ε
ζ(z) = u(z) ∀ z ∈ {z10 } × [ẑ20 , ∞),
ζ(z) ≥ u(z) +
ζ(z) > u(z) ∀ z ∈ {z10 } × (−∞, ẑ20 ).
Using ζ(z10 , ẑ20 −ε) > u(z10 , ẑ20 −ε) and continuity, we can find η ∈ (0, εβ −1 ) such that ζ(z1 , z20 −
ε) > u(z1 , z20 − ε) + η for all z1 ∈ [z10 − η, z10 + η]. Hence,
{
(βη)2 /(4ε)
if |z1 − z10 | = η, z2 ∈ R
ζ(z1 , z2 ) − u(z1 , z2 ) ≥
η
if z2 = ẑ20 − ε, |z1 − z10 | ≤ η.
Finally, set η̂ = min{(λ2 + µ2 )/2, η/(2ε), (βη)2 /(8ε2 )} and consider the function
ζ̂(z) = ζ(z) + (z2 − ẑ20 )η̂
in D := (z10 − η, z10 + η) × (ẑ20 − ε, ẑ20 + ε).
Note that ζ̂(z) > u(z) on the boundary ∂D of D. In addition, ζ̂(ẑ 0 ) = u(ẑ 0 ). Now set
m := maxD {u(z) − ζ̂(z)}. Then m ≥ 0. Let ẑ ∈ D be the point such that u(ẑ) − ζ̂(ẑ) = m,
then ẑ ∈ D since u < ζ̂ on ∂D. Thus, [ζ̂(z) + m] − u(z) ≥ [ζ̂(ẑ) + m] − u(ẑ) = 0 for every
z ∈ D. Hence, by definition of viscosity solution, we have
min{−A[ζ̂ + m](ẑ),
B(∇ζ̂(ẑ)} ≤ 0.
However, for every z ∈ D,
p1 < ∂z1 ζ̂(z) < q1 ,
∂z2 ζ̂(z) = 1 − µ2 + η̂ ∈ (1 − µ2 , 1 − λ2 ),
D2 ζ̂(z) = −
β2
e1 ⊗ e1 .
2ε
20
Xinfu Chen and Min Dai
This implies that B(∇ζ̂(ẑ)) > 0. Hence, we must have −A[ζ̂(ẑ) + m](ẑ) ≤ 0. However, since
z10 ̸= 0, we have −A[ζ̂(ẑ) + m](ẑ) → ∞ as ε ↘ 0; this contradicts −A[ζ̂(ẑ) + m](ẑ) ≤ 0.
Similarly, we can derive a contradiction in the second case (ii) p2 = q2 = 1 + λ2 .
The contradiction shows that ∂1 u(z1 , z2 ) is singleton if z1 ̸= 0. Now for each fixed z2 ∈ R,
consider the one dimensional function g(t) = u(t, z2 ). By Lemma 5.2 (4), ∂g(t) = ∂1 u(t, z2 ) is
1 ,z2 )
:= g ′ (z1 )
singleton if t ̸= 0. Hence, g ∈ C 1 (R \ {0}), so the classical partial derivative ∂u(z
∂z1
exists. Since any limit point of ∂1 u(ẑ) as ẑ → z is in ∂1 u(z), we conclude that limẑ→z
∂u(z1 ,z2 )
∂z1
∂u(z)
∂z1
∂u(ẑ)
∂z1
=
if z1 ̸= 0. Hence,
∈
\ ({0} × R)). As u is Lipschitz continuous, we also
know that z1 ∂z1 u ∈ C(R). This completes the proof of Theorem 2.3.
C(R2
Appendix A. Basic Theory of The Finite Horizon Problem. In this section, we follow the
standard technique connecting the value function Φ to the HJB equation; that is, we prove
Theorem 2.1.
A.1. The Case of No Risky Assets.
First we establish a useful lower bound of Φ(x, 0, t) by considering the category of strategies
that do not use risky assets; that is, we consider the strategies where ys ≡ 0, Ls ≡ 0, Ms ≡ 0
for all s ∈ [t, T ]. Writing (cs , xs ) as (c(s), x(s)), we have dx(s) = [rx(s) − κc(s)]ds. Subject
to x(t) = x we obtain
∫ T
r(T −t)
er(T −s) c(s)κds.
(A.1)
x(T ) = xe
−
t
Thus, for any consumption strategy c, the total utility can be written as
∫ T
∫
r(T −t) +γκ T er(T −s) c(s)ds−β(T −t)
x,t
t
.
e−β(s−t)−γc(s) κds − Ke−γxe
(A.2)
J0 [c] := −
t
We want to find an optimal consumption strategy that maximizes J0x,t .
r(T −t) −β(T −t)
When κ = 0, we have J0x,t [c] = −Ke−γxe
.
Next consider the case κ > 0. The first variation of J0x,t can be calculated by
⟨ δJ x,t [c]
0
δc
⟩
J x,t [c + hζ] − J0x,t [c]
, ζ := lim 0
h→0
h
∫ T
{
}
= κγ
e−β(s−t) ζ(s) e−γc(s) − Ke−γx(T )+(r−β)(T −s) ds,
t
where x(T ) is as given in (A.1). Hence, if c∗ is a critical point of J0x,t , i.e., δJ0x,t [c∗ ]/δc = 0,
∗
∗
then e−γc (s) − Ke−γx (T )+(r−β)(T −s) = 0, where x∗ (T ) is as in (A.1) with c replaced by c∗ .
Thus c∗ (s) = x∗ (T ) − [(r − β)(T − s) + ln K]/γ. Using the definition of x∗ (T ) we then obtain
(A.3)
c∗ (s) = xξ(τ ) +
(r − β)[s − t − b(τ )] − Z(τ ) ln K
γ
∀ s ∈ [t, T ],
where τ = T − t and ξ, Z, b are defined in (2.7) which satisfy (2.8)-(2.10). Note that J0x,t is a
concave functional, so c∗ is the global maximizer.
Multi-Asset Investment and Consumption with Transaction Costs
21
We have proved the following result:
Lemma A.1.For each x ∈ R, the linear function c∗ defined in (A.3), where τ = T − t and
ξ, Z, and b are as in (2.7), is the global maximizer of J0x,t defined in (A.2):
J0x,t [c] ≤ J0x,t [c∗ ].
The strategy that liquidates all risky assets at time t gives the estimate
x+ℓ(y),t
Φ(x, y, t) ≥ Φ(x + ℓ(y), 0, t) ≥ J0
[c∗ ]
= −e−γξ(τ )[x+ℓ(y)]+(r−β)b(τ )−ln ξ(τ )+Z(τ ) ln K .
Then we have the following corollary:
Corollary A.2.There is the following lower bound for Φ:
Φ(x, y, t) ≥ −e−γξ(τ )[x+ℓ(y)]+(r−β)b(τ )−ln ξ(τ )+Z(τ ) ln K .
(A.4)
A.2. Separation of Investment and Consumption. The following result of separation of
variables is similar to the one obtained by Davis et al. (1993):
Lemma A.3. Let τ = T − t and ξ be as defined in (2.7). Then
(A.5)
Φ(x, y, t) = e−γξ(τ )x Φ(0, y, t)
∀ (x, y, t) ∈ R × Rn × (−∞, T ].
Proof. Let S = (C, L, M ) be an investment-consumption strategy for initial position
(xt , yt ) = (0, y), resulting in the subsequent portfolio {(xs , ys )}s∈[t,T ] . For another initial
position (x, y) at time t, we consider the investment-consumption strategy S̃ = (C̃, L̃, M̃ )
defined by
(L̃, M̃ ) ≡ (L, M ),
c̃s = cs + ξ(τ )x,
τ := T − t.
Denote the corresponding portfolio (starting from (x, y) at time t) by {(x̃s , ỹs )}s∈[t,T ] . Then
ỹs = ys and x̃s = xs + x̂(s) where x̂(s) is the solution of dx̂(s) = [rx̂(s) − κξ(τ )x]ds subject
to x̂(t) = x. Solving this initial value problem for x̂ gives
x̂(T ) =
κξ(τ )x (
κξ(τ )x ) rτ
+ x−
e = ξ(τ )x
r
r
by the definition of ξ(τ ) in (2.7). It then follows that
J(S̃, t) = −Ke
−γ x̂(T )
U (xT , yT )e
−β(T −t)
∫
−
T
e−γξ(τ )x e−γcs −β(T −s) κ ds = e−γξ(τ )x J(S, t).
t
The relation between S and S̃ is 1-1 and onto, so taking the supremum yields (A.5).
22
Xinfu Chen and Min Dai
A.3. Super-Solution.
Let φ(x, y, t) be a smooth function in R × Rn × (−∞, T ] satisfying ∂x φ > 0. Let S be an
investment-consumption strategy in At . By Ito’s formula and (2.1),
φ(xt , yt , t)
= φ(xT , yT , T )e−β(T −t) −
∫
T
t
= φ(xT , yT , T )e−β(T −t) +
∫
T
(
)
d φ(xs , ys , s)e−β(s−t)
∫
V (cs )e−β(s−t) (κds) −
t
T
e−β(s−t)
t
∑
aij yi ∂yi φdBsj
i,j
∫ T
]
]
[
[
T
−β(s−t)
φ
dMsic
+
e
(−1
+
µ
)∂
φ
+
∂
+
e−β(s−t) (1 + λi )∂x φ − ∂yi φ dLic
i x
yi
s
t
t
∫ T
∫ T
(
)
[ (
) (
)]
−
e−β(s−t) ∂t φ + Lφ ds +
e−β(s−t) V ∗ ∂x φ − V (cs ) − cs ∂x φ κds
t
t
∑
−β(s−t)
+
e
[φ(xs− , ys− , s−) − φ(xs , ys , s)],
∫
t≤s≤T
where Lic and M ic are the continuous part of Li and M i respectively,
{
}
V ∗ (q) := max V (c) − cq
∀ q > 0,
c∈R
(A.6)
(
)
∑
1∑
αi yi ∂yi φ + rx∂x φ − βφ + κV ∗ ∂x φ
Lφ :=
σij yi yj ∂yi yj φ +
2
i
ij
Now suppose φ satisfies φ(·, T ) ≥ KU (·) and
(A.7)
(−1 + µi )∂x φ + ∂yi φ ≥ 0.
(1 + λi )∂x φ − ∂yi φ ≥ 0,
−∂t φ − Lφ ≥ 0,
Combination of (A.7) and ∂x φ > 0 leads to
φ(xs− , ys− , s−) − φ(xs , ys , s) ≥ 0.
Now assume S = (C, L, M ) is an admissible strategy. Taking the expectation we obtain
φ(x, y, t) ≥
Ex,y
t
∫
[
−β(T −t)
KU (xT , yT )e
+
T
]
V (cs )e−β(s−t) (κds) .
t
Taking the supreme we obtain the following:
Lemma A.4. Suppose φ is a smooth function on R × Rn × (−∞, T ] satisfying φx > 0,
φ(·, T ) ≥ KU (·), and (A.7). Then Φ(x, y, t) ≤ φ(x, y, t) for all (x, y, t) ∈ R × Rn × [0, T ].
We call such φ a super-solution. Due to the separation property (A.5), we seek a supersolution of the form
φ(x, y, t) = −e−γξ(τ )x+(r−β)b(τ )−ln ξ(τ )+Z(τ ) ln K−ϕ(z,τ ) ,
τ = T − t,
z = γξ(τ )y,
Multi-Asset Investment and Consumption with Transaction Costs
23
where ξ, Z and b are defined in (2.7). We can compute
{
[ ξ′
]
−∂t φ − Lφ = |φ| γx(ξ ′ − rξ + κξ 2 ) +
+ κξ − β − (r − β)(b′ + κξb)
ξ
′
∑
ξ
−(Z ′ + κξZ) ln K +
zi ∂zi ϕ
ξ
i
}
∑
1∑
σij zi zj (∂zi zj ϕ − ∂zi ϕ∂zj ϕ) −
αi zi ∂zi ϕ + κξϕ
+∂τ ϕ −
2
i,j
i
(
)
= |φ| ∂τ ϕ − Aτ [ϕ] ,
where we have used (2.8)-(2.10) in the last equality and the definition of Aτ in (2.14). Thus,
(A.7) can be written as
{
}
min ∂τ ϕ − Aτ [ϕ], B(∇ϕ) ≥ 0 in Rn × (0, ∞).
Since ξ(0) = 1 = Z(0) and b(0) = 0, the condition φ(·, T ) ≥ KU (·) can be written as
∑
ϕ(z, 0) ≥ ℓ(z) =
ℓi (zi ),
i
where ℓ and ℓi are as defined in (2.4).
A.4. Upper Bound.
Let k = (k0 , k1 , · · · , kn ) be a constant vector satisfying
k0 ≥ 0,
1 − µi ≤ ki ≤ 1 + λi
∀ i.
Consider the function
ϕ̄(k; z, τ ) := k0 b(τ ) +
∑
ki zi .
i
∑
¯
It is easy to see that ϕ̄(k; z, 0) =
i ki zi ≥ ℓ(z), 1 + λi − ∂zi ϕ = 1 + λi − ki ≥ 0 and
−1 + µi + ∂¯zi ϕ = −1 + µi + ki ≥ 0 for each i. Also,
ϕ̄τ − Aτ [ϕ̄] = (b′ + kξb)k0 +
∑
1∑
(zi ki )σij (zj kj ) −
(αi − r)(zi ki )
2
i,j
i
1∑
= k0 +
(zi ki − mi )σij (zj kj − mj ) − A0 ,
2
i,j
where {mj }, A0 are as in (2.5). Since (σij )n×n is positive-definite, taking k0 = A0 we have
ϕ̄ − Aτ ϕ̄ ≥ 0. Hence, by Lemma A.4,
(A.8)
Φ(x, y, t) ≤ −e−γξ(τ )x+(r−β)b(τ )+Z(τ ) ln K−ln ξ(τ )−ϕ̄(k,z,τ ) .
We are now ready to complete the proof of Theorem 2.1.
24
Xinfu Chen and Min Dai
A.5. Proof of Theorem 2.1. By (A.4), Φ(x, y, t) > −∞. Also, from (A.8) we see that
Φ(x, y, t) < 0 for every (x, y, t) ∈ R × Rn × (−∞, T ]. Hence, we can define
(A.9)
ψ(z, τ ) = (r − β)b(τ ) + Z(τ ) ln K − ln ξ(τ ) − ln Φ(0, [γξ(τ )]−1 z, T − τ ).
Then by Lemma A.3, we obtain (2.6). Now we establish properties of ψ.
(1) Let x ∈ R, y ∈ Rn and ŷ ∈ Rn . Stating at position (x, ŷ) at time t, one can immediately
liquidate ŷ − y amount of risky asset holding to reach the position (x + ℓ(ŷ − y), y) at time
t+. Hence, we have
Φ(x, ŷ, t) ≥ Φ(x + ℓ(ŷ − y), y, t) = e−ℓ(γξ(τ )(ŷ−y)) Φ(x, y, t).
In terms of (2.6), this implies that ψ(ẑ, τ ) ≥ ψ(z, τ ) + ℓ(ẑ − z). Thus, we obtain (2.11).
(2) Combination of the estimate (A.4) (with y = z/[γξ(τ )]) and (2.6) yields the left hand
we use (A.8) with k0 := A0
side inequality of (2.12). To show the right hand side inequality,
∑
and (2.6) to derive that ψ(z, τ ) ≤ ϕ̄(k, z, τ ) = A0 b(τ ) + i ki zi whenever ki ∈ [1 − µi , 1 + λi ]
for all i. Hence,
)
(
∑
ψ(z, τ ) ≤
min
A0 b(τ ) +
ki zi = A0 b(τ ) + ℓ(z).
1−µj ≤kj ≤1+λj ∀ j
i
(3) Thanks to the linearity of transaction costs and the concavity of the exponential utility
function, we immediately obtain the concaveness of Φ(·, ·, t). The concaveness of ψ(·, τ ) follows
by noting (A.9) and the fact that the function ln(·) is concave and increasing.
(4) Once we know the basic regularity (e.g. Φ is Lipschitz continuous), we can use the
dynamic programming principle [Pham (2009)] and follow the routine technique detailed by
Davis et al. (1993) and Shreve and Soner (1994), to show that ψ is a viscosity solution of
(2.13). Here we do not need the Standing Assumption 2.3 in Shreve and Soner (1994) since
we are working on finite horizon problem and we have already shown the (local) boundedness
of Φ. This completes the proof of Theorem 2.1.
Appendix B. The Asymptotic Behavior as T → ∞.
This section is devoted to the proof of Theorem 2.2. We always assume κ > 0.
We begin with the following estimate:
Lemma B.1. For any z ∈ Rn and τ ≥ 0,
(B.1)
0 ≤ ψ(z, τ ) − z · ∂ψ(z, τ ) ≤ A0 b(τ ).
Proof. The assertion holds for z = 0 by (2.12).
Fix z ∈ Rn \ {0}. Consider the function
f (s) := ψ(sz, τ ),
s ∈ [0, ∞).
This is a concave and Lipschitz continuous function in one space dimension. Hence, f ′ (s) is a
decreasing function. Set
p∞ = lim
s→∞
f (s) − f (0)
.
s
Multi-Asset Investment and Consumption with Transaction Costs
25
In view of (2.11) and the homogeneity ℓ(sz) = sℓ(z) for s > 0, we find that p∞ ≥ ℓ(z). As
f ′ (s) is a decreasing function, by L’Hôpital’s rule, f ′ (s) ↘ p∞ as s → ∞. Consequently, for
any s > 0,
0 ≤ f (0) ≤ f (s) + f ′ (s)(0 − s) ≤ f (s) − p∞ s
≤ A0 b(τ ) + ℓ(sz) − p∞ s = A0 b(τ ),
by (2.12) and the definition f (s) = ψ(sz, τ ). Note that as super-differential, z·∂ψ(z, τ ) ⊂ f ′ (1).
Hence, setting s = 1 we obtain the assertion of the lemma.
Proof of Theorem 2.2. In what follows, we construct viscosity sub and supersolutions
to bound the viscosity solution.
For any constant h > 0 and smooth function W (·), consider the function ψ̂(z, τ ) :=
ψ(z, τ + h) + W (τ ). One can calculate
ψτ (z, τ + h) − Aτ +h ψ(z, τ + h)
= ψ̂τ − Aτ +h ψ̂ − W ′ (τ ) − κξ(τ + h)W (τ )
= ψ̂τ − Aτ ψ̂ + [κξ(τ ) − κξ(τ + h)](z · ∂ ψ̂ − ψ̂) − W ′ (τ ) − κξ(τ + h)W (τ )
= ψ̂τ − Aτ ψ̂ + f W ,
where
f W (z, τ ) := −[W ′ (τ ) + κξ(τ )W (τ )]
+κ[ξ(τ ) − ξ(τ + h)](z · ∂ψ(z, τ + h) − ψ(z, τ + h)).
Hence, equation (2.13) with (τ, ψ) replaced by (τ + h, ψ(·, τ + h)) can be written as
(B.2)
{
}
min ψ̂τ − Aτ ψ̂ + f W , B(∇ψ̂(z, τ )) = 0.
One can show by the definition of viscosity solution that if ψ is a viscosity solution of (2.13),
then ψ̂ is a viscosity solution of (B.2).
(i) Suppose 0 < r ≤ κ. Then ξ ′ (τ ) = (r − κ)ξ 2 e−rτ ≤ 0.
1. Setting W = 0 and using the first inequality of (B.1) we find that
f W (z, τ ) = κ[ξ(τ ) − ξ(τ + h)](z · ∂ψ(z, τ + h) − ψ(z, τ + h)) ≤ 0.
This implies from (B.2) that ψ̂(z, τ ) := ψ(z, τ + h) satisfies
{
}
min ψ̂τ − Aτ ψ̂, B(∇ψ̂(z, τ )) ≥ 0.
Also ψ̂(z, 0) = ψ(z, h) ≥ ℓ(z) = ψ(z, 0). Hence, ψ̂(z, τ ) := ψ(z, τ + h) is a (viscosity)
super-solution, so by comparison principle we have ψ(z, τ ) ≤ ψ(z, τ + h).
26
Xinfu Chen and Min Dai
2. Setting W = −A0 [b(τ + h) − b(τ )] and using b′ + κξb = 1 and the second inequality in
(B.1), we obtain,
[
]
f W (z, τ ) = κ[ξ(τ ) − ξ(τ + h)] z · ∂ψ(z, τ + h) − ψ(z, τ + h) + A0 b(τ + h) ≥ 0.
Thus, ψ̂(z, τ ) := ψ(z, τ + h) − A0 [b(τ + h) − b(τ )] satisfies
}
{
min ψ̂τ − Aτ ψ̂, B(∇ψ̂(z, τ )) ≤ 0.
Also, by (2.12), ψ̂(z, 0) = ψ(z, h) − A0 b(h) ≤ ℓ(z). Hence, ψ̂ is a viscosity subsolution,
so ψ(z, τ ) ≥ ψ(z, τ + h) − A0 [b(τ + h) − b(τ )].
In conclusion, for every h > 0,
(B.3)
0 ≤ ψ(z, τ + h) − ψ(z, τ ) ≤ A0 [b(τ + h) − b(τ )].
Sending h ↘ 0 we obtain
0 ≤ ψτ (z, τ ) ≤ A0 b′ (τ ) = O(1)(1 + τ )e−rτ .
(B.4)
Thus, u(z) := limτ →∞ ψ(z, τ ) exists, and by (2.12), 0 ≤ u − ℓ ≤ A0 b(∞) = A0 /r. Finally,
sending h → ∞ we obtain from (B.3) that
0 ≤ u(z) − ψ(z, τ ) ≤ A0 [b(∞) − b(τ )] = O(1)(1 + τ )e−rτ .
This proves (2.16). Sending τ → ∞ in (2.13) and using (B.4), we find that u is a Lipschitz
continuous and concave viscosity solution of (2.17).
(ii) Suppose r ≥ κ. Then ξ ′ = (r − κ)ξ 2 e−rτ ≥ 0.
1. Setting W = −A0 b(h)Z(τ ) and using Z ′ (τ ) + κξ(τ )Z(τ ) = 0 we derive that
[
]
f W (z, τ ) = [κξ(τ ) − κξ(τ + h)] z · ∂ψ(z, τ + h) − ψ(z, τ + h) ≥ 0.
Also, ψ̂(z, 0) = ψ(z, h)−A0 b(h) ≤ ℓ(z). Hence, ψ̂ = ψ(z, τ )−A0 b(h)Z(τ ) is a viscosity
subsolution, so ψ(z, τ + h) − A0 b(h)Z(τ ) ≤ ψ(z, τ ).
2. Set W = A0 [b(h)Z(τ ) − b(τ + h) + b(τ )]. We derive that
[
]
f W (z, τ ) = [κξ(τ ) − κξ(τ + h)] z · ∂ψ(z, τ + h) − ψ(z, τ + h) + A0 b(τ + h) ≤ 0.
Also, ψ̂(z, 0) = ψ(z, h) ≥ ψ(z, 0) (by (2.12)). Hence, ψ̂ is a viscosity super-solution,
and ψ(z, τ ) ≤ ψ(z, τ + h) + A0 [b(h)Z(τ ) − b(τ + h) + b(τ )].
In summary, we have
(B.5)
A0 [b(τ + h) − b(τ ) − b(h)Z(τ )] ≤ ψ(z, τ + h) − ψ(z, τ ) ≤ A0 b(h)Z(τ ).
Sending h ↘ 0 we obtain
A0 [b′ (τ ) − Z(τ )] ≤ ψτ (z, τ ) ≤ A0 Z(τ ).
Multi-Asset Investment and Consumption with Transaction Costs
27
In particular, this implies that |ψτ | = O(1)(1 + τ )e−rτ . Consequently, u = limτ →∞ ψ(z, τ )
exists. Also, sending h → ∞ we obtain from (B.5) that
]
[1
1
A0
A0 − b(τ ) − Z(τ ) ≤ u(z) − ψ(z, τ ) ≤
Z(τ ).
r
r
r
This implies that |ψ(z, τ ) − u(z)| = O(1)(1 + τ )e−rτ . Finally, sending τ → ∞ in (2.13), we
find that u is a viscosity solution of (2.17). This completes the proof of Theorem 2.2.
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