CHARACTERIZATION OF OPTIMAL STRATEGY ... - Dept of Maths, NUS

CHARACTERIZATION OF OPTIMAL STRATEGY FOR MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS XINFU CHEN

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. Keywords: portfolio selection, optimal investment and consumption, transaction costs, multiple risky assets, shape of trading and no-trading regions.

1. Introduction We consider the optimal investment and consumption decision of a risk-averse 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)]. Date: First version: March 2012; This version: November 2012. 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

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Since the seminal work of Magill and Constantinides (1976), portfolio selection with transaction 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)], 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 placed in Appendix. 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 1

There do exist many papers working on perturbation analysis or numerical solutions for the multiple risky-asset 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. Shape of trading and no-trading regions with CARA utility shapes of trading regions are extremely 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 . 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. 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 (Section 2), where the expected utility is from not only the intermediate consumption but also the terminal wealth (i.e., bequest). Section 3 is devoted to some basic properties of the value function associated with the finite horizon problem. In Section 4 we derive the infinite horizon problem as the limit of the finite horizon problem. In Section 5, we show the C 1 continuity of the resulting value function that is useful for analyzing the optimal strategy. The main results are presented in section 6. 2. Problem Formulation Consider a portfolio consisting of one risk-free 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 X dSti = αi dt + aij dWtj Sti j=1

for i = 1, · · · , n,

2It 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.

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where {Wt1 , · · · , Wtn }t>0 is a standard n-dimensional Wiener process, r > 0 is the constant bank rate, αi > 0 is the expected return rates of the i-th risky asset, and (aij )n×n is a 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 λdLis . 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  X X  dxs = (rxs − κcs )ds − (1 + λi )dLis + (1 − µi )dMsi ,    i i X (2.1) i j i i i  = y (α ds + a dW ) + dL − dM , i = 1, · · · , n. dy  i ij s s s s   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 ³ ´ (2.2) dLis > 0, dMsi > 0, sup kcs kL∞ + kxs kL∞ + kys kL∞ < ∞. t6s6T

where {ys }t6s6T is the solution of the second set of equations in (2.1) subject to constant initial conditions. We denote by At all the admissible strategies. Here we remark that the “optimal strategy” may not be attained, but can be approximated, by the admissible strategies constrained by (2.2).4 2.2. The Merton’s Problem. Given concave utilities U (x, y) for the terminal portfolio, V (c) for consumption, a discount factor constant β > 0, and positive dimensionless constant weight K, we consider the measure of quality of an investment-consumption strategy S defined by Z T −β(T −t) V (cs )e−β(s−t) κds, J(S, t) := KU (xT , yT )e + (2.3) t

3The 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) should be replaced by r < κ. If the issue of unit for κ and K is neglected, we can simply write K = 1 − κ. 4For a given type of investor, we may expand the set of admissible strategies such that the “optimal strategy” is admissible. However, as mentioned later, constructing the “optimal strategy” requires certain regularity of the boundary of the no-trading region, which is not covered by the present paper.

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

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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 ∈ R × Rn , t 6 T,

S∈At

where Ex,y t is the expectation under the condition (xt , yt ) = (x, y). In this paper, we consider the exponential utility V (c) := −e−γc ,

U (x, y) := V (x + `(y)),

where `(y) is the liquidation value of the holdings in the risky assets: ½ X (1 − µi )yi if yi > 0, `(y) = `i (yi ), `i (yi ) = (1 + λi )yi if yi < 0.

(2.4)

i

Notice that `i (·) is a concave function and `i (yi ) =

min

1−µi 6k61+λi

{kyi } = min{(1 − µi )yi , (1 + λ)yi }

∀ yi ∈ R.

3. Basic Properties of Φ 3.1. The Case of No Risky Assets. Here 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 Z T (3.1) x(T ) = xer(T −t) − er(T −s) c(s)κds. t

Thus, for any consumption strategy c ∈ L∞ , the total utility can be written as Z T R 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 J0 [c] := − .

(3.2)

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 D δJ x,t [c] E J x,t [c + hζ] − J0x,t [c] 0 ,ζ := lim 0 h→0 δc h Z T n o e−β(s−t) ζ(s) e−γc(s) − Ke−γx(T )+(r−β)(T −s) ds, = κγ t

where x(T ) is as given in (3.1). 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 (3.1) 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 c∗ (s) = xξ(τ ) + where τ = T − t and rerτ ξ(τ ) = , r + κerτ − κ

(r − β)[s − t − b(τ )] − Z(τ ) ln K γ Z(τ ) = ξe−rτ ,

b(τ ) =

∀ s ∈ [t, T ],

κ(erτ − 1 − rτ ) + r2 τ . r(r + κerτ − κ)

(3.3)

(3.4)

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It is easy to verify that ξ 0 = (r − κξ)ξ

on [0, ∞),

ξ(0) = 1;

(3.5)

Z 0 = −κξZ

on [0, ∞),

Z(0) = 1;

(3.6)

b = −κξb + 1

on [0, ∞),

b(0) = 0.

(3.7)

0

Note that J0x,t is a concave functional, so c∗ is the global maximizer. We have proved the following result: Lemma 3.1. For each x ∈ R, the linear function c∗ defined in (3.3), where τ = T − t and ξ, Z, and b are as in (3.4), is the global maximizer of J0x,t defined in (3.2): J0x,t [c] 6 J0x,t [c∗ ]

∀c ∈ L∞ .

The strategy that liquidates all risky assets at time t gives the estimate x+`(y),t

Φ(x, y, t) > Φ(x + `(y), 0, t) > J0

[c∗ ]

= −e−γξ(τ )[x+`(y)]+(r−β)b(τ )−ln ξ(τ )+Z(τ ) ln K . Then we have the following corollary: Corollary 3.2. We have the following lower bound for Φ: Φ(x, y, t) > −e−γξ(τ )[x+`(y)]+(r−β)b(τ )−ln ξ(τ )+Z(τ ) ln K .

(3.8)

3.2. Separation of Investment and Consumption. Lemma 3.3. Let τ = T − t and ξ be as defined in (3.4). Then Φ(x, y, t) = e−γξ(τ )x Φ(0, y, t)

∀ (x, y, t) ∈ R × Rn × (−∞, T ].

(3.9)

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 ˜ L, ˜ M ˜) position (x, y) at time t, we consider the investment-consumption strategy S˜ = (C, defined by ˜ M ˜ ) ≡ (L, M ), (L, c˜s = cs + ξ(τ )x, τ := T − t. Denote the corresponding portfolio (starting from (x, y) at time t) by {(˜ xs , y˜s )}s∈[t,T ] . Then y˜s = ys and x ˜s = xs + x ˆ(s) where x ˆ(s) is the solution of dˆ x(s) = [rˆ x(s) − κξ(τ )x]ds subject to x ˆ(t) = x. Solving this initial value problem for x ˆ gives κξ(τ )x ³ κξ(τ )x ´ rτ x ˆ(T ) = + x− e = ξ(τ )x r r by the definition of ξ(τ ) in (3.4). It then follows that Z T −γ x ˆ(T ) −β(T −t) ˜ J(S, t) = −Ke U (xT , yT )e − e−γξ(τ )x e−γcs −β(T −s) κ ds = e−γξ(τ )x J(S, t). t

The relation between S and S˜ is 1-1 and onto, so taking the supremum yields (3.9).

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MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

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3.3. 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)

Z −β(T −t)

= ϕ(xT , yT , T )e

T

− t

Z −β(T −t)

= ϕ(xT , yT , T )e Z

T

+ t

Z

´ ³ d ϕ(xs , ys , s)e−β(s−t) Z

T

+ t

−β(s−t)

V (cs )e

(κds) − Z

e−β(s−t) (1 + λi )∂x ϕ − ∂yi ϕ dLic s + ³

´

Z

e−β(s−t)

t

i

h

T

T

t

X

aij yi ∂yi ϕdBsj

i,j

i e−β(s−t) (−1 + µi )∂x ϕ + ∂yi ϕ dMsic h

h

³ ´ ³ ´i e−β(s−t) ∂t ϕ + Lϕ ds + e−β(s−t) V ∗ ∂x ϕ − V (cs ) − cs ∂x ϕ κds t t X −β(s−t) + e [ϕ(xs− , ys− , s−) − ϕ(xs , ys , s)], T

T

−

t6s6T

where Lic and M ic are the continuous part of Li and M i respectively, n o V ∗ (q) := max V (c) − cq ∀ q > 0, c∈R

³ ´ X 1X Lϕ := σij yi yj ∂yi yj ϕ + αi yi ∂yi ϕ + rx∂x ϕ − βϕ + κV ∗ ∂x ϕ , 2 ij i X σij := aik ajk .

(3.10)

k

Now suppose ϕ satisfies ϕ(·, T ) > KU (·), −∂t ϕ − Lϕ > 0, and (1 + λi )∂x ϕ − ∂yi ϕ > 0,

(−1 + µi )∂x ϕ + ∂yi ϕ > 0.

(3.11)

Combination of (3.11) and ∂x ϕ > 0 leads to ϕ(xs− , ys− , s−) − ϕ(xs , ys , s) > 0. Taking the expectation we obtain ϕ(x, y, t) >

Ex,y t

h

Z −β(T −t)

KU (xT , yT )e

T

+ t

i V (cs )e−β(s−t) (κds) .

Taking the supremum we obtain the following: Lemma 3.4. Suppose ϕ is a smooth function on R × Rn × (−∞, T ] satisfying ϕx > 0, ϕ(·, T ) > KU (·) and (3.11). Then Φ(x, y, t) 6 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 (3.9), 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,

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where ξ, Z and b are defined in (3.4). We can compute n h ξ0 i −∂t ϕ − Lϕ = |ϕ| γx(ξ 0 − rξ + κξ 2 ) + + κξ − β − (r − β)(b0 + κξb) ξ 0 X ξ −(Z 0 + κξZ) ln K + zi ∂zi φ ξ i o X 1X σij zi zj (∂zi zj φ − ∂zi φ∂zj φ) − αi zi ∂zi φ + κξφ +∂τ φ − 2 i,j i ³ ´ τ = |ϕ| ∂τ φ − A [φ] , where we have used (3.5)-(3.7) in the last equality, and h i X 1X Aτ [φ] := σij zi zj ∂zi zj φ − ∂zi φ∂zj φ + (αi − r)zi ∂zi φ + κξ [z · ∇φ − φ] 2 i,j

(3.12)

i

where the dependence of Aτ on τ is via ξ = ξ(τ ). Define ∀ p = (p1 , · · · , pn ) ∈ Rn .

B(p) := min min{1 + λi − pi , −1 + µi + pi } i

Then (3.11) can be written as © ª min ∂τ φ − Aτ [φ], B(∇φ) > 0

(3.13)

in Rn × (0, ∞).

Since ξ(0) = 1 = Z(0) and b(0) = 0, the condition ϕ(·, T ) > KU (·) can be written as X φ(z, 0) > `(z) = `i (zi ), i

where ` and `i are as defined in (2.4). 3.4. Upper Bound. Let k = (k0 , k1 , · · · , kn ) be a constant vector satisfying k0 > 0,

1 − µi 6 ki 6 1 + λi

∀ i.

Consider the function ¯ z, τ ) := k0 b(τ ) + φ(k;

X

ki zi .

i

¯ z, 0) = P ki zi > φ0 (z), 1 + λi − ∂¯z φ = 1 + λi − ki > 0 and It is easy to see that φ(k; i i −1 + µi + ∂¯zi φ = −1 + µi + ki > 0 for each i. Also, X X ¯ = (b0 + kξb)k0 + 1 φ¯τ − Aτ [φ] (zi ki )σij (zj kj ) − (αi − r)(zi ki ) 2 i,j

= k0 +

1X 2

i

(zi ki − mi )σij (zj kj − mj ) − A0 ,

i,j

where, denoting by (σ ij )n×n the inverse matrix of (σij )n×n , X 1X mj := σ ji (αi − r) ∀ j, A0 := mi σij mj . 2 i

i,j

(3.14)

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

9

Here m := (m1 , · · · , mn ) is the optimal strategy for the Merton’s problem without transaction costs, being the solution of the linear system X σij mj = αi − r ∀ i = 1, · · · , n. j

Since (σij )n×n is positive-definite, taking k0 = A0 we have φ¯ − Lτ φ¯ > 0. Hence, by Lemma 3.4, ¯

Φ(x, y, t) 6 −e−γξ(τ )x+(r−β)b(τ )+Z(τ ) ln K−ln ξ(τ )−φ(k,z,τ ) .

(3.15)

We are now ready to show the following: Theorem 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(γξ(τ )y,τ ) ,

(3.16)

where τ = T − t and ξ, Z, b are defined in (3.4). In addition, (1) for each τ > 0, ψ(·, τ ) is Lipschitz continuous: `(ˆ z − z) 6 ψ(ˆ z , τ ) − ψ(z, τ ) 6 −`(z − zˆ)

∀ z, zˆ ∈ Rn ;

(3.17)

(2) for A0 defined in (3.14) with (σ ij )n×n being the inverse of (σij )n×n , ∀ z ∈ Rn , τ > 0;

`(z) 6 ψ(z, τ ) 6 `(z) + A0 b(τ )

(3) for each τ > 0, ψ(·, τ ) is concave; (4) ψ is a viscosity solution of  n o  min ∂τ ψ − Aτ [ψ], B(∇ψ) = 0 in Rn × (0, ∞),  ψ(·, 0) = `(z) on Rn × {0},

(3.18)

(3.19)

where Aτ and B are defined in (3.12) and (3.13). Proof. By (3.8), Φ(x, y, t) > −∞. Also, from (3.15) we see that Φ(x, y, t) < 0 for every (x, y, t) ∈ R × Rn × (−∞, T ]. Hence, we can define ¯ ¯ (3.20) 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 3.3, we obtain (3.16). Now we establish properties of ψ. (1) Let x ∈ R, y ∈ Rn and yˆ ∈ Rn . Stating at position (x, yˆ) at time t, one can immediately liquidate yˆ−y amount of risky asset holding to reach the position (x+`(ˆ y −y), y) at time t+. Hence, we have Φ(x, yˆ, t) > Φ(x + `(ˆ y − y), y, t) = e−`(γξ(τ )(ˆy−y)) Φ(x, y, t). In terms of (3.16), this implies that ψ(ˆ z , τ ) > ψ(z, τ ) + `(ˆ z − z). Thus, we obtain (3.17). (2) Combination of the estimate (3.8) (with y = z/[γξ(τ )]) and (3.16) yields the left hand side inequality of (3.18). To show the right hand side inequality, we use (3.15) with k0 := A0 ¯ z, τ ) = A0 b(τ ) + P ki zi whenever ki ∈ [1 − µ, 1 + λ] and (3.16) to derive that ψ(z, τ ) 6 φ(k, i for all i. Hence, ³ ´ X ψ(z, τ ) 6 min A0 b(τ ) + ki zi = A0 b(τ ) + `(z). 1−µj 6kj 61+λj ∀ j

i

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(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 (3.20) and the fact that the function ln(·) is concave and increasing. (4) That ψ is a viscosity solution of (3.19) follows by a dynamical programming principle [see, for example, Shreve and Soner (1994)]. This completes the proof of Theorem 1. ¤

4. The Asymptotic Behavior as T → ∞ Since we are interested in the infinite horizon (i.e., T = ∞) problem, let us consider the asymptotic behavior of ψ(·, τ ) as τ = T − t → ∞. From now on we always assume κ > 0. For any function f defined on Rn , we define its super-differential by ∂f (z) = {p ∈ Rn : f (ˆ z ) 6 f (z) + p · (ˆ z − z) ∀ zˆ ∈ Rn }.

(4.1)

We shall use the following fact. Lemma 4.1. Suppose f is a concave function on Rn . Define its super-differential by (4.1). Then the following holds: (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 h ∂i f (z) = ∂g(0) =

g(h) − g(0) g(0) − g(−h) i , lim . h&0 h&0 h h lim

Most of the conclusion of the Lemma should be well-known. For completeness, we provide the full proof in Appendix. 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). We begin with the following estimate: Lemma 4.2. For any z ∈ Rn and τ > 0, 0 6 ψ(z, τ ) − z · ∂ψ(z, τ ) 6 A0 b(τ ). Proof. The assertion holds for z = 0 by (3.18). Fix z ∈ Rn \ {0}. Consider the function f (s) := ψ(sz, τ ),

s ∈ [0, ∞).

(4.2)

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

11

This is a concave and Lipschitz continuous function in one space dimension. Hence, f 0 (s) is a decreasing function. Set f (s) − f (0) . s→∞ s

p∞ = lim

In view of (3.17) and the homogeneity `(sz) = s`(z) for s > 0, we find that p∞ > `(z). As f 0 (s) is a decreasing function, by L’Hˆopital’s rule, f 0 (s) & p∞ as s → ∞. Consequently, for any s > 0, 0 6 f (0) 6 f (s) + f 0 (s)(0 − s) 6 f (s) − p∞ s 6 A0 b(τ ) + `(sz) − p∞ s = A0 b(τ ), by (3.18) and the definition f (s) = ψ(sz, τ ). Note that as super-differential, z · ∂ψ(z, τ ) ⊂ f 0 (1). Hence, setting s = 1 we obtain the assertion of the Lemma. ¤ Theorem 2. Suppose κ > 0. Then there exist a function u and a constant M such that |∂τ ψ(z, τ )| + |ψ(z, τ ) − u(z)| 6 M (1 + rτ )e−rτ

∀ τ > 0, z ∈ Rn .

(4.3)

In addition, u is a Lipschitz continuous concave viscosity solution of the following equation min{−A[u], B(∇u)} = 0,

A0 r

06u−`6

in Rn

(4.4)

where `(z) and B are as defined in (2.4) and (3.13) respectively, and A[φ] :=

X 1X σij zi zj (∂zi zj φ − ∂zi φ∂zj φ) + αi zi ∂zi φ − rφ. 2 i,j

(4.5)

i

The estimate (4.3) implies that the finite horizon value function ψ and its time-derivative ∂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 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)].5 It is also worthwhile pointing out that the solution u to the equation (4.4) can be formally regarded as the value function associated with the infinite horizon utility maximization problem.6

Proof of Theorem 2. Keeping in mind that there are comparison principles for viscosity solutions, in the sequel, we treat the viscosity solution ψ as if it were a classical solution. 5It should be admitted that the model with the CARA utility discussed here is less plausible than the

one with the CRRA utility. 6As mentioned in the introduction part, we first consider the finite horizon problem because the terminal utility of bequest can prevent unlimited consumption. If we directly consider the infinite horizon problem, some technical conditions should be given [see, e.g., Liu (2004)].

12

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ˆ τ ) := For any constant h > 0 and smooth function W (·), consider the function ψ(z, ψ(z, τ + h) + W (τ ). One can calculate ψτ (z, τ + h) − Aτ +h ψ(z, τ + h) = ψˆτ − Aτ +h ψˆ − W 0 (τ ) − κξ(τ + h)W (τ ) ˆ − W 0 (τ ) − κξ(τ + h)W (τ ) = ψˆτ − Aτ ψˆ + [κξ(τ ) − κξ(τ + h)](z · ∂ ψˆ − ψ) = ψˆτ − Aτ ψˆ + f W , where f W (z, τ ) := −[W 0 (τ ) + κξ(τ )W (τ )] +κ[ξ(τ ) − ξ(τ + h)](z · ∂ψ(z, τ + h) − ψ(z, τ + h)). Hence, equation (3.19) with (τ, ψ) replaced by (τ + h, ψ(·, τ + h)) can be written as ª © ˆ τ )) = 0. min ψˆτ − Aτ ψˆ + f W , B(∇ψ(z, (4.6) (i) Suppose 0 < r 6 κ. Then ξ 0 (τ ) = (r − κ)ξ 2 e−rτ 6 0. (1) Setting W = 0 and using the first inequality of (4.2) we find that f W (z, τ ) = κ[ξ(τ ) − ξ(τ + h)](z · ∂ψ(z, τ + h) − ψ(z, τ + h)) 6 0. ˆ τ ) := ψ(z, τ + h) satisfies This implies from (4.6) that ψ(z, © ª ˆ B(∇ψ(z, ˆ τ )) > 0. min ψˆτ − Aτ ψ, ˆ 0) = ψ(z, h) > `(z) = ψ(z, 0). Hence, ψ(z, ˆ τ ) := ψ(z, τ + h) is a superAlso ψ(z, solution, so by comparison principle we have ψ(z, τ ) 6 ψ(z, τ + h). (2) Setting W = −A0 [b(τ + h) − b(τ )] and using b0 + κξb = 1 and the second inequality in (4.2), 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. ˆ τ ) := ψ(z, τ + h) − A0 [b(τ + h) − b(τ )] satisfies Thus, ψ(z, © ª ˆ B(∇ψ(z, ˆ τ )) 6 0. min ψˆτ − Aτ ψ, ˆ 0) = ψ(z, h) − A0 b(h) 6 `(z). Hence, ψˆ is a subsolution, so Also, by (3.18), ψ(z, ψ(z, τ ) > ψ(z, τ + h) − A0 [b(τ + h) − b(τ )]. In conclusion, for every h > 0, 0 6 ψ(z, τ + h) − ψ(z, τ ) 6 A0 [b(τ + h) − b(τ )].

(4.7)

Sending h & 0 we obtain 0 6 ψτ (z, τ ) 6 A0 b0 (τ ) = O(1)(1 + τ )e−rτ .

(4.8)

Thus, u(z) := limτ →∞ ψ(z, τ ) exists, and by (3.18), 0 6 u − ` 6 A0 b(∞) = A0 /r. Finally, sending h → ∞ we obtain from (4.7) that 0 6 u(z) − ψ(z, τ ) 6 A0 [b(∞) − b(τ )] = O(1)(1 + τ )e−rτ . This proves (4.3). Sending τ → ∞ in (3.19) and using (4.8), we find that u is a Lipschitz continuous and concave viscosity solution of (4.4). (ii) Suppose r > κ. Then ξ 0 = (r − κ)ξ 2 e−rτ > 0.

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

13

(1) Setting W = −A0 b(h)Z(τ ) and using Z 0 (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. ˆ 0) = ψ(z, h) − A0 b(h) 6 φ0 (z). Hence, ψˆ = ψ(z, τ ) − A0 b(h)Z(τ ) is a Also, ψ(z, subsolution, so ψ(z, τ + h) − A0 b(h)Z(τ ) 6 ψ(z, τ ). (2) Set W = A0 [b(h)Z(τ ) − b(τ + h) + b(τ )]. One derive that £ ¤ f W (z, τ ) = [κξ(τ ) − κξ(τ + h)] z · ∂ψ(z, τ + h) − ψ(z, τ + h) + A0 b(τ + h) 6 0. ˆ 0) = ψ(z, h) > ψ(z, 0) (by (3.18)). Hence, ψˆ is a super-solution, and Also, ψ(z, ψ(z, τ ) 6 ψ(z, τ + h) + A0 [b(h)Z(τ ) − b(τ + h) + b(τ )]. In summary, we have A0 [b(τ + h) − b(τ ) − b(h)Z(τ )] 6 ψ(z, τ + h) − ψ(z, τ ) 6 A0 b(h)Z(τ ).

(4.9)

Sending h & 0 we obtain A0 [b0 (τ ) − Z(τ )] 6 ψτ (z, τ ) 6 A0 Z(τ ). 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 (4.9) that h1 i 1 A0 A0 − b(τ ) − Z(τ ) 6 u(z) − ψ(z, τ ) 6 Z(τ ). r r r This implies that |ψ(z, τ ) − u(z)| = O(1)(1 + τ )e−rτ . Finally, sending τ → ∞ in (3.19), we find that u is a viscosity solution of (4.4). This completes the proof of Theorem 2. ¤ 5. 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.7 The rigorous analysis in this section needs C 1 regularity of u whose proof is very technical, and therefore is given in the next section. Regarding the shape, we have the following result. Theorem 3. Let (i, ˘ı) = (1, 2) or (2, 1). Define Bi := {z | ∂zi u(z) = 1 + λi }, Si := {z | ∂zi u(z) = 1 − µi }, Ni := {z | 1 − µi < ∂zi u(z) < 1 + λi }, 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) there are bounded functions l˘ı± (·) such that Bi = {(z1 , z2 ) | z˘ı ∈ R, zi 6 l˘ı− (z˘ı )}, Si = {(z1 , z2 ) | z˘ı ∈ R, zi > l˘ı+ (z˘ı )}, Ni = {(z1 , z2 ) | l˘ı− (z˘ı ) < zi < l˘ı+ (z˘ı )}.

(5.1)

(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. 7We consider the two dimensional case because of notational simplicity. We believe that all results of this section remain valid in the higher dimensional case.

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MIN DAI

(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. ± 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 > l2 (z2 )}, + SB = [s− 1 , ∞) × (−∞, b2 ], − − NB = {(z1 , z2 ) | z1 ∈ (b− 1 , s1 ), z2 6 l1 (z1 )}, − BB = (−∞, b− 1 ] × (−∞, b2 ], − − BN = {(z1 , z2 ) | z2 ∈ (b− 2 , s2 ), z1 6 l2 (z2 )}, − BS = (−∞, b+ 1 ] × [s2 , ∞), + + NS = {(z1 , z2 ) | z1 ∈ (b+ 1 , s1 ), z2 > l1 (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± := {(l2± (z2 ), z2 ) | z2 ∈ (b± 2 , s2 )},

± Γ1± := {(z1 , l1± (z1 )) | z1 ∈ (b± 1 , s1 )}.

These four curves connect each other only at their tips:  ± ±  l (b ) = limzi &b± li± (zi ), li± (s± li± (zi ),  i ) = limzi %s± i i  i i + + + + + − − − (b+ (l2+ (s+ 2 ), s2 ) = (s1 , l1 (s1 )), 1 , l1 (b1 )) = (l2 (s2 ), s2 ),    (l− (b− ), b− ) = (b− , l− (b− )), + + + − − (s− 1 , l1 (s1 )) = (l2 (b2 ), b2 ). 1 1 1 2 2 2

(5.2)

± Each function li± is constant outside (b± i , si ): ± li± (zi ) = li± (s± i ) ∀ zi 6 b i ,

± li± (zi ) = li± (s± i ) ∀ zi > si .

(5.3)

As emphasized in the introduction part, Theorem 3 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 out of 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± , so we are unable to construct the controlled portfolio process as a reflected diffusion as in Davis and Norman (1990) and Shreve and Soner (1994). Regarding the location, we have the following Theorem 4. Let (m1 , m2 ) and A0 be as given in (3.14) with n = 2, and s √ √ 2(A0 − rh) σ12 , M (h) := . σ1 = σ11 , σ2 = σ22 , ρ = √ σ11 σ22 1 − ρ2

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

Also, define the ellipse C(k1 , k2 , h) and constant cij by ¯ 1X n o ¯ C(k1 , k2 , h) := (z1 , z2 ) ¯ σij (ki zi − mi )(kj zj − m) = A0 − rh , 2

15

(5.4)

i,j

cij

=

lim

{u(z) − `(z)},

(−1)i z1 →∞,(−1)j z2 →∞

i, j = 1, 2.

(1) The no-trading region NT is contained in the set [ D := C(k1 , k2 , h).

(5.5)

(5.6)

1−µi Lu(z) = f (z) >

1X σij (zi ∂zi u − mi )(zj ∂zj u − mj ) − A0 + ru(0), 2 i,j

where we have used (5.7) in the last inequality. Thus, z is on or inside of the ellipse C(∂z1 u, ∂z2 u, u(0)). Since ∂zi u(z) ∈ (1 − µi , 1 + λi ) we see that z ∈ D defined in (5.6). The first assertion the of Theorem 4 thus follows. ¤ In the sequel, we shall use the following important fact: Lemma 5.1. 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 6 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) 6 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. ¤

5.2. 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 . z1 →±∞

Lemma 5.2. and

z2 →±∞

(5.10)

(1) With ki+ := 1 − µi and ki− := 1 + λi , vi± defined in (5.10) is concave

n o u(z1 , z2 ) 6 min v2± (z2 ) + k1± z1 , v1± (z1 ) + k2± z2

∀ z ∈ R2 .

± ± ± ± (2) There are intervals (b± i , si ) and functions li defined on (bi , si ) such that  if zi > s±  = 1 − µi i , ±0 ± vi (zi ) ∈ (1 − µi , 1 + λi ) if zi ∈ (b± i , si ),  ± = 1 + λi if z 6 bi ,

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

 + v2 (z2 ) + (1 − µ1 )z1      v − (z2 ) + (1 + λ1 )z1 2 u(z1 , z2 ) =  v1+ (z1 ) + (1 − µ2 )z2     − v1 (z1 ) + (1 + λ2 )z2

17

+ if z1 > l2+ (z2 ), z2 ∈ (b+ 2 , s2 ), − if z1 6 l2− (z2 ), z2 ∈ (b− 2 , s2 ), + if z2 > l1+ (z1 ), z1 ∈ (b+ 1 , s1 ), − if z2 6 l1− (z1 ), z1 ∈ (b− 1 , s1 ),

∂z1 u(z1 , z2 ) > 1 − µ1

+ if z1 < l2+ (z2 ), z2 ∈ (b+ 2 , s2 ),

∂z1 u(z1 , z2 ) < 1 + λ1

− if z1 > l2− (z2 ), z2 ∈ (b− 2 , s2 ),

∂z2 u(z1 , z2 ) > 1 − µ2

+ if z2 < l1+ (z1 ), z1 ∈ (b+ 1 , s1 ),

∂z2 u(z1 , z2 ) < 1 + λ2

− if z2 > l1− (z1 ), z1 ∈ (b− 1 , s1 ).

(3) Define ± li± (s± i ) = lim li (zi ), zi %s± i

± li± (b± i ) = lim li (zi ), zi &bi

± ± li∗ (si ) = lim li± (zi ), zi %s± i

± ± li∗ (bi ) = lim li± (zi ). zi &bi

Then + + + + + + + + [l2+ (s+ 2 ), l2∗ (s2 )] × {s2 } ∪ {s1 } × [l1 (s1 ), l1∗ (s1 )] ⊂ ∂NT ∩ SS, − − − − − + + + [l2+ (b+ 2 ), l2∗ (b2 )] × {b2 } ∪ {s1 } × [l1 (s1 ), l1∗ (s1 )]

⊂ ∂NT ∩ SB,

+ + + + {b+ 1 } × [l1 (b1 ), l1∗ (b1 )] − − + − {b− 1 } × [l1 (b1 ), l1∗ (s1 )]

⊂ ∂NT ∩ BS,

[l2+ (s− 2 ), + − [l2 (b2 ),

+ − l2∗ (s2 )] × {s− 2} − + − l2∗ (b2 )] × {b2 }

∪ ∪

⊂ ∂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 6 u(z) − `(z) 6 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 0 (z2 ) < 1 + λ2 . Since ∂z2 u(z1 , z2 ) → v 0 (z2 ) as z1 → ∞, ∂z2 u(z1 , z2 ) ∈ (1 − µ2 , 1 + λ2 ) for all z1 À 1. As NT is bounded, we must have ∂z1 u(z1 , z2 ) = 1 − µ1 for all z1 À 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 5.1, 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 0 (z2 ) ∈ (1 − µ2 , 1 + λ2 ) for all z1 ∈ [l(z2 ), ∞). Thus, for small positive ε, B(∇u(z)) > 0 on [l(z2 ) − 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 0 = 1 + λ2 on (−∞, b],

1 + λ2 > v 0 > 1 − µ2 on (b, s),

v 0 = 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 0 (s)) = (1 − µ1 , 1 − µ2 ) for all z ∈ [l(s), ∞) × {s}. Hence, [l(s), ∞) × {s} ∈ SS.

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Note that for each z2 < s and z1 ∈ [l(s), ∞), we have Z ∞ [∂z1 u(ξ, s) − ∂z1 u(ξ, z2 )]dξ u(z1 , s) − u(z1 , z2 ) = v(s) − v(z2 ) − z1 Z s Z ∞ 0 = 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 . Recalling (s, l) = (s+ 2 , l2 ), we have

∂z2 u > 1 − µ2 on [l2+ (s), ∞) × (−∞, s+ 2 ).

(5.11)

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 5.2.

¤

5.3. The Intersection of ∂NT with B1 ∩ B2 , S1 ∩ S2 and Bi ∩ Sj . In this subsection we prove the following: Lemma 5.3. Let cij and C(k1 , k2 , h) be defined as in (5.5) and (5.4). (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 5.1, u(z) = π(z 0 , z) on ([z10 , ∞) × {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) 6 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 ) 6 lim f (z 0 + se1 + se2 ) = f (z 0 ). s&0

s&0

Using u = c22 + z · ∇u(z0 ) on [z10 , ∞) × [z20 , ∞) we find that f (z 0 ) = f22 (z 0 ) > 0 where 1X f22 (z) := σij (zi [1 − µi ] − mi )(zj [1 − µj ] − mj ) − A0 + rc22 ∀ z ∈ R2 . 2 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) 6 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) 6 0.

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

19

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 ellpise 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 6

(1 − µ1 )z10 − m1 6 1, M (c22 )/σ1

−ρ 6

(1 − µ2 )z20 − m2 6 1. M (c22 )/σ2

(iii) Now we show that ∂NT ∩ SS is a singleton, by a contradiction argument. Suppose z 0 = (z10 , z20 ) ∈ ∂NT ∩ SS, zˆ0 = (ˆ z10 , zˆ20 ) ∈ ∂NT ∩ SS, and zˆ0 6= z 0 . Then both z 0 and zˆ0 lies on the upper-right part of the ellipse C(1 − µ1 , 1 − µ2 , c22 ). Exchanging the roles of zˆ0 and z 0 , we can assume that z20 < zˆ20 and z10 > zˆ10 . Note that u(z) = π(ˆ z 0 , z) for all z ∈ 0 0 0 0 0 [ˆ z1 , ∞) × (ˆ z2 , ∞) and u(z) = π(z , z) for all z ∈ [z1 , ∞) ∩ [z2 , ∞). Hence, π(z 0 , z) = π(ˆ z 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, 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(·) 6 π(z 0 , ·) on Rn . We derive that SS is a convex set. Consequently, it contains L, the line segment connecting z 0 and zˆ0 . Next for each s ∈ R, denote z s = (ˆ z10 , z20 ) + (s, s). For s < 0, z s 6∈ SS since otherwise it 0 0 would imply [ˆ z1 + s, ∞) × [z2 + s, ∞) ∈ SS, contradicting z 0 ∈ ∂NT. Hence, there exists s∗ > 0 such that z ∗ = (ˆ z10 + 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 6∈ SS since SS is convex. Also ∂z1 u(z s ) > 1 − µ1 since by Lemma 5.1, ∂z1 u(z s ) = 1 − µ1 would imply that ∇u(z s ) = ∇u(ˆ z ) = ∇u(z0 ) where zˆ 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 6∈ C. Consider case (b) s∗ = 0. First of all ∂z1 u(z1 , z20 ) > 1−µ1 for all z1 < zˆ10 since otherwise it would imply ∇u(z1 , z20 ) = ∇u(z 0 ) and thus [z1 , ∞) × [z20 , ∞) ∈ SS contradicting zˆ0 ∈ ∂NT. Similarly, ∂z2 u(ˆ z10 , z2 ) > 1 − µ2 for every z2 < z20 . Now for every small positive ε, consider the closed set Dε := [ˆ z10 − ε, zˆ10 ] × [z20 − ε, z20 ] \ (ˆ z10 − ε/2, zˆ10 ] × (z20 − ε/2, z20 ]. If z ∗ = (z1∗ , z2∗ ) ∈ SS ∩ Dε we would have z1∗ < zˆ10 and z2∗ < z20 so z 0 is an interior point of [z1∗ , ∞) × [z2∗ , ∞) ⊂ SS, a contradiction. Thus, SS ∩ Dε = ∅. Consequently, the closed sets 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 6= ∅. Consequently, (ˆ z10 , z20 ) ∈ ∂NT ∩ SS, but this is impossible since (ˆ z10 , 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. 5.4. Completion of the Proof of Theorems 3, 4. By Lemmas 5.2 and 5.3, we see that the limits in (5.2) exist, and the limits satisfy the ± matching condition stated in (5.2). We extend li± from (b± i , si ) to R by (5.3).

20

XINFU CHEN

MIN DAI

+ + + By the definition of l1+ on (b+ 1 , s1 ) we know that when z1 ∈ (b1 , s1 ), ∂z2 u(z1 , z2 ) > 1 − µ2 + if and only if z1 < l1 (z1 ). Also, in view of (5.11) and the matching (5.2), we derive that + + + when z1 ∈ [s+ 1 , ∞), ∂z2 u(z1 , z2 ) > 1 − µ2 if and only if z2 < l1 (s1 ) = l1 (z1 ). Similarly, we + + + can show that when z1 ∈ (−∞, b+ 1 ], ∂z2 u(z1 , z2 ) < 1 − µ2 if and only if z2 < l1 (b1 ) = l1 (z1 ). Thus,

S2 := {z | ∂z2 u = 1 − µ2 } = {(z1 , z2 ) | z1 ∈ R, z2 > l1+ (z1 )}. Similarly, we can show the other equations in (5.1). The rest assertions of Theorems 3 and 4 thus follow from Lemmas 5.2 and 5.3. 6. C 1 Regularity In this section we show that the viscosity solution u of the infinite horizon problem is except on the coordinates planes where the elliptic operator A is degenerate. The C 1 continuity plays a critical role in analyzing the optimal strategy, where a key step is to derive the continuity of f (·) as defined in (5.9). We begin with recalling the definition of a viscosity solution: C1

Definition 1. A function u defined on Rn is called a viscosity solution of (4.4) if u is continuous, u − ` ∈ 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 n o min − A[ζ](z 0 ), B(∇ζ(z 0 )) 6 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) 6 0 = ζ(z 0 ) − u(z 0 ) for each z ∈ Bε (z 0 ), then n o min − A[ζ](z 0 ), B(∇ζ(z 0 )) > 0. One can show that viscosity solution of (4.4) is unique. Since the solution is the limit ψ(·, τ ) that is concave, we see that viscosity solution of (4.4) is concave. We shall use this fact to prove the C 1 regularity of u. Theorem 5. Let u be the solution of (4.4). Then zi ∂zi u ∈ C(Rn ) for each i = 1, · · · , n; Q consequently, u ∈ C 1 (Ω) where Ω = {(z1 , · · · , zn ) ∈ Rn | i zi 6= 0}. Proof. For illustration, we consider only the two space dimensional case. First we show that ∂1 u(z1 , z2 ) is a singleton if z1 6= 0. We use a contradiction argument. Suppose the assertion is not true. Then there exist z 0 = (z10 , z20 ) with z10 6= 0 and p = (p1 , p2 ) ∈ ∂u(z 0 ) and q = (q1 , q2 ) ∈ ∂u(z 0 ) with 1 − µ1 6 p1 < q1 6 1 + λ1 . By the definition of super-differential, we have u(z) 6 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 ) +

p+q [(q − p) · (z − z 0 )]2 · (z − z 0 ) − 2 4ε

(6.1)

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

21

in Qε := {z : |(q−p)·(z−z 0 )| 6 ε}. By considering separately the cases 0 6 (q−p)·(z−z 0 ) 6 ε and −ε 6 (q − p) · (z − z 0 ) < 0, we find that ζ(z) −

[(q − p) · (z − z 0 )]2 4ε

p+q [(q − p) · (z − z 0 )]2 · (z − z 0 ) − 2 2ε 0 0 0 > u(z ) + min{p · (z − z ), q · (z − z )} > u(z) = u(z 0 ) +

(6.2)

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 ))} 6 0.

It is easy to calculate, 1 1 ∇ζ(z0 ) = (p + q), D2 ζ(z0 ) = − (q − p) ⊗ (q − p). 2 2ε 0 Since z1 6= 0 and q1 − p1 > 0, when ε is sufficiently small, we have −A[ζ](z0 ) > 0. Thus, we mush have B( 21 (p + q)) 6 0. Since 1 − µ1 6 p1 < q1 6 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 z > z20 . We define zˆ20 = inf{z2 6 z20 | u(z10 , z2 ) = u(z 0 ) + (1 − µ2 )(z2 − z20 )}. Since u(z10 , z2 ) 6 O(1) + (1 + λ2 )z2 for z2 6 0, we see that zˆ20 > −∞. In addition, set zˆ0 = (z10 , zˆ20 ) we have ( u(z10 , z2 ) = u(ˆ z 0 ) + (1 − µ2 )(z2 − zˆ20 ) ∀ z2 > zˆ20 (6.3) u(z10 , z2 ) < u(ˆ z 0 ) + (1 − µ2 )(z2 − zˆ20 ) ∀ z2 < zˆ02 . From the definition of ζ and the fact that p2 = q2 = 1 − µ2 , we obtain from (6.2) that, setting β = q1 − p1 , [β(z1 − z10 )]2 ∀ z ∈ [z10 − εβ −1 , z10 + εβ −1 ] × R, 4ε ζ(z) = u(z) ∀ z ∈ {z10 } × [ˆ z20 , ∞),

ζ(z) > u(z) +

ζ(z) > u(z) ∀ z ∈ {z10 } × (−∞, zˆ20 ). Using ζ(z10 , zˆ20 − ε) > u(z10 , zˆ20 − ε) and continuity, we can find η ∈ (0, εβ −1 ) such that ζ(z1 , z02 − ε) > u(z1 , z02 − 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 = zˆ20 − ε, |z1 − z10 | 6 η. Finally, set ηˆ = min{(λ2 + µ2 )/2, η/(2ε), (βη)2 /(8ε2 )} and consider the function 0 0 ˆ = ζ(z) + (z2 − zˆ0 )ˆ ζ(z) z20 − ε, zˆ20 + ε). 2 η in D := (z1 − η, z1 + η) × (ˆ

ˆ ˆ z 0 ) = u(ˆ Note that ζ(z) > u(z) on the boundary ∂D of D. In addition, ζ(ˆ z 0 ). Now set ˆ ˆ z ) = m, m := maxD {u(z) − ζ(z)}. Then m > 0. Let zˆ ∈ D be the point such that u(ˆ z ) − ζ(ˆ

22

XINFU CHEN

MIN DAI

ˆ + m] − u(z) > [ζ(ˆ ˆ z ) + m] − u(ˆ then zˆ ∈ D since u < ζˆ on ∂D. Thus, [ζ(z) z ) = 0 for every z ∈ D. Hence, by definition of viscosity solution, we have min{−A[ζˆ + m](ˆ z ),

ˆ z )} 6 0. B(∇ζ(ˆ

However, for every z ∈ D, ˆ < q1 , p1 < ∂z1 ζ(z)

ˆ = 1 − µ2 + ηˆ ∈ (1 − µ2 , 1 − λ2 ), ∂z2 ζ(z)

ˆ =− D2 ζ(z)

β2 e1 ⊗ e1 . 2ε

ˆ z )) > 0. Hence, we must have −A[ζ(ˆ ˆ z )+m](ˆ This implies that B(∇ζ(ˆ z ) 6 0. However, since 0 ˆ z ) + m](ˆ ˆ z ) + m](ˆ z1 6= 0, we have −A[ζ(ˆ z ) → ∞ as ε & 0; this contradicts −A[ζ(ˆ z ) 6 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 6= 0. Now for each fixed z2 ∈ R, consider the one dimensional function g(t) = u(t, z2 ). By Lemma 4.1 (4), ∂g(t) = ∂1 u(t, z2 ) 1 ,z2 ) is singleton if t 6= 0. Hence, g ∈ C 1 (R \ {0}), so the classical partial derivative ∂u(z := ∂z1 0 g (z1 ) exists. Since any limit point of ∂1 u(ˆ z ) as zˆ → z is in ∂1 u(z), we conclude that ∂u(ˆ z) ∂u(z1 ,z2 ) ∂u(z) limzˆ→z ∂z1 = if z1 6= 0. Hence, ∂z1 ∈ C(R2 \ ({0} × R)). As u is Lipschitz ∂z1 continuous, we also know that z1 ∂z1 u ∈ C(R). This completes the proof. ¤

7. Appendix 7.1. Proof of Lemma 4.1. (1) For each y ∈ Rn , sending k → ∞ in f (y) 6 f (zk ) + pk · (y − zk ) we obtain f (y) 6 f (z) + p · (y − z), so p ∈ ∂f (z). (2) Let ρ be a smooth non-negative function supported in the unit ball having unit mass. Set ρε (z) = ρ(z/ε) and fε = ρε ∗ f . Then fε is smooth and concave. Hence, ∂fε (z) = {∇fε (z)} where ∇ is the classical gradient. In the expression fε (y) 6 fε (z)+∇fε (z)·(y −z), setting y = z − ∇fε (z)/|∇fε (z)| we find that |∇fε (z)| 6 max |fε (z) − fε (y)| 6 y∈B1 (z)

max

|f (y 1 ) − f (y 2 )|

∀ ε ∈ (0, 1].

y 1 ,y 2 ∈B2 (z)

Let p be a limit point of ∇fε (ˆ z ) as ε → 0 and zˆ → z, i.e., there exists a sequence {εj , zj } such that as j → ∞, (zj , εj , ∇fεj (zj )) −→ (z, 0, p). Then for any y ∈ Rn , sending j → ∞ in fεj (y) 6 fεj (zj ) + ∇fεj (zj ) · (y − zj ) we obtain f (y) 6 f (z) + p · (y − z). This means that p ∈ ∂f (z), so ∂f (z) is non-empty. It is easy to show that ∂f (z) is convex and compact. (3) Suppose ∂f (z) = {p} is a singleton. Then limzˆ→z,ε&0 ∇fε (ˆ z ) = p since any limit point of ∇fε (ˆ z ) is in ∂f (z). Hence, for any β > 0, there exist r > 0 and ε0 > 0 such that |∇fε (ˆ z ) − p| 6 β for all zˆ ∈ Br (z) and ε ∈ (0, ε0 ]. This implies that ¯ ¯Z 1 ¯ ¯ [∇fε (z + θ[y − x]) − p]dθ · (y − z)¯ 6 β|y − x| |fε (y) − fε (z) − p · (y − z)| = ¯ 0

for every y ∈ Br (z) and ε ∈ (0, ε0 ]. Sending ε & 0 we obtain |f (y) − f (z) − p · (y − z)| 6 β|y − x|. Thus, f is differentiable at z and p = ∇f (z).

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

23

(4) If ∂f (z) is singleton for every z in a neighborhood of z 0 , then ∂f (z) = {∇f (z)} for z in that neighborhood. As every limit point of ∇f (ˆ z ) as zˆ → z is in ∂f (z), we must have limzˆ→z ∇f (ˆ z ) = ∇f (z). Hence, f is C 1 in that open neighborhood of z 0 . (5) By definition, it is clear that ∂i f (z) ⊂ ∂g(0). We prove the reverse inclusion. Since ∞ g(·) is a one dimensional concave function, there are two sequences {tk }∞ k=1 and {τk }k=1 ∞ with −1/k < τk < 0 < tk < 1/k such that g is differentiable at {tk }∞ k=1 ∪ {τk }k=1 and ∂g(0) = [a, b] where a = limk→∞ g 0 (tk ) and b = limk→∞ g 0 (τk ). Let pk ∈ ∂f (z + τk ei ). Then pk · ei = g 0 (τk ). Let {pkj } be a subsequence such that pkj converges to a limit as kj → ∞. Then p ∈ ∂f (z). In addition, p · ei = limj→∞ pkj · ei = limk→∞ g 0 (τk ) = b. Similarly, one can find q ∈ ∂f (z) such that q · ei = limj→∞ g 0 (tk ) = a. As ∂f (z) is convex, we conclude that ∂i f (z) = {p · ei | p ∈ ∂f (z)} ⊃ [a, b] = ∂g(0). This completes the proof. 7.2. Extension to the CRRA utility. Now let us examine the case with the CRAR utility, namely, 1 V (c) = cγ , γ < 1, γ 6= 0, γ for which we require that the liquidated wealth be non-negative: X x+ `i (yi ) > 0. i

Note that we can directly consider the infinite horizon problem with the CRAR utility and the above solvency constraint. Let Φ (x, y1 , y2 ) be the associated value function which satisfies (cf. Dai and Zhong (2010)) ½ ¾ min −LΦ, min [(1 + λi ) ∂x Φ − ∂yi Φ] , min [− (1 − µi ) ∂x Φ + ∂yi Φ] = 0, (7.1) i

P

i

in x + i `i (yi ) > 0, where L is as given in (3.10). For illustration, we still consider the case of two risky assets. The homogeneity of the utility function allows us to make the following transformation: yi zi = , i = 1, 2, x + y1 + y2 1 ϕ (z1 , z2 ) ≡ Φ (1 − z1 − z2 , z1 , z2 ) = Φ (x, y1 , y2 ) . (x + y1 + y2 )γ Then (7.1) reduces to " " ( # #) X X b min λi γϕ − min −Aϕ, =0 (δik + λi zk ) ∂zk ϕ , min µi γϕ − (−δik + µi zk ) ∂zk ϕ i

in D = {(z1 , z2 ) : 1 − Now we define

P

Bi =

i

k

i [zi

− `i (zi )] > 0} , where the expression of Ab is omitted.

( (z1 , z2 ) ∈ D : λi γϕ − (

Si =

k

(z1 , z2 ) ∈ D : µi γϕ −

X k

X k

Ni = D ∩ Bic ∩ Sic .

) (δik + λi zk ) ∂zk ϕ = 0 , ) (−δik + µi zk ) ∂zk ϕ = 0 ,

24

XINFU CHEN

MIN DAI z2 1 µ2

2

z

1

N1∩ S2

−µ

2

1

1

z

=0

µ 1−

B ∩S

2

2

−µ

z

1

1+ λ

1

z

2

=0

S ∩S 1

B ∩N 1

2

2

N1∩ N2 S ∩N 1

2

− λ11

1 µ1

B1∩ B2

N1∩ B2

z1

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 2. Shape of trading and no-trading regions with CRRA utility 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 take as an example the region S1 ∩S2 . Recall the proof for the CARA utility case in which 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,

MULTI-ASSET INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

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which implies ∂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 developed in the CARA utility case to obtain the desired result. We point out that most results derived earlier with the CARA utility can be extended to the CRRA utility case.

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