Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem

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1 To appear in Journal of Differential Equations Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem Min Dai Dept of Math, National University of Singapore, Singapore Fahuai Yi Dept of Math, South China Normal University, Guangzhou 5063, China Abstract This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well developed theory of obstacle problem to attack the problem. The C 2, regularity of the value function is proven and the behaviors of the free boundaries are completely characterized. Keywords: optimal investment, transaction costs, finite horizon, double obstacle problem, free boundary, singular stochastic control, portfolio selection Introduction Merton 97 pioneered in applying continuous-time stochastic models to the study of financial markets. In the absence of transaction costs, he showed that an optimal investment problem can be formulated as a Hamilton-Jacobi-Bellman HJB equation that allows an explicit solution for a constant relative risk aversion CRRA investor. The corresponding optimal investment policy is to keep a constant fraction of total wealth in each asset during the whole investment period. To implement the policy, the investor would have to indulge in incessant trading which is completely unrealistic in the face of transaction costs and violates the conventional largely buyand-hold investment strategy as well. To overcome the shortages, Magil and Constantinides 976 introduced proportional transaction costs to Merton s model. They provided a fundamental insight that there is a no-trading region in the presence of transaction costs and the no-trading region must be a wedge. But, their This project is partially supported by Singapore MOE AcRF grant No. R and NUS RMI grant No. R /646, NNSFs of China No and No , NSF of Guangdong Province No and University Special Research Fund for Ph.D. Program The authors thank Lishang Jiang, Zuoquan Xu, Xunyu Zhou and seminar participants at Hong Kong University of Science and Technology, Chinese University of Hong Kong, Jilin University and Quantitative Methods in Finance Conference 2005 Sydney for helpful discussion and comments. All errors are our own. matdm@nus.edu.sg. Fax: fhyi@scnu.edu.cn.

2 2 argument is heuristic at best. In terms of a rigorous mathematical analysis, Davis and Norman 990 showed that for an infinite horizon investment and consumption with transaction costs, the optimal policies are determined by the solution of a free boundary problem, where the free boundaries correspond to the optimal buying and selling policies. Relying on the concept of viscosity solutions to HJB equations, Shreve and Soner 994 fully characterized the infinite horizon optimal policies. Using a martingale approach, Cvitanic and Karatzas 996 proved the existence of an optimal solution to the portfolio optimization problem with transaction costs. Other existence results can be found in Bouchard 2002, Guasoni 2002 and Guasoni and Schachermayer Akian, Menaldi and Sulem 996 and Kabanov and Kluppelberg 2004 considered a multi-asset investment-consumption model with transaction costs. This paper concerns the finite horizon optimal investment with transaction costs, and aims to provide a theoretical analysis of the optimal investment policies. From the angle of stochastic control, it is a singular control problem for the displacement of the state variables due to transaction costs might not be continuous. It can be shown that the value function is governed by a timedependent HJB equation involving gradient constraints which leads to two time-dependent free boundaries. Due to the convexity of utility functions, the value function is expected to be twice continuously differentiable in the spatial direction. Since such regularity of the value function generally leads to free boundary conditions, this regularity property, first observed by Benes et al. 980, is called the principle of smooth fit and plays a critical role in the study of singular control problems. For an infinite horizon problem, Davis and Norman 990 and Shreve and Soner 994 proved the regularity property using an ordinary differential equation approach and the viscosity solution approach, respectively. However, for a finite horizon problem, the resulting free boundaries change through time such that it is hard to follow their approaches. One paper related to the present work is Liu and Loewenstein 2002 where the authors adopted an indirect method. They first considered an optimal investment problem with a stochastic time horizon following Erlang distribution. For this tractable problem, they derived some analytical properties on the optimal investment policies. They then extended those results to the situation of a deterministic time horizon using the fact that the optimal investment policies of the Erlang distributed case converge to those of the deterministic time case. They obtained some interesting characterization of optimal investment policies i.e. free boundaries. However, their results are incomplete and some are not sharp. In addition, their approach cannot be extended to including the consumption term. We will attack the problem directly by virtue of a partial differential equation PDE approach. Our key idea is to establish a link between the singular control problem and the obstacle problem. More precisely, we will show that the spatial partial derivative of the value function is the solution to a double obstacle problem. Since the solution to an obstacle problem is once continuously differentiable in the spatial direction, the above link will immediately yield the desired principle of smooth fit. The link also enables us to make use of the well-developed theory of obstacle problem to study the present problem and the behaviors of the resulting free boundaries can then be completely characterized. The rest of this paper is arranged as follows. In the next section, we present the model formulation. In Section 3, we formally derive the parabolic double obstacle problem regarding the spatial partial derivative of the value function, and study the existence and regularity of solution. Section 4 is devoted to the analysis of the behaviors of the free boundaries i.e. optimal investment policies. The equivalence between the double obstacle problem and the original problem is proven in Section 5. To examine the asymptotic behaviors of the free boundaries as time to maturity goes to infinity, we study the stationary solution to the double obstacle problem in Section 6. The paper ends with conclusive remark in Section 7.

3 3 2 Formulation of the model We take into account the optimal investment problem with transaction costs and a finite horizon T. Except for notational changes, the model formulation is that of Liu and Loewenstein Most notations are from Davis and Norman 990 and Shreve and Soner The asset market Suppose that there are only two assets available for investment: a riskless asset bank account and a risky asset stock. Their prices, denoted by P 0 t and P t, respectively, evolve according to the following equations: dp 0t = rp 0t dt, dp t = P t [αdt + σdb t ], where r > 0 is the constant riskless rate, α > r and σ > 0 are constants called the expected rate of return and the volatility, respectively, of the stock. The process {B t ; t > 0} is a standard Brownian motion on a filtered probability space S, F, {F t } t 0, P with B 0 = 0 almost surely. We assume F = F, the filtration {F t } t 0 is right-continuous and each F t contains all null sets of F. Assume that a CRRA investor holds X t and Y t in bank and stock respectively, expressed in monetary terms. In the presence of transaction costs, the equations describing their evolution are dx t = rx t dt + λdl t + µdm t 2. dy t = αy t dt + σy t db t + dl t dm t, 2.2 where L t and M t are right-continuous with left hand limits, nonnegative, and nondecreasing {F t } t 0 -adapted processes with L 0 = M 0 = 0, representing cumulative dollar values for the purpose of buying and selling stock respectively. The constants λ [0, and µ [0, appearing in these equations account for proportional transaction costs incurred on purchase and sale of stock respectively. 2.2 The investor s problem Due to transaction costs, the investor s net wealth in monetary terms at time t is { Xt + µy W t = t if Y t 0, X t + + λy t if Y t < 0. Since it is required that the investor s net wealth be positive, following Davis and Norman 990, we define the solvency region S = { x, y R 2 : x + + λy > 0, x + µy > 0 }. Assume that the investor is given an initial position in S. An investment strategy L, M is admissible for x, y starting from s [0, T if X t, Y t given by with X s = x and Y s = y is in S for all t [s, T ]. We let A s x, y be the set of admissible investment strategies. The investor s problem is to choose an admissible strategy so as to maximize the expected utility of terminal wealth, that is, sup E x,y 0 [UW T ] L,M A 0 x,y

4 4 subject to Here E x,y t denotes the conditional expectation at time t given that initial endowment X t = x, Y t = y, and the utility function { W γ UW = γ if γ <, γ 0, log W if γ = 0. We define the value function by ϕx, y, t = sup E x,y t [UW T ], x, y S, t [0, T L,M A tx,y 2.3 Merton s result with no transaction costs If λ = µ = 0, the problem reduces to the case of no transaction costs in which the wealth process W t can be chosen as state variable, and Y t be control variable cf. [23]. Consequently, the control problem becomes a classical one and allows an explicit expression of the value function: e ϕx, y, t = α r 2 γr+ 2σ 2 γ T t x+y γ γ if γ <, γ 0, r + α r2 T t + logx + y if γ = 0. 2σ 2 Correspondingly, the optimal policy is to keep a constant proportion of wealth in the the bank account and the stock, namely, Here x M is called Merton line. 2.4 HJB equation x y = α r γσ2 α r = x M. 2.3 Throughout the rest of this paper, we take into account the case of transaction costs, namely, λ + µ > 0. The problem is indeed a singular control problem for the displacement of the state variables X t, Y t due to control effort might be discontinuous. The value function is shown to be the viscosity solution to the following HJB equation cf. [] or [25] with the terminal condition min { ϕ t L ϕ, µϕ x + ϕ y, + λϕ x ϕ y } = 0, x, y S, t [0, T ϕx, y, T = { U x + µy if y > 0, U x + + λy if y 0, where L ϕ = 2 σ2 y 2 ϕ yy + αyϕ y + rxϕ x. The uniqueness of solution of problem is proven by Davis et al. 993 in the sense of viscosity solution. Due to the homotheticity of the utility function, we have for any positive constant ρ, ϕρx, ρy, t = { ρ γ ϕx, y, t if γ <, γ 0, ϕx, y, t + log ρ if γ =

5 5 It is well known that under the assumption α > r, short selling is always suboptimal. Hence, we only need to consider y > 0. Denote V x, t = ϕx,, t. Then, it follows from 2.6 that for y > 0, { y γ V x y, t if γ <, γ 0, ϕx, y, t = V x y, t + log y if γ = 0. Accordingly, are reduced to { min { Vt L V, x + µv x + γv, x + + λv x γv } = 0 in Ω, or V x, T = γ x + µγ, { min { Vt L 2 V, x + µv x +, x + + λv x } = 0 in Ω, V x, T = logx + µ, where Ω = µ, + [0, T, L V = 2 σ2 x 2 V xx + β 2 xv x + β V with β = γ α 2 σ2 γ and β 2 = α r σ 2 γ, and L 2 V = 2 σ2 x 2 V xx α r σ 2 xv x + α 2 σ2. In the following, we will concentrate on the problem 2.7 and the problem A parabolic double obstacle problem if γ <, γ if γ = 0, 2.8 In this section, we will formally derive and study the parabolic double obstacle problem regarding the spatial partial derivative of the value function. 3. Derivation Let us first take into account the case of γ 0, γ <. Eq 2.7 can be rewritten as V t L V = 0 if x++λ < Vx γv < x+ µ V t L V 0 if Vx γv = Vx x++λ or γv = x+ µ 3. V x, T = γ x + µγ Consider the transformation wx, t = logγv. γ Apparently w x = Vx γv. It is easy to see that wx, t satisfies w t L 3 w = 0 if x++λ < w x < x+ µ, w t L 3 w 0 if w x = x++λ or w x = wx, T = log x + µ, x+ µ, 3.2

6 6 where L 3 w = 2 σ2 x 2 w xx + γw 2 x + β2 xw x + γ β = 2 σ2 x 2 w xx + γw 2 x α r σ 2 γ xw x + α 2 σ2 γ. It is worth pointing out that 3.2 reduces to 2.8 when γ = 0. Hence the following arguments cover the case of γ = 0. Set vx, t = w x x, t. Formally we have x L 3w = 2 σ2 x 2 w xxx + 2γw x w xx + σ 2 x w xx + γwx 2 + β2 xw xx + β 2 w x = 2 σ2 x 2 v xx α r 2 γσ 2 xv x α r γσ 2 v + γσ 2 x 2 vv x + xv 2 = Lv. We then postulate that v satisfies the following parabolic double obstacle problem: v t Lv = 0 if x++λ < v < x+ µ, v t Lv 0 if v = x++λ, x+ µ, v t Lv 0 if v = vx, T = x+ µ in Ω. Here x+ µ correspond to lower and upper obstacles, respectively. We stress that v t Lv 0 on the lower obstacle and v t Lv 0 on the upper obstacle, which has a clear physical interpretation cf. [3]. x++λ and Remark It is well known that an optimal stopping time problem can be described as an obstacle problem variational inequality problem cf. [24]. To some extent, our approach is identical to finding a connection between the singular control problem and the optimal stopping time problem pursued by Karatzas and Shreve 984. The proof of the equivalence between the double obstacle problem 3.4 and the original problem 3.2 is deferred to Section 5. Let us first study the double obstacle problem 3.4. For later use, we define { } SR = x, t Ω : vx, t =, BR = NT = x + µ { x, t Ω : vx, t = x + + λ { x, t Ω : }, x + + λ < vx, t < x + µ In finance, the three regions defined above stand for the selling region, buying region and no transaction region, respectively. }.

7 7 3.2 Existence and regularity of solution to problem 3.4 We aim to prove the existence and regularity of solution to the double obstacle problem. One technical difficulty is that the upper obstacle is infinite on the boundary x = µ. To avoid the singularity, we confine ourselves within Ω = {x > x, 0 < t < T }, where x > µ is sufficiently close to µ, and the following boundary condition will be imposed on x = x : vx, t = x, t [0, T µ Later we will see that 3.5 is indeed true because {x x } is contained in SR when x x s, defined in Section 4. Proposition 3. The double obstacle problem 3.4 has a unique solution vx, t W 2, p Ω N \{ x < δ}, for any δ > 0, < p < +, where Ω N is any bounded set in Ω. Moreover, vx, t C NT, 3.6 v t x, t 0, 3.7 and where v0, t = v0, t = µ when α r γ σ2 0, 3.8 { e α r γσ2 T t µ for t < t T, +λ for 0 t t when α r γ σ 2 > 0, 3.9, t = T α r γσ 2 log + λ µ. 3.0 The main difficulty of the proof lies in the degeneracy of the operator L at x = 0. Before providing a proof, we would like to give its sketch. Thanks to the Fichera criterion cf. [0], we are able to deal with the problem in {x < x < 0} and in {x > 0} independently in order to avoid the degeneracy of the operator L at x = 0, and no boundary value is required on x = 0. The standard regularity of solution cf. [3], and 3.7 can be deduced from maximum principle. Regarding 3.6, we only need to show the smoothness on {x = 0} NT. For γ = 0 in which case the operator L is linear, it can be directly obtained by the hypoellipticity of the operator L cf. [22]. In the case of γ < and γ 0, a careful analysis will be made due to the nonlinearity of the operator L. penalty method can be adopted to show the W 2, p 3.3 The proof of Proposition 3. We will only confine our attention to {x < x < 0}, and the case of {x < x < 0} is similar. By transformation x = e y and uy, t = vx, t, 3.4 and 3.5 become u t L y u = 0 if e y ++λ < u < u t L y u 0 if u = e y ++λ, u t L y u 0 if u = e y + µ, uy, T = e y + µ, uy, t = e y + µ, e y + µ, < y < y, 0 t < T, 3.

8 8 where y = log x, and L y u = 2 σ2 u yy α r 32 γσ2 u y α r γσ 2 u γσ 2 e y u u y + u. First we prove the comparison principle for the problem 3.. Lemma 3.2 Let u i, i =, 2, satisfy u it L y u i = 0 if e y ++λ < u i < u it L y u i 0 if u i = e y ++λ, u it L y u i 0 if u i = u i y, T = ψ i y, u i y, t = e y + µ, e y + µ, e y + µ, < y < y, 0 t < T, 3.2 Assume that a ψ i y = e y + µ ; b ψ iy is bounded; c for any N > 0, u i y, t W 2, p N, y [0, T ; d y u is bounded. If ψ y ψ 2 y, then u y, t u 2 y, t. Proof: Denote N = {y, t : u y, t < u 2 y, t, < y < y, 0 t < T }. We use the method of contradiction. Suppose not. Then N must be a nonempty open set, and It follows that u < Let w = u u 2, which satisfies e y + µ, u 2 > e y + + λ in N. u t L y u 0, u 2t L y u 2 0 in N. { wt L y w 0 in N, w = 0 on p N, where p N is the parabolic boundary of N, and L y w = 3 2 σ2 w yy α r 2 γ σ 2 γσ 2 e y [u 2 w y + u y + u + u 2 w]. w y α r γ σ 2 w Since w is bounded and all coefficients in the above equation are bounded as well, applying the maximum principle, we have w 0 in N, namely u u 2 0 in N, which contradicts the definition of N. Since the interval, y is unbounded, we confine our attention to 3. in a finite domain N, y [0, T with N > 0, namely, u N t L y u N = 0 if e y ++λ < un < e y + µ, u N t L y u N 0 if u N = e y ++λ, u N t L y u N 0 if u N = e y + µ, u N y, T = e y + µ, u N y, t = e y + µ, un N, t = e N + µ, where a boundary condition on y = N is imposed. N < y < y, 0 t < T, 3.3

9 9 Lemma 3.3 For any N > 0 given, the problem 3.3 has a solution u N y, t W 2, p N, y [0, T, < p < +, and u N t u N W 2, p Ñ,y [0,T c, 3.5 where Ñ < N and c depends only on Ñ but is independent of N. Moreover u N L y M, 3.6 N,y [0,T where M is independent of N. Proof: Following Friedman 982, we consider a penalty approximation of the problem 3.3: u N,ε t L y u N,ε + β ε u N,ε e y ++λ + γ ε u N,ε e y + µ = 0, u y, T = e y + µ, N < y < y, u N,ε y, t = e y + µ, un,ε N, t = e N + µ, 0 t < T, 3.7 where β ε ξ 0, γ ε ξ 0, β ε ξ = 0 if ξ ε, γ ε ξ = 0 if ξ ε, β ε 0 = c, c > 0, γ ε 0 = c 2, c 2 > 0, β ε ξ 0, γ ε ξ 0, β ε ξ 0, γ ε ξ 0 with constants c and c 2 to be chosen later. For any ε > 0 given, it is not hard to show by the fixed point theorem that the above semi-linear problem has a unique solution u N,ε Wp 2, N, y [0, T. Next we want to prove Let Then Notice = e y + + λ un,ε g y = t L y g y + β ε g y g y = e y + µ in N, y [0, T. 3.8 e y + µ. e y e y + µ 2, g y = e2y + µ e y e y + µ 3. e y + γ + + λ ε g y µ [ α r e y e y + µ 3 + α r γ σ 2 µ ] λ + µ +β ε + γ ε 0 e y + µ e y + + λ µ2 γ σ 2 x + µ 3 + β ε λ + µ e y + µ e y + + λ e y + µ + γ ε 0, in N, y [0, T.

10 0 When ε is sufficiently small, β ε Then e y + µ λ+µ e y + µ e y ++λ is a supersolution, and thus 0. Take c 2 = γ ε 0 = µ2 γ σ 2 x + µ 3. u N,ε e y + µ in N, y [0, T. In the same way, we can choose c = β ε 0 = α r γ σ 2 + λ 2 x + + λ 3 such that e y ++λ is a subsolution. So, we obtain 3.8. Due to 3.8, we get c β ε u N,ε e y 0 and 0 γ + + λ ε u N,ε e y c 2. + µ We then deduce from 3.7 that u N,ε W 2, c, where c is independent of ε because p N,y [0,T both c and c 2 are independent of ε. Using Wp 2, interior estimate, we further have for any Ñ < N, u N,ε W 2, p Ñ,y [0,T c, where c depends on Ñ but is independent of ε and N. Due to 3.8, we infer u N,ε t t=t 0. Differentiating the equation in 3.7 w.r.t. t, we get an equation that u N,ε t satisfies. Applying the maximum principle, we then deduce u N,ε t 0, which gives 3.4 by letting ε 0. Now we prove 3.6. Since the bound of u N,ε and the C 2 norm of terminal value are independent of N and ε, we obtain by the Wp 2, interior estimate u N,ε W 2, p y,y [0,T M, for any y 0, where M is independent of N and ε. Applying the embedding theorem, we have L y,y [0,T M. u N,ε y Since y is arbitrary, it follows u N,ε y L N,y [0,T M, which yields 3.6 by letting ε 0. The proof is complete. In Lemma 3.3, we let N go to +, which immediately leads to the existence of solution to the problem 3.. The uniqueness of solution can be deduced from the comparison principle Lemma 3.2. Therefore, we arrive at the following lemma.

11 Lemma 3.4 The problem 3. has a unique solution uy, t, uy, t Wp 2, N, y [0, T, for any N > 0, u t 0, u y L,y [0,T M. Now we come back to the problem 3.4 with 3.5. By Lemma 3.4, 3.4 with 3.5 has a unique solution vx, t W Ω 2, {x < δ}, for any δ > 0. Similarly we can show that vx, t W 2, p p Ω {x > δ}, for any δ > 0. So, we have the desired W 2, p regularity. 3.7 also follows from Lemma 3.4. In the following, we will prove 3.6. As mentioned earlier, we only need to show the C smoothness on {x = 0} NT if it is non-empty. Let us first prove vx, t C NT {x x 0}. 3.9 Assume that NT {x x 0} is non-empty. It is sufficient to prove that for any compact set E NT {x x 0}, all derivatives of v are bounded on E. Notice that E may contain a part of line x= 0. Let us start from the first equation in 3.. Observe that the bound of the coefficient γσ 2 e y u is independent of y, and so are the bound of uy, t and the C norm of the initial value function. As a result, we infer by the C θ,θ/2 estimate u C θ,θ/2 E y c, where E y denotes the counterpart of E w.r.t. y, and c depends only on the C norm of the initial value function and the bounds of uy, t and of those coefficients appeared in L y. Using the C 2+θ,+θ/2 estimate, we have u C 2+θ,+θ/2 E c. y We can further use the bootstrap argument to get the boundedness of any order partial derivative of uy, t w.r.t. y. We assert that v x = e y u y is bounded. Indeed, denote ũy, t = e y u y, which satisfies ũ t 2 σ2 ũ yy + α r 5 2 γ σ 2 ũ y + 2 α r 3 2 γ σ 2 ũ ũy, T = e y + µ 2. = γσ 2 uu yy + u 2 y + 3uu y + u 2 in E y, Because both the right-hand side terms of the above equation and the terminal value are bounded, we deduce that ũ is bounded, so is v x. Using the same argument, we can show the boundedness of v xx = e 2y u yy u y as well as of any order partial derivatives in x. 3.9 then follows. Due to 3.9, we obtain an ordinary differential inequality by letting x 0 in 3.4 v t 0, t + α r γ σ 2 v0, t = 0 if +λ < v 0, t < µ, v t 0, t + α r γ σ 2 v0, t 0 if v 0, t = +λ, v t 0, t + α r γ σ 2 v0, t 0 if v 0, t = µ, 0 t < T v0, T = µ, t [0, T. In the same way, we can show vx, t C NT {x 0} and obtain the same differential inequality 3.20 by letting x 0 +. This implies the continuity of vx, t across NT {x = 0}.

12 2 Solving 3.20 gives , from which we infer that NT {x = 0} = when α r γσ 2 0, and NT {x = 0} = {x = 0, t < t < T } when α r γσ 2 > 0. Assume that NT {x = 0} = i.e., α r γσ 2 > 0. We differentiate the first equation of 3.4 w.r.t. x and let x 0 + and x 0 respectively. Then we obtain the same ordinary differential equation and terminal condition for v x 0 +, t and v x 0, t in NT {x = 0}, which implies the continuity of v x x, t across NT {x = 0}. Using the same argument we can show that any order derivative of vx, t is continuous across NT {x = 0}. So, vx, t C NT and the proof is complete. 4 Characterization of free boundaries This section is devoted to the theoretical analysis of free boundaries. A double obstacle problem usually gives rise to two free boundaries. We will first show that each free boundary can be expressed as a single-value function of time t. Then we will examine the properties of the free boundaries. To begin with, we introduce a lemma which will play a critical role in the existence proof of free boundaries. Lemma 4. Let vx, t be the solution to the double obstacle problem 3.4. Then v x + v 2 0 in Ω. Proof: It is clear that v x + v 2 = 0 in BR and SR. So, the rest is to show v x + v 2 0 in NT. Denote px, t = v x x, t and qx, t = v 2 x, t. It is not hard to check that and p t 2 σ2 x 2 p xx + α r 3 γ σ 2 xp x + 2α 2r 3 2γ σ 2 p = γσ 2 4xvv x + x 2 v 2 x + x 2 vv xx + v 2 in NT q t 2 σ2 x 2 q xx + α r 2 γ σ 2 xq x + 2α 2r 2 2γ σ 2 q = σ 2 x 2 v 2 x + γσ 2 2x 2 v 2 v x + 2xv 3 in NT. Let Hx, t = v x x, t + v 2 x, t = px, t + qx, t. Then H x, t satisfies namely, H t 2 σ2 x 2 H xx + α r 3 γ σ 2 xh x + 2α 2r 3 2γ σ 2 H = σ 2 x 2 v 2 x σ 2 xq x σ 2 q + γσ 2 2x 2 v 2 v x + 2xv 3 + 4xvv x + x 2 v 2 x + x 2 vv xx + v 2 = σ 2 x 2 v 2 x + 2xvv x + σ 2 v 2 + γσ 2 [ x 2 v v x + v 2 x + 2xv v 2 + v x + xvx + v 2] = γ σ 2 xv x + v 2 + γσ 2 [ x 2 vh x + 2xvH ] in NT, H t 2 σ2 x 2 H xx + α r 3 γ σ 2 γσ 2 xv xh x + 2α 2r 3 2γ σ 2 2γσ 2 xv H = γ σ 2 xv x + v 2 0 in NT. Obviously Hx, t = 0 on the parabolic boundary of NT. Applying the maximum principle yields the desired result.

13 3 Remark 2 Assume that we already have w x = v, then it is easy to verify that Lemma 4. is equivalent to ϕ xx ϕ yy ϕ 2 xy 0, which implies the concavity of the original value function ϕx, y, t in x and y. Note that the concavity is obvious from the definition of ϕx, y, t for the dynamics are linear and the utility function is concave. However, we emphasize that at the moment the relation w x = v has not yet been established, so we cannot utilize the concavity to prove Lemma 4.. Now we can prove the existence of two free boundaries. Theorem 4.2 There are two monotonically increasing functions x s t : [0, T ] [ µ, + ] and x b t : [0, T ] [ µ, + ], such that SR = {x, t Ω : x x s t, t [0, T } and Moreover, BR = {x, t Ω : x x b t, t [0, T }. x s t < x b t for all t [0, T. 4. Proof: Notice v x It follows from the above lemma = v x + x + + λ v 0. x x + + λ x + + λ 2 v x + v 2. As a consequence, if x, t BR, i.e., vx, t = x ++λ, then for any x 2 > x, 0 vx 2, t x λ vx, t x + + λ = 0, from which we infer vx 2, t = x 2 ++λ, i.e., x 2, t BR. This indicates the connection of each t-section of BR. The existence of x b t as a single-value function follows. To prove the existence of x s t, we instead consider vx, t = x + µ 2 vx, t. Notice BR= {x, t Ω : vx, t = x + µ} and [v x + µ] = x x ] [x + µ 2 v x + µ v + x + µ 2 v x + = 2 x + µ x + µ = [x + µ v ] 2 + x + µ 2 v x + v 2 0. The desired result then follows. The monotonicity of x s t and x b t can be similarly deduced by virtue of v = v = v t 0. t x + + λ t x + µ x + µ 2 4. is clear because SR BR=.

14 4 Remark 3 The monotonicity of x s t and x b t indicates that the shorter the maturity, the less the chance of buying risky asset and the more the chance of selling risky asset. This is consistent with the investment criterion that younger investors should allocate a greater share of wealth to stocks than older investors. In finance, x s t and x b t stand for the optimal selling and buying boundaries, respectively. In what follows we study their behaviors. To begin with, we study the selling boundary x s t. Theorem 4.3 Let x s t be the optimal selling boundary in Theorem 4.2. Then i x s t µ x M, 4.2 where x M is defined in 2.3, and ii x s T = lim t T x s t = µx M ; 4.3 x s t 0 when α r γσ 2 = 0, 4.4 x s t > 0 when α r γσ 2 < 0, 4.5 x s t < 0 when α r γσ 2 > 0; 4.6 iii x s t is continuous. Moreover, x s t C [0, T. Proof: For any x, t SR, 0 t L x + µ Then = = L x + µ µ x + µ 3 [ α r x + µ α r γσ 2 ]. 4.7 x α r γσ2 µ = µx M, α r which implies 4.2. To prove the rest part, we will utilize 3.7 many times. Let us first consider 4.3. Suppose not, we then have x s T < µx M. For any x 0 x s T, µx M, applying the equation v t Lv = 0 at t = T gives v t t=t, x=x0 = Lv t=t, x=x0 = L x+ µ < 0, where the last x=x0 strict inequality is due to 4.7. This, however, contradicts is a corollary of 4.2. Now we prove 4.5. Again we use the method of contradiction. Suppose not. Since x s T > 0 when α r γσ 2 < 0, there must be a t < T such that x s t 0 for t t. Then the equation 3.4 on x = 0 reduces to { v t + α r γ σ 2 v x=0 0 for t < t v0, t = x+ µ for t t T. So, v t 0, t α r γ σ 2 v 0, t < 0 for t < t, which also contradicts 3.7. So 4.5 follows.

15 5 Combining 4.2 and 4.5, we get 0 < x s t µ x M = µ α r γσ2 α r when α r γσ 2 < 0. We then obtain 4.4 by letting α r γσ 2 0. At last, let us prove the continuity of x s t. Suppose not, i.e., there is a time t < T at which x s t is discontinuous. Then v x, t = x + µ, x [ x s t, x s t + ] Applying the equation v t Lv = 0 on t = t, x x s t, x s t +, we have v t x, t = L x + µ = µ [ α r x + µ α r γ σ 2 ] x + µ 3 < 0, which is again in contradiction with 3.7. So, we obtain the continuity of x s t. Thanks to 3.7, we can take advantage of the same arguments as in Friedman 975 to obtain x s t C [0, T. The proof is complete. Remark 4 From , we deduce x s t 0 i.e. NT {x > 0} if and only if α r γσ 2 0, which indicates that the leverage is always suboptimal when α r γσ 2 0. This conclusion is the same as in the absence of transaction costs. Now we move on to the buying boundary x b t. Theorem 4.4 Let x b t be the optimal buying boundary in Theorem 4.2. Denote t 0 = T α r log + λ. 4.8 µ Then, i where x M is defined in 2.3, and x b t + λ x M, 4.9 and ii x b t = if and only if t 0 t < T ; 4.0 x b t > 0 when α r γσ x b t > 0 for t t, T, x b t = 0, x b t < 0 for t 0, t when α r γσ 2 > Here t is defined in 3.0. iii x b t is continuous.

16 6 Proof: 4.9 is a counterpart of 4.2 and can be similarly deduced by t L x + + λ +λ [ = α r x + +λ α r γσ 2 ] x + + λ 3 0. Combination of 4., 4.4 and 4.5 yields 4., and 4.2 can be directly inferred from 3.9. The proof of the continuity of x b t is the same as that of x s t. Hence, what remains is to prove 4.0. By the transformation z = the problem 3.4 becomes x x + + λ 2, ṽz, t = vx, t, x + + λ x + + λ + λ ṽ t Lṽ = 0 if 0 < ṽ < ṽ t Lṽ 0 if ṽ = 0, ṽ t Lṽ 0 if ṽ = ṽz, T = λ+µ µ+λ+µz, λ+µ µ+λ+µz, λ+µ µ+λ+µz. µ λ+µ < z <, 0 t < T, 4.3 Here Lṽ = 2 σ2 z 2 z 2 ṽ zz α r 2 γσ 2 + 3σ 2 z z zṽ z It is easy to verify Define Clearly α r γσ 2 2α r 2 γσ 2 z 3σ 2 z 2 ṽ σ 2 z + α r γσ 2 + γσ 2 z[ + zṽ] [z zṽ z + 2zṽ + ]. ṽ z = v x + v 2 2 v x + + λ { z b t = sup z [ λ + µ }, ] : ṽz, t > 0, t [0, T. µ z b t = In order to prove 4.0, it suffices to show x b t x b t + + λ. z b t = if and only if t [t 0, T. Note that at z =, the problem 4.3 is reduced to ṽ t, t α r ṽ, t + α r = 0 if 0 < ṽ, t < λ+µ +λ, ṽ t, t α r ṽ, t + α r 0 if ṽ, t = 0, ṽ t, t α r ṽ, t + α r 0 if ṽ, t = λ+µ +λ, ṽ, T = λ+µ +λ, 0 t < T,

17 7 whose solution is ṽ, t = max α rt t µ e + λ, 0 = { e α rt t µ +λ when t 0 < t T, 0 when 0 t t 0. Combining with 4.4, we immediately obtain z b t = for t t 0, T and z b t < for t [0, t 0. So, we only need to further prove z b t 0 =. Using the strong maximum principle, we can deduce v x x, t 0 + v 2 x, t 0 < 0 for µ < x < + and thus ṽ z z, t 0 < 0 for z <, which implies ṽz, t 0 > 0 for z <. Hence, we must have z b t 0 =. This completes the proof. Remark and 4.4 in Theorem 4.3 and part i in Theorem 4.4 are also obtained by Liu and Loewenstein 2002 in terms of another approach. 5 Equivalence This section is devoted to the equivalence between the double obstacle problem 3.4 and the original problem 3.2. Theorem 5. Let vx, t be the solution to the double-obstacle problem 3.4. Define where At = T t wx, t = At + log x s t + µ + x x st rx 2 + α + r µx + α 2 σ2 γ µ 2 x + µ 2 Then wx, t is the solution to the problem 3.2. Moreover, vξ, tdξ, 5. x=xsτ dτ. 5.2 wx, t C 2, Ω\F, 5.3 where F is the intersection of the free boundaries and the line x = 0, i.e., F = {0, t x s t = 0 or x b t = 0, t [0, T }. Remark 6 We exclude the set F on which some partial derivatives of vx, t or wx, t are discontinuous because of the degeneracy of the differential operator L or L 3. Thanks to Theorem ,, if α r γ σ 2 < 0, F = {x = 0}, if α r γ σ 2 = 0, 0, t, if α r γ σ 2 > 0, where t is defined in 3.0. Proof of Theorem 5.: Since vx, t = x+ µ for x x st, it is not hard to get by 5. wx, t = At + logx + µ, x x s t. 5.4 Clearly wx, t satisfies the terminal condition. Therefore, to prove that wx, t is the solution to the problem 3.2, it suffices to show { wt L 3 w 0 in SR and BR, 5.5 w t L 3 w = 0 in NT.

18 8 Observe According to the definition of At, we claim w x x, t = vx, t. 5.6 w t L 3 w = 0 on x = x s t. 5.7 Indeed, because of 5.6, L 3 w x=xst = 2 σ2 x 2 v x + γv 2 α r σ 2 γ xv + α 2 σ2 γ x=xst = 2 σ2 x s t 2 x s t + µ 2 + γ x s t + µ 2 α r σ 2 γ x s t x s t + µ + α 2 σ2 γ [ = x s t + µ 2 rx s t 2 + α + r µx s t + α ] 2 σ2 γ µ 2 = A t = w t x s t, t, 5.8 where the last equality is due to 5.4. Furthermore, due to 3.3, 5.6 and the fact that vx, t is the solution to the problem 3.4, we have x w t L 3 w 0, w x = x + µ x w t L 3 w = 0 if x s t < x < x b t i.e. in NT, x w t L 3 w 0, w x = x + + λ if x x st i.e. in SR, if x x bt i.e. in BR. Combining with 5.7, we then deduce 5.5. Now we prove 5.3. By Proposition 3., v C,0 Ω\ F and then w C 2,0 Ω\F. What remains is to show w t C Ω\F. Owing to 5., w t x, t = A x t + st x s t + µ v x st, t x st + = A t + = A t maxminx,xb t,x st x st maxminx,xb t,x st x st x x st v t ξ, t dξ = A t v t ξ, t dξ maxminx,xb t,x st x st Lvξ, tdξ dl 3 w = A t + L 3 w xst L 3w maxminx,xb t,x st = L 3 w maxminx,xb t,x st, 5.9 which implies the continuity of w t. The proof is complete. 6 A semi-explicit stationary solution to the obstacle problem In this section, we aim to investigate the asymptotic behavior of x s t and x b t as T t + through the stationary double obstacle problem, which can be written as a stationary free boundary

19 9 problem: Here and clearly Lv = 0, x x s,, x b,, 6. v x s, = x s, + µ, v x s, = x s, + µ 2, 6.2 v x b, = x s, = x b, + + λ, v x b, = x b, + + λ lim x s t and x b, = lim x bt T t + T t + x s, < x b, < 0 when α r γ σ 2 > 0; 0 < x s, < x b, when α r γ σ 2 < 0. We will reduce the equation 6. to a Riccati equation, through which we can find a semiexplicit solution to problem if α r γ σ 2 0. The main result is summarized as follows. Theorem 6. Assume α r γ σ 2 0. The solution of the stationary problem takes the form C x + gx if x s, < x < x b,, v x = x+ µ if x x s,, x++λ if x x b,, where xs, β xs, x s, + µ g x = x C x s, µ C γx s, + γx if β, 6.4 β + β + x x s, + µ x g x = + γx log if β =, 6.5 C x s, µ C x s, x s, = a a + k µ, x b, = a a + k k + λ, 6.6 a = α r γ σ2 2, γσ2 6.7 β = γ a 2γC, 6.8 C = 2 k a 2 2 k a 2 + γ + a + γ + 4 γ k γ a k 2 2, 6.9 and k is the root to the following equation in, 2 when a > 0, or in 2, when a < 0, a + k γ k β+ + β+ k a+c γ a + k β+ + = + λ if β, 6.0 k µ k a+c a + k k a + k exp γ k a + C k a + C = + λ if β =. 6. µ k

20 20 Let us give a remark before the proof. Remark 7 For α r γ σ 2 > 0 i.e. a > 0, using 6.6 and noticing k, 2, we immediately obtain [ γ σ 2 ] x s t 2 α r γ σ 2 µ. 6.2 Liu and Loewenstein 2002 obtained another estimate: [ ] γ σ 2 x s t 2 α r µ, 6.3 which is obviously not sharp because we have shown x s t 0 when α r γ σ 2 0. For α r γ σ 2 > 0, our estimate 6.2 is also better than 6.3. Proof of Theorem 6.. Define Note that Owing to 6., x2 γ x2 f x = γ Lv = 2 v x + γv x 2 + axv x. γ σ2 d dx fx. v x + γv x 2 + axv x = f x s,, x x s,, x b,. 6.4 This is a Riccati equation that we need to solve subject to the free boundary conditions Applying the boundary condition 6.2, we have fx s, = x2 s, γ x s, + µ 2 + γ ax s, x s, + µ 2 + x s, + µ = a + x2 s, + a µ x s, x s, + µ Similarly, the boundary condition 6.3 gives Due to 6.4, f x s, = f x b, or fx b, = a + x2 b, + a + λ x b, x b, + + λ 2. a + x 2 s, + a µ x s, x s, + µ 2 = a + x2 b, + a + λ x b, x b, + + λ To illustrate method, we only consider a > 0 in which case x s, < x b, < 0 and x b, + λx M = a a+2 + λ by Theorem 4.4. So a + x b, + a + λ a + 2x b, + a + λ 0. Then, it follows from 6.6 that a + x s, + a µ 0, namely x s, a µ. a +

21 2 By Theorem 4.3, we know x s, x M µ = a µ. a + 2 Therefore x s, must take the form of x s, = a µ, 6.7 a + k where k [, 2] is to be determined. Substituting this into 6.5, we have fx s, = 2 a + a a+k µ 2 a a a+k µ2 2 a a+k µ + µ k = k 2 a In virtue of 6.6 and 6.8, x b, satisfies the quadratic equation which has two solutions a + x 2 b, + a + λx b, = k k 2 a 2 x b, + + λ 2, a a + k k + λ or a + λ. a + k Because the latter is less than x s, = a a+k µ remember x s, < x b,, we choose the former, namely, a x b, = a + k k + λ. 6.9 Now we come back to 6.4. Due to 6.8, we are able to rewrite 6.4 as x2 γ v + γv 2 + axv = k k 2 a 2, x x s,, x b, This is a Riccati equation which has a special solution C x. Here C satisfies the quadratic algebraic equation γ γ C2 a + C k γ k 2 a 2 = 0, solving which we get 6.9 the smaller root is chosen. Then, the general solution of the Riccati equation 6.20 in x s,, x b, is where g x satisfies v x = C x + gx, 6.2 g x + β g x x γ = 0 or x β g x = γx β Integrating the equation, we obtain xs, β x g x s, γ g x = x β+ x s, gxs, x s, + γ log x x s,, if β =. + γ β+x, if β, 6.23

22 22 Let us focus on β. At x = x b,, β xs, g x b, = g x s, γ β + x s, + γ β + x b,, or equivalently, x b, γ β+ gxs, x s, γ β+ gx b, x b, β+ Thanks to 6.2 and boundary conditions 6.2 and 6.3, we have g x s, x s, = gx b, x b, = = x b, x s, x s, x C, s, + µ 6.25 x b, x b, ++λ C Substitution of 6.25 into 6.23 gives 6.4. Combining with 6.7 and 6.9, we obtain 6.0. When β =, we can obtain 6.5 and 6. using a similar argument. The proof is complete. 7 Conclusion In this paper, we study the optimal investment problem for a CRRA investor who faces a finite horizon and transaction costs. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. Using an elegant transformation, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries which correspond to the optimal buying and selling boundaries, respectively. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C 2, regularity of the value function is proven. Another purpose of the paper is to characterize the free boundaries i.e. optimal investment policies. Relying on the double obstacle problem, the behaviors of the free boundaries can be completely characterized. In addition to the results obtained by Liu and Loewenstein 2002, we show that the free boundaries increase with time and their behaviors depend sensitively on the relative magnitude of α r and γ σ 2. When the maturity goes to infinity, the asymptotic behaviors of the free boundaries are determined by the solution of a Riccati equation with free boundary conditions for which a semi-explicit solution is gained. Our approach can be generalized to a larger class of problems. For example, it can be extended to including the consumption term see Dai et al Also, the approach can be used to deal with the infinite horizon problem discussed by Davis and Normal 990 and Shreve and Soner 994. However, it seems to us that it is not straightforward to extend our approach to the case of multiple risky assets or general utility functions, and more efforts should be made to tackle a general setting. References [] Akian, M., J. L. Menaldi and A. Sulem 996: On an Investment-Consumption Model with Transaction Costs, SIAM Journal of Control and Optimization, 34,

23 23 [2] Benes, V. E., L. A. Shepp and H. S. Witsenhausen 980: Some Solvable Stochastic Control Problems, Stochastics, 4, [3] Bouchard B. 2002: Utility Maximization on the Real Line under Proportional Transaction Costs, Finance and Stochastics, 64, [4] Cvitanic, J., and I. Karatzas 996: Hedging and Portfolio Optimization under Transaction Costs: A Martingale Approach, Mathematical Finance, 62, [5] Dai, M., L.S. Jiang, P.F. Li, and F.H. Yi 2007: Finite Horizon Optimal Investment and Consumption with Transaction Costs, Working Paper, National University of Singapore. [6] Dai, M., and Y. K. Kwok 2005: Optimal Policies of Call with Notice Period Requirement for American Warrants and Convertible Bonds, Asia Pacific Financial Markets, 24: [7] Dai, M., Y. K. Kwok and L.X. Wu 2004: Optimal Shouting Policies of Options with Strike Reset Rights, Mathematical Finance, 43, [8] Davis, M. H. A., and A. R. Norman 990: Portfolio Selection with Transaction Costs, Mathematics of Operations Research, 5, [9] Davis, M. H. A., V. G. Panas, and T. Zariphopoulou 993: European Option Pricing with Transaction Costs, SIAM Journal of Control and Optimization, 32, [0] Fichera, G. 956: Sulle Equazioni Differenziali Lineari Ellittico-Paraboliche Del Secondo Ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I., 5, -30. [] Fleming, W. H., and H. M. Soner 2006: Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer-Verlag, New York. [2] Friedman, A. 975: Parabolic Variational Inequality in One Space Dimension and Smoothness of the Free Boundary, Journal of Functional Analysis, 83, [3] Friedman, A. 982: Variational Principles and Free-Boundary Problems, Wiley, New York. [4] Gennotte, G., and A. Jung 994: Investment Strategies under Transaction Costs: The Finite Horizon Case, Management Science, 40, [5] Guasoni P. 2002: Optimal Investment with Transaction Costs and without Seminartingales, The Annals of Applied Probability, 24, [6] Guasoni P., and W. Schachermayer 2004: Necessary Conditions for the Existence of Utility Maximizing Strategies under Transaction Costs, Statistics and Decisions, 222, [7] Kabanov Y., and C. Kluppelberg 2004: A Geometric Approach to Portfolio Optimization in Models with Transaction Costs, Finance and Stochastics, 8, [8] Karatzas, I., and S.E. Shreve 984: Connections Between Optimal Stopping and Singular Stochastic Control I: Monotone Follower Problems, SIAM J. Control and Optimization, 226, [9] Karatzas, I., and S.E. Shreve 998: Methods of Mathematical Finance. Springer-Verlag, New York.

24 24 [20] Liu, H., and M. Loewenstein 2002: Optimal Portfolio Selection with Transaction Costs and Finite Horizons, Review of Financial Studies, 53, [2] Magill, M. J. P., and G. M. Constantinides 976: Portfolio Selection with Transaction Costs, Journal of Economic Theory, 3, [22] Matsuzawa, T. 973: On Some Degenerate Parabolic Equations II, Nagoya Math. J., 52, [23] Merton, R. C. 97: Optimal Consumption and Portfolio Rules in a Continuous Time Model, Journal of Economic Theory, 3, [24] Shiryaev, A.N. 978: Optimal Stopping Rules. Springer-Verlag, New York. [25] Shreve, S. E., and H. M. Soner 994: Optimal Investment and Consumption with Transaction Costs, Annals of Applied Probability, 4, [26] Soner, H.M. and S.E. Shreve 99: A Free Boundary Problem Related to Singular Stochastic Control: The Parabolic Case, Communication in Partial Differential Equations, 62&3, [27] Yong, J., and X. Zhou 999: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York. [28] Wilmott, P., J. Dewynne, and S. Howison 995: The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press.