An Optimal Consumption-Investment Model with Constraint on Consumption
aa r X i v : . [ q -f i n . P M ] A p r An Optimal Consumption-Investment Model withConstraint on Consumption
Zuo Quan Xu ∗ and Fahuai Yi † November 22, 2013
Abstract
A continuous-time consumption-investment model with constraint is consideredfor a small investor whose decisions are the consumption rate and the allocationof wealth to a risk-free and a risky asset with logarithmic Brownian motion fluc-tuations. The consumption rate is subject to an upper bound constraint whichlinearly depends on the investor’s wealth and bankruptcy is prohibited. The in-vestor’s objective is to maximize total expected discounted utility of consumptionover an infinite trading horizon. It is shown that the value function is (second or-der) smooth everywhere but a unique possibility of (known) exception point andthe optimal consumption-investment strategy is provided in a closed feedback formof wealth, which in contrast to the existing work does not involve the value func-tion. According to this model, an investor should take the same optimal investmentstrategy as in Merton’s model regardless his financial situation. By contrast, theoptimal consumption strategy does depend on the investor’s financial situation: heshould use a similar consumption strategy as in Merton’s model when he is in a badsituation, and consume as much as possible when he is in a good situation.
Keywords:
Optimal consumption-investment model, constrained viscosity solu-tion, free boundary problem, stochastic control in finance, constraint consumption
The publication of the monumental 1952 article
Portfolio Selection and the 1959 bookof the same title by Harry M. Markowitz (1952, 1959) heralded the beginning of modernfinance. To develop a general theory of portfolio choice, Samuelson (1969) and Merton ∗ Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong. This authoracknowledges financial supports from Hong Kong Early Career Scheme (No. 533112), Hong Kong GeneralResearch Fund (No. 529711) and Hong Kong Polytechnic University. Email: [email protected] . † School of Mathematical Sciences, South China Normal University, Guangzhou, China. The projectis supported by NNSF of China(No.11271143 and No.11371155) and University Special Research Fundfor Ph.D. Program of China (20124407110001 and 20114407120008). Email: [email protected] . consumption-investmentstrategy ) are the consumption rate and the allocation of wealth to risk-free and risky as-sets over time. According to Merton (1975), studying this type of problems is the naturalstarting point for the development of a theory of finance.Samuelson and Merton’s pioneering papers prompted researchers to contribute a con-siderable volume of new work on the subject in various directions. The literature hasextensively covered the optimal consumption-investment problems in the financial mar-kets that are subject to constraints and market imperfections. For example, the bookauthored by Sethi (1997) summarized the research conducted by Sethi and his collabo-rators on the optimal consumption-investment problems under various constraints suchas bankruptcy prohibited, subsistence consumption requirement, borrowing prohibited,and random coefficients market. Fleming and Zariphopoulou (1991) considered the opti-mal consumption-investment problem with borrowing constraints. Cvitani and Karatzas(1992, 1993) considered the scenario in which the investment strategy of an investor isrestricted to take values in a given closed convex set. Zariphopoulou (1994) consideredthe problem under the constraint that the amount of money invested in a risky assetmust not exceed an exogenous function of the wealth, and bankruptcy is prohibited atany time. Elie and Touzi (2008) considered the optimal consumption-investment problemwith the constraint that the wealth process never falls below a fixed fraction of its runningmaximum. Davis and Norman (1990), Zariphopoulou (1992), Shreve and Soner (1994),Akian, Menaldi, and Sulem (1996), and Dai and Yi (2009) considered proportional trans-action costs in the study of optimal consumption-investment problems. These optimalconsumption-investment models focus on the constraints on the wealth process and theinvestment strategy.Bardhan (1994) considered the optimal consumption-investment problem with con-straint on the consumption rate and the wealth. The constraint is that the investor mustconsume at a minimal (constant) rate throughout the investment period, which is knownas the subsistence consumption requirement, and must maintain their wealth over a lowboundary at all times. However, in financial practice, an upper boundary constraint onthe consumption rate typically exists in addition to the subsistence consumption require-ment. An example of such scenario is an investment firm with cash flow commitmentsthat is subject to regulatory capital constraints. No study in the extant literature hasconsidered an upper boundary constraint on the consumption rate in the theory of optimalconsumption-investment in intertemporal economies.Harry Markowitz, a Nobel laureate in economics, stated, “It remains to be seen whetherthe introduction of realistic investor constraints is an impenetrable barrier to analysis, ora golden opportunity for someone with a novel approach; and whether progress in thisdirection will come first from discrete or from continuous-time models,” in the forewordof the book by Sethi (1997). Research on the optimal consumption-investment problemthat considers the upper constraint on the consumption rate is scant, although exten-sive research has been conducted on the problem involving other constraints, such as no2ankruptcy or limits on the amount of money borrowed. Consequently, this researchtopic has not been sufficiently explored. This motivated us to investigate the optimalconsumption-investment problems with constraint on consumption rate.In this paper, we consider a continuous-time consumption-investment model with anupper bound constraint on the consumption rate, which linearly depends on the amountof wealth of an investor at any time. The problem is considered in a standard Black-Scholes market with a risk-free and a risky asset over an infinite trading horizon. Wemake the usual assumption that shorting is allowed but bankruptcy is prohibited in themarket. We will primarily use techniques derived from the theories of free boundaryand viscosity solution in the field of differential equations to solve the problem (See e.g.,Crandall and Lions (1983), Lions (1983), Fleming and Soner (1992), Dai, Xu and Zhou(2010), Dai and Xu (2011), Chen and Yi (2012)). As is well-known, the value functionis the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman(HJB) equation. Using this fact, we first prove that the viscosity solution of the equationis smooth everywhere but a unique possibility of (known) exception point. The detaileddescriptions of an unconstrained and a constrained trading regions are then provided.Finally, we derive the optimal consumption-investment strategy in a closed feedback formof wealth. In contrast to the existing models, the optimal strategy explicitly given in ourmodel does not involve the value function. The result shows that an investor should usea similar optimal consumption-investment strategy as in the unconstraint case when hisfinancial situation is bad and should consume at the maximum possible rate when hissituation is good.The paper is organized as follows. We formulate a continuous-time optimal consumption-investment model with constraint on the consumption rate in Section 2. A case withoutconstraint is studies in Section 3. In Section 4, the associated Hamilton-Jacobi-Bellmanequation to the problem is introduced and a case with homogeneous constraint is investi-gated. Using the techniques in the theory of viscosity solution, we show some propertiesof the value function of the problem in Section 5. The descriptions of an unconstrainedand a constrained trading regions are provided in Section 6. Finally, we derive the optimalconsumption-investment strategy in a closed feedback form of wealth in Section 7. Weconclude the paper in Section 8. We consider a standard Black-Scholes financial market with two assets: a bond and astock. The price of the bond is driven by an ordinary differential equation (ODE) d P t = rP t d t, where r is the risk-free interest rate. The price of the stock is driven by a stochasticdifferential equation (SDE): d S t = αS t d t + σS t d W t , α is the mean return rate of the stock, σ is the volatility of the stock, and W ( · ) is a standard one-dimensional Brownian motion on a given complete probability space (Ω , F , P ) . We denote by {F t = σ ( W s , s t ) , t > } the filtration generated by theBrownian motion. The interest rate r , the mean rate of return α , and the volatility σ areassumed to be constant with r > , σ > , and µ := α − r > . There are no transactionfees or taxes and shorting is also allowed in the market.Let us consider a small investor in the market. The investor’s trading will not affectthe market prices of the two assets. His trading strategy is self-financing meaning thatthere is no incoming or outgoing cash flow during the whole ivestment period. Then it iswell-known that the wealth process of the investor is driven by an SDE: ( d X t = ( rX t + π t µ − c t ) d t + π t σ d W t ,X = x, (1)where x > is the initial endowment of the investor, π t is the amount of money investedin the stock at time t , c t > is the consumption rate at time t . In this paper, we assumethat no bankruptcy is allowed, that is X t > , ∀ t > , (2)almost surely (a.s.). The target of the investor is to choose the best consumption-investment strategy ( c ( · ) , π ( · )) , which is subject to certain constraints specified below,to maximize the total expected (discounted) utility from consumption over an infinitetrading horizon maximize E (cid:20)Z ∞ e − βt U ( c t ) d t (cid:21) , (3)where U : R + R + is the utility function of the investor, which is strictly increasing,and β > is a constant discounting factor. In this paper, we consider risk-verse investoronly, this is equivalent to say U ( · ) is concave.The consumption-investment strategy ( c ( · ) , π ( · )) is required to satisfy the followingintegrability constraint E (cid:20)Z T e − βt ( π t + c t ) d t (cid:21) < ∞ , ∀ T > , (4)in which case, SDE (1) admits a unique solution X ( · ) satisfying E (cid:20)Z T e − βt | X t | d t (cid:21) < ∞ , ∀ T > . If no other constraint on the consumption rate and investment strategy exists, problem(3) becomes the classical Merton (1971)’s consumption-investment problem. However, inpractice, constraint on the consumption rate always exists; for example, the consumptionrate cannot be too low because an investor has basic needs, which are the minimal amount4f resources necessary required for long-term physical well-being; this is the so-called thesubsistence consumption requirement. Another practical example is when the managerof a fund requests a fixed salary and a proportion of the managed wealth as a bonus.However, most of the wealth still belongs to the owner, and consequently, the managercannot take excessive amounts from the total wealth. These scenarios motivated us toconsider an upper constraint on the consumption rate.In this paper, specifically, we assume that the consumption rate is upper bounded bya time-invariant linear function of wealth X t at any time: c t kX t + ℓ, t > , (5)where k and ℓ are nonnegative constants, at least one of which is positive.Denote the value function by V ( x ) := sup ( c ( · ) ,π ( · )) E (cid:20)Z ∞ e − βt U ( c t ) d t (cid:21) . (6)where the consumption-investment strategy ( c ( · ) , π ( · )) is subject to the constraints (2),(4) and (5).Same as Merton (1971)’s model, we focus on the constant relative risk aversion (CRRA)type utility function U ( x ) = x p p , x > (7)for some constant < p < . It is well-known that logarithmic utility function can betreated as a limit case of CRRA type utility function as log( x ) = lim p → x p − p , so the resultsof this paper can be extended to cover logarithmic utility function. We first recall the well-known result of Merton (1971) for the scenario without constraint.Define θ := µ σ (1 − p ) > , and κ := β − p ( θ + r )1 − p . Theorem 3.1 If κ > and there is no constraint on the consumption rate, i.e., k = + ∞ or ℓ = + ∞ , then the optimal consumption-investment strategy for problem (6) is given by ( c t , π t ) = (cid:18) κX t , µσ (1 − p ) X t (cid:19) , t > , nd the optimal value is V ∞ ( x ) = 1 p κ p − x p . (8)The optimal value V ∞ ( x ) = p κ p − x p will serve as an upper bound for the optimalvalue in scenarios with constraint. We adopt the viscosity solution approach in differential equations to solve problem (6).Let us start with proving some basic properties of the value function.
Proposition 4.1 If κ > , then the value function V ( · ) of problem (6) satisfies V ( x ) p κ p − x p , x > . (9) Moreover, V ( · ) is continuous, increasing, and concave on [0 , + ∞ ) with V (0) = 0 . Proof . Both the set of admissible controls and the optimal value of problem (6) areincreasing in ℓ and consequently, an upper bound of the optimal value is given by thescenario ℓ = + ∞ . So the inequality (9) follows from (8).If the initial endowment of problem (6) is 0, then the unique admissible consumption-investment strategy is ( c ( · ) , π ( · )) ≡ (0 , , so V (0) = 0 and consequently, V ( · ) is con-tinuous at 0 from the right by (9). By the definition of V ( · ) , it is not hard to proveits the concavity and monotonicity. We leave the details to the interested readers. Thecontinuity of V ( · ) on (0 , + ∞ ) follows from its finiteness and concavity. (cid:3) With this proposition, using the theory of viscosity solution in differential equations(See Crandall and Lions (1983), Lions (1983), Fleming and Soner (1992)), we can provethat
Theorem 4.2 If κ > , then the value function V ( · ) of problem (6) is the unique viscositysolution of its associated HJB equation βV ( x ) − sup π (cid:18) σ π V xx ( x ) + πµV x ( x ) (cid:17) − sup c kx + ℓ (cid:16) U ( c ) − cV x ( x ) (cid:19) − rxV x ( x )= βV ( x ) + µ σ V x ( x ) V xx ( x ) + ( c ( x ) − rx ) V x ( x ) − c p ( x ) p = 0 , x > , (10) in the class of increasing concave functions on [0 , + ∞ ) with V (0) = 0 , where c ( x ) := min n ( V x ( x )) p − , kx + ℓ o , x > . Proof . Standard proof (See e.g., Zariphopoulou (1992, 1994)). We leave the details tothe interested readers. (cid:3) .1 A Case with Homogeneous Constraint We first consider the scenario with a homogeneous constraint on the consumption rate.The results will be useful in studying general scenarios in the following sections.
Theorem 4.3 If k > , ℓ = 0 , and κ > , then the optimal consumption-investmentstrategy for problem (6) is given by ( c t , π t ) = (cid:18) min { κ, k } X t , µσ (1 − p ) X t (cid:19) , t > , (11) and the optimal value is V ( x ) = min { κ, k } p p ( κ (1 − p ) + min { κ, k } p ) x p = ( k p p ( κ (1 − p )+ kp ) x p , k < κ ; p κ p − x p , k > κ. (12) Proof . Suppose κ > . Let V ( · ) defined as in (12). Then c ( x ) = min n ( V x ( x )) p − , kx + ℓ o = min n ( V x ( x )) p − , kx o = min ( min { κ, k } pp − ( κ (1 − p ) + min { κ, k } p ) p − , k ) x = min { κ, k } x, where we used the fact that min { κ, k } pp − ( κ (1 − p ) + min { κ, k } p ) p − > min { κ, k } pp − (min { κ, k } (1 − p ) + min { κ, k } p ) p − = min { κ, k } = k, when k < κ . It is easy to check that V ( · ) and c ( · ) satisfy HJB equation (10). Because V ( · ) is increasing and concave with V (0) = 0 , by Theorem (4.2), V ( · ) is the value function ofproblem (6). It is easy to verify that the value (12) is achieved by taking the consumption-investment strategy (11). (cid:3) Corollary 4.4 If k > κ > and ℓ > , then the optimal consumption-investment strategyfor problem (6) is given by ( c t , π t ) = (cid:18) κX t , µσ (1 − p ) X t (cid:19) , t > , (13) and the optimal value is V ( x ) = 1 p κ p − x p . (14) If κ and ℓ > , then problem (6) is ill-possed, i.e., its optimal value is infinity. roof . Both the set of admissible controls and the optimal value are increasing in ℓ andconsequently, the scenario ℓ = + ∞ gives an upper bound (8), V ( x ) V ∞ ( x ) = p κ p − x p .It is easy to verify that the upper bound p κ p − x p is achieved by taking the consumption-investment strategy (13).If κ goes down to , then the optimal value V ( x ) = p κ p − x p goes to infinity. Becausethe optimal value of problem (6) is decreasing in β , we conclude that V ( x ) = + ∞ if κ and ℓ > . (cid:3) By Theorem 4.3 and Corollary 4.4, we only need to study the scenario κ > k > , ℓ > , which are henceforth assumed unless otherwise specified. Remark 1
We will not study the scenario κ > k = 0 and ℓ > , because it can be treatedeasily by a similar argument as follows. We will address this issue again at the end of thepaper. Theorem 5.1
The value function V ( · ) of problem (6) is in C [0 , + ∞ ) ∩ C (0 , + ∞ ) if r k ; and in C [0 , + ∞ ) ∩ C ((0 , + ∞ ) \{ x e } ) if r > k , where x e := ℓr − k , (15) is the unique possibility of exception point, in which case, V x ( x e − ) ( kx e + ℓ ) p − and V ( x e ) = βp ( kx e + ℓ ) p . Proof . It is proved that V ( · ) ∈ C [0 , + ∞ ) in Proposition 4.1. Note V ( · ) is increasingand concave, so we can define the right and left derivatives as V x ( x ± ) := lim ε → V ( x ± ε ) − V ( x ) ± ε > , for all x > . Moreover, both V x ( ·± ) are decreasing functions and V x ( x +) V x ( x − ) < + ∞ for all x > .Now we show that V ( · ) is continuously differentiable on (0 , + ∞ ) \{ x e } . By Darboux’sTheorem, it is sufficient to show that V ( · ) is differentiable on (0 , + ∞ ) \{ x e } , which isequivalent to V x ( x − ) = V x ( x +) for all positive x = x e .8er absurdum, suppose V x ( x +) < V x ( x − ) for some x > . Let ξ be any numbersatisfying V x ( x +) < ξ < V x ( x − ) . Define φ ( x ) = V ( x ) + ξ ( x − x ) − N ( x − x ) , where N is any large positive number. Then by the concavity of V ( · ) , V ( x ) V ( x ) + V x ( x − )( x − x ) = φ ( x ) + ( V x ( x − ) − ξ )( x − x ) + N ( x − x ) < φ ( x ) , if < x − x < N ( V x ( x − ) − ξ ); and V ( x ) V ( x ) + V x ( x +)( x − x ) = φ ( x ) + ( V x ( x +) − ξ )( x − x ) + N ( x − x ) < φ ( x ) , if < x − x < N ( ξ − V x ( x +)) . Therefore, V ( x ) = φ ( x ) and V ( x ) < φ ( x ) in a neighbourhood of x . By Theorem 4.2, V ( · ) is a viscosity solution of HJB (10), noting φ ( · ) ∈ C (0 , + ∞ ) , so > βφ ( x ) − sup π (cid:0) σ π φ xx ( x ) + πµφ x ( x ) (cid:1) − sup c kx + ℓ ( U ( c ) − cφ x ( x )) − rx φ x ( x )= βV ( x ) − µ ξ σ N − sup c kx + ℓ ( U ( c ) − cξ ) − rx ξ = βV ( x ) − µ ξ σ N − g ( ξ ) , where g ( ξ ) := sup c kx + ℓ ( U ( c ) − cξ ) + rx ξ, < ξ < + ∞ . Letting N → + ∞ , we get g ( ξ ) > βV ( x ) , (16)for all ξ ∈ ( V x ( x +) , V x ( x − )) .On the other hand, because V ( · ) is concave, it is second order differentiable almosteverywhere, and consequently, there exists a sequence { x n : n > } going up to x suchthat V ( · ) is both first and second order differentiable at each x n . By Theorem 4.2, βV ( x n ) − sup π (cid:0) σ π V xx ( x n ) + πµV x ( x n ) (cid:1) − sup c kx n + ℓ ( U ( c ) − cV x ( x n )) − rx n V x ( x n ) βV ( x n ) − sup c kx n + ℓ ( U ( c ) − cV x ( x n )) − rx n V x ( x n )= βV ( x n ) − g ( V x ( x n )) + r ( x − x n ) V x ( x n ) . So g ( V x ( x n )) βV ( x n ) + r ( x − x n ) V x ( x n ) . g ( · ) is convex on (0 , + ∞ ) , so it is continuous on (0 , + ∞ ) . Hence g ( V x ( x − )) = lim n → + ∞ g ( V x ( x n )) lim n → + ∞ ( βV ( x n ) + r ( x − x n ) V x ( x n )) = βV ( x ) . (17)Similarly, we have g ( V x ( x +)) βV ( x ) . (18)Noting that g ( · ) is convex on (0 , + ∞ ) and (16), max { g ( V x ( x − )) , g ( V x ( x +)) } > g ( ξ ) > βV ( x ) , V x ( x +) < ξ < V x ( x − ) . (19)By (17), (18), and (19), we conclude that g ( ξ ) = βV ( x ) for all ξ ∈ [ V x ( x +) , V x ( x − )] .Note g ( ξ ) = sup c kx + ℓ ( U ( c ) − cξ ) + rx ξ = ( U ( kx + ℓ ) − ( kx + ℓ − rx ) ξ, if ξ ( kx + ℓ ) p − ; (cid:16) p − (cid:17) ξ pp − + rx ξ, if ξ > ( kx + ℓ ) p − . Therefore, g ( · ) is a constant on [ V x ( x +) , V x ( x − )] if and only if kx + ℓ − rx = 0 and V x ( x − ) ( kx + ℓ ) p − . It can only happen in the scenario r > k , x = x e and V x ( x e − ) ( kx e + ℓ ) p − , in which case, βV ( x e ) = g ( ξ ) = p ( kx e + ℓ ) p . The proof is com-plete. (cid:3) Although V ( · ) may not be differentiable at x e when r > k , we can still define V x ( x e − ) .From now on, we denote V x ( x e ) := V x ( x e − ) unless otherwise specified. Proposition 6.1
The value function V ( · ) of problem (6) satisfies the following properties:(a). V ( x ) /x p is decreasing, and hence xV x ( x ) pV ( x ) , x > (b). we have k p p ( κ (1 − p ) + kp ) x p V ( x ) p κ p − x p , x > (c). V ( · ) is strictly concave on (0 , + ∞ ) and V x ( · ) is strictly decreasing on (0 , + ∞ ) ; d). we have kκ (1 − p ) + kp κ p − x p − V x ( x ) κ p − x p − , x > . Proof . We first consider the scenario x = x e .(a). Let V ( x, ℓ ) denote the value function V ( x ) with constraint (5). Given the form ofCRRA type utility function (7), the dynamics (1) and constraint (5), a standardargument can show that V ( · , · ) is homogeneous of degree p , i.e., V ( λx, λℓ ) = λ p V ( x, ℓ ) , λ > . Letting λ = x − , V (1 , x − ℓ ) = x − p V ( x, ℓ ) , the property (a) follows from V (1 , x − ℓ ) is decreasing in x .(b). The upper bound is given by (9). The lower bound given by the scenario ℓ = 0 is(12).(c). Note V x ( · ) is decreasing by the concavity of V ( · ) . Suppose it is not strictly decreasing.Then V x ( x ) = A , x ∈ ( x , x ) for some constant A > and ( x , x ) ⊂ (0 , + ∞ ) .It follows that V xx ( x ) = 0 , x ∈ ( x , x ) . If A = 0 , because V ( · ) is concave andincreasing, V x ( x ) = 0 , x ∈ ( x , + ∞ ) which contradicts the property (b). Suppose A > . Applying HJB equation (10), βV ( x ) − sup π (cid:0) σ π V xx ( x ) + πµV x ( x ) (cid:1) − sup c kx + ℓ ( U ( c ) − cV x ( x )) − rxV x ( x ) = 0 ,x < x < x , we get βV ( x ) = sup π ( πµA ) + sup c kx + ℓ ( U ( c ) − cA ) + rxA = + ∞ , x < x < x , which contradicts the property (b).(d). The upper bound follows from the the properties (a) and (b). Note V ( · ) is concaveand apply the property (b), V x ( x ) > V ( x + y ) − V ( x ) y > y (cid:18) k p p ( κ (1 − p ) + kp ) ( x + y ) p − p κ p − x p (cid:19) , x > , y > . y = κ − kk x in the above inequality, V x ( x ) > k ( κ − k ) x (cid:18) k p p ( κ (1 − p ) + kp ) (cid:16) κk (cid:17) p x p − p κ p − x p (cid:19) = k ( κ − k ) x (cid:18) κκ (1 − p ) + kp − (cid:19) p κ p − x p = kκ − k (cid:18) κp − kpκ (1 − p ) + kp (cid:19) p κ p − x p − = kκ (1 − p ) + kp κ p − x p − , x > . Thus the property (d) is proved.For the scenario x = x e , all the properties can be proved by a limit argument. The proofis complete. (cid:3) Define an unconstrained trading region U and a constrained trading region C as follows: U := { x > V x ( x ) p − < kx + ℓ } , C := { x > V x ( x ) p − > kx + ℓ } . One of the main results of this paper is providing detailed descriptions of these two regions.It follows from Theorem (4.2) that βV ( x ) + µ σ V x ( x ) V xx ( x ) − rxV x ( x ) + (cid:16) − p (cid:17) V x ( x ) pp − = 0 , x ∈ U ; (20) βV ( x ) + µ σ V x ( x ) V xx ( x ) + ( kx + ℓ − rx ) V x ( x ) − p ( kx + ℓ ) p = 0 , x ∈ C . (21)Define η := (cid:18) kκ (1 − p ) + kp (cid:19) p − κ > κ. Proposition 6.2
We have (cid:18) ℓκ − k , + ∞ (cid:19) ⊆ C , (22) and (cid:18) , ℓη − k (cid:19) ⊆ U . (23)12 roof . By the property (d) in Proposition 6.1, we have V x ( x ) κ p − x p − , x > , and hence, V x ( x ) p − > κx > kx + ℓ, if x ∈ (cid:18) ℓκ − k , + ∞ (cid:19) , thus (22) follows.Similarly, we have V x ( x ) > kκ (1 − p ) + kp κ p − x p − = ( ηx ) p − , x > , and hence, V x ( x ) p − ηx < kx + ℓ, if x ∈ (cid:18) , ℓη − k (cid:19) , thus (23) follows. (cid:3) Corollary 6.3 If κ > r > k , then x e ∈ C . If r > η , then x e ∈ U . Proof . If κ > r > k , then x e = ℓr − k > ℓκ − k . Similarly, notting η > κ > k , if r > η , then r > k , and x e = ℓr − k < ℓη − k . The claim follows from the above result. (cid:3)
Now we are ready to provide the detailed descriptions of the regions U and C . Theorem 6.4 If κ > k + r , then there exists a constant x ∗ ∈ (cid:20) ℓη − k , ℓκ − k (cid:21) (24) such that U = (0 , x ∗ ) , (25) and C = [ x ∗ , + ∞ ) . (26)13 roof . In order to prove the claim, we first derive the formula of solution in the un-constrained region U , although the problem does not admit a closed form solution on (0 , + ∞ ) .Let Z( · ) : ( c , c )
7→ U be determined by V x (Z( c )) = c p − , c < c < c . (27)By Corollary 6.3, x e / ∈ U , so V x ( · ) is continuous and strictly decreasing on U . Thus Z( · ) is well-defined and strictly increasing. It follows that V xx (Z( c )) Z ′ ( c ) = ( p − c p − , c < c < c . (28)Applying (27) and (28), equation (20) becomes βV (Z( c )) − θc p Z ′ ( c ) − rc p − Z( c ) + p − p c p = 0 , c < c < c . differentiating with respect to c , βV x (Z( c )) Z ′ ( c ) − θ ( c p Z ′′ ( c ) + pc p − Z ′ ( c )) − r ( c p − Z ′ ( c )+ ( p − c p − Z( c )) + ( p − c p − = 0 . Applying (27) again and eliminating c p − , βc Z ′ ( c ) − θ ( c Z ′′ ( c ) + pc Z ′ ( c )) − r ( c Z ′ ( c ) + ( p −
1) Z( c )) + ( p − c = 0 . Now we obtain an ordinary differential equation for Z( · ) : L Z = 0 , c < c < c , (29)where L Z := − θc Z ′′ ( c ) + ( β − θp − r ) c Z ′ ( c ) + r (1 − p ) Z( c ) − (1 − p ) c. Now we are ready to prove (25). Per absurdum, suppose that besides the originalinterval (0 , x ∗ ) , the unconstrained region U contains another bounded interval by (22).That is, there exist x and x such that x ∗ < x < x < + ∞ , ( x , x ) ⊆ U , V x ( x ) p − = kx + ℓ, V x ( x ) p − = kx + ℓ, where the last two identities are from the continuity of V x ( · ) and Corollary 6.3. Let c = Z − ( x ) and c = Z − ( x ) . Then recalling (27), Z( c ) = x = V x ( x ) p − − ℓk = V x (Z( c )) p − − ℓk = c − ℓk > , (30) Z( c ) = x = V x ( x ) p − − ℓk = V x (Z( c )) p − − ℓk = c − ℓk > . (31)14hus c > c > ℓ, if c < c < c . We now confirm c c − ℓk is a supersolution of ODE (29) with boundary conditions (30)and (31). In fact, c c − ℓk satisfies boundary conditions (30) and (31), thus we only needto confirm L (cid:0) c − ℓk (cid:1) > . Note L (cid:18) c − ℓk (cid:19) = ( β − θp − r ) ck + r (1 − p ) c − ℓk − (1 − p ) c = (cid:18) β − θp − r + r (1 − p )1 − p − k (cid:19) (1 − p ) ck − r (1 − p ) ℓk = (cid:18) κ (1 − p )1 − p − k (cid:19) (1 − p ) ck − r (1 − p ) ℓk = ( κ − k ) (1 − p ) ck − r (1 − p ) ℓk > ( κ − k ) (1 − p ) ℓk − r (1 − p ) ℓk > , where we used the assumption κ > k + r in the last inequality. Thus we proved c − ℓk is asupersolution of ODE (29) with boundary conditions (30) and (31). Therefore, Z( c ) c − ℓk , c c c , and consequently, V x (Z( c )) p − = c > k Z( c ) + ℓ that contradicts Z( c ) ∈ U , c < c < c .Thus we proved (25). By the definitions of U and C , (26) follows immediately.The claim (24) follows from (22) and (23). (cid:3) Theorem 7.1
Suppose κ > k + r . If k > r , then V xx ( · ) ∈ C (0 , + ∞ ) . If k < r , then V xx ( · ) ∈ C (cid:0) (0 , + ∞ ) \{ x e } (cid:1) , where x e defined in (15) is the unique possibility of exceptionpoint. Our main idea to prove the above result is to consider the dual function of the valuefunction V ( · ) . Making dual transformation v ( y ) := max x> ( V ( x ) − xy ) , y > . (32)Then v ( · ) is a finite decreasing convex function on (0 , + ∞ ) . Since V x ( · ) is strictly de-creasing, we denote the inverse function of V x ( x ) = y by I ( y ) = x. (33)15y the property (d) in Proposition 6.1, I ( · ) is decreasing and mapping (0 , + ∞ ) to itself.From (32), v ( y ) = [ V ( x ) − xV x ( x )] (cid:12)(cid:12)(cid:12) x = I ( y ) = V ( I ( y )) − yI ( y ) . (34)Differentiating with respect to y , v y ( y ) = V x ( I ( y )) I ′ ( y ) − yI ′ ( y ) − I ( y ) = − I ( y ) , (35) v yy ( y ) = − I ′ ( y ) = − V xx ( I ( y )) , (36)Inserting (35) into (34), V ( I ( y )) = v ( y ) − yv y ( y ) . Making the transformation (33), applying (34), (35), (36), and V x ( x ) = y , HJB equation(10) becomes β ( v ( y ) − yv y ( y )) − µ σ y v yy ( y ) + yd ( y ) + ryv y ( y ) − p d p ( y ) = 0 , y > , (37)where d ( y ) := min n y p − , ℓ − kv y ( y ) o . Equation (37) is quasilinear ODE, which degenerate at y = 0 . It follows that v ( y ) ∈ C (0 , + ∞ ) ∩ C ∞ ((0 , + ∞ ) \{ y ∗ } ) , where y ∗ = V x ( x ∗ ) and x ∗ is defined in Theorem 6.4.Theorem 7.1 will follow from the following two propositions. Proposition 7.2
Suppose κ > k + r . Let x ∗ be defined as in Theorem 6.4. If k > r , then V xx ( · ) ∈ C [ x ∗ + ∞ ) . If k < r , then V xx ( · ) ∈ C (cid:0) [ x ∗ + ∞ ) \{ x e } (cid:1) , where x e defined in (15) is the unique possibility of exception point. Proof . By (36), to prove V xx ( · ) ∈ C (cid:0) [ x ∗ + ∞ ) \{ x e } (cid:1) is equivalent to prove v yy ( y ) > for all y ∈ (0 , y ∗ ] \{ y e } , where y e = V x ( x e − ) .Suppose there exists a point < y < y ∗ such that v yy ( y ) = 0 , which is the minimumvalue of v yy ( · ) by the convexity of v ( · ) . It follows that v yyy ( y ) = 0 . Differentiating (37)with respect to y yields β ( − yv yy ( y )) − µ σ (2 yv yy ( y ) + y v yyy ( y )) + ℓ − kv y ( y ) − kyv yy ( y )+ r ( v y ( y ) + yv yy ( y )) + k ( ℓ − kv y ( y )) p − v yy ( y ) = 0 , < y < y ∗ . (38)16pplying v yy ( y ) = 0 and v yyy ( y ) = 0 , we get ( k − r ) v y ( y ) = ℓ which is equivalent to ( r − k ) x = ℓ where x = I ( y ) . Hence x = x e is the unique possibility of exception pointwhich can only happen in the scenario r > k .It remains to show v yy ( y ∗ ) > . If v yy ( y ∗ ) = 0 which is the minimum value of v yy ( · ) .It follows that v yyy ( y ∗ − ) . By (38), it follows ( k − r ) v y ( y ∗ ) > ℓ which is impossibleif k > r because v ( · ) is decreasing. If r > k , then ( r − k ) x ∗ > ℓ , x ∗ > x e which is alsoimpossible because x e ∈ C by Corollary 6.3. (cid:3) Proposition 7.3
Suppose κ > k + r . Let x ∗ be defined as in Theorem 6.4. Then V xx ( · ) ∈ C (0 , x ∗ ] . Proof . It is proved that v yy ( y ∗ ) > in the proof of Proposition 7.2. Suppose thereexists a point y > y ∗ such that v yy ( y ) = 0 , which is the minimum value of v yy ( · ) by theconvexity of v ( · ) . It follows that v yyy ( y ) = 0 . Differentiating (37) with respect to y yields β ( − yv yy ( y )) − µ σ (2 yv yy ( y ) + y v yyy ( y )) + r ( v y ( y ) + yv yy ( y )) + y p − = 0 , y > y ∗ . Applying v yy ( y ) = 0 and v yyy ( y ) = 0 , we get rv y ( y ) = − y p − which is equivalent to V x ( x ) p − = rx where x = I ( y ) . However, by the property (d) in Proposition (6.1), wehave V x ( x ) p − > κx > rx . The proof is complete. (cid:3) Before proving the global continuity of the first order derivative of the value function,we recall a result in convex analysis.
Lemma 7.4
Let h ( · ) be a finite concave function on (0 , + ∞ ) . Define its convex dual b h ( y ) := max x> ( h ( x ) − xy ) , y > . Let y = inf { y > b h ( y ) < + ∞} . Then the following two statements are equivalent:1. b h ( · ) is strictly convex on ( y , + ∞ ) .2. h ( · ) is continuous differentiable on (0 , + ∞ ) . Proof . ′′ ⇒ ′′ : Per absurdum, suppose h ( · ) is not differentiable at some x > ,then h ( x ) − h ( x ) y ( x − x ) , ∀ x > , for all y ∈ [ h x ( x +) , h x ( x − )] . Then it follows h ( x ) − yx h ( x ) − yx , ∀ x > , and hence b h ( y ) = h ( x ) − yx , y ∈ [ h x ( x +) , h x ( x − )] . This contradicts that b h ( · ) is strictlyconvex. Therefore, h ( · ) is differentiable. Because h x ( · ) is increasing, by the Darboux’sTheorem, h x ( · ) is also continuous. 17 ′ ⇒ ′′ : Because h ( · ) is continuous differentiable on (0 , + ∞ ) , b h ( h ′ ( x )) = h ( x ) − h ′ ( x ) x, x > . For any b > a > y , we need to show b h (( a + b ) / < b h ( a ) + b h ( b ) . Let < x < x suchthat b = h ′ ( x ) > h ′ ( x ) = a . Let y satisfy h ′ ( y ) = ( h ′ ( x ) + h ′ ( x )) = ( a + b ) , then x < y < x . It is sufficient to show b h ( h ′ ( y )) < b h ( h ′ ( x )) + b h ( h ′ ( x )) , i.e., h ( y ) − h ′ ( y ) y < h ( x ) − h ′ ( x ) x + h ( x ) − h ′ ( x ) x . (39)Because h ( · ) is concave, h ( y ) − h ( x ) ( y − x ) h ′ ( x ) ,h ( y ) − h ( x ) ( y − x ) h ′ ( x ) . If both of them are identities, then h ( · ) is linear on [ x , x ] , and h ′ ( x ) = h ′ ( x ) , acontradiction. So h ( y ) − h ( x ) − h ( x ) < y ( h ′ ( x ) + h ′ ( x )) − h ′ ( x ) x − h ′ ( x ) x = 2 h ′ ( y ) y − h ′ ( x ) x − h ′ ( x ) x , which is equivalent to the desired inequality (39). (cid:3) Corollary 7.5
Suppose κ > k + r . The value function V ( · ) of problem (6) is in C [0 , + ∞ ) ∩ C (0 , + ∞ ) . Proof . It is proved that v yy ( y ) > if y = V x ( x e − ) in the proofs of Proportions 7.2 and7.3. This implies that v ( · ) is a strictly convex function on (0 , + ∞ ) . Consequently, V ( · ) is continuous differentiable on (0 , + ∞ ) by Lemma 7.4. (cid:3) To give an explicit optimal consumption-investment strategy for problem (6), we derivethe formula of Z( · ) in the unconstrained region U , although we cannot obtain a closedform solution on (0 , + ∞ ) , but it is adequate for our purpose. Proposition 7.6
Suppose κ > k + r . Let Z( · ) be defined as (27) and x ∗ be defined as inTheorem 6.4, then Z( c ) = 1 κ c − κ (( k − κ ) x ∗ + ℓ ) (cid:18) ckx ∗ + ℓ (cid:19) λ , < c kx ∗ + ℓ. (40)18 roof . Let c ∗ = Z − ( x ∗ ) . Because V x ( · ) is in C (0 , + ∞ ) and (27), c ∗ = V x (Z( c ∗ )) p − = V x ( x ∗ ) p − = kx ∗ + ℓ. (41)The general solution of the corresponding homogeneous equation is Bc λ + Bc λ , where B and B are constants, λ > λ are two roots of function f ( λ ) = θλ ( λ −
1) + ( r − β + pθ ) λ + r ( p − . Note f (1) = − β + p ( θ + r ) < and f (+ ∞ ) = + ∞ . It follows that λ > and λ < . Notea particular solution to the inhomogeneous equation (29) is κ c . Thus the general solutionto equation (29) is given by Z( c ) = 1 κ c − Bc λ − Bc λ , < c c ∗ . Because
Z(0+) = 0 and λ < , we conclude that B = 0 , and hence Z( c ) = 1 κ c − Bc λ , < c c ∗ . By Z( c ∗ ) = x ∗ and (41), we obtain B = κ (( k − κ ) x ∗ + ℓ )( kx ∗ + ℓ ) − λ and Z( c ) = 1 κ c − κ (( k − κ ) x ∗ + ℓ ) (cid:18) ckx ∗ + ℓ (cid:19) λ , < c c ∗ = kx ∗ + ℓ. The proof is complete. (cid:3)
The main result of the paper is stated as follows.
Theorem 7.7
Suppose κ > k + r . Let x ∗ be defined as in Theorem 6.4. The optimalconsumption-investment strategy ( c ∗ ( · ) , π ∗ ( · )) for problem (6) is given by a closed feedbackform of wealth: ( c ∗ t , π ∗ t ) = ( c ∗ ( X t ) , π ∗ ( X t )) , t > , where c ∗ ( x ) = ( Z − ( x ) , < x < x ∗ ; kx + ℓ, x > x ∗ , and π ∗ ( x ) = µσ (1 − p ) x, x > , and Z − ( · ) is the inverse function of Z( · ) defined in (40) . Proof . It is evident that c ∗ ( x ) = ( V x ( x ) p − , < x < x ∗ ; kx + ℓ, x > x ∗ . We only need to show V x ( x ) p − = Z − ( x ) which follows from (27). (cid:3) Concluding Remarks
As mentioned in Remark 1, the scenario κ > k = 0 and ℓ > can be treated by ourargument. In fact, in this scenario, both U and C are clearly intervals as V x ( · ) is decreasing.Moreover, ODE (37) can be solved separately in the two regions. So we will not only havean explicit optimal consumption-investment strategy in a feedback form, but also havean explicit expression of the optimal value. We leave the details to the interested readers.As you may see, the problem is still open in the scenario κ < k + r . We will continuouswork on this scenario and hope to fill the gap in the near future although the scenario isless likely to happen in real financial practice. References [1]
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