Continuous-Time Mean-Variance Portfolio Selection with Constraints on Wealth and Portfolio
CContinuous-Time Mean-Variance Portfolio Selectionwith Constraints on Wealth and Portfolio
Xun Li ∗ and Zuo Quan Xu † July 27, 2015
Abstract
We consider continuous-time mean-variance portfolio selection with bankruptcyprohibition under convex cone portfolio constraints. This is a long-standing anddifficult problem not only because of its theoretical significance, but also for its prac-tical importance. First of all, we transform the above problem into an equivalentmean-variance problem with bankruptcy prohibition without portfolio constraints.The latter is then treated using martingale theory. Our findings indicate that wecan directly present the semi-analytical expressions of the pre-committed efficientmean-variance policy without a viscosity solution technique but within a generalframework of the cone portfolio constraints. The numerical simulation also shedslight on results established in this paper.
Keywords: continuous-time, mean-variance portfolio selection, bankruptcy prohibi-tion, convex cone constraints, efficient frontier, HJB equation
Since Markowitz [17] published his seminal work on the mean-variance portfolio selectionsixty years ago, the mean-risk portfolio selection framework has become one of the most ∗ Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China.This author acknowledges financial supports from Research Grants Council of Hong Kong under grantsNo. 520412 and 15209614. E-mail: [email protected]. † Corresponding author. Department of Applied Mathematics, The Hong Kong Polytechnic University,Hong Kong, China. This author acknowledges financial supports from Hong Kong Early Career Scheme(No. 533112) and Hong Kong General Research Fund (No. 529711). E-mail: [email protected]. a r X i v : . [ q -f i n . P M ] J u l rominent topics in quantitative finance. Recently, there has been increasing interestin studying the dynamic mean-variance portfolio problem with various constraints, aswell as addressing their financial applications. Typical contributions include Bielecki,Jin, Pliska and Zhou [1], Cui, Gao, Li and Li [3], Cui, Li and Li [4], Czichowsky andSchweizer [8], Heunis [11], Hu and Zhou [12], Duffie and Richardson [9],Labbé and Heunis[14], Li and Ng [15], Zhou and Li [22], and Li, Zhou and Lim [16]. The dynamic mean-variance problem can be treated in a forward-looking way by starting with the initialstate. In some financial engineering problems, however, one needs to study stochasticsystems with constrained conditions, such as cone-constrained policies. This naturallyresults in a continuous-time mean-variance portfolio selection problem with constraintsfor the wealth process , (see [1]), and constraints for the policies , (see [12] and [16]), fora given expected terminal target. To the best of our knowledge, despite active researchefforts in this direction in recent years, there has been little progress in the continuous-time mean-variance problem with the mixed restriction of both bankruptcy prohibitionand convex cone portfolio constraints. This paper aims to address this long-standingand notoriously difficult problem, not only for its theoretical significance, but also forits practical importance. New ideas, significantly different from those developed in theexisting literature, establish a general theory for stochastic control problems with mixedconstraints for state and control variables.Li, Zhou and Lim [16] considered a continuous-time mean-variance problem with no-shorting constraints, while Cui, Gao, Li and Li [3] developed its counterpart in discrete-time. Bielecki, Jin, Pliska and Zhou [1] paved the way for investigating continuous-timemean-variance with bankruptcy prohibition using a martingale approach. Sun and Wang[19] introduced a market consisting of a riskless asset and one risky portfolio under con-straints such as market incompleteness, no-shorting, or partial information. Labbé andHeunis [14] employed a duality method to analyze both the mean-variance portfolio selec-tion and mean-variance hedging problems with general convex constraints. In particular,Czichowsky and Schweizer [8] further studied cone-constrained continuous-time mean-variance portfolio selection problem with the price processes being semimartingales. Cui,Gao, Li and Li [3] and Cui, Li and Li [4] derived explicit optimal semi-analytical mean-variance policies for discrete-time markets under no-shorting and convex cone constraints,respectively. Also, Föllmer and Schied [10] and Pham and Touzi [18] showed that in aconstrained market, no arbitrage opportunity is equivalent to the existence of a super-martingale measure, under which the discounted wealth process of any admissible policyis a supermartingale, (see Carassus, Pham and Touzi [2] for a situation with upper boundson proportion positions). In particular, Xu and Shreve in their two-part papers [20, 21]investigated a utility maximization problem with no-shorting constraints using dualityanalysis. Recently, Heunis [11] carefully considered to minimize the expected value of2 general quadratic loss function of the wealth in a more general setting where there isa specified convex constraint on the portfolio over the trading interval, together with aspecified almost-sure lower-bound on the wealth at close of trade.The existing theory and methods cannot directly handle the continuous-time mean-variance problem with the mixed restriction of both bankruptcy prohibition and convexcone portfolio constraints. Therefore, we introduce a Hamilton-Jacobi-Bellman (HJB)equation to analyze this problem. Based on our analysis, we find out that the marketprice of risk in policy is actually independent of the wealth process. This important findingallows us to overcome the difficulty of the original problem and also makes the similarcontinuous-time financial investment problem both interesting and practical. Hence, wecan construct a transformation to tackle the model presented above using linear-quadraticconvex optimization technique. Finally, we discuss the equivalent problem using the modelof Bielecki, Jin, Pliska and Zhou [1] without the viscosity solution technique developed in[16].This paper is organized as follows. In Section 2, we formulate the continuous-timemean-variance problem with the mixed restriction of a bankruptcy prohibition and con-vex cone portfolio constraints. In Section 3, we transform our mean-variance probleminto an equivalent mean-variance problem with a bankruptcy prohibition without convexcone portfolio constraints. We derive the pre-committed continuous-time efficient mean-variance policy for our problem using the model proposed by Bielecki, Jin, Pliska and Zhou[1] in Section 4. In Section 5, we discuss properties of the continuous-time mean-varianceproblems with different constraints. In Section 6, we present a numerical simulation toillustrate results established in this paper. Finally, we summarize and conclude our workin Section 7. We make use of the following notation: z + : the transformation of vector z with every component z + i = max { z i , } ; z − : the transformation of vector z with every component z − i = max {− z i , } ; M (cid:48) : the transpose of any matrix or vector M ; (cid:107) M (cid:107) : (cid:113)(cid:80) i,j m ij for any matrix or vector M = ( m ij ) ; R m : m dimensional real Euclidean space; R m + : the subset of R m consisting of elements with nonnegative components.3he underlying uncertainty is generated on a fixed filtered complete probability space (Ω , F , P , {F t } t (cid:62) ) on which is defined a standard {F t } t (cid:62) -adapted m -dimensional Brow-nian motion W ( · ) ≡ ( W ( · ) , · · · , W m ( · )) (cid:48) . Given a probability space (Ω , F , P ) with afiltration {F t | a (cid:54) t (cid:54) b } ( −∞ < a < b (cid:54) + ∞ ) and a Hilbert space H with the norm (cid:107) · (cid:107) H , we can define a Banach space L F ( a, b ; H ) = (cid:40) ϕ ( · ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( · ) is an F t -adapted, H -valued measurableprocess on [ a, b ] and (cid:107) ϕ ( · ) (cid:107) F < + ∞ (cid:41) with the norm (cid:107) ϕ ( · ) (cid:107) F = (cid:18) E (cid:20)(cid:90) ba (cid:107) ϕ ( t, ω ) (cid:107) H d t (cid:21)(cid:19) . Consider an arbitrage-free financial market where m + 1 assets are traded continuouslyon a finite horizon [0 , T ] . One asset is a bond , whose price S ( t ) , t (cid:62) , evolves accordingto the differential equation (cid:40) d S ( t ) = r ( t ) S ( t ) d t, t ∈ [0 , T ] ,S (0) = s > , where r ( t ) is the interest rate of the bond at time t . The remaining m assets are stocks ,and their prices are modeled by the system of stochastic differential equations (cid:40) d S i ( t ) = S i ( t ) { b i ( t ) d t + (cid:80) mj =1 σ ij ( t ) d W j ( t ) } , t ∈ [0 , T ] ,S i (0) = s i > , where b i ( t ) is the appreciation rate of the i -th stock and σ ij ( t ) is the volatility coefficient attime t . Denote b ( t ) := ( b ( t ) , · · · , b m ( t )) (cid:48) and σ ( t ) := ( σ ij ( t )) . We assume throughout that r ( t ) , b ( t ) and σ ( t ) are given deterministic, measurable, and uniformly bounded functionson [0 , T ] . In addition, we assume that the non-degeneracy condition on σ ( · ) , that is, y (cid:48) σ ( t ) σ ( t ) (cid:48) y (cid:62) δy (cid:48) y, ∀ ( t, y ) ∈ [0 , T ] × R m , (1)is satisfied for some scalar δ > . Also, we define the excess return vector B ( t ) = b ( t ) − r ( t ) , where = (1 , , · · · , (cid:48) is an m -dimensional vector.Suppose an agent has an initial wealth x > and the total wealth of his position attime t (cid:62) is X ( t ) . Denote by π i ( t ) , i = 1 , · · · , m, the total market value of the agent’s4ealth in the i th stock at time t . We call π ( · ) := ( π ( · ) , · · · , π m ( · )) (cid:48) ∈ L F (0 , T ; R m ) aportfolio. We will consider self-financing portfolios in this paper. Then it is well-knownthat X ( t ) , t (cid:62) , follows (see, e.g., [22]) (cid:40) d X ( t ) = [ r ( t ) X ( t ) + π ( t ) (cid:48) B ( t )] d t + π ( t ) (cid:48) σ ( t ) d W ( t ) ,X (0) = x . (2)An important restriction considered in this paper is the convex cone portfolio con-straints, that is π ( · ) ∈ C , where C = (cid:8) π ( · ) : π ( · ) ∈ L F (0 , T ; R m ) , C ( t ) (cid:48) π ( t ) ∈ R k + , ∀ t ∈ [0 , T ] (cid:9) , (3)and C ( · ) : [0 , T ] (cid:55)→ R m × k is a given deterministic and measurable function. Anotherimportant yet relevant restriction considered in this paper is the prohibition of bankruptcy,i.e., we required that X ( t ) (cid:62) , ∀ t ∈ [0 , T ] . (4)On the other hand, borrowing from the money market (at the interest rate r ( · ) ) is stillallowed; that is, the money invested in the bond π ( · ) = X ( · ) − (cid:80) mi =1 π m ( · ) has noconstraint. Definition 1
A portfolio π ( · ) is called an admissible control (or portfolio) if π ( · ) ∈ C andthe corresponding wealth process X ( · ) defined in (2) satisfies (4) . In this case, the process X ( · ) is called an admissible wealth process, and ( X ( · ) , π ( · )) is called an admissible pair. Remark 1
In view of the boundedness of σ ( · ) and the non-degeneracy condition (1) , wehave that π ( · ) ∈ L F ( a, b ; R m ) if and only if σ ( · ) (cid:48) π ( · ) ∈ L F ( a, b ; R m ) . The latter is oftenused to define the admissible process in the literature, for instance, Bielecki, Jin, Pliskaand Zhou [1]. Remark 2
It is easy to show that both the set of all admissible controls and the set of alladmissible wealth processes are convex. In particular, the set of expected terminal wealths (cid:8) E [ X ( T )] : X ( · ) is an admissible process (cid:9) is an interval. Mean-variance portfolio selection refers to the problem of, given a favorable meanlevel d , finding an allowable investment policy, (i.e., a dynamic portfolio satisfying all the5onstraints), such that the expected terminal wealth E [ X ( T )] is d while the risk measuredby the variance of the terminal wealth Var ( X ( T )) = E [ X ( T ) − E [ X ( T )]] = E [ X ( T ) − d ] is minimized.We impose throughout this paper the following assumption: Assumption 1
The value of the expected terminal wealth d satisfies d (cid:62) x e (cid:82) T r ( s ) d s . Remark 3
Assumption 1 states that the investor’s expected terminal wealth d should beno less than x e (cid:82) T r ( s ) ds which coincides with the amount that he/she would earn if all ofthe initial wealth is invested in the bond for the entire investment period. Clearly, this isa reasonable assumption, for the solution of the problem under d < x e (cid:82) T r ( s ) ds is foolishfor rational investors. Definition 2
The mean-variance portfolio selection problem is formulated as the follow-ing optimization problem parameterized by d : min Var ( X ( T )) = E [ X ( T ) − d ] , subject to E [ X ( T )] = d,π ( · ) ∈ C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies equation (2) . (5) An optimal control satisfying (5) is called an efficient strategy, and ( Var ( X ( T )) , d ) , where Var ( X ( T )) is the optimal value of (5) corresponding to d , is called an efficient point. Theset of all efficient points, when the parameter d runs over all possible values, is called theefficient frontier. In the current setting, the admissible controls belong to a convex cone, so the expectationof the final outcome may not be arbitrary. Denote by V ( d ) the optimal value of problem(5). Denote (cid:98) d = sup (cid:8) E [ X ( T )] : X ( · ) is an admissible process (cid:9) . (6)Taking π ( · ) ≡ , we see that X ( t ) = x e (cid:82) t r ( s ) d s is an admissible process, so (cid:98) d (cid:62) E [ X ( T )] = x e (cid:82) T r ( s ) d s . The following nontrivial example shows that it is possible that (cid:98) d = x e (cid:82) T r ( s ) d s . 6 xample 1 Let B ( t ) = − C ( t ) χ , where χ is any positive vector of appropriate dimension.Then for any admissible control π ( · ) ∈ C , we have π ( · ) (cid:48) B ( · ) = − π ( · ) (cid:48) C ( · ) χ (cid:54) . Therefore,by (2) , d ( E [ X ( t )]) = ( r ( t ) E [ X ( t )] + E [ π ( t ) (cid:48) B ( t )]) d t (cid:54) r ( t ) E [ X ( t )] d t, which implies E [ X ( T )] (cid:54) x e (cid:82) T r ( s ) d s . Hence (cid:98) d = x e (cid:82) T r ( s ) d s . Theorem 1
Assume that (cid:98) d = x e (cid:82) T r ( s ) d s . Then the optimal value of problem (5) is 0. Proof . From Assumption 1 and with (cid:98) d = x e (cid:82) T r ( s ) d s , we obtain that the only possiblevalue of d is x e (cid:82) T r ( s ) d s . Note that ( X ( t ) , π ( t )) ≡ ( x e (cid:82) t r ( s ) d s , is an admissible pairsatisfying the constraint of problem (5), so V ( d ) (cid:54) E [ X ( T ) − d ] = E [ x e (cid:82) T r ( s ) d s − d ] = 0 .The proof is complete. (cid:3) From now on we assume (cid:98) d > x e (cid:82) T r ( s ) d s . Denote D = (0 , (cid:98) d ) and D + = (cid:2) x e (cid:82) T r ( s ) d s , (cid:98) d (cid:1) .In this case both D and D + are nonempty intervals. Lemma 1
The value function V ( · ) is convex on D and strictly increasing on D + . Proof . Let ( X ( · ) , π ( · )) and ( (cid:101) X ( · ) , (cid:101) π ( · )) be any two admissible pairs such that d = E [ X ( T )] and d = E [ (cid:101) X ( T )] are different and both in D . For any < α < , define ( (cid:98) X ( · ) , (cid:98) π ( · )) = (cid:0) αX ( · ) + (1 − α ) (cid:101) X ( · ) , απ ( · ) + (1 − α ) (cid:101) π ( · ) (cid:1) . Then ( (cid:98) X ( · ) , (cid:98) π ( · )) satisfies (2)such that (cid:98) π ( · ) ∈ C , (cid:98) X ( · ) (cid:62) and E [ (cid:98) X ( T )] = αd + (1 − α ) d ∈ D , that is, ( (cid:98) X ( · ) , (cid:98) π ( · )) isan admissible pair. Therefore, V ( αd + (1 − α ) d ) (cid:54) Var ( (cid:98) X ( T )) = Var ( αX ( · ) + (1 − α ) (cid:101) X ( · )) (cid:54) α Var ( X ( T )) + (1 − α ) Var ( (cid:101) X ( T )) , where we used the convexity of square function to get the last inequality. Because ( X ( · ) , π ( · )) and ( (cid:101) X ( · ) , (cid:101) π ( · )) are arbitrary chosen, we conclude that V ( αd + (1 − α ) d ) (cid:54) αV ( d ) + (1 − α ) V ( d ) . This establishes the convexity of V ( · ) .Taking π ( · ) ≡ , we see that X ( T ) = x e (cid:82) T r ( s ) d s . This clearly implies that V (cid:16) x e (cid:82) T r ( s ) d s (cid:17) = 0 . It is known that if there are no portfolio constraints (i.e. C ( t ) = 0 for all t ∈ [0 , T ] ), thenthe optimal value is positive on D + (see [1]), so V ( · ) must be positive on D + . Using theconvexity, we conclude that V ( · ) is strictly increasing on D + . (cid:3) orollary 1 The value function V ( · ) is finite and continuous on D . Since problem (5) is a convex optimization problem, the mean constraint E [ X ( T )] = d can be dealt with by introducing a Lagrange multiplier. As well-known, then mean-variance portfolio selection problem (5) is meaningful only when d ∈ D + . We will focuson this case.Because V ( · ) is convex on D and strictly increasing at any d ∈ (cid:0) x e (cid:82) T r ( s ) d s , (cid:98) d (cid:1) , thereis a constant λ > such that V ( x ) − λx (cid:62) V ( d ) − λd for all x ∈ D , where the factor in front of the multiplier λ is introduced just for convenience. In this way the portfolioselection problem (5) is equivalent to the following problem min E [ X ( T ) − d ] − λ ( E [ X ( T )] − d ) , subject to (cid:40) π ( · ) ∈ C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies equation (2) , (7)or equivalently, min E [ X ( T ) − ( d + λ )] , subject to (cid:40) π ( · ) ∈ C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies equation (2)in the sense that these problems have exactly the same optimal pair if one of them admitsone.We plan to use dynamic programming to study the aforementioned problems, so wedenote by ˆ V ( t, x ) the optimal value of problem min E [ X ( T ) − ( d + λ ) | F t , X ( t ) = x ] , subject to (cid:40) π ( · ) ∈ C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies equation (2) (8) Lemma 2
The function (cid:98) V ( t, · ) is strictly decreasing and convex on (cid:0) , ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:3) for each fixed t ∈ [0 , T ] . Proof . The proof is similar to that of Lemma 1. We leave the proof for the interestedreaders. (cid:3)
Remark 4
If the initial wealth X ( t ) = x is too big, then as well-known the mean-varianceportfolio selection problem (8) is not meaningful. This make us focus on the small initialsin (cid:0) , ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:3) . emma 3 If X ( · ) is a feasible wealth process with X ( t ) = 0 for some t ∈ [0 , T ] , then X ( s ) = 0 for all s ∈ [ t, T ] . Proof . Since X ( · ) is a feasible wealth process, we have X ( s ) (cid:62) , for all s ∈ [ t, T ] .If P ( X ( s ) > is positive for some s ∈ [ t, T ] , then this would lead to an arbitrageopportunity. (cid:3) Lemma 4
We have that (cid:98) V ( t,
0) = ( d + λ ) and (cid:98) V (cid:16) t, ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:17) = 0 for all t ∈ [0 , T ] . Proof . If X ( t ) = 0 , then X ( T ) = 0 by Lemma 3. Hence, (cid:98) V ( t,
0) = ( d + λ ) .Suppose X ( t ) = ( d + λ ) e − (cid:82) Tt r ( s ) d s . Then taking π ( · ) ≡ , we obtain that X ( T ) = d + λ ,so (cid:98) V (cid:16) t, ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:17) (cid:54) E [ X ( T ) − ( d + λ )] = 0 . The proof is complete. (cid:3) Since the Riccati equation approach to solve problem (8) is not applicable in this case,we consider the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This is thefollowing partial differential equation: L v = 0 , ( t, x ) ∈ S ,v (cid:16) t, ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:17) = 0 , v ( t,
0) = ( d + λ ) , (cid:54) t (cid:54) T,v ( T, x ) = (cid:0) x − ( d + λ ) (cid:1) , < x < d + λ, (9)where L v = v t ( t, x ) + inf π ∈C t (cid:110) v x ( t, x )[ r ( t ) x + π (cid:48) B ( t )] + 12 v xx ( t, x ) π (cid:48) σ ( t ) σ ( t ) (cid:48) π (cid:111) , S = (cid:110) ( t, x ) : 0 (cid:54) t < T, < x < ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:111) , and C t = { z ∈ R m : C ( t ) (cid:48) z ∈ R k + } . We need the following technical result.
Lemma 5
Suppose problem (9) admits a solution v ∈ C , ( S ) which is convex in thesecond argument. Then v (cid:54) ( d + λ ) on S . roof . By the convexity of v in the second argument, we have, for each ( t, x ) ∈ S , v ( t, x ) (cid:54) max (cid:110) v ( t, , v (cid:16) t, ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:17)(cid:111) = ( d + λ ) . The proof is complete. (cid:3)
Now we are ready to establish the following result:
Theorem 2
Suppose problem (9) admits a solution v ∈ C , ( S ) which is convex in thesecond argument. Then (cid:98) V = v on S . Proof . Without loss of generality, we shall show (cid:98) V (0 , x ) = v (0 , x ) .Let ( X ( · ) , π ( · )) be the corresponding optimal pair. Define τ = inf (cid:110) t ∈ [0 , T ] : X ( t ) = 0 or X ( t ) = ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:111) ∧ T, (10) τ N = sup (cid:26) t ∈ [0 , T ] : (cid:90) t (cid:107) v x ( s, X ( s )) π ( s ) (cid:48) σ ( s ) (cid:107) d s (cid:54) N (cid:27) ∧ T. (11)Applying Itô’s Lemma to v ( t, X t ) yields v ( τ ∧ τ N , X ( τ ∧ τ N ))= (cid:90) τ ∧ τ N (cid:18) v t ( t, X ( t )) + v x ( t, X ( t ))[ r ( t ) X ( t ) + π ( t ) (cid:48) B ( t )] + 12 v xx ( t, X ( t )) π ( t ) (cid:48) σ ( t ) σ ( t ) (cid:48) π ( t ) (cid:19) d t + (cid:90) τ ∧ τ N v x ( t, X ( t )) π ( t ) (cid:48) σ ( t ) d W ( t ) + v (0 , x ) (cid:62) (cid:90) τ ∧ τ N L v ( t, X ( t )) d t + (cid:90) τ ∧ τ N v x ( t, X ( t )) π ( t ) (cid:48) σ ( t ) d W ( t ) + v (0 , x )= (cid:90) τ ∧ τ N v x ( t, X ( t )) π ( t ) (cid:48) σ ( t ) d W ( t ) + v (0 , x ) . Taking expectation of both sides, we obtain E [ v ( τ ∧ τ N , X ( τ ∧ τ N ))] (cid:62) E (cid:20)(cid:90) τ ∧ τ N v x ( t, X ( t )) π ( t ) (cid:48) σ ( t ) d W ( t ) + v (0 , x ) (cid:21) = v (0 , x ) . Because v is continuous, and τ ∧ τ N and X ( τ ∧ τ N ) are both uniformly bounded, letting N → ∞ and applying the dominated convergence theorem, we obtain E [ v ( τ, X ( τ ))] (cid:62) v (0 , x ) . (12)If X ( τ ) = 0 , then by Lemma 3, we have X ( T ) = 0 . Using Lemma 5 yields E [ v ( T, X ( T )) |F τ ] { X ( τ )=0 } = ( d + λ ) { X ( τ )=0 } (cid:62) v ( τ, X ( τ )) { X ( τ )=0 } . (13)10f X ( τ ) = ( d + λ ) e − (cid:82) Tτ r ( s ) d s , then v ( τ, X ( τ )) = 0 . This trivially leads to E [ v ( T, X ( T )) |F τ ] { X ( τ )=( d + λ ) e − (cid:82) Tτ r ( s ) d s } (cid:62) v ( τ, X ( τ )) { X ( τ )=( d + λ ) e − (cid:82) Tτ r ( s ) d s } . (14)If < X ( τ ) < ( d + λ ) e − (cid:82) Tτ r ( s ) d s , then τ = T by the definition. Hence, E [ v ( T, X ( T )) |F τ ] { Lemma 6 Suppose A ∈ R m × k , B ∈ R m , D ∈ R m × m and D D (cid:48) is positive definite, and C = { z ∈ R m : A (cid:48) z ∈ R k + } . Then, for each fixed scalar α > , the following two convexoptimization problems min z ∈ C z (cid:48) D D (cid:48) z − α B (cid:48) z (16) and min z ∈ R m z (cid:48) D D (cid:48) z − α ¯ z (cid:48) D D (cid:48) z (17) have the same optimal solution α ¯ z and the same optimal value − α ¯ z D (cid:48) D ¯ z , where ¯ z = argmin z ∈ C (cid:107) D (cid:48) z − D − B (cid:107) . (18)11 roof . Because C is a cone, it is sufficient to study the case α = 1 . From the definition, ¯ z solves min z ∈ C z (cid:48) D D (cid:48) z − B (cid:48) z. (19)By the definition of C , the above problem is equivalent to min z ∈ R m , A (cid:48) z ∈ R k + z (cid:48) D D (cid:48) z − B (cid:48) z. Introducing a Lagrange multiplier yields the following unconstrained problem min z ∈ R m z (cid:48) D D (cid:48) z − B (cid:48) z − ν (cid:48) A (cid:48) z. (20)Problems (19) and (20) should have the same unique solution ¯ z and optimal value, forsome ν ∈ R k . Since ( D D (cid:48) ) − ( B + A ν ) is the unique solution to problem (20), we have ¯ z = ( D D (cid:48) ) − ( B + A ν ) and D D (cid:48) ¯ z = B + A ν . Therefore, the optimal solution and theoptimal value of the above problem (20) are the same as those of the following uncon-strained problem min z ∈ R m z (cid:48) D D (cid:48) z − ¯ z (cid:48) D D (cid:48) z. (21)Note that α = 1 , so the above problem is actually the same as problem (17). The proofis complete. (cid:3) Remark 5 By Lemma 2, we know that the solution (cid:98) V ( t, · ) to problem (9) is strictlydecreasing and convex on (cid:0) , ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:3) for t ∈ [0 , T ] . Therefore, v x ( t, x ) < and v xx ( t, x ) > for t ∈ [0 , T ] . We now return to the dynamic problem (9). Let ¯ z ( t ) := argmin z ∈C t (cid:107) σ ( t ) (cid:48) z − σ ( t ) − B ( t ) (cid:107) . (22)By Lemma 6 with α = − v x ( t,x ) v xx ( t,x ) > , the infimum in the HJB equation (9) is attained by π = − v x ( t, x ) v xx ( t, x ) ¯ z ( t ) ∈ C t . (23)Moreover, the HJB equation (9) is equivalent to v t ( t, x ) + inf π ∈ R m (cid:110) v x ( t, x )[ r ( t ) x + π (cid:48) (cid:98) B ( t )] + v xx ( t, x ) π (cid:48) σ ( t ) σ ( t ) (cid:48) π (cid:111) = 0 , ( t, x ) ∈ S ,v (cid:16) t, ( d + λ ) e − (cid:82) Tt r ( s ) d s (cid:17) = 0 , v ( t, 0) = ( d + λ ) , (cid:54) t (cid:54) T,v ( T, x ) = (cid:0) x − ( d + λ ) (cid:1) , < x < d + λ, (24)12here (cid:98) B ( t ) := σ ( t ) σ ( t ) (cid:48) ¯ z ( t ) . In fact, the solution to the above HJB equation also is the value function associated withthe following problem min E [ X ( T ) − ( d + λ ) | F t , X ( t ) = x ] , subject to (cid:40) π ( · ) ∈ (cid:98) C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies the following equation (27) , (25)where (cid:98) C = (cid:8) π ( · ) : π ( · ) ∈ L F (0 , T ; R m ) (cid:9) , (26)and (cid:40) d X ( t ) = [ r ( t ) X ( t ) + π ( t ) (cid:48) (cid:98) B ( t )] d t + π ( t ) (cid:48) σ ( t ) d W ( t ) ,X (0) = x . (27)Removing the Lagrange multiplier, the above problem has the same optimal control asthe following mean-variance problem without constraints on the portfolio: min Var ( X ( T )) = E [ X ( T ) − ˜ d ] , subject to E [ X ( T )] = ˜ d,π ( · ) ∈ (cid:98) C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies equation (27) , (28)for some ˜ d . Because the optimal solution to the above problem (28) is also optimal toproblem (5), the mean of the optimal terminal wealth should be the same, that is to say ˜ d = d . Therefore, we conclude that problem (5) and problem min Var ( X ( T )) = E [ X ( T ) − d ] , subject to E [ X ( T )] = d,π ( · ) ∈ (cid:98) C and X ( · ) (cid:62) , ( X ( · ) , π ( · )) satisfies equation (27) , (29)have the same optimal solution.The above mean-variance with bankruptcy prohibition problem was fully solved in[1]. Consequently, so is our problem (5). Moreover, these two problems have the sameefficient frontier.It is interesting to note that the market price of risk (cid:98) θ ( · ) = σ ( · ) − (cid:98) B ( · ) does not dependon the wealth process X ( · ) . This important feature allows us to give a linear feedbackpolicy in X ( · ) at or before the terminal time.13 Optimal Portfolio The result of the martingale pricing theory states that the set of random terminal payoffsthat can be generated by feasible trading strategies corresponds to the set of nonnegative F T -measurable random payoffs X ( T ) which satisfy the budget constraint E [ φ ( T ) X ( T )] (cid:54) x . Therefore, the dynamic problem (29), of choosing an optimal trading strategy π ( · ) , isequivalent to the static problem of choosing an optimal payoff X ( T ) : min Var ( X ( T )) = E [ X ( T ) − d ] , subject to E [ X ( T )] = d, E [ φ ( T ) X ( T )] = x ,X ( T ) (cid:62) , (30)where φ ( · ) is the state price density , or stochastic discount factor , defined by (cid:40) d φ ( t ) = φ ( t ) {− r ( t ) d t − (cid:98) θ ( t ) (cid:48) d W ( t ) } ,φ (0) = 1 , (31)and (cid:98) θ ( t ) = σ ( t ) − (cid:98) B ( t ) = σ ( t ) (cid:48) ¯ z ( t ) . The above static optimization problem (30) was solved in [1]. The optimal random ter-minal payoff is X ∗ ( T ) = ( µ − γφ ( T )) + , (32)where x + = max { x, } , µ > , γ > and ( µ, γ ) ∈ R solves the system of equations (cid:40) E [( µ − γφ ( T )) + ] = d, E [ φ ( T )( µ − γφ ( T )) + ] = x . (33)That is, µN (cid:16) ln (cid:0) µγ (cid:1) + (cid:82) T [ r ( s )+ | (cid:98) θ ( s ) | ] d s √ (cid:82) T | (cid:98) θ ( s ) | ds (cid:17) − γe − (cid:82) T r ( s ) d s N (cid:16) ln (cid:0) µγ (cid:1) + (cid:82) T [ r ( s ) − | (cid:98) θ ( s ) | ] d s √ (cid:82) T | (cid:98) θ ( s ) | ds (cid:17) = d,µN (cid:16) ln (cid:0) µγ (cid:1) + (cid:82) T [ r ( s ) − | (cid:98) θ ( s ) | ] d s √ (cid:82) T | (cid:98) θ ( s ) | ds (cid:17) − γe − (cid:82) T [ r ( s ) −| (cid:98) θ ( s ) | ] d s N (cid:16) ln (cid:0) µγ (cid:1) + (cid:82) T [ r ( s ) − | (cid:98) θ ( s ) | ] d s √ (cid:82) T | (cid:98) θ ( s ) | ds (cid:17) = x e (cid:82) T r ( s ) d s , (34)where N ( y ) = √ π (cid:82) y −∞ e − t d t is the cumulative distribution function of the standardnormal distribution. 14he investor’s optimal wealth is given by the stochastic process X ∗ ( t ) = E (cid:20) φ ( T ) φ ( t ) X ∗ ( T ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = f ( t, φ ( t )) , (35)where f ( t, y ) = µN (cid:0) − d ( t, y ) (cid:1) e − (cid:82) Tt r ( s ) d s − γN (cid:0) − d ( t, y ) (cid:1) ye − (cid:82) Tt [2 r ( s ) −| (cid:98) θ ( s ) | ] d s , (36)and d ( t, y ) := ln (cid:0) γµ y (cid:1) + (cid:82) Tt [ − r ( s )+ | (cid:98) θ ( s ) | ] d s √ (cid:82) Tt | (cid:98) θ ( s ) | ds , d ( t, y ) := d ( t, y ) − (cid:113)(cid:82) Tt | (cid:98) θ ( s ) | d s. Applying Itô’s lemma to f ( · , φ ( · )) yields d X ∗ ( t ) = d f ( t, φ ( t )) = {· · · } d t + γ ˆ θ ( t ) N (cid:0) − d ( t, φ ( t )) (cid:1) φ ( t ) e − (cid:82) Tt [2 r ( s ) −| (cid:98) θ ( s ) | ] ds d W ( t ) . Comparing this to the wealth evolution equation (27), we obtain the efficient portfolio π ∗ ( t ) = γ ( σ ( t ) σ ( t ) (cid:48) ) − (cid:98) B ( t ) N (cid:0) − d ( t, φ ( t )) (cid:1) φ ( t ) e − (cid:82) Tt [2 r ( s ) −| (cid:98) θ ( s ) | ] d s = − ( σ ( t ) σ ( t ) (cid:48) ) − (cid:98) B ( t ) (cid:104) X ∗ ( t ) − µN (cid:0) − d ( t, φ ( t )) (cid:1) e − (cid:82) Tt r ( s ) d s (cid:105) . (37) Remark 6 The above results for the efficient portfolio and the associated wealth processwere first derived in Bielecki et al. [1] and give a complete solution to the mean-varianceportfolio selection problem with bankruptcy prohibition (29). Based on the above analysis, we have the following main theorem. Theorem 3 Assume that (cid:82) T | (cid:98) θ ( s ) | d s > . Then there exists a unique efficient portfoliofor (5) corresponding to any given d (cid:62) x e (cid:82) T r ( s ) d s . Moreover, the efficient portfolio isgiven by (37) and the associated wealth process is expressed by (35). The main result of the constrained mean-variance portfolio selection model with bankruptcyprohibition derived in Section 4 is quite surprising but important. The mean-varianceportfolio selection model, like many other stochastic optimization models, is based onaveraging over all the possible random scenarios. We now discuss the model in terms ofhow its different constraints could guide real investment in practice.15 .1 Bankruptcy Prohibition with Unconstrained Portfolio The mean-variance unconstrained portfolio problem with bankruptcy prohibition is aninteresting but practically relevant model. In this case, k = m and π ( · ) ∈ L F (0 , T ; R m ) .It follows from (22) that ¯ z ( t ) = argmin z ∈ R m (cid:107) σ ( t ) (cid:48) z − σ ( t ) − B ( t ) (cid:48) (cid:107) = ( σ ( t ) σ ( t ) (cid:48) ) − B ( t ) (cid:48) . Therefore, (cid:98) B ( t ) = σ ( t ) σ ( t ) (cid:48) ¯ z ( t ) = B ( t ) . Proposition 1 Assume that (cid:82) T | (cid:98) θ ( s ) | d s > . Then there exists a unique efficient port-folio for this mean-variance model corresponding to any given d (cid:62) x e (cid:82) T r ( s ) d s . Moreover,the efficient portfolio is given by (37) and the associated wealth process is expressed by(35), where (cid:98) B ( t ) = B ( t ) and (cid:98) θ ( t ) = σ ( t ) − B ( t ) . The proof of Proposition 1 can be found in Bielecki et al. [1]. The mean-variance portfolio problem with mixed bankruptcy and no-shorting constraintsis another interesting and challenging model. In this case, k = m and π ( · ) ∈ L F (0 , T ; R m + ) .It follows from (22) that ¯ z ( t ) = argmin z ∈ R m + (cid:107) σ ( t ) (cid:48) z − σ ( t ) − B ( t ) (cid:48) (cid:107) = ( σ ( t ) σ ( t ) (cid:48) ) − ( B ( t ) + λ ( t )) (cid:48) , where λ ( t ) := argmin y ∈ R m + (cid:107) σ ( t ) − y + σ ( t ) − B ( t ) (cid:48) (cid:107) . (38)Therefore, (cid:98) B ( t ) = σ ( t ) σ ( t ) (cid:48) ¯ z ( t ) = B ( t ) + λ ( t ) . Proposition 2 Assume that (cid:82) T | (cid:98) θ ( s ) | d s > . Then there exists a unique efficient port-folio for this mean-variance model corresponding to any given d (cid:62) x e (cid:82) T r ( s ) d s . Moreover,the efficient portfolio is given by (37) and the associated wealth process is expressed by(35), where (cid:98) B ( t ) = B ( t ) + λ ( t ) and (cid:98) θ ( t ) = σ ( t ) − ( B ( t ) + λ ( t )) . .3 No-shorting Constraint without Bankruptcy Prohibition The mean-variance portfolio problem with no-shorting constraints is also an importantmodel in financial investment. In this case, k = m and π ( · ) ∈ L F (0 , T ; R m + ) . We againhave (cid:98) B ( t ) = B ( t ) + λ ( t ) , where λ ( t ) is determined by (38).In particular, d ( t, φ ( t )) = −∞ and d ( t, φ ( t )) = −∞ , that is, N (cid:0) − d ( t, φ ( t )) (cid:1) = N (cid:0) − d ( t, φ ( t )) (cid:1) = 1 . The investor’s optimal wealth is the stochastic process X ∗ ( t ) = µe − (cid:82) Tt r ( s ) d s − γφ ( t ) e − (cid:82) Tt [2 r ( s ) −| (cid:98) θ ( s ) | ] d s (39)and π ∗ ( t ) = − ( σ ( t ) σ ( t ) (cid:48) ) − (cid:98) B ( t ) (cid:104) X ∗ ( t ) − µe − (cid:82) Tt r ( s ) d s (cid:105) , (40)where µ = E [ φ ( T ) ] d − x E [ φ ( T )] Var ( φ ( T )) = d − x e (cid:82) T [ r ( s ) −| (cid:98) θ ( s ) | ] d s − e − (cid:82) T | (cid:98) θ ( s ) | d s , (41) γ = E [ φ ( T )] d − x Var ( φ ( T )) = (cid:0) d − x e (cid:82) T r ( s ) d s (cid:1) e (cid:82) T [ r ( s ) −| (cid:98) θ ( s ) | ] d s − e − (cid:82) T | (cid:98) θ ( s ) | d s . (42)where (cid:98) θ ( t ) = σ ( t ) − ( B ( t ) + λ ( t )) . Proposition 3 Assume that (cid:82) T | (cid:98) θ ( s ) | d s > . Then there exists a unique efficient port-folio for this mean-variance model corresponding to any given d (cid:62) x e (cid:82) T r ( s ) d s . Moreover,the efficient portfolio is given by (39) and the associated wealth process is expressed by(41). Another version of the proof of Proposition 3 can be found in Li, Zhou and Lim [16]. In this section, a numerical example with constant coefficients is presented to demonstratethe results in the previous section. Let m = 3 . The interest rate of the bond and the17ppreciation rate of the m stocks are r = 0 . and ( b , b , b ) (cid:48) = (0 . , . , . (cid:48) ,respectively, and the volatility matrix is σ = . . . − . . . . Then we have σ − = . − . . . − . . and B = ( b − r, b − r, b − r ) (cid:48) = (0 . , . , . (cid:48) . Hence, θ := σ − B = (0 . , . , . (cid:48) . In addition, we suppose that the initial prices of the stocks are ( S (0) , S (0) , S (0)) =(1 , , (cid:48) and the initial wealth is X (0) = 1 . In this subsection, we determine the optimal portfolio and the corresponding wealth pro-cess in Subsection 5.1 for the above market data. According to (34), we obtain thenumerical results µ = 1 . and γ = 0 . . Hence, the wealth process (35) can beexpressed by X ∗ ( t ) = µN (cid:0) − d ( t, φ ( t )) (cid:1) e − r ( T − t ) − γN (cid:0) − d ( t, φ ( t )) (cid:1) φ ( t ) e − [2 r −| ˆ θ | ]( T − t ) , (43)where d ( t, φ ( t )) = ln (cid:0) γµ φ ( t ) (cid:1) +[ − r + | ˆ θ | ]( T − t ) √ | ˆ θ | ( T − t ) , d ( t, φ ( t )) = d ( t, φ ( t )) − (cid:113) | ˆ θ | ( T − t ) ,φ ( t ) = e − [ r + | ˆ θ | ]( T − t ) − ˆ θ ( W ( T ) − W ( t )) , ˆ θ = θ = σ − B = (0 . , . , . (cid:48) . The efficient portfolio is given by π ∗ ( t ) = − ( σσ (cid:48) ) − (cid:98) B (cid:104) X ∗ ( t ) − µN (cid:0) − d ( t, φ ( t )) (cid:1) e − r ( T − t ) (cid:105) , (44)where ( σσ (cid:48) ) − (cid:98) B = ( σσ (cid:48) ) − B = (3 . , − . , . (cid:48) . In particular, the policy of investing in the second stock π ∗ ( t ) is negative.18 .2 Bankruptcy Prohibition with No-shorting Constraint From Subsection 6.1, we see that there exists a shorting case in policy (44). Using (38),we obtain the following λ to re-construct the no-shorting policy λ := argmin y ∈ R m + (cid:107) σ − y + σ − B (cid:48) (cid:107) = (0 , . , (cid:48) . (45)Hence, (cid:40) ˆ θ := σ − (cid:98) B = σ − ( B + λ ) = (0 . , . , . (cid:48) , ( σσ (cid:48) ) − (cid:98) B = ( σσ (cid:48) ) − ( B + λ ) = (2 . , , . (cid:48) . (46)According to (34), we obtain the numerical results µ = 1 . and γ = 0 . . Hence,the wealth process (35) can be expressed by X ∗ ( t ) = µN (cid:0) − d ( t, φ ( t )) (cid:1) e − r ( T − t ) − γN (cid:0) − d ( t, φ ( t )) (cid:1) φ ( t ) e − [2 r −| ˆ θ | ]( T − t ) , (47)where d ( t, φ ( t )) = ln (cid:0) γµ φ ( t ) (cid:1) +[ − r + | ˆ θ | ]( T − t ) √ | ˆ θ | ( T − t ) , d ( t, φ ( t )) = d ( t, φ ( t )) − (cid:113) | ˆ θ | ( T − t ) ,φ ( t ) = e − [ r + | ˆ θ | ]( T − t ) − ˆ θ ( W ( T ) − W ( t )) . The π ∗ ( t ) = − ( σσ (cid:48) ) − (cid:98) B (cid:104) X ∗ ( t ) − µN (cid:0) − d ( t, φ ( t )) (cid:1) e − r ( T − t ) (cid:105) . (48)Note that the policy in (48) is non-negative, that is, this is a no-shorting policy. In this subsection, we present the optimal no-shorting policy without the bankruptcyprohibition of Subsection 5.3 and its corresponding wealth process. According to (41), wefind the numerical results µ = 1 . and γ = 0 . . Hence, the wealth process (35) canbe expressed by X ∗ ( t ) = µe − r ( T − t ) − γφ ( t ) e − [2 r −| ˆ θ | ]( T − t ) (49)and π ∗ ( t ) = − ( σσ (cid:48) ) − (cid:98) B (cid:104) X ∗ ( t ) − µe − r ( T − t ) (cid:105) , (50)19here ˆ θ and ( σσ (cid:48) ) − (cid:98) B are given by (46), and φ ( t ) = e − [ r + | ˆ θ | ]( T − t ) − ˆ θ ( W ( T ) − W ( t )) . Note that the policy in (50) is non-negative, i.e., this is a no-shorting policy. However,its corresponding wealth (49) is possibly negative. We shall further discuss this point bysimulation results in Subsection 6.4. In this subsection, we further analyze using simulation how the properties of the optimalportfolio strategies (44), (48) and (50) change according to the given target wealth d =1 . X (0) , and compare their wealth processes (43), (47) and (49).The value of boundaryPolicy of Bankruptcy Prohibition with Unconstrained Portfolio20olicy of Bankruptcy Prohibition with No-shorting ConstraintPolicy of No-shorting Constraint without Bankruptcy ProhibitionDifferent Wealth Processes21omparision of Wealth Processes We have studied the continuous-time mean-variance portfolio selection with mixed restric-tions of bankruptcy prohibition ( constrained state ) and convex cone portfolio constraints( constrained controls ). The main contribution of the paper is that we developed semi-analytical expressions for the pre-committed efficient mean-variance policy without theviscosity solution technique. A natural extension of our result to continuous-time linear-quadratic cone constrained controls with constrained states is straightforward, at leastconceptually. On the other hand, if the rates of all market coefficients are random, theproblem becomes more complicated. 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