Mean-variance-utility portfolio selection with time and state dependent risk aversion
aa r X i v : . [ q -f i n . P M ] A ug Mean-variance-utility portfolio selection with time and statedependent risk aversion
Ben-Zhang Yang a , Xin-Jiang He b , and Song-Ping Zhu b ∗ a. School of Mathematics, Sichuan University, Chengdu 610064, P.R. Chinab. School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia Abstract.
Under mean-variance-utility framework, we propose a new portfolio selectionmodel, which allows wealth and time both have influences on risk aversion in the processof investment. We solved the model under a game theoretic framework and analyticallyderived the equilibrium investment (consumption) policy. The results conform with thefacts that optimal investment strategy heavily depends on the investor’s wealth and futureincome-consumption balance as well as the continuous optimally consumption process ishighly dependent on the consumption preference of the investor.
Keywords:
Mean-variance portfolio problem; Utility; Optimal investment and con-sumption; Equilibrium; State dependent risk aversion;
Since Markowitz’s pioneering work on a static portfolio selection model [19], mean-variance problemhas become one of the most important tools in finance to achieve a balance between uncertain returnsand risks. Under mean-variance framework, there are several models have been proposed and developedto address investment problems, which have attracted a lot of attention from both academic researchersand market practitioners [18, 21, 22, 23].Within the complete market setting, various pre-commitment (or time-inconsistent) results have beenpresented for the variance-minimizing policy using martingale methods, given that the expected terminalwealth is equal to a certain level (see [1, 3, 10, 11, 26]). In an incomplete market, Cochrane [9] derivedthe optimal investment policy that minimizes the ”long-term” variance of portfolio returns subject tothe constraint that the long-term mean of portfolio returns equals to a pre-specified target level. Thisapproach has also been applied in futures trading strategies by Du ffi e and Richardson [12] through settinga mean-variance objective at the initial date. They obtained a pre-commitment solution, which also ∗ Corresponding author. E-mail address: [email protected] ff ensen [14] worked under the Black-Scholes framework without the pre-commitment assumption, and showed that the optimal strategy derived for a mean-standard deviationinvestor is to take no risk at all. Bj¨ork et al. [5] further considered mean-variance optimization problemsunder a game theoretic framework, and the optimal strategies were derived in the context of sub-gameperfect Nash equilibrium.Recently, researchers started to incorporate consumption choices into the mean-variance problem,investigating the optimal investment-consumption problem together with the mean-variance criterion.For example, Kronborg and Ste ff ensen [13] directly added the accumulated consumption to the terminalwealth to formulate an “adjusted” terminal wealth, and tried to maximize the adjusted terminal wealthover time under the mean-variance framework. Christiansen and Ste ff ensen [8] further considered thesame optimization problem with deterministic consumption and investment to avoid a series of di ffi cul-ties. Unfortunately, the optimal consumption strategy derived under this particular model has causeda probably absurd conclusion that investor could suddenly be required to switch his / her optimal con-sumption strategy from consuming as much as possible to as little as possible. To obtain a rationalconsumption policy, Yang et. al [24] proposed a new portfolio selection model which simultaneouslymaximize the terminal wealth and accumulated consumption utility subject to a mean variance criterioncontrolling the final risk of the portfolio. The analytically derived policy performs the continuous influ-ence of investors’ consumption preference on the optimal consumption strategy and represents a moreeconomically rational investment / consumption behavior.Unfortunately, the optimal amount of moment to invest in is not dependent of wealth under themodel setting [24], which means that for a given risk aversion degree a rich or poor investor optimallyinvest the same amount of the money in stocks. In fact, the investor will change his investment policyaccording to the update of her / his wealth in the case of multi-stage investment. Inspired this, we proposea new portfolio selection model which allows the risk aversion depend on present wealth and time,therefore the progression of wealth and time phasing can have impacts on the varying risk aversion. Thenewly formulated optimization problem still preserves the analytical tractability under a continuous-timegame theoretic framework, and the analytical optimal continuous investment and consumption strategiesderived in the sense of equilibrium [4, 5] admit intuitive economic explanation.The rest of this paper is organized as follows. Section 2 proposes the new portfolio selection problem.In Section 3, we analytically derive the optimal strategies based on the definition of the equilibriumstrategy. Some concluding remarks are given in the last section.2 The portfolio selection problem
We now assume that we work under the standard Black-Scholes market, where an investor has access toa risk-free bank account and a stock whose dynamics can be specified as dM ( t ) = rM ( t ) dt , M (0) = , dS ( t ) = µ S ( t ) dt + σ S ( t ) dB ( t ) , S (0) = s > . (1)Here, r > µ and σ are constants, and it is assumed that µ > r . The process B ( t ) is a standard Brownianmotion on the probability space ( Ω , F , P ) with the filtration σ { B ( s ); 0 ≤ s ≤ t } , ∀ t ∈ [0 , T ].Let L F (0 , T ; R ) denote the set of all R -valued, measurable stochastic process f ( t ) adapted to { F t } t ≥ such that E (cid:20)R T f ( t ) dt (cid:21) < ∞ . We also assume that the investor in this market needs to make investmentdecisions on a finite time horizon [0 , T ], and he / she allocates a proportion π ( t ) and 1 − π ( t ) of his wealthinto the stock and bank account, respectively, at time t . Let X π ( t ) be the wealth of the investor at time t following the investment strategy π ( · ) with an initial wealth of x at time 0. We assume that the investorpossesses a continuous deterministic income rate l ( t ), and chooses a non-negative consumption rate c ( t ).Under these assumptions, the dynamic of the investor’s wealth can be derived as dX c ,π ( t ) = [( r + π ( t )( µ − r )) X c ,π ( t ) + l ( t ) − c ( t )] dt + π ( t ) σ X c ,π ( t ) dB ( t ) , t ∈ [0 , T ) , X (0) = x > . (2)In this paper, by introducing a time and sate dependent risk aversion function [5, 13], we propose ageneral portfolio selection model: at a given time t , the investor attempt to achieve the following objectivemax c ( · ) ,π ( · ) E (cid:16) e − δ ( T − t ) X ( T ) (cid:17) − γ x + K ( c ) ( t , x )) Var (cid:16) e − δ ( T − t ) X ( T ) (cid:17) + β E Z Tt e − ρ ( s − t ) U ( c ( s )) ds ! s . t . c ( · ) , π ( · ) ∈ L F (0 , T ; R ) , ( X ( · ) , c ( · ) , π ( · )) satis f y Equation (2) , (3)where K ( c ) ( t , x ) : = E " Z Tt e − r ( s − t ) (cid:0) l ( s ) − c ( s , X c ,π ( s )) (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( t ) = x (4)is the time- t financial value of future labor income net of consumption, where γ > β > ffi cient of total utility of consumption.Obviously, the model (3) involves the optimization of the mutual objective of expected return, riskand consumption utility. It is necessary for the investor to consider selecting a set of appropriate invest-ment and consumption strategies to achieve the goal of both maximizing return and minimizing risk aswell as maximizing consumption utility. It should be highlighted that, except for being relevant to theoriginal constant risk-aversion coe ffi cient γ , the preference of risk tolerance is also dependent with thetime and the investor’s wealth. If the investor’s future income and expenditure is relatively excellent, he3ill reduce the corresponding risk aversion degree, and will tend to invest more in the stock to obtain thepotential outcomes.We would like to point out that the model (3) includes several known models as special cases. In fact,if we remove the consumption component, the model degenerates into the one studied in [8, 13]; if wedo not consider the time and sate dependent risk aversion function, the model becomes the one reportedin [24]; if δ is set to 0, the time and sate dependent risk aversion and the consumption component are notbe taken into account, then the model becomes the classical mean-variance model (see [2, 16, 27]). We shall solve the optimal portfolio selection problem (3) under a game theoretic framework, which wasintroduced in [4, 5] and developed by [13, 24]. The equilibrium strategy under the continuous-time gametheoretic equilibrium for the problem (3) can be defined as follows.
Definition 2.1.
Consider a strategy ( c ∗ , π ∗ ) and a fixed point ( c , π ) . For a fixed number h > and aninitial point ( t , x ) , we define the strategy ( e c h , e π h ) as ( e c h ( s ) , e π h ( s )) = ( c , π ) , for t ≤ s < t + h , ( c ∗ ( s ) , π ∗ ( s )) , for t + h ≤ s < T . (5) If lim h → inf 1 h (cid:16) f c ∗ ,π ∗ ( t , x , y c ∗ ,π ∗ , z c ∗ ,π ∗ , w c ∗ ,π ∗ ) − f e c h , e π h ( t , x , y e c h , e π h , z e c h , e π h , w e c h , e π h ) (cid:17) ≥ for all ( c , π ) ∈ R + × R , where f is an optimal value function andy c ,π : = y c ,π ( t , x ) = E h e − δ ( T − t ) X c ,π ( T ) (cid:12)(cid:12)(cid:12) X ( t ) = x i , z c ,π : = z c ,π ( t , x ) = E (cid:20) (cid:16) e − δ ( T − t ) X c ,π ( T ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( t ) = x (cid:21) , w c ,π : = w c ,π ( t , x ) = E " Z Tt e − ρ ( s − t ) U ( c ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( t ) = x , (7) then ( c ∗ , π ∗ ) is an equilibrium strategy. If we denote ( c ∗ , π ∗ ) as the equilibrium strategy satisfying Definition 2.1, and let V be the the corre-sponding value function with the equilibrium strategy, we can obtain V ( t , x ) = f c ,π ( t , x , y c ∗ ,π ∗ , z c ∗ ,π ∗ , w c ∗ ,π ∗ ) . (8)Clearly, our problem is to search for the corresponding optimal strategies and the optimal value function f : [0 , T ] × R → R as a C , , , , function of the form f c ∗ ,π ∗ ( t , x , y c ,π , z c ,π , w c ,π ) = y − ψ ( t , x )2 ( z − y ) + β w , ( c , π ) ∈ A , (9)where ψ ( t , x ) = γ x + K ( c ) ( t , x ) and A is the class of admissible strategies.Before we are able to present the optimal solution, some preliminaries need to be outlined. Asreported in studies [13, 24], we can establish an extension of the HJB equation for the characterization4f the optimal value function and the corresponding optimal strategy, so that the stochastic problemcan be transformed into a system of deterministic di ff erential equations and a deterministic point-wiseminimization problem. We introduce the following two lemmas. Due to the length limitation, we are notprepared to prove the following lemmas and recommend interested readers to refer to the literature [24]. Lemma 2.1.
Suppose there exist three functions Y = Y ( t , x ) , Z = Z ( t , x ) and W = W ( t , x ) such that Y t ( t , x ) = − [( r + π ( µ − r )) x + l − c ] Y x ( t , x ) − π σ x Y xx ( t , x ) + δ Y ( t , x ) , Y ( T , x ) = x , (10) Z t ( t , x ) = − [( r + π ( µ − r )) x + l − c ] Z x ( t , x ) − π σ x Z xx ( t , x ) + δ Z ( t , x ) , Z ( T , x ) = x , (11) and W t ( t , x ) = − [( r + π ( µ − r )) x + l − c ] W x ( t , x ) − π σ x W xx ( t , x ) − e − ρ t U ( c ) , W ( T , x ) = , (12) where ( c , π ) is an arbitrary admissible strategy. Then,Y ( t , x ) = y c ,π ( t , x ) , Z ( t , x ) = z c ,π ( t , x ) , W ( t , x ) = w c ,π ( t , x ) , (13) where y c ,π , z c ,π and w c ,π are given by (7) . Lemma 2.2.
If there exists a function F = F ( t , x ) such that F t = inf c ,π ∈A ( − [( r + π ( µ − r )) x + l − c ]( F x − Q ) − π σ x ( F xx − U ) + J ) , F ( T , x ) = f c ,π ( T , x , x , x , , (14) where Q = f c ∗ ,π ∗ x ,U = f c ∗ ,π ∗ xx + f c ∗ ,π ∗ yy ( F (1) x ) + + f c ∗ ,π ∗ zz ( F (2) x ) + f c ∗ ,π ∗ ww ( F (3) ) + f c ∗ ,π ∗ xy F (1) x + f c ∗ ,π ∗ xz F (2) x + f c ∗ ,π ∗ xw F (3) x + f c ∗ ,π ∗ yz F (1) x F (2) x + f c ∗ ,π ∗ yw F (1) x F (3) x + f c ∗ ,π ∗ zw F (2) x F (3) x (15) and J = f c ∗ ,π ∗ t + f c ∗ ,π ∗ y δ F (1) + f c ∗ ,π ∗ z δ F (2) − f c ∗ ,π ∗ w e − ρ t U ( c ( t )) . (16) with F (1) = y c ∗ ,π ∗ ( t , x ) , F (2) = z c ∗ ,π ∗ ( t , x ) , F (3) = w c ∗ ,π ∗ ( t , x ) , then F ( t , x ) = V ( t , x ) , where V is the optimal value function defined by (8) . Determination of optimal strategy
In this section, we present the optimal solutions to the optimal portfolio selection problem (3) based onthe results derived in the previous section, and some detailed discussions are provided to illustrate thebehaviour of the optimal strategies.
Lemma 3.1.
The optimal policy for the optimal value function (14) can be solved as π ∗ = − µ − rx σ F (1) x + ψ F (1) F (1) x − ψ F (2) x + β F (3) x F (1) xx + ψ F (1) F (1) xx − ψ F (2) xx + β F (3) xx (17) and c ∗ = [ U ′ ] − β e − ρ t R ( t , x ) ! , (18) where [ f ] − ( · ) is the inverse function of f andR ( t , x ) = F (1) x + ψ F (1) F (1) x − ψ F (2) x + β F (3) x + γ x + K ( c ) ) (1 + K c ∗ x ( t , x )) (cid:16) F (2) − ( F (1) ) (cid:17) . (19) Proof.
A candidate strategy for the optimal value function (14) can be derived by simply di ff erentiating(14) with respect to π and c , respectively. Therefore, the optimal strategy π ∗ should satisfy ∂∂π − π ( µ − r ) x ( F x − Q ) − π σ x ( F xx − U ) ! = . (20)A further simplification then yields π ∗ = − µ − rx σ F x − QF xx − U . (21)Recall the corresponding objective form f ( t , x , y , z , w ) = y − ψ ( t , x )2 ( z − y ) + β w , (22)where ψ ( t , x ) = γ x + K ( c ) ( t , x ) . Substituting (22) into (15) and (16) gives F x − Q = F (1) x + ψ F (1) F (1) x − ψ F (2) x + β F (3) x , F xx − U = F (1) xx + ψ F (1) F (1) xx − ψ F (2) xx + β F (3) xx . (23)Similarly, we can also obtain J = δ F (1) − ( ψδ + ψ t (cid:16) F (2) − ( F (1) ) (cid:17) − β e − ρ t U ( c ) . (24)By characterizing as the solution to a Feynman-Kac PDE, we can obtain ψ t = − γ ( x + K ( c ) ) rK ( c ) − l + c − ( rx + l − c ) K ( c ) x − π σ x K ( c ) xx ! . (25)Inserting (25) into (24), we have the new form of J as follows J = δ F (1) − γδ x + K ( c ) (cid:16) F (2) − ( F (1) ) (cid:17) + γ x + K ( c ) ) (cid:18) rK ( c ) − l + c − ( rx + l − c ) K ( c ) x − π σ x K ( c ) xx (cid:19) × (cid:16) F (2) − ( F (1) ) (cid:17) − β e − ρ t U ( c ) . (26)By substituting (26) into optimal value function (14) and di ff erentiating with respect to c , we then arriveat the optimal consumption strategy c ∗ defined as (18). This completes the proof. (cid:3)
6o obtain a more explicit form of the optimal policy, we search for solutions where F (1) , F (2) and F (3) are tractable. We report the new derived forms below for the optimal solutions given in Lemma 3.1,and we also verify the new solutions are well-defined. Theorem 3.1.
The optimal investment and consumption strategies for model (3) are π ∗ ( t ) x = µ − r σ γ f ( t ) (cid:16) a ( t ) + γ ( a ( t ) − f ( t )) (cid:17) ( x + K ( c ∗ ) ( t )) (27) provided that γ f ( t ) x + K ( c ∗ ) ( t ) > , (28) and c ∗ ( t ) = [ U ′ ] − (cid:18) a ( t ) + γ a ( t ) − f ( t )) (cid:19) (29) respectively, and the optimal objective value isF ( t , x ) = a ( t ) (cid:16) x + K c ∗ ( t ) (cid:17) − γ (cid:16) f ( t ) − a ( t ) (cid:17) (cid:16) x + K c ∗ ( t ) (cid:17) + β Z Tt e − ρ s U ( c ∗ ( s )) ds . (30) where a ( t ) and f ( t ) are given byda ( t ) dt = − ( r − δ ) + µ − r σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) ! a ( t ) (31) andd f ( t ) dt = − ( r − δ ) + µ − r σ γ f ( t ) (cid:16) a ( t ) + γ ( a ( t ) − f ( t )) (cid:17)! + µ − r σ γ f ( t ) (cid:16) a ( t ) + γ ( a ( t ) − f ( t )) (cid:17) f ( t ) (32) with initial conditions a ( T ) = f ( T ) = .Proof. To obtain an explicit solution for this optimal portfolio selection problem, we assume that F (1) , F (2) and F (3) can be written in the following form: F (1) ( t , x ) = a ( t )( x + K c ∗ ( t )) + b ( t ) , F (2) ( t , x ) = f ( t )( x + K c ∗ ( t )) + g ( t )( x + K c ∗ ( t )) + h ( t ) , F (3) ( t , x ) = p ( t )( x + K c ∗ ( t )) + q ( t ) , (33)where a , b , f , g , h , p and q are deterministic functions of time. The candidate for the optimal consumptionstrategy c is assumed to be independent of wealth, which implies that K c ∗ ( t ) = Z Tt e − r ( s − t ) ( l ( s ) − c ∗ ( s )) ds . (34)We also assume that a ( t ) b ( t ) = g ( t )2 , h ( t ) = b ( t ) . (35)Substituting (33) into (17) and (18) can yield the new forms (27) and (29). Now insert (27) and (29) into(10) and include the terminal conditions to get da ( t ) dt = − ( r − δ ) + µ − r σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) ! a ( t ) (36)7nd db ( t ) dt = δ b ( t ) (37)with terminal conditions a ( T ) = b ( T ) =
0, respectively. In the same way, substituting (27) and(29) into (11) yields d f ( t ) dt = − ( r − δ ) + µ − r σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) ! a ( t ) + µ − r σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) f ( t ) dg ( t ) dt = − r + ( µ − r ) σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) ! g ( t ) + ρ g ( t ) , (38)and dh ( t ) dt = δ h ( t ) , (39)with terminal conditions f ( T ) = g ( T ) = h ( T ) =
0. By inserting (27) and (29) into (12), we then have d p ( t ) dt = − ( r − δ ) + µ − r σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) ! p ( t ) (40)and dq ( t ) dt = e − ρ t U ( c ∗ ( t )) dt (41)with terminal conditions p ( T ) = q ( T ) = b ( t ) = g ( t ) = h ( t ) = p ( t ) =
0, which guarantees theassumptions (35). Besides, q ( t ) = R Tt e − ρ s U ( c ∗ ( s )) ds .In addition, the optimal value function F can be also derived as F ( t , x ) = F (1) − γ x + K ( c ∗ ) ( t )) (cid:16) F (2) − ( F (1) ) (cid:17) + β F (3) = a ( t ) (cid:16) x + K c ∗ ( t ) (cid:17) + b ( t ) − γ (cid:16) f ( t ) − a ( t ) (cid:17) (cid:16) x + K c ∗ ( t ) (cid:17) + β (cid:16) p ( t )( x + K c ∗ ( t )) + q ( t ) (cid:17) = a ( t ) (cid:16) x + K c ∗ ( t ) (cid:17) − γ (cid:16) f ( t ) − a ( t ) (cid:17) (cid:16) x + K c ∗ ( t ) (cid:17) + β Z Tt e − ρ s U ( c ∗ ( s )) ds . (42)This completes the proof. (cid:3) Remark 3.1.
It follows (2) and (34) thatd (cid:16) X c ∗ ,π ∗ ( t ) + K ( c ∗ ) ( t ) (cid:17) = r + ( µ − r ) σ γ f ( t ) ( a ( t ) + γ ( a ( t ) − f ( t ))) ! (cid:16) X c ∗ ,π ∗ ( t ) + K ( c ∗ ) ( t ) (cid:17) dt + µ − r σ γ f ( t ) (cid:16) a ( t ) + γ ( a ( t ) − f ( t ))) (cid:17) (cid:16) X c ∗ ,π ∗ ( t ) + K ( c ∗ ) ( t ) (cid:17) dB ( t ) . (43) Therefore,X c ∗ ,π ∗ ( t ) + K ( c ∗ ) ( t ) = ( x + K ( c ∗ ) (0)) exp (cid:20) Z t r + ( µ − r ) σ γ f ( s ) ( a ( s ) + γ ( a ( s ) − f ( s ))) −
12 ( µ − r ) σ γ f ( s ) ( a ( s ) + γ ( a ( s ) − f ( s ))) ! ds + Z t µ − r σ γ f ( s ) (cid:16) a ( s ) + γ ( a ( s ) − f ( s ))) (cid:17) dB ( s ) (cid:21) . (44) Since the initial condition ensures x + K ( c ∗ ) (0) > and f is proved to be strictly positive in (50) below,we conclude the condition (28) for the optimal investment strategy is fulfilled. emark 3.2. The system composed of PDEs (10) , (11) and (12) has a unique global solution. In fact,by replacing the integral interval to [ t , T ] and taking conditional expectation at t in (44) , we haveE h X ( c ∗ ,π ∗ ) ( T ) (cid:12)(cid:12)(cid:12) X ( t ) = x i = ( x + K c ∗ ( t )) exp Z Tt [ r + ( µ − r ) ˜ π ∗ ( s )] ds ! (45) and E h ( X ( c ∗ ,π ∗ ) ( T )) (cid:12)(cid:12)(cid:12) X ( t ) = x i = ( x + K c ∗ ( t )) exp Z Tt [ r + ( µ − r ) ˜ π ∗ ( s ) + σ ( ˜ π ∗ ( s )) ] ds ! , (46) where ˜ π ∗ ( s ) = µ − r σ γ f ( t ) (cid:16) a ( t ) + γ ( a ( t ) − f ( t )) (cid:17) . (47) Comparing (33) with (45) and (46) yieldsa ( t )( x + K c ∗ ( t )) + b ( t ) = E h e − δ ( T − t ) X ( c ∗ ,π ∗ ) ( T ) (cid:12)(cid:12)(cid:12) X ( t ) = x i = ( x + K c ∗ ( t )) exp Z Tt [( r − δ ) + ( µ − r ) ˜ π ∗ ( s )] ds ! (48) and f ( t )( x + K c ∗ ( t )) + g ( t )( x + K c ∗ ( t )) + h ( t ) = E h ( e − δ ( T − t ) X ( c ∗ ,π ∗ ) ( T )) (cid:12)(cid:12)(cid:12) X ( t ) = x i = ( x + K c ∗ ( t )) exp Z Tt [( r − δ ) + ( µ − r ) ˜ π ∗ ( s ) + σ ( ˜ π ∗ ( s )) ] ds ! . (49) Collecting terms we obtaina ( t ) = exp Z Tt [( r − δ ) + ( µ − r ) ˜ π ∗ ( s )] ds ! , f ( t ) = exp Z Tt [( r − δ ) + ( µ − r ) ˜ π ∗ ( s ) + σ ( ˜ π ∗ ( s )) ] ds ! , (50) and b ( t ) = g ( t ) = h ( t ) = . Substituting (50) into (47) leads to ˜ π ∗ ( t ) = µ − r σ γ (cid:18) e − R Tt [( r − δ ) + ( µ − r )˜ π ∗ ( s ) + σ (˜ π ∗ ( s )) ] ds + γ e − R Tt σ (˜ π ∗ ( s )) ] ds − γ (cid:19) . (51) By designing the algorithm as ˜ π ( t ) = and ˜ π n + ( t ) = µ − r σ γ (cid:18) e − R Tt [( r − δ ) + ( µ − r )˜ π n ( s ) + σ (˜ π n ( s )) ] ds + γ e − R Tt σ (˜ π n ( s )) ] ds − γ (cid:19) for n ≥ on [0 , T ] , we can prove that the sequence { ˜ π n } converges to the solution ˜ π ∗ , which verifies theuniqueness of the optimal investment strategy. In this paper, we introduced the time and state dependent risk aversion into the mean-variance-utilityportfolio selection problem and a new portfolio selection model embraces is proposed. We solved themodel under a game theoretic framework and analytically derived the continuous equilibrium investment(consumption) policy. The results perform economically reasonable implication that optimal investmentstrategy heavily depends on the investor’s current wealth and future income-consumption balance. Inaddition, the continuous optimally consumption process shows high dependence on the investor’s con-sumption preference. 9 eferences [1] I. Bajeux-Besnainou, R. Portait, Dynamic Asset Allocation in a Mean-Variance Framework,
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