Multi-time state mean-variance model in continuous time
MMulti-time state mean-variance model incontinuous time ∗ Shuzhen Yang †‡ Abstract : The objective of the continuous time mean-variance model is tominimize the variance (risk) of an investment portfolio with a given mean at ter-minal time. However, the investor can stop the investment plan at any time beforethe terminal time. To solve this problem, we consider minimizing the variances ofthe investment portfolio in the multi-time state. The advantage of this multi-timestate mean-variance model is that we can minimize the risk of the investment port-folio within the investment period. To obtain the optimal strategy of the model, weintroduce a sequence of Riccati equations, which are connected by a jump bound-ary condition. Based on this equations, we establish the relationship between themeans and variances in the multi-time state mean-variance model. Furthermore,we use an example to verify that the variances of the multi-time state can a ff ectthe average of Maximum-Drawdown of the investment portfolio. ∗ Keywords : mean-variance; multi-time state; stochastic control.
MSC2010 subject classification : 91B28; 93E20; 49N10. OR / MS subject classification : Finance / portfolio; dynamic programming / optimal control. † Shandong University-Zhong Tai Securities Institute for Financial Studies, Shandong Univer-sity, PR China, ([email protected]). ‡ This work was supported by the National Natural Science Foundation of China (GrantNo.11701330) and Young Scholars Program of Shandong University. a r X i v : . [ q -f i n . M F ] D ec Introduction
To balance the return (mean) and risk (variance) in a single-period portfolio se-lection model, Markowitz (1952, 1959) proposed the mean-variance model. Sincethen, many related works focused on these topics. Under some mild assumptions,Merton (1972) solved the single-period problem analytically. Richardson (1989)studied a mean-variance model in which a single stock with a constant risk-freerate is introduced. Dynamic asset allocation in a mean-variance framework wasstudied by Bajeux-Besnainou and Portait (1998). Li and Ng (2000) embeddedthe discrete-time multi-period mean-variance problem within a multi-objectiveoptimization framework and obtained an optimal strategy. By extending the em-bedding technique introduced in Li and Ng (2000) and applying the results fromthe stochastic LQ control in the continuous time case, Zhou and Li (2000) inves-tigated an optima pair for the continuous-time mean-variance problem. Furtherresults in the mean-variance problem include those with bankruptcy prohibition,transaction costs, and random parameters in an complete and incomplete markets(see Bielecki et al. (2005); Dai et al. (2010); Lim (2004); Lim and Zhou (2002);Xia (2005)).The pre-committed strategies in the aforementioned multi-period and contin-uous time cases, di ff ered from those of the single-period case. For further details,see (Kydland and Prescott, 1997). Basak and Chabakauri (2010) adopted a gametheoretic approach to study the time inconsistency in the mean-variance modeland Bj ¨ork et al. (2014) studied the mean-variance problem with state dependentrisk aversion.In the financial market, for a given terminal time T , Y π ( T ) represents a portfo-lio asset with strategy π ( · ), while E [ Y π ( T )] and Var( Y π ( T )) = E (cid:0) Y π ( T ) − E [ Y π ( T )] (cid:1) represent the mean and variance of Y π ( T ), respectively. In the classical mean-2ariance model, we want to minimize the variance of the portfolio asset Var( Y π ( T ))for a given mean E [ Y π ( T )] = L , where L is a constant. The investor can stopthe investment plan at an uncertain horizon time τ before the terminal time T ,where τ ≤ T . Therefore, there are many related works on the mean-varianceportfolio model with an uncertain horizon time. Martellini and Uroˇsevi´c (2006)considered static mean-variance analysis with an uncertain time horizon. Yi et al.(2008) studied the mean-variance model of a multi-period asset-liability manage-ment problem under uncertain exit time. Furthermore, see (Wu et al., 2011; Yaoand Ma, 2010; Yu, 2013) for additional studies in this vein. However, in the lit-erature of mean-variance model under uncertain or random exit time, we alwayssuppose that the uncertain horizon time τ satisfies a distribution (or a conditionaldistribution) and investigate the related mean-variance model at time τ .However, in general, we do not know the information of τ at initial time t = Ω , F , P ), notice that for a given partition 0 = t < t < · · · < t N = T of interval [0 , T ] and ω ∈ Ω , there exists i ∈ { , , · · · , N − } such that τ ( ω ) ∈ [ t i , t i + ]. To reduce the variance of the portfolio asset Y π ( · ) at τ ∈ (0 , T ], we consider minimizing the variances of the portfolio asset at multi-time state ( Y π ( t ) , Y π ( t ) , · · · , Y π ( t N )) with constraint on means of multi-time state( Y π ( t ) , Y π ( t ) , · · · , Y π ( t N )). Therefore, we introduce the following multi-time statemean-variance model: J ( π ( · )) = N (cid:88) i = E (cid:0) Y π ( t i ) − E [ Y π ( t i )] (cid:1) , (1.1)with constraint on the multi-time state mean, E [ Y π ( t i )] = L i , i = , , · · · , N . (1.2)In this multi-time state mean-variance model, we can minimize the risk of the in-vestment portfolio within the multi-time ( t , t , · · · , t N ). It should be noted that the3ulti-time state ( Y π ( t ) , Y π ( t ) , · · · , Y π ( t N )) of the investment portfolio can a ff ectthe value of each other, and we cannot solve the multi-time state mean-variancemodel via one classical Riccati equation directly. To obtain the optimal strategyof the multi-time state mean-variance model, we introduce a sequence of Riccatiequations, which are connected by a jump boundary condition (see equations (3.5)and (3.6)). Based on this sequence of Riccati equations, we investigate an optimalstrategy (see Theorem 3.1) and establish the relationship between the means andvariances of this multi-time state mean-variance model (see Lemma 3.1).The Maximum-Drawdown of the asset Y π ( · ) is an important index to evaluatea strategy in the investment portfolio model, where the Maximum-Drawdown ofthe asset Y π ( · ) is defined in the interval [0 , h ] , h ≤ T , byMD hY π = esssup { z | z = Y π ( t ) − Y π ( s ) , ≤ t ≤ s ≤ h } . Based on simulation results of the multi-time state mean-variance model (see sub-section 4.2), we can see that the constrained condition (1.2) can a ff ect the averageof MD hY π of the portfolio asset Y π ( · ) (see Figure 3). The work is most closely re-lated to the study of (Yang, 2018), in which the author established the necessaryand su ffi cient conditions for stochastic di ff erential systems with multi-time statecost functional.The remainder of this paper is organized as follows. In Section 2, we formulatethe multi-time state mean-variance model. Then, in Section 3, we investigatean optimal strategy and establish the relationship between multi-time state meanand variance for the proposed model. In Section 4, based on the main results ofSection 3, we compare the multi-time state mean-variance model with classicalmean-variance model. Finally, we conclude the paper in Section 5.4 Multi-time state mean-variance model
Let W be a d -dimensional standard Brownian motion defined on a complete fil-tered probability space ( Ω , F , P ; {F ( t ) } t ≥ ), where {F ( t ) } t ≥ is the P -augmentationof the natural filtration generated by W . We suppose the existence of one risk-freebond asset and n risky stock assets that are traded in the market, where the bondsatisfies the following equation: d R ( t ) = r ( t ) R ( t )d t , t > , R (0) = a > , and the i ’th (1 ≤ i ≤ n ) stock asset is described by d R i ( t ) = b i ( t ) R i ( t )d t + R i ( t ) d (cid:88) j = σ i j ( t )d W j ( t ) , t > , R i (0) = a i > , where r ( · ) ∈ R is the risk-free return rate of the bond, b ( · ) = ( b ( · ) , · · · , b n ( · )) ∈ R n is the expected return rate of the risky asset, and σ ( · ) = ( σ ( · ) , · · · , σ n ( · )) (cid:62) ∈ R n × d is the corresponding volatility matrix. Given initial capital x > γ ( · ) = ( γ ( · ) , · · · , γ n ( · )) ∈ R n , where γ i ( · ) = b i ( · ) − r ( · ) , ≤ i ≤ n . The investor’s wealth Y π ( · ) satisfies d Y π ( t ) = (cid:2) r ( t ) Y π ( t ) + γ ( t ) π ( t ) (cid:62) (cid:3) d t + π ( t ) σ ( t )d W ( t ) , Y π (0) = y , (2.1)where π ( · ) = ( π ( · ) , · · · , π n ( · )) ∈ R n is the capital invested in the risky asset R ( · ) = ( R ( · ) , · · · , R n ( · )) ∈ R n and π ( · ) is the capital invested in the bond. Thus, we have Y π ( · ) = n (cid:88) i = π i ( · ) .
5n this study, we consider the following multi-time state mean-variance model: J ( π ( · )) = N (cid:88) i = E (cid:0) Y π ( t i ) − E [ Y π ( t i )] (cid:1) , (2.2)with constraint on the multi-time state mean, E [ Y π ( t i )] = L i , i = , , · · · , N , (2.3)where 0 = t < t < · · · < t N = T . The set of admissible strategies π ( · ) is definedas: A = (cid:26) π ( · ) : π ( · ) ∈ L F [0 , T ; R n ] (cid:27) , where L F [0 , T ; R n ] is the set of all square integrable measurable R n valued {F t } t ≥ adaptive processes. If there exists a strategy π ∗ ( · ) ∈ A that yields the minimumvalue of the cost functional (2.2), then we say that the multi-time state mean-variance model (2.2) is solved.We make the following assumptions to obtain the optimal strategy for the pro-posed model (2.2): H : r ( · ) , b ( · ) and σ ( · ) are bounded deterministic continuous functions. H : r ( · ) , γ ( · ) > σ ( · ) σ ( · ) (cid:62) > δ I , where δ > I is theidentity matrix of S n , S n is the set of symmetric matrices. In this section, we investigate an optimal strategy π ( · ) for the problem definedin (2.2), with a constraint on the multi-time state mean (2.3). Here, we describehow to construct an optimal strategy for (2.2) with constrained condition (2.3).Similar to (Zhou and Li, 2000), we introduce the following multi-time state6ean-variance problem: minimizing the cost functional, J ( π ( · )) = N (cid:88) i = (cid:18) µ i Y π ( t i )) − E [ Y π ( t i )] (cid:19) . (3.1)To solve the cost functional (3.1), we employ the following model: J ( π ( · )) = N (cid:88) i = E [ µ i Y π ( t i ) − λ i Y π ( t i )] , (3.2)where µ i > λ i ∈ R , i = , , · · · , N . For given µ i > , i = , , · · · , N , wesuppose π ∗ ( · ) is an optimal strategy of cost functional (3.1). Based on Theorem3.1 of (Zhou and Li, 2000), taking λ i = + µ i E [ Y π ∗ ( t i )] , i = , , · · · , N , wecan show that π ∗ ( · ) is an optimal strategy of cost functional (3.2). It should benoted that, we cannot solve the cost functional (3.2) by applying the embeddingtechnique of (Zhou and Li, 2000) for the multi-time state mean-variance modelsvia the classical Riccati equation directly. This is because the value Y π ( t i ) cana ff ect Y π ( t i + ), for i = , , · · · , N − ρ i = λ i µ i , z π i ( t ) = Y π ( t ) − ρ i , t ≤ t i , i = , , · · · , N ,β ( t ) = γ ( t )[ σ ( t ) σ ( t ) (cid:62) ] − γ ( t ) (cid:62) , t ≤ T . Thus, the cost functional (3.2) is equivalent to J ( π ( · )) = N (cid:88) i = E [ µ i z π i ( t i ) ] , (3.3)where z π i ( · ) satisfies d z π i ( t ) = (cid:2) r ( t ) z π i ( t ) + γ ( t ) π ( t ) (cid:62) + ρ i r ( t ) (cid:3) d t + π ( t ) σ ( t )d W ( t ) , z π i ( t i − ) = Y π ( t i − ) − ρ i , t i − < t ≤ t i . (3.4)7ow, we construct a new sequence of Riccati equations that are connected bya jump boundary condition, in which the jump boundary condition can o ff set theinteraction of Y π ( t i + ) and Y π ( t i ), for i = , , · · · , N −
1. We first introduce asequence of deterministic Riccati equations: d P i ( t ) = (cid:2) β ( t ) − r ( t ) (cid:3) P i ( t )d t , P i ( t i ) = µ i + P i + ( t i ) , t i − ≤ t < t i , i = , , · · · , N , (3.5)and related equations, d g i ( t ) = (cid:2) ( β ( t ) − r ( t )) g i ( t ) − ρ i r ( t ) P i ( t ) (cid:3) d t , g i ( t i ) = g i + ( t i ) + P i + ( t i )( ρ i − ρ i + ) , t i − ≤ t < t i , i = , , · · · , N , (3.6)where P N + ( t N ) = , g N + ( t N ) = , ρ N + =
0. Furthermore, by a simple calcula-tion, we can obtain, g i ( t ) P i ( t ) = g i ( t i ) P i ( t i ) e − (cid:82) tit r ( s )d s + ρ i (1 − e − (cid:82) tit r ( s )d s ) , t i − ≤ t ≤ t i i = , , · · · , N , which is used to obtain the following results. Theorem 3.1.
Let Assumptions H and H hold, there exists an optimal strategy π ∗ ( · ) for cost functional (3.3), where the optimal strategy π ∗ ( · ) is given as follows: π ∗ ( t ) = γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − (cid:2) ( ρ i − g i ( t i ) P i ( t i ) ) e − (cid:82) tit r ( s )d s − Y ∗ ( t ) (cid:3) , t i − < t ≤ t i , where Y ∗ ( t ) = z π ∗ ( t ) + ρ i , t i − < t ≤ t i and i = , , · · · , N. Proof : For any given i ∈ { , , · · · , N } , t i − < t ≤ t i , applying Itˆo formula to8 π i ( t ) P i ( t ) and z π i ( t ) g i ( t ), respectively, we have12 d z π i ( t ) P i ( t ) = (cid:26) z π i ( t ) P i ( t ) (cid:2) r ( t ) z π i ( t ) + γ ( t ) π ( t ) (cid:62) + ρ i r ( t ) (cid:3) + z π i ( t ) (cid:2) β ( t ) − r ( t ) (cid:3) P i ( t ) + P i ( t ) π ( t ) σ ( t ) σ ( t ) (cid:62) π ( t ) (cid:62) (cid:27) d t + z π i ( t ) P i ( t ) π ( t ) σ ( t )d W ( t ) = (cid:26) z π i ( t ) P i ( t ) (cid:2) γ ( t ) π ( t ) (cid:62) + ρ i r ( t ) (cid:3) + z π i ( t ) β ( t ) P i ( t ) + P i ( t ) π ( t ) σ ( t ) σ ( t ) (cid:62) π ( t ) (cid:62) (cid:27) d t + z π i ( t ) P i ( t ) π ( t ) σ ( t )d W ( t )and d z π i ( t ) g i ( t ) = (cid:26) g i ( t ) γ ( t ) π ( t ) (cid:62) + g i ( t ) ρ i r ( t ) + z π i ( t ) (cid:2) β ( t ) g i ( t ) − ρ i r ( t ) P i ( t ) (cid:3)(cid:27) d t + g i ( t ) π ( t ) σ ( t )d W ( t ) . We add the above two equations together and integrate from t i − to t i , it follows9hat E (cid:20) P i ( t i )2 z π i ( t i ) − P i ( t i − )2 z π i ( t i − ) + z π i ( t i ) g i ( t i ) − z π i ( t i − ) g i ( t i − ) (cid:21) = E (cid:20) µ i + P i + ( t i )2 z π i ( t i ) − P i ( t i − )2 z π i ( t i − ) + z π i ( t i ) (cid:2) g i + ( t i ) + P i + ( t i )( ρ i − ρ i + ) (cid:3) − z π i ( t i − ) g i ( t i − ) (cid:21) = E (cid:20) µ i + P i + ( t i )2 z π i ( t i ) − P i ( t i − )2 (cid:2) z π i − ( t i − ) + ρ i − − ρ i (cid:3) + z π i ( t i ) (cid:2) g i + ( t i ) + P i + ( t i )( ρ i − ρ i + ) (cid:3) − (cid:2) z π i − ( t i − ) + ρ i − − ρ i (cid:3) g i ( t i − ) (cid:21) = E (cid:20) µ i z π i ( t i ) − (cid:0) ρ i − − ρ i (cid:1) P i ( t i − )2 − ( ρ i − − ρ i ) g i ( t i − ) + P i + ( t i )2 z π i ( t i ) + P i + ( t i )( ρ i − ρ i + ) z π i ( t i ) + z π i ( t i ) g i + ( t i ) − P i ( t i − )2 z π i − ( t i − ) − P i ( t i − )( ρ i − − ρ i ) z π i − ( t i − ) − z π i − ( t i − ) g i ( t i − ) (cid:21) = E (cid:90) t i t i − (cid:26) P i ( t ) π ( t ) σ ( t ) σ ( t ) (cid:62) π ( t ) (cid:62) + γ ( t ) π ( t ) (cid:62) ( z π i ( t ) P i ( t ) + g i ( t )) + z π i ( t ) β ( t ) P i ( t ) + z π i ( t ) β ( t ) g i ( t ) + g i ( t ) ρ i r ( t ) (cid:27) d t = E (cid:90) t i t i − (cid:26)(cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) σ ( t ) P i ( t ) σ ( t ) (cid:62) (cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) (cid:62) − γ ( t )( P i ( t ) σ ( t ) σ ( t ) (cid:62) ) − γ ( t ) (cid:62) g i ( t ) + g i ( t ) ρ i r ( t ) (cid:27) d t , the third equality is derived by the following results, z π i ( t i − ) = Y π ( t i − ) − ρ i = Y π ( t i − ) − ρ i − + ρ i − − ρ i = z π i − ( t i − ) + ρ i − − ρ i , z π ( t ) = y , ρ = . Thus, we have E (cid:20) µ i z π i ( t i ) − (cid:0) ρ i − − ρ i (cid:1) P i ( t i − )2 − ( ρ i − − ρ i ) g i ( t i − ) + P i + ( t i )2 z π i ( t i ) + P i + ( t i )( ρ i − ρ i + ) z π i ( t i ) + z π i ( t i ) g i + ( t i ) − P i ( t i − )2 z π i − ( t i − ) − P i ( t i − )( ρ i − − ρ i ) z π i − ( t i − ) − z π i − ( t i − ) g i ( t i − ) (cid:21) = E (cid:90) t i t i − (cid:26)(cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) σ ( t ) P i ( t ) σ ( t ) (cid:62) (cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) (cid:62) − γ ( t )( P i ( t ) σ ( t ) σ ( t ) (cid:62) ) − γ ( t ) (cid:62) g i ( t ) + g i ( t ) ρ i r ( t ) (cid:27) d t , (3.7)Adding i on both sides of equation (3.7) from 1 to N , it follows that N (cid:88) i = E (cid:20) µ i z π i ( t i ) − (cid:0) ρ i − − ρ i (cid:1) P i ( t i − )2 − ( ρ i − − ρ i ) g i ( t i − ) + P i + ( t i )2 z π i ( t i ) + P i + ( t i )( ρ i − ρ i + ) z π i ( t i ) + z π i ( t i ) g i + ( t i ) − P i ( t i − )2 z π i − ( t i − ) − P i ( t i − )( ρ i − − ρ i ) z π i − ( t i − ) − z π i − ( t i − ) g i ( t i − ) (cid:21) = N (cid:88) i = E (cid:20) µ i z π i ( t i ) − (cid:0) ρ i − − ρ i (cid:1) P i ( t i − )2 − ( ρ i − − ρ i ) g i ( t i − ) (cid:21) − E (cid:20) P ( t )2 z π ( t ) + P ( t )( ρ − ρ ) z π ( t ) + z π ( t ) g ( t ) (cid:21) = N (cid:88) i = E (cid:90) t i t i − (cid:26)(cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) σ ( t ) P i ( t ) σ ( t ) (cid:62) (cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) (cid:62) − γ ( t )( P i ( t ) σ ( t ) σ ( t ) (cid:62) ) − γ ( t ) (cid:62) g i ( t ) + g i ( t ) ρ i r ( t ) (cid:27) d t , (3.8)11nd thus E (cid:20) N (cid:88) i = µ i z π i ( t i ) (cid:21) = N (cid:88) i = E (cid:90) t i t i − (cid:26)(cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) σ ( t ) σ ( t ) (cid:62) (cid:2) π ( t ) + γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π i ( t ) + g i ( t ) P i ( t ) ) (cid:3) (cid:62) − γ ( t )( P i ( t ) σ ( t ) σ ( t ) (cid:62) ) − γ ( t ) (cid:62) g i ( t ) + g i ( t ) ρ i r ( t ) (cid:27) d t + N (cid:88) i = E (cid:20)(cid:0) ρ i − − ρ i (cid:1) P i ( t i − )2 + ( ρ i − − ρ i ) g i ( t i − ) (cid:21) + E (cid:20) P ( t )2 z π ( t ) + P ( t )( ρ − ρ ) z π ( t ) + z π ( t ) g ( t ) (cid:21) . (3.9)Based on the representation of E (cid:2) N (cid:88) i = µ i z π i ( t i ) (cid:3) , we can obtain an optimal strat-egy π ∗ ( · ) for J ( π ( · )), for t ∈ ( t i − , t i ] , i = , , · · · , N , π ∗ ( t ) = − γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − ( z π ∗ i ( t ) + g i ( t ) P i ( t ) ) . Note that g i ( t ) P i ( t ) = g i ( t i ) P i ( t i ) e − (cid:82) tit r ( s )d s + ρ i (1 − e − (cid:82) tit r ( s )d s ) , t i − < t ≤ t i , where g i ( t i ) P i ( t i ) = g i + ( t i ) + P i + ( t i )( ρ i − ρ i + ) µ i + P i + ( t i ) , i = , , · · · , N , which leads to π ∗ ( t ) = γ ( t )( σ ( t ) σ ( t ) (cid:62) ) − (cid:2)(cid:0) ρ i − g i ( t i ) P i ( t i ) (cid:1) e − (cid:82) tit r ( s )d s − Y ∗ ( t ) (cid:3) , t i − < t ≤ t i , where Y ∗ ( t ) = z π ∗ ( t ) + ρ i , t i − < t ≤ t i and i = , , · · · , N .This completes the proof. (cid:3) π ∗ ( · ), d Y ∗ ( t ) = (cid:2) r ( t ) Y ∗ ( t ) + γ ( t ) π ∗ ( t ) (cid:62) (cid:3) d t + π ∗ ( t ) σ ( t )d W ( t ) , Y ∗ (0) = y . (3.10) E [ Y ∗ ( · )] and E [ Y ∗ ( · ) ] satisfy the following linear ordinary di ff erential equations: d E [ Y ∗ ( t )] = (cid:20) ( r ( t ) − β ( t )) E [ Y ∗ ( t )] + (cid:0) ρ i − g i ( t i ) P i ( t i ) (cid:1) e − (cid:82) tit r ( s )d s β ( t ) (cid:21) d t , Y ∗ (0) = y , t i − < t ≤ t i , i = , , · · · , N , (3.11)and d E [ Y ∗ ( t ) ] = (cid:20) (2 r ( t ) − β ( t )) E [ Y ∗ ( t ) ] + (cid:0) ρ i − g i ( t i ) P i ( t i ) (cid:1) e − (cid:82) tit r ( s )d s β ( t ) (cid:21) d t , Y ∗ (0) = y , t i − < t ≤ t i , i = , , · · · , N . (3.12)In the following, we investigate the e ffi cient frontier of the multi-time statemean-variance Var( Y ∗ ( t i )) and E [ Y ∗ ( t i )]. Lemma 3.1.
Let Assumptions H and H hold, the relationship of Var( Y ∗ ( t i )) and E [ Y ∗ ( t i )] is given as follows: Var( Y ∗ ( t i )) = Var( Y ∗ ( t i − )) e (cid:82) titi − [2 r ( t ) − β ( t )]d t + (cid:18) E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − r ( t )d t (cid:19) e (cid:82) titi − β ( t )d t − , (3.13) where i = , , · · · , N. Proof : Combining equations (3.11) and (3.12), we have for i = , , · · · , N , E [ Y ∗ ( t i )] = E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t + (cid:0) ρ i − g i ( t i ) P i ( t i ) (cid:1)(cid:0) − e − (cid:82) titi − β ( t )d t (cid:1) , (3.14)13nd E [ Y ∗ ( t i ) ] = E [ Y ∗ ( t i − ) ] e (cid:82) titi − [2 r ( t ) − β ( t )]d t + (cid:0) ρ i − g i ( t i ) P i ( t i ) (cid:1) (cid:0) − e − (cid:82) titi − β ( t )d t (cid:1) . (3.15)By equation (3.14), we have ρ i − g i ( t i ) P i ( t i ) = E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t − e − (cid:82) titi − β ( t )d t . Plugging ρ i − g i ( t i ) P i ( t i ) into equation (3.15), it follows that E [ Y ∗ ( t i ) ] = E [ Y ∗ ( t i − ) ] e (cid:82) titi − [2 r ( t ) − β ( t )]d t + (cid:18) E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t (cid:19) − e − (cid:82) titi − β ( t )d t , and thus Var( Y ∗ ( t i ))(1 − e − (cid:82) titi − β ( t )d t ) = (cid:0) E [ Y ∗ ( t i − ) ] − (cid:2) E Y ∗ ( t i − ) (cid:3) (cid:1) e (cid:82) titi − [2 r ( t ) − β ( t )]d t (1 − e − (cid:82) titi − β ( t )d t ) + [ E Y ∗ ( t i − )] (cid:0) e (cid:82) titi − [2 r ( t ) − β ( t )]d t − e (cid:82) titi − [2 r ( t ) − β ( t )]d t (cid:1) + (cid:18) E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t (cid:19) + ( e − (cid:82) titi − β ( s )d s − (cid:2) E Y ∗ ( t i ) (cid:3) = (cid:0) E [ Y ∗ ( t i − ) ] − (cid:2) E Y ∗ ( t i − ) (cid:3) (cid:1) e (cid:82) titi − [2 r ( t ) − β ( t )]d t (1 − e − (cid:82) titi − β ( t )d t ) + [ E Y ∗ ( t i − )] e (cid:82) titi − [2 r ( t ) − β ( t )]d t + [ E Y ∗ ( t i )] e − (cid:82) titi − β ( t )d t − E [ Y ∗ ( t i )] E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t = (cid:0) E [ Y ∗ ( t i − ) ] − (cid:2) E Y ∗ ( t i − ) (cid:3) (cid:1) e (cid:82) titi − [2 r ( t ) − β ( t )]d t (1 − e − (cid:82) titi − β ( t )d t ) + e − (cid:82) titi − β ( t )d t (cid:18) E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − r ( t )d t (cid:19) , Y ∗ ( t i )) = Var( Y ∗ ( t i − )) e (cid:82) titi − [2 r ( t ) − β ( t )]d t + (cid:18) E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − r ( t )d t (cid:19) e (cid:82) titi − β ( t )d t − . This completes the proof. (cid:3)
Remark 3.1.
Specially, for i = , one obtains Var( Y ∗ ( t )) = (cid:18) E [ Y ∗ ( t )] − ye (cid:82) t t r ( t )d t (cid:19) e (cid:82) t t β ( t )d t − , which is the same as the e ffi cient frontier in (Zhou and Li, 2000). It should be noted that the optimal strategy π ∗ ( · ) of cost functional (3.3) de-pends on the parameters µ = ( µ , · · · , µ N ) , λ = ( λ , · · · , λ N ) ∈ R N . We want toshow that there exist λ and µ such that the optimal strategy π ∗ ( · ) of cost functional(3.3) is an optimal strategy of cost functional (3.2). Theorem 3.2.
Let Assumptions H , H hold, andL i − L i − e (cid:82) titi − r ( t )d t > , i = , , · · · , N ; (cid:2) + P i + ( t i ) ρ i + − g i + ( t i ) (cid:3) (1 − e − (cid:82) titi − β ( t )d t ) > (cid:2) L i − L i − e (cid:82) titi − [ r ( t ) − β ( t )]d t (cid:3) P i + ( t i ) , i = , , · · · , N − , (3.16) where L = y. There exists λ ∗ = ( λ ∗ , λ ∗ , · · · , λ ∗ N ) , µ = ( µ , µ , · · · , µ N ) ∈ R N whichare determined by λ ∗ i = + µ i E [ Y ∗ ( t i )] , ρ i = λ ∗ i µ i , i = , , · · · , N , (3.17) such that the optimal strategy π ∗ ( · ) of cost functional (3.3) is an optimal strategyof cost functional (3.2). roof : By Theorem 3.1, an optimal strategy of model (3.2) can be solved by (3.3),let λ ∗ i = + µ i E [ Y ∗ ( t i )] , ρ i = λ ∗ i µ i , i = , , · · · , N . (3.18)Note that E [ Y ∗ ( t i )] depends on λ ∗ i . To solve the parameters λ ∗ i , i = , , · · · , N , byequation (3.14), we first consider the case i = N , E [ Y ∗ ( t N )] = E [ Y ∗ ( t N − )] e (cid:82) tNtN − [ r ( t ) − β ( t )]d t + λ ∗ N µ N (cid:2) − e − (cid:82) tNtN − β ( t )d t (cid:3) (3.19)and λ ∗ N = + µ N E [ Y ∗ ( t N − )] e (cid:82) tNtN − [ r ( t ) − β ( t )]d t + λ ∗ N (cid:2) − e − (cid:82) tNtN − β ( t )d t (cid:3) . Thus, we have λ ∗ N = e (cid:82) tNtN − β ( t )d t + µ N E [ Y ∗ ( t N − )] e (cid:82) tNtN − r ( t )d t . (3.20)Based on the representation of λ ∗ N , by equation (3.19), we have E [ Y ∗ ( t N )] = e (cid:82) tNtN − β ( t )d t − µ N + E [ Y ∗ ( t N − )] e (cid:82) tNtN − r ( t )d t , which indicates that µ N = e (cid:82) tNtN − β ( t )d t − E [ Y ∗ ( t N )] − E [ Y ∗ ( t N − )] e (cid:82) tNtN − r ( t )d t . (3.21)Based on constrained condition (2.3) of E [ Y ∗ ( t N )] = L N , E [ Y ∗ ( t N − )] = L N − andcondition (3.16), we can solve λ ∗ N and µ N > i = N −
1. By equations (3.14) and(3.18), one obtains λ ∗ N − = + µ N − E [ Y ∗ ( t N − )] , [ Y ∗ ( t N − )] − E [ Y ∗ ( t N − )] e (cid:82) tN − tN − [ r ( t ) − β ( t )]d t = (cid:0) ρ N − − g N − ( t N − ) P N − ( t N − ) (cid:1)(cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) = (cid:0) λ ∗ N − µ N − − g N ( t N − ) + P N ( t N − )( ρ N − − ρ N ) µ N − + P N ( t N − ) (cid:1)(cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) = λ ∗ N − + P N ( t N − ) ρ N − g N ( t N − ) µ N − + P N ( t N − ) (cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) , and thus E [ Y ∗ ( t N − )] − E [ Y ∗ ( t N − )] e (cid:82) tN − tN − [ r ( t ) − β ( t )]d t = λ ∗ N − + P N ( t N − ) ρ N − g N ( t N − ) µ N − + P N ( t N − ) (cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) . (3.22)It follows that, λ ∗ N − = + µ N − E [ Y ∗ ( t N − )] = + P N ( t N − ) ρ N − g N ( t N − ) u N − + P N ( t N − ) (cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) µ N − + µ N − E [ Y ∗ ( t N − )] e (cid:82) tN − tN − [ r ( t ) − β ( t )]d t + µ N − λ ∗ N − u N − + P N ( t N − ) (cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) . Note that the coe ffi cient of λ ∗ N − is I ∗ N − = P N ( t N − ) + µ N − e − (cid:82) tN − tN − β ( t )d t µ N − + P N ( t N − ) > , which indicates that there exists a unique solution for λ ∗ N − : λ ∗ N − = P N ( t N − ) + µ N − P N ( t N − ) + µ N − e − (cid:82) tN − tN − β ( t )d t + µ N − (cid:18) P N ( t N − ) ρ N − g N ( t N − ) P N ( t N − ) + µ N − e − (cid:82) tN − tN − β ( t )d t (cid:2) − e − (cid:82) tN − tN − β ( t )d t (cid:3) + P N ( t N − ) + u N − P N ( t N − ) + µ N − e − (cid:82) tN − tN − β ( t )d t E [ Y ∗ ( t N − )] e (cid:82) tN − tN − [ r ( t ) − β ( t )]d t (cid:19) . (3.23)17ombining equations (3.22) and (3.23), we have, µ N − = (cid:2) + P N ( t N − ) ρ N − g N ( t N − ) (cid:3) ( e (cid:82) tN − tN − β ( t )d t − E [ Y ∗ ( t N − )] − E [ Y ∗ ( t N − )] e (cid:82) tN − tN − r ( t )d t − (cid:2) E [ Y ∗ ( t N − )] − E [ Y ∗ ( t N − )] e (cid:82) tN − tN − [ r ( t ) − β ( t )]d t (cid:3) P N ( t N − ) e (cid:82) tN − tN − β ( t )d t E [ Y ∗ ( t N − )] − E [ Y ∗ ( t N − )] e (cid:82) tN − tN − r ( t )d t . (3.24)Again, based on constrained condition (2.3) of E [ Y ∗ ( t N − )] = L N − , E [ Y ∗ ( t N − )] = L N − and condition (3.16), we can solve λ ∗ N − and µ N − > i = N −
1, we can solve λ ∗ i , µ i , i = , , · · · , N − N − λ ∗ i = P i + ( t i ) + µ i P i + ( t i ) + µ i e − (cid:82) titi − β ( t )d t + µ i (cid:18) P i + ( t i ) ρ i + − g i + ( t i ) P i + ( t i ) + µ i e − (cid:82) titi − β ( t )d t (cid:2) − e − (cid:82) titi − β ( t )d t (cid:3) + P i + ( t i ) + u i P i + ( t i ) + µ i e − (cid:82) titi − β ( t )d t E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t (cid:19) , (3.25)and µ i = (cid:2) + P i + ( t i ) ρ i + − g i + ( t i ) (cid:3) ( e (cid:82) titi − β ( t )d t − E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − r ( t )d t − (cid:2) E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − [ r ( t ) − β ( t )]d t (cid:3) P i + ( t i ) e (cid:82) titi − β ( t )d t E [ Y ∗ ( t i )] − E [ Y ∗ ( t i − )] e (cid:82) titi − r ( t )d t . (3.26)Therefore, the optimal strategy π ∗ ( · ) of cost functional (3.3) is an optimal strategyof cost functional (3.2). This completes the proof. (cid:3) Remark 3.2.
The conditions (3.16) and (3.18) guarantee that cost functional (3.2)has an optimal strategy with the parameters λ ∗ and µ . However, we haven’t giventhe condition for L i , i = , , · · · , N to guarantee µ i > , i = , , · · · , N whichsatisfies conditions (3.16). In the following section, we give the condition for , L to guarantee µ , µ > for the case N = and solve λ ∗ = ( λ ∗ , λ ∗ ) , µ = ( µ , µ ) by E [ Y π ( t )] = L , E [ Y π ( t )] = L . In this section, we consider a simple example with N = λ ∗ = ( λ , λ ) , µ = ( µ , µ ) , ( E [ Y ∗ ( t )] , E [ Y ∗ ( t )]) , and variance (Var[ Y ∗ ( t )] , Var[ Y ∗ ( t )]).Furthermore, we compare our multi-time state mean-variance model with classi-cal mean-variance model. We suppose there are two assets, one bond and one stock, which are traded inthe market. Let d = , n = , N =
2, the bond satisfies, d R ( t ) = r ( t ) R ( t )d t , t > , R (0) = a > , and the stock asset is described by, d S ( t ) = b ( t ) S ( t )d t + σ ( t ) S ( t )d W ( t ) , t > , S (0) = s > . Our target is to minimize the following multi-time sate mean-variance problem: J ( π ( · )) = (cid:88) i = (cid:18) µ i Y π ( t i )) − E [ Y π ( t i )] (cid:19) , (4.1)19nd a tractable auxiliary problem is given as follows:ˆ J ( π ( · )) = (cid:88) i = E [ µ i Y π ( t i ) − λ i Y π ( t i )] . (4.2)Based on the results in Theorem 3.2 and formulas (3.20) and (3.25), we set λ ∗ = e (cid:82) t t β ( t )d t + µ E [ Y ∗ ( t )] e (cid:82) t t r ( t )d t ; λ ∗ = P ( t ) + µ P ( t ) + µ e − (cid:82) t β ( t )d t + µ (cid:18) P ( t ) ρ − g ( t ) P ( t ) + µ e − (cid:82) t β ( t )d t (cid:2) − e − (cid:82) t β ( t )d t (cid:3) + P ( t ) + µ P ( t ) + µ e − (cid:82) t β ( t )d t ye (cid:82) t [ r ( t ) − β ( t )]d t (cid:19) . The optimal strategy of model (4.2) is given as follows: π ∗ ( t ) = b ( t ) − r ( t ) σ ( t ) (cid:20)(cid:0) λ ∗ i µ i − g i ( t i ) P i ( t i ) (cid:1) e − (cid:82) tit r ( t )d t − Y ∗ ( t ) (cid:21) , t i − < t ≤ t i , i = , , and E [ Y ∗ ( t )] = e (cid:82) t t β ( t )d t − µ + E [ Y ∗ ( t )] e (cid:82) t t r ( t )d t ; E [ Y ∗ ( t )] = ye (cid:82) t [ r ( t ) − β ( t )]d t + (cid:0) λ ∗ µ − P ( t )( ρ − ρ ) + g ( t ) P ( t ) + µ (cid:1)(cid:0) − e − (cid:82) t β ( t )d t (cid:1) ; β ( t ) = (cid:18) b ( t ) − r ( t ) σ ( t ) (cid:19) , t ≤ t , (4.3)and ( P ( t ) , g ( t )) satisfies the following Riccati equations, d P ( t ) = (cid:2) β ( t ) − r ( t ) (cid:3) P ( t )d t , P ( t ) = µ + P ( t ) , t ≤ t < t , (4.4)20nd related equations, d g ( t ) = (cid:2) ( β ( t ) − r ( t )) g ( t ) − ρ r ( t ) P ( t ) (cid:3) d t , g ( t ) = g ( t ) + P ( t )( ρ − ρ ) , t ≤ t < t , (4.5)where P ( t ) = , g ( t ) = , ρ =
0. By a simple calculation, we can obtain that P ( t ) = µ e (cid:82) t t [2 r ( t ) − β ( t )]d t ; λ ∗ µ − g ( t ) P ( t ) = λ ∗ µ ; λ ∗ µ − g ( t ) P ( t ) = λ ∗ + λ ∗ e (cid:82) t t [ r ( t ) − β ( t )]d t µ + µ e (cid:82) t t [2 r ( t ) − β ( t )]d t . (4.6)Combining formulas (4.3) and (4.6), it follows that E [ Y ∗ ( t )] = ye (cid:82) t [ r ( t ) − β ( t )]d t + λ ∗ + λ ∗ e (cid:82) t t [ r ( t ) − β ( t )]d t µ + µ e (cid:82) t t [2 r ( t ) − β ( t )]d t (cid:2) − e − (cid:82) t β ( t )d t (cid:3) ; E [ Y ∗ ( t )] = e (cid:82) t t β ( t )d t − µ + E [ Y ∗ ( t )] e (cid:82) t t r ( t )d t ; λ ∗ = e (cid:82) t t β ( t )d t + µ E [ Y ∗ ( t )] e (cid:82) t t r ( t )d t ; λ ∗ = P ( t ) + µ P ( t ) + µ e − (cid:82) t β ( t )d t + µ (cid:18) P ( t ) ρ − g ( t ) P ( t ) + µ e − (cid:82) t β ( t )d t (cid:2) − e − (cid:82) t β ( t )d t (cid:3) + P ( t ) + µ P ( t ) + µ e − (cid:82) t β ( t )d t ye (cid:82) t [ r ( t ) − β ( t )]d t (cid:19) . (4.7)In the following, we set T = , y = t =
1, and t =
2. Let r ( t ) = r , b ( t ) = , σ ( t ) = σ, β ( t ) = β , where 0 ≤ t ≤ T . From formulas (4.7), we have E [ Y ∗ (1)] = e r − β + λ ∗ + λ ∗ e r − β µ + µ e r − β (cid:0) − e − β (cid:1) ; E [ Y ∗ ( t )] = e β − µ + E [ Y ∗ ( t )] e r ; λ ∗ = e β + µ E [ Y ∗ (1)] e r ; λ ∗ = µ (cid:18) λ ∗ ( e r − e r − β ) µ e r + µ + µ e r − β + µ e r µ e r + µ (cid:19) + µ e r + µ e β µ e r + µ . (4.8) Remark 4.1.
Let E [ Y ∗ (1)] = L , E [ Y ∗ (2)] = L , andL > L e r > e r ;( L − L e r ) e β > ( L − e r ) e r . (4.9) Note that, the condition L > L e r > e r guarantees that the constraints on themean values E [ Y ∗ (1)] = L , E [ Y ∗ (2)] = L are bigger than the return which isinvested into the risk-free asset bond, while the condition ( L − L e r ) e β > ( L − e r ) e r guarantees the parameter µ > in technique. Applying formulas (4.8), by a simple calculation, one obtains µ = e β − L − L e r ; λ ∗ = L e β − L e r L − L e r ; µ = ( e β − e β + λ ∗ e r ) − ( L e β − e r ) e r µ ( L − e r ) e β ; λ ∗ = µ (cid:18) λ ∗ ( e r − e r − β ) µ e r + µ + µ e r − β + µ e r µ e r + µ (cid:19) + µ e r + µ e β µ e r + µ . (4.10)22ased on Theorem 3.2, applying the formula (4.6), we can obtain the related op-timal strategy for the multi-time state mean-variance model (4.1) with the con-straints on means E [ Y ∗ (1)] = L , E [ Y ∗ (2)] = L , π ∗ ( t ) = b − r σ (cid:20) λ ∗ + λ ∗ e r − β µ + µ e r − β e r ( t − − Y ∗ ( t ) (cid:21) , ≤ t ≤ b − r σ (cid:20) λ ∗ µ e r ( t − − Y ∗ ( t ) (cid:21) , < t ≤ . (4.11)Thus, E [ Y ∗ ( · )] satisfies E [ Y ∗ ( t )] = e ( r − β ) t + λ ∗ + λ ∗ e r − β µ + µ e r − β [ e r ( t − − e ( r − β ) t − r ] , ≤ t ≤ E [ Y ∗ (1)] e ( r − β )( t − + λ ∗ µ [ e r ( t − − e r ( t − − β ( t − ] , < t ≤ , and from Lemma 3.1, the variance of Y ∗ ( · ) at t = , t = Y ∗ (1)) = (cid:18) E [ Y ∗ (1)] − e r (cid:19) e β − Y ∗ (2)) = Var( Y ∗ (1)) e r − β + (cid:18) E [ Y ∗ (2)] − E [ Y ∗ (1)] e r (cid:19) e β − . (4.12)Now, we show the results of case N = E [ Y (2)] = L ; µ = e β − L − e r ; λ ∗ = L e β − e r L − e r , where the related optimal strategy is π ( t ) = b − r σ (cid:20) λ ∗ µ e r ( t − − Y ( t ) (cid:21) , ≤ t ≤ . (4.13)23he mean E [ Y ( · )] and variance Var( Y ( · )) satisfy E [ Y ( t )] = e ( r − β ) t + λ ∗ e r ( t − µ [1 − e − β t ];Var( Y ( t )) = (cid:0) E [ Y ( t )] − e rt (cid:1) e β t − , ≤ t ≤ , (4.14)where E [ Y (2)] = L . Based on formulas (4.12) and (4.14), we have the followingcomparison results for (Var( Y ∗ (1)) , Var( Y ∗ (2))) and (Var( Y (1)) , Var( Y (2))): Corollary 4.1.
Suppose L and L satisfy condition (4.9), one obtains Var( Y ∗ (1)) < Var( Y (1));Var( Y ∗ (2)) > Var( Y (2)) . Furthermore, if L + e r − β + e r − β + e r + e − r ≤ L < L + e r − β e r − β + e r , we have Var( Y ∗ (1)) + Var( Y ∗ (2)) < Var( Y (1)) + Var( Y (2)) . Proof : By equality (4.14), we have E [ Y (1)] = L e β − r + e r e β + . Applying the condition ( L − L e r ) e β > ( L − e r ) e r in (4.9), one obtains e r < E [ Y ∗ (1)] = L < L e β − r + e r e β + = E [ Y (1)] , it follows that, Var( Y ∗ (1)) < Var( Y (1)) .
24y formula (4.12), we haveVar( Y ∗ (2)) = (cid:0) L − e r (cid:1) e β − e r − β + (cid:0) L − L e r (cid:1) e β − = [ e r − β + e r ] L − e r [ L + e r − β ] L + L + e r − β e β − = e r − β + e r e β − (cid:20) L − L e β − r + e r e β + (cid:21) + (cid:0) L − e r (cid:1) e β − . From equality (4.14), one obtainsVar( Y (2)) = (cid:0) L − e r (cid:1) e β − . It follows that, Var( Y ∗ (2)) > Var( Y (2)) . Furthermore, we haveVar( Y ∗ (1)) + Var( Y ∗ (2)) = (cid:0) L − e r (cid:1) e β − e r − β + + (cid:0) L − L e r (cid:1) e β − = [ e r − β + e r + L − e r [ L + e r − β + L + L + e r − β + e r e β − . It follows that Var( Y ∗ (1)) + Var( Y ∗ (2)) admits the minimum values at L = L + e r − β + e r − β + e r + e − r , Again, applying condition (4.9), we have L + e r − β + e r − β + e r + e − r < L + e r − β e r − β + e r = E [ Y (1)] . Notice that, if L = L + e r − β e r − β + e r , Y ∗ (1)) + Var( Y ∗ (2)) = Var( Y (1)) + Var( Y (2)) . Thus if L + e r − β + e r − β + e r + e − r ≤ L < L + e r − β e r − β + e r , we have Var( Y ∗ (1)) + Var( Y ∗ (2)) < Var( Y (1)) + Var( Y (2)) . This completes the proof. (cid:3)
Let r = . , b = . , σ = . , β = .
16, we show the simulation resultsof the case N =
2, and case N =
1, where case N = E [ Y ∗ ( · )] and E [ Y ( · )] M ean v a l ue The mean value of Y * (t)The mean value of Y (t) In Figure 1, we take L = e r , L = e . r which satisfies conditions (4.9). Theexpectations of Y ∗ ( · ) and Y ( · ) are given as follows, respectively, E [ Y ∗ ( t )] = e ( r − β ) t + λ ∗ + λ ∗ e r − β µ + µ e r − β [ e r ( t − − e ( r − β ) t − r ] , ≤ t ≤ E [ Y ∗ (1)] e ( r − β )( t − + λ ∗ µ [ e r ( t − − e r ( t − − β ( t − ] , < t ≤ , and E [ Y ( t )] = e ( r − β ) t + λ ∗ e r ( t − µ [1 − e − β t ] , ≤ t ≤ . From conditions (4.9), we obtain E [ Y ∗ (1)] = L < E [ Y (1)] and thus, E [ Y ∗ ( t )] < E [ Y ( t )] , < t < . t = , t = Y ∗ ( · ) and Y ( · ) Y * ( t ) The mean value of Y * (t) Y ( t ) The mean value of Y (t)
In Figure 2, we plot the values Y ∗ ( · ) and Y ( · ) in pathwise. The left one showsthat the pathwise of the function Y ∗ ( · ) along with E [ Y ∗ ( · )], while the right oneshows that of Y ( · ). We can see that the variance of Y ∗ (1) is bigger than thatof Y ∗ (1), and the variance of Y ∗ (2) is almost the same as that of Y ∗ (2). Thesephenomena verify the results of Corollary 4.1. In addition, In Figure 1, we cansee that E [ Y ∗ ( t )] < E [ Y ( t )] , < t ≤
1, while Figure 2 shows that the variance of Y ∗ ( · ) is smaller than that of Y ( · ) before time 1.28igure 3: The average of Maximum-Drawdown of Y ∗ ( · ) along with θ θ M D y Maximum Drawdown of Y * with t Maximum Drawdown of Y * with t In Figure 3, we plot the function of E [MD Y ∗ ] along with θ ∈ [1 . , . hY ∗ = esssup { z | z = Y ∗ ( t ) − Y ∗ ( s ) , ≤ t ≤ s ≤ h } , < h ≤
2, and L = e θ r , E [ Y (1)] = e . r . We can see that E [MD t Y ∗ ] is decreasing with θ ∈ [1 . , . θ ∈ [2 . , . L ∈ [ e . r , e . r ], increasing with L ∈ [ e . r , e . r ], while E [MD t Y ∗ ] is increasing with L ∈ [ e . r , e . r ], where t = , t =
2. 29
Conclusion
For given 0 = t < t < · · · < t N = T , to reduce the variance of the mean-variance model at the multi-time state ( Y π ( t ) , · · · , Y π ( t N )), we propose a multi-time state mean-variance model with a constraint on the multi-time state meanvalue. In the proposed model, we solve the multi-time state mean-variance modelby introducing a new sequence of Riccati equations.Our main results are as follows: • We can use the multi-time state mean-variance model to manage the risk ofthe investment portfolio along the multi-time 0 = t < t < · · · < t N = T . • A new sequence of Riccati equations which are connected by a jump bound-ary condition are introduced, based on which we find an optimal strategy forthe multi-time state mean-variance model. • Furthermore, the relationship of the means and variances of this multi-time state mean-variance model is established and is similar to the classicalmean-variance model. • An example is employed to show that minimizing the variances for multi-time state can a ff ect the average value of Maximum-Drawdown of the in-vestment portfolio. References
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