Optimal mean-variance portfolio selection

Transcription

1 Math Finan Econ 2017) 11: DOI /s Optimal mean-variance portfolio selection Jesper Lnd Pedersen 1 Goran Peskir 2 Received: 26 November 2015 / Accepted: 20 May 2016 / Pblished online: 20 Jne 2016 The Athors) This article is pblished with open access at Springerlink.com Abstract Assming that the wealth process X is generated self-financially from the given initial wealth by holding its fraction in a risky stock whose price follows a geometric Brownian motion with drift μ R and volatility σ>0) and its remaining fraction 1 in a riskless bond whose price componds exponentially with interest rate r R), and letting P t,x denote a probability measre nder which X takes vale x at time t, westdythe dynamic version of the nonlinear mean-variance optimal control problem sp E t,x X t T ) c Var t,x t X T ) where t rns from 0 to the given terminal time T > 0, the spremm is taken over admissible controls, andc > 0 is a given constant. By employing the method of Lagrange mltipliers we show that the nonlinear problem can be redced to a family of linear problems. Solving the latter sing a classic Hamilton-Jacobi-Bellman approach we find that the optimal dynamic control is given by t, x) = δ 2c σ 1 x eδ2 r)t t) where δ = μ r)/σ. The dynamic formlation of the problem and the method of soltion are applied to the constrained problems of maximising/minimising the mean/variance sbject to the pper/lower bond on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself. B Goran Peskir goran@maths.man.ac.k Jesper Lnd Pedersen jesper@math.k.dk 1 Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark 2 School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

2 138 Math Finan Econ 2017) 11: Keywords Nonlinear optimal control Static optimality Dynamic optimality Mean-variance analysis The Hamilton Jacobi Bellman eqation Martingale Geometric Brownian motion Markov process Mathematics Sbject Classification Primary 60H30, 60J65 Secondary 49L20, 91G80 JEL Classification C61 G11 1 Introdction Imagine an investor who has an initial wealth which he wishes to exchange between a risky stock and ariskless bond in a self-financing manner dynamically in time so as to maximise his retrn and minimise his risk at the given terminal time. In line with the mean-variance analysis of Markowitz 11 where the optimal portfolio selection problem of this kind was solved in a single period model see e.g. Merton 12 and the references therein) we will identify the retrn with the expectation of the terminal wealth and the risk with the variance of the terminal wealth. The qadratic nonlinearity of the variance then moves the reslting optimal control problem otside the scope of the standard optimal control theory see e.g. 5) which may be viewed as dynamic programming in the sense of solving the Hamilton Jacobi Bellman HJB) eqation and obtaining an optimal control which remains optimal independently from the initial and hence any sbseqent) vale of the wealth. Conseqently the reslts and methods of the standard/linear optimal control theory are not directly applicable in this new/nonlinear setting. The prpose of the present paper is to develop a new methodology for solving nonlinear optimal control problems of this kind and demonstrate its se in the optimal mean-variance portfolio selection problem stated above. This is done in parallel to the novel methodology for solving nonlinear optimal stopping problems that was recently developed in 13 when tackling an optimal mean-variance selling problem. Assming that the stock price follows a geometric Brownian motion and the bond price componds exponentially, we first consider the constrained problem in which the investor aims to maximise the expectation of his terminal wealth XT over all admissible controls representing the fraction of the wealth held in the stock) sch that the variance of XT is bonded above by a positive constant. Similarly the investor cold aim to minimise the variance of his terminal wealth XT over all admissible controls sch that the expectation of XT is bonded below by a positive constant. A first application of Lagrange mltipliers implies that the Lagrange fnction Lagrangian) for either/both constrained problems can be expressed as a linear combination of the expectation of XT and the variance of X T with opposite signs. Optimisation of the Lagrangian over all admissible controls ths yields the central optimal control problem nder consideration. De to the qadratic nonlinearity of the variance we can no longer apply standard/linear reslts of the optimal control theory to solve the problem. Conditioning on the size of the expectation we show that a second application of Lagrange mltipliers redces the nonlinear optimal control problem to a family of linear optimal control problems. Solving the latter sing a classic HJB approach we find that the optimal control depends on the initial point of the controlled wealth process in an essential way. This spatial inconsistency introdces a time inconsistency in the problem that in trn raises the qestion whether the optimality obtained is adeqate for practical prposes. We refer to this optimality as the static optimality Definition 1) to distingish it from the dynamic optimality Definition 2) in which each new position of the controlled wealth process yields a new optimal

3 Math Finan Econ 2017) 11: control problem to be solved pon overrling all the past problems. This in effect corresponds to solving infinitely many optimal control problems dynamically in time with the aim of determining the optimal control in the sense that no other control applied at present time cold prodce a more favorable vale at the terminal time). While the static optimality has been sed in the paper by Strotz 21 nder the name of pre-commitment as far as we know the dynamic optimality has not been stdied in the nonlinear setting of optimal control before. In Sect. 4 below we give a more detailed accont of the mean-variance reslts and methods on the static optimality starting with the paper by Richardson 19. Optimal controls in all these papers are time inconsistent in the sense described above. This line of papers ends with the paper by Basak and Chabakari 1 where a time-consistent control is derived that corresponds to the Strotz s approach of consistent planning 21 realised as the sbgame-perfect Nash eqilibrim the optimality concept refining Nash eqilibrim proposed by Selten in 1965). We show that the dynamic formlation of the nonlinear optimal control problem admits a simple closed-form soltion Theorem 3) in which the optimal control no longer depends on the initial point of the controlled wealth process and hence is time consistent. Remarkably we also verify that this control yields the expected terminal vale which i) coincides with the expected terminal vale obtained by the statically optimal control Remark 4) and moreover ii) dominates the expected terminal vale obtained by the sbgame-perfect Nash eqilibrim control in the sense of Strotz s consistent planning ) derived in 1 Sect. 4). Closed-form soltions to the constrained problems are then derived sing the soltion to the nconstrained problem Corollaries 5 and 7). These reslts are of both theoretical and practical interest. In the first problem we note that the optimal wealth exhibits a dynamic compliance effect Remark 6) and in the second problem we observe that the optimal wealth solves a meander type eqation of independent interest Remark 8). In both problems we verify that the expected terminal vale obtained by the dynamically optimal control dominates the expected terminal vale obtained by the statically optimal control. The novel problems and methodology of the present paper sggest a nmber of avenes for frther research. Firstly, we work within the transparent setting of one-dimensional geometric Brownian motion in order to illstrate the main ideas and describe the new methodology withot nnecessary technical complications. Extending the reslts to higher dimensions and more general diffsion/markov processes appears to be worthy of frther consideration. Secondly, for similar tractability reasons we assme that i) nlimited short-selling and borrowing are permitted, ii) transaction costs are zero, iii) the wealth process may take both positive and negative vales of nlimited size. Extending the reslts nder some of these constraints being imposed is also worthy of frther consideration. In both of these settings it is interesting to examine to what extent the reslts and methods laid down in the present paper remain valid nder any of these more general or restrictive hypotheses. 2 Formlation of the problem Assme that the riskless bond price B solves db t = rb t dt 2.1) with B 0 = b for some b > 0wherer R is the interest rate, and let the risky stock price S follow a geometric Brownian motion solving ds t = μs t dt + σ S t dw t 2.2)

4 140 Math Finan Econ 2017) 11: with S 0 = s for some s > 0whereμ R is the drift, σ > 0 is the volatility, and W is a standard Brownian motion defined on a probability space, F, P). Note that a niqe soltion to 2.1) isgivenbyb t = be rt and recall that a niqe strong soltion to 2.2) is given by S t = s expσ W t +μ σ 2 /2))t) for t 0. Consider the investor who has an initial wealth x 0 R which he wishes to exchange between B and S in a self-financing manner with no exogenos infsion or withdrawal of wealth) dynamically in time p to the given horizon T > 0. It is then well known see e.g. 2, Chapter 6) that the investor s wealth X solves dx t = r 1 t )+μ t ) X t dt + σ t X t dw t 2.3) with Xt 0 = x 0 where t denotes the fraction of the investor s wealth held in the stock at time t, T for 0, T ) given and fixed. Note that i) t < 0 corresponds to short selling of the stock, ii) t > 1 corresponds to borrowing from the bond, and iii) t 0, 1 corresponds to a long position in both the stock and the bond. To simplify the exposition we will assme that the control in 2.3) isgivenby t = t, Xt ) where t, x) t, x) x is a continos fnction from 0, T R into R for which the stochastic differential eqation 2.3) nderstood in Itô s sense has a niqe strong soltion X meaning that the soltion X to 2.3) is adapted to the natral filtration of W and if X is another soltion to 2.3) of this kind then X and X are eqal almost srely). We will call controls of this kind admissible in the seqel. Recalling that the natral filtration of S coincides with the natral filtration of W we see that admissible controls have a natral financial interpretation as they are obtained as deterministic measrable) fnctionals of the observed stock price. Moreover, adopting the convention that t, 0) 0 := lim 0 =x 0 t, x) x we see that the soltion X to 2.3) cold take both positive and/or negative vales after passing throgh zero when the latter limit is different from zero as is the case in the main reslts below). This convention corresponds to re-expressing 2.3) in terms of the total wealth t Xt held in the stock as opposed to its fraction t which we follow throghot note that the essence of the wealth eqation 2.3) remains the same in both cases). We do always identify t, 0) with t, 0) 0howeversincex t, x) may not be well defined a. Note that the reslts to be presented below also hold if the set of admissible controls is enlarged to inclde discontinos and path dependent controls that are adapted to the natral filtration of W, or even controls which are adapted to a larger filtration still making W a martingale so that 2.3) has a niqe weak soltion X meaning that the soltion X to 2.3) is adapted to the larger filtration and if X is another soltion to 2.3) of this kind then X and X are eqal in law). Since these extensions follow along the same lines and needed modifications of the argments are evident, we will omit frther details in this direction and focs on the set of admissible controls as defined above. For a given admissible control we let P t,x denote the probability measre defined on the canonical space) nder which the soltion X to 2.3) takes vale x at time t for t, x) 0, T R. Note that X is a strong) Markov process with respect to P t,x for t, x) 0, T R. Consider the optimal control problem V t, x) = sp Et,x XT ) c Var t,xxt ) 2.4) where the spremm is taken over all admissible controls sch that E t,x XT )2 < for t, x) 0, T R and c > 0 is a given and fixed constant. A sfficient condition for the T latter expectation to be finite is that E t,x t 1+ 2 s )X s )2 ds < and we will assme in the seqel that all admissible controls by definition satisfy that condition as well.

5 Math Finan Econ 2017) 11: De to the qadratic nonlinearity of the second term in the expression Var t,x XT ) = E t,x XT )2 E t,x XT )2 it is evident that the problem 2.4) falls otside the scope of the standard/linear optimal control theory for Markov processes see e.g. 5). Moreover, we will see below that in addition to the static formlation of the nonlinear problem 2.4)wherethe maximisation takes place relative to the initial point t, x) which is given and fixed, one is also natrally led to consider a dynamic formlation of the nonlinear problem 2.4)inwhich each new position of the controlled process t, Xt )) t 0,T yields a new optimal control problem to be solved pon overrling all the past problems. We believe that this dynamic optimality is of general interest in the nonlinear problems of optimal control as well as nonlinear problems of optimal stopping as discssed in 13). The problem 2.4) seeks to maximise the investor s retrn identified with the expectation of XT and minimise the investor s risk identified with the variance of X T pon applying the control. This identification is done in line with the mean-variance analysis of Markowitz 11. Moreover, we will see in the proof below that the problem 2.4) is obtained by optimising the Lagrangian of the constrained problems V 1 t, x) = V 2 t, x) = sp E t,x X : Var t,x XT ) α T ) 2.5) inf Var t,xx : E t,x XT ) β T ) 2.6) respectively, where is any admissible control, and α 0, ) and β R are given and fixed constants. Solving 2.4) we will therefore be able to solve 2.5)and2.6) as well. Note that the constrained problems have transparent interpretations in terms of the investor s retrn and the investor s risk as discssed above. We now formalise definitions of the optimalities allded to above. Recall that all controls throghot refer to admissible controls as defined/discssed above. Definition 1 Static optimality). A control is statically optimal in 2.4)fort, x) 0, T R given and fixed, if there is no other control v sch that E t,x XT v ) c Var t,xxt v )>E t,xx T ) c Var t,xx T ). 2.7) A control is statically optimal in 2.5) fort, x) 0, T R given and fixed, if Var t,x X T ) α and there is no other control v satisfying Var t,xxt v ) α sch that E t,x XT v )>E t,xx T ). 2.8) A control is statically optimal in 2.6) fort, x) 0, T R given and fixed, if E t,x X T ) β and there is no other control v satisfying E t,xxt v ) β sch that Var t,x XT v )<Var t,xx T ). 2.9) Note that the static optimality refers to the optimality relative to the initial point t, x) which is given and fixed. Changing the initial point may yield a different optimal control in the nonlinear problems since the statically optimal controls may and generally will depend on the initial point in an essential way cf. 21). This stands in sharp contrast with standard/linear problems of optimal control where in view of dynamic programming the HJB eqation) the optimal control does not depend on the initial point explicitly. This is a key difference between the static optimality in nonlinear problems of optimal control and the standard optimality in linear problems of optimal control cf. 5).

6 142 Math Finan Econ 2017) 11: Definition 2 Dynamic optimality). A control is dynamically optimal in 2.4), if for every given and fixed t, x) 0, T R and every control v sch that vt, x) = t, x), there exists a control w satisfying wt, x) = t, x) sch that E t,x XT w ) c Var t,xxt w )>E t,xxt v ) c Var t,xxt v ). 2.10) A control is dynamically optimal in 2.5), if for every given and fixed t, x) 0, T R and every control v sch that vt, x) = t, x) with Var t,x XT v ) α, there exists a control w satisfying wt, x) = t, x) with Var t,x XT w ) α sch that E t,x XT w )>E t,xxt v ). 2.11) A control is dynamically optimal in 2.6), if for every given and fixed t, x) 0, T R and every control v sch that vt, x) = t, x) with E t,x XT v ) β, there exists a control w satisfying wt, x) = t, x) with E t,x XT w ) β sch that Var t,x XT w )<Var t,xxt v ). 2.12) Dynamic optimality above is nderstood in the strong sense. Replacing the strict ineqalities in 2.10) 2.12) by ineqalities wold yield dynamic optimality in the weak sense. Note that the dynamic optimality corresponds to solving infinitely many optimal control problems dynamically in time where each new position of the controlled process t, X t )) t 0,T yields a new optimal control problem to be solved pon overrling all the past problems. The optimal decision at each time tells s to exert the best control among all possible controls. While the static optimality remembers the past throgh the initial point) the dynamic optimality completely ignores it and only looks ahead. Nonetheless it is clear that there is a strong link between the static and dynamic optimality the latter being formed throgh the beginnings of the former as shown below) and this will be exploited in the proof below when searching for the dynamically optimal controls. In the case of standard/linear optimal control problems for Markov processes it is evident that the static and dynamic optimality coincide nder mild reglarity conditions de to the fact that dynamic programming the HJB eqation) is applicable. This is not the case for the nonlinear problems of optimal control considered in the present paper as it will be seen below. 3 Soltion to the problem In this section we present soltions to the problems formlated in the previos section. We first focs on the nconstrained problem. Theorem 3 Consider the optimal control problem 2.4) where X solves 2.3) with Xt 0 = x 0 nder P t0,x 0 for, x 0 ) 0, T R given and fixed. Recall that B solves 2.1), S solves 2.2), and we set δ = μ r)/σ for μ R, r R and σ>0. We assme throghot that δ = 0 and r = 0 the cases δ = 0 or r = 0 follow by passage to the limit when the non-zero δ or r tends to 0). A) The statically optimal control is given by s t, x) = δ σ 1 x x 0 e rt ) x + 1 2c eδ2 T ) rt t) 3.1)

7 Math Finan Econ 2017) 11: for t, x), T R. The statically optimal controlled process is given by Xt s = x 0 e rt t0) + 1 2c eδ2 r)t t) e δ2 t ) e δw t W t0 ) δ2 2 t ) B t = x Bt ) 1 δ2/r Bt ) δ2/r Bt ) δ1/σ +δ σ)/2r) St ) δ/σ B t0 2c B T B t0 B t0 S t0 for t, T. The static vale fnction V s := EXT s ) c VarX T s ) is given by V s, x 0 ) = x 0 e rt t0) + 1 e δ2 T ) 1 4c for, x 0 ) 0, T R. B) The dynamically optimal control is given by d t, x) = δ 2c σ 1 x eδ2 r)t t) for t, x), T R. The dynamically optimal controlled process is given by Xt d = x 0 e rt t0) + 1 t 2c eδ2 r)t t) e δ2 t ) 1 + δ e δ2 t s) dw s B t = x 0 + δ Bt ) 1 δ2/r σ 2 2r ) Bt ) δ2/r 1 B t0 2cσ B T 2δ 2 B t0 St ) t + log + δ 2 Bt ) δ2/r Ss ) log ds S t0 B s S t0 for t, T. The dynamic vale fnction V d := EXT d ) c VarX T d ) is given by V d, x 0 ) = x 0 e rt t0) + 1 e δ2 T ) 1 2c 4 e2δ2 T ) ) 3.3) 3.4) 3.5) 3.6) for, x 0 ) 0, T R. Proof We assme throghot that the process X solves the stochastic differential eqation 2.3) with X = x 0 nder P t0,x 0 for, x 0 ) 0, T R given and fixed where is any admissible control as defined/discssed above. To simplify the notation we will drop the sbscript zero from and x 0 in the first part of the proof below. A): Note that the objective fnction in 2.4) reads E t,x X T ) c Var t,xx T ) = E t,xx T ) + c E t,x X T ) 2 c Et,x X T ) 2 3.7) where the key difficlty is the qadratic nonlinearity of the middle term on the right-hand side. To overcome this difficlty we will condition on the size of E t,x XT ). This yields V t, x) = sp sp Et,x X M R : E t,x XT )=M T ) c Var t,xxt ) = sp sp E t,x XT ) + c E t,x XT ) 2 c Et,x X T ) 2 M R : E t,x XT )=M = sp M R M + cm 2 c inf : E t,x X T )=M E t,x X T ) )

8 144 Math Finan Econ 2017) 11: Hence to solve 3.8) and ths 2.4) we need to solve the constrained problem V M t, x) = inf E : E t,x XT )=M t,x X T ) 2 3.9) for M R given andfixedwhere is any admissible control. 1. To tackle the problem 3.9) we will apply the method of Lagrange mltipliers. For this, define the Lagrangian as follows L t,x,λ)= E t,x X T ) 2 λ E t,x X T ) M 3.10) for λ R and let λ denote the optimal control in the nconstrained problem L t,x λ,λ):= inf L t,x,λ) 3.11) pon assming that it exists. Sppose moreover that there is λ=λm, t, x) R sch that E t,x X λ ) T = M. 3.12) It then follows from 3.10) 3.12)that E t,x X λ ) 2 T = Lt,x λ,λ) E t,x X T ) ) for any admissible control sch that E t,x XT ) = M. This shows that λ satisfying 3.11) and 3.12) is optimal in 3.9). 2. To tackle the problem 3.11) with 3.12) we consider the optimal control problem V λ t, x) = inf E t,x X T ) 2 λ XT 3.14) where is any admissible control. This is a standard/linear problem of optimal control see e.g. 5) that can be solved sing a classic HJB approach. For the sake of completeness we present key steps in the derivation of the soltion. From 3.14) combined with 2.3) we see that the HJB system reads Vt λ + r1 )+μ ) xvx λ σ 2 2 x 2 Vxx λ = ) inf R V λ T, x) = x 2 λx 3.16) on 0, T R. Making the ansatz that Vxx λ > 0 and minimising the qadratic fnction of over R in 3.15) wefindthat = δ 1 Vx λ σ x Vxx λ. 3.17) Inserting 3.17) back into 3.15) yields V λ t + rxv λ x δ2 2 Seeking the soltion to 3.18) of the form and making se of 3.16) wefindthat V λ x )2 V λ xx = ) V λ t, x) = at)x 2 + bt)x + ct) 3.19) a t) = δ 2 2r)at) & b t) = δ 2 r)bt) & c t) = δ2 4 b 2 t) at) 3.20)

9 Math Finan Econ 2017) 11: on 0, T with at ) = 1, bt ) = λand ct ) = 0. Solving 3.20) nder these terminal conditions we obtain at) = e δ2 2r)T t) & bt) = λe δ2 r)t t) & ct) = λ2 1 e δ2 T t) ) Inserting 3.21) into3.19) and calclating 3.17)wefindthat t, x) = δ 1 x λ σ x 2 e rt t). 3.22) Applying Itô s formla to the process Z defined by Z t = K e rt ) X t 3.23) wherewesetk := λ/2)e rt ) and making se of 2.3)wefindthat dz t = δ 2 Z t dt δ Z t dw t 3.24) with Z t0 = K x 0 nder P t0,x 0. Solving the linear eqation 3.24) explicitly we obtain the following closed form expression X t = e rt ) K K x 0 )e δw t W t0 ) 3δ2 2 t ) 3.25) for t, T. The process X defined by 3.25) is a niqe strong soltion to the stochastic differential eqation 2.3) obtained by the control from 3.22) and yielding the vale fnction V λ given in 3.19) combined with 3.21) above. It is then a matter of rotine to apply Itô s formla to V λ composed with t, Xt v ) for any admissible control v and sing 3.15)+3.16) verify that the candidate control from 3.22) is optimal in 3.14) as envisaged these argments are displayed more explicitly in 3.36) 3.37)below). 3. Having solved the problem 3.14) we still need to meet the condition 3.12). For this, we find from 3.25)that ) E t0,x 0 X T = x0 e δ2 r)t ) + λ 1 e δ2 T ). 3.26) 2 To realise 3.12) we need to identify 3.26) with M. This yields λ = 2 M x 0e δ2 r)t ) 3.27) 1 e δ2 T ) for δ = 0. Note that the case δ = 0 is evident since in this case λ = 0 is optimal in 3.14) for every λ R and hence the ineqality in 3.13) holds for every admissible control while from 2.3) we also easily see that 3.26) with δ = 0) holds for every admissible control so that we only have one M possible in 3.8) and that is the one given by 3.26) with δ = 0). This shows that 3.1) 3.3) are valid when δ = 0 and we will therefore assme in the seqel that δ = 0. Moreover, from 3.25) we also find that E t0,x 0 X T ) 2 = x0 2 e δ2 2r)T ) + λ2 1 e δ2 T ). 3.28) 4 Note that this expression can also be obtained from 3.14) and3.26) pon recalling 3.19) with 3.21) above. Inserting 3.27)into3.28) and recalling 3.13) we see that 3.9)isgiven by V M, x 0 ) = x0 2 e δ2 2r)T M ) x0 e δ2 r)t ) ) ) 1 e δ2 T )

10 146 Math Finan Econ 2017) 11: for δ = 0. Inserting 3.29) into3.8)weget V, x 0 ) = sp M + cm 2 c x0 2 e δ2 2r)T M ) x0 e δ2 r)t ) ) 2 ) + M R 1 e δ2 T ) 3.30) for δ = 0. Note that the fnction of M to be maximised on the right-hand side is qadratic with the coefficient in front of M 2 strictly negative when δ = 0. This shows that there exists a niqe maximm point in 3.30) that is easily fond to be given by M = x 0 e rt ) + 1 2c Inserting 3.31) into3.27) wefindthat e δ2 T ) ) λ = 2x 0 e rt ) + 1 c eδ2 T ). 3.32) Inserting 3.32) into3.22) we establish the existence of the optimal control in 2.4) thatis given by 3.1) above. Moreover, inserting 3.32) into3.25) we obtain the first identity in 3.2). The second identity in 3.2) then follows pon recalling the closed form expressions for B and S stated following 2.2) above. Finally, inserting 3.31) into3.30) we obtain 3.3) and this completes the first part of the proof. B): Identifying with t and x 0 with x in the statically optimal control s from 3.1)we obtain the control d from 3.4). We claim that this control is dynamically optimal in 2.4). For this, take any other admissible control v sch that v, x 0 ) = d, x 0 ) and set w = s. Then w, x 0 ) = d, x 0 ) and we claim that V w, x 0 ) := E t0,x 0 X w T ) c Var,x 0 X w T )>E,x 0 X v T ) c Var,x 0 X v T ) =: V v, x 0 ) 3.33) pon noting that V w, x 0 ) eqals V, x 0 ) since w is statically optimal in 2.4). 4. To verify 3.33) setm v := E t0,x 0 X v T ) and first consider the case when M v = M where M is given by 3.31) above. Using 3.9)+3.29)and3.30)+3.31) we then find that V v, x 0 ) = M v + cmv 2 c E,x 0 X v T ) 2 M v + cmv 2 cv M v, x 0 ) M v + cmv x 2 c 0 2 e δ2 2r)T ) Mv x 0 e δ2 r)t ) ) 2 ) + 1 e δ2 T ) < M + cm x 2 c 0 2 e δ2 2r)T ) M x 0 e δ2 r)t ) ) 2 ) + 1 e δ2 T ) = V w, x 0 ) 3.34) for δ = 0 where the strict ineqality follows since M is the niqe maximm point of the qadratic fnction as pointed ot following 3.30) above. The case δ = 0 is exclded since then as pointed ot following 3.27) above we only have M possible in 3.8) sothatm v wold be eqal to M. This shows that 3.33) is satisfied when M v = M as claimed. Next consider the case when M v = M. We then claim that V λ v t, x) := E,x 0 X v T ) 2 λ X v T > V λ, x 0 ) 3.35)

11 Math Finan Econ 2017) 11: where V λ is defined in 3.14)andλ is given by 3.32) above. For this, note that sing 3.16) and applying Itô s formla we get XT v )2 λ XT v = V λ T, XT v ) = V λ, x 0 ) T + V λ t s, X v s ) + r 1 vs, X v s )) +μvs, X v s ) X v s V λ x s, X v s ) σ 2 v 2 s, X v s )X v s )2 V λ xx s, X v s ) ds + M T 3.36) where M t := t σ vs, Xs v) X s v V λ x s, Xs v) dw s is a continos local martingale nder T P t0,x 0 for t, T. Using that E t0,x vt 2)X t v)2 dt < it is easily seen from 2.3) by means of Jensen s and Brkholder Davis Gndy s ineqalities that E t0,x 0 max t0 t T Xt v)2 < and hence pon recalling 3.19)+3.21) above with λ in place of λ) it follows by Hölder s ineqality that E t0,x 0 M, M T < so that M is a martingale. Taking E t0,x 0 on both sides of 3.36) we therefore get V λ v t, x) = V λ, x 0 ) T + E t0,x 0 V λ t s, X v s ) + r 1 vs, X v s )) +μvs, X v s ) X v s V λ x s, X v s ) σ 2 v 2 s, X v s )X v s )2 V λ xx s, X v s ) ds 3.37) where the integrand is non-negative de to 3.15) with λ in place of λ). Since d, x 0 ) = s, x 0 ) and v, x 0 ) = d, x 0 ) we see that v, x 0 ) = w, x 0 ). Assming x 0 = 0 by the continity of v and w it then follows that vs, x) = ws, x) for all s, x) R ε :=, +ε x 0 ε, x 0 +ε for some ε>0 small enogh sch that +ε T as well. Moreover, since wt, x) is the niqe minimm point of the continos fnction on the left-hand side of 3.15) with λ in place of λ) evalated at t, x) for every t, x) 0, T R, weseethat this ε>0 can be chosen small enogh so that V λ t + r1 v)+μv ) xv λ x σ 2 v 2 x 2 V λ xx β>0 3.38) on R ε for some β>0 given and fixed. Setting τ ε := inf { s, +ε s, X v s )/ R ε } we see by 3.37)and3.38)that V λ v t, x) V λ, x 0 ) + β E t0,x 0 τ ε )>V λ, x 0 ) 3.39) where in the first ineqality we se that the integrand in 3.37) is non-negative as pointed ot above and in the final strict) ineqality we se that τ ε > with P t0,x 0 -probability one de to the continity of X v. The argments remain also valid when x 0 = 0 pon recalling that v, 0) and w, 0) are identified with v, 0) 0andw, 0) 0 in this case. From 3.39) we see that 3.35) holds as claimed. Recalling from 3.10) 3.13) thatv λ, x 0 ) = V M, x 0 ) λ M as well as that M v = M by hypothesis we see from 3.35)that E t0,x 0 X v T ) 2 > V M, x 0 ). 3.40) It follows therefore from 3.8)that V w, x 0 ) = M + cm 2 cv M, x 0 )>M + cm 2 c E,x 0 X v T ) 2 = E t0,x 0 XT v ) c Var,x 0 XT v ) = V v, x 0 ). 3.41)

12 148 Math Finan Econ 2017) 11: This shows that 3.33) holds when M v = M as well and hence we can conclde that the control d from 3.4) is dynamically optimal as claimed. 5. Applying Itô s formla to e rt t) Xt d wherewesetx d := X d and making se of 2.3) we easily find that the first identity in 3.5) is satisfied. Integrating by parts and recalling the closed form expressions for B and S stated following 2.2) above we then establish that the second identity in 3.5) also holds. From the first identity in 3.5) weget ) E t0,x 0 X d T = x0 e rt t0) + 1 e δ2 T ) ) 2c Var t0,x 0 XT d ) = 1 8c 2 e 2δ2 T ) ) From 3.42)and3.43) we obtain 3.6) and this completes the proof. Remark 4 The dynamically optimal control d from 3.4) by its natre rejects any past point, x 0 ) to measre its performance so that althogh the static vale V s, x 0 ) by its definition dominates the dynamic vale V d, x 0 ) this comparison is meaningless from the standpoint of the dynamic optimality. Another isse with a plain comparison of the vales V s t, x) and V d t, x) for t, x), T R is that the optimally controlled processes X s and X d may never come to the same point x at the same time t so that the comparison itself may be nreal. A more dynamic way that also makes more sense in general is to compare the vale fnctions composed with the controlled processes. This amonts to look at V s t, Xt s) and V d t, Xt d) for t, T and pay particlar attention to t becoming the terminal vale T. Note that V s T, XT s ) = X T s and V dt, XT d ) = X T d so that to compare E,x 0 Vs T, XT s ) and E t0,x 0 Vd T, XT d ) is the same as to compare E t0,x 0 XT s ) and E,x 0 XT d ). It is easily seen from 3.2)and3.5) that the latter two expectations coincide. We can therefore conclde that E t0,x 0 Vs T, X s T ) = E t0,x 0 X s T ) = E,x 0 X d T ) = E,x 0 Vd T, X d T ) 3.44) for all, x 0 ) 0, T R. This shows that the dynamically optimal control d is as good as the statically optimal control s from this static standpoint as well with respect to any past point, x 0 ) given and fixed). In addition to that however the dynamically optimal control d is time consistent while the statically optimal control s is not. Note also from 3.4) that the amont of the dynamically optimal wealth d t, x) x held in the stock at time t does not depend on the amont of the total wealth x. Thisis consistent with the fact that the risk/cost in 2.4) is measred by the variance applied at a constant rate c) which is a qadratic fnction of the terminal wealth while the retrn/gain is measred by the expectation applied at a constant rate too) which is a linear fnction of the terminal wealth. The former therefore penalises stochastic movements of the large wealth more severely than what the latter is able to compensate for and the investor is discoraged to hold larger amonts of his wealth in the stock. Ths even if the total wealth is large in modls) it is still dynamically optimal to hold the same amont of wealth d t, x) x in the stock at time t as when the total wealth is small in modls). The same optimality behavior has been also observed for the sbgame-perfect Nash eqilibrim controls cf. Sect. 4). We now trn to the constrained problems. Note in the proofs below that the nconstrained problem above is obtained by optimising the Lagrangian of the constrained problems. Corollary 5 Consider the optimal control problem 2.5) where X solves 2.3) with X = x 0 nder P t0,x 0 for, x 0 ) 0, T R given and fixed. Recall that B solves 2.1), S solves 2.2), and we set δ = μ r)/σ for μ R, r R and σ>0. We assme throghot that

13 Math Finan Econ 2017) 11: δ = 0 and r = 0 the cases δ = 0 or r = 0 follow by passage to the limit when the non-zero δ or r tends to 0). A) The statically optimal control is given by s t, x) = δ 1 x 0 e rt t0) x + α eδ2 T ) rt t) σ x e δ2 T ) 1 for t, x), T R. The statically optimal controlled process is given by Xt s = x 0 e rt t0) + α eδ2 r)t t) e δ2 t ) e δw t W t0 ) δ2 2 t ) e δ2 T ) 1 α = x 0 B t B t0 + BT B t0 ) δ 2 /r 1 Bt B T ) 1 δ2/r for t, T. The static vale fnction V 1 s V 1 s, x 0 ) = x 0 e rt ) + α for, x 0 ) 0, T R. B) The dynamically optimal control is given by 3.45) Bt ) δ2/r Bt ) δ1/σ +δ σ)/2r) St ) δ/σ B t0 B t0 S t0 := EXT s ) is given by 3.46) e δ2 T ) ) d t, x) = α δ 1 e δ2 r)t t) σ x e δ2 T t) 1 for t, x), T R. The dynamically optimal controlled process is given by Xt d = x 0 e rt t0) + 2 α e e rt t) δ2 T ) 1 e δ2 T t) 1 + δ t e δ2t s) dw s 2 e δ2 T s) 1 = x 0 B t B t0 + + δ α σ B t B T 2r +δ1 σ)) σ2 δ 2 BT ) δ 2 /r B t St ) BT ) log + δ2 t BT δ 2 /r S B t 1 t0 2 BT B t ) δ2/r 1 B s ) δ2/r BT ) δ 2 /r ) B s 2 BT ) δ 2 /r ) log 3/2 B s ) BT ) δ2/r 1 B t0 ) ds S t0 Ss 3.49) for t, T ). The dynamic vale fnction Vd 1 := lim t T EXt d ) is given by Vd 1, x 0 ) = x 0 e rt t0) + 2 α e δ2 T ) ) for, x 0 ) 0, T ) R. Proof We assme throghot that the process X solves the stochastic differential eqation 2.3) with X = x 0 nder P t0,x 0 for, x 0 ) 0, T R given and fixed where is any admissible control as defined/discssed above. To simplify the notation we will drop the sbscript zero from and x 0 in the first part of the proof below.

14 150 Math Finan Econ 2017) 11: A): Note that we can think of 3.7) as the essential part of) the Lagrangian for the constrained problem 2.5)definedby L t,x, c) = E t,x X T ) c Var t,x X T ) α 3.51) for c > 0. By the reslt of Theorem 3 we know that the control s given in 3.1) is optimal in nconstrained problem L t,x c, c) := sp L t,x, c) 3.52) for c > 0. Sppose moreover that there exists c = cα, t, x) >0 sch that Var t,x X c ) T = α. 3.53) It then follows that E t,x X c T ) = L t,x c, c) E t,xxt ) c Var t,x XT ) α E t,x XT ) 3.54) for any admissible control sch that Var t,x XT ) α. This shows that the control c from 3.1) with c = cα, t, x) >0 is statically optimal in 2.5). To realise 3.53) note that taking E t0,x 0 in 3.2) and making se of 3.3) wefindthat Var t0,x 0 X c ) 1 T = 4c 2 e δ2 T ) ) Setting this expression eqal to α yields c = 1 2 α e δ2 T ) ) By 3.53) and3.54) we can then conclde that the control c is statically optimal in 2.5). Inserting 3.56) into3.1) and3.2) we obtain 3.45) and3.46) respectively. Taking E t0,x 0 in 3.46) we obtain 3.47) and this completes the first part of the proof. B): Identifying with t and x 0 with x in the statically optimal control s from 3.45) we obtain the control d from 3.48). We claim that this control is dynamically optimal in 2.5). For this, take any other admissible control v sch that v, x 0 ) = d, x 0 ) and set w = s.thenw, x 0 ) = d, x 0 ) and 3.33) holds with c from 3.56). Using that Var t0,x 0 XT w ) = α by 3.55)and3.56) we see that 3.33) yields E t0,x 0 XT w )>E,x 0 XT v ) + c α Var t0,x 0 XT v )) E t0,x 0 XT v ) 3.57) whenever Var t0,x 0 XT v ) α. This shows that the control d from 3.48) is dynamically optimal in 2.5) as claimed. Applying Itô s formla to e rt t) Xt d wherewesetx d := X d and making se of 2.3) we easily find that the first identity in 3.49) is satisfied. Integrating by parts and recalling the closed form expressions for B and S stated following 2.2) above we then establish that the second identity in 3.49) also holds. From the first identity in 3.49) weget ) E t0,x 0 Xt d = x 0 e rt t0) + 2 α e rt t) e δ2 T ) 1 e δ2 T t) ) for t, T ). Letting t T in 3.58) we obtain 3.50) and this completes the proof.

15 Math Finan Econ 2017) 11: Remark 6 A dynamic compliance effect). From 3.47) and3.50) we see that the dynamic vale Vd 1, x 0 ) strictly dominates the static vale Vs 1t 0, x 0 ). To see why this is possible note that sing 3.49) wefindthat ) Var t0,x 0 Xt e d ) = α e 2rT t) δ2 T ) e δ2 T t) e δ2 T ) 1 +log 3.59) e δ2 T t) 1 for t, T ) with δ = 0. This shows that Var t0,x 0 Xt d ) as t T so that the static vale Vs 1t 0, x 0 ) can indeed be exceeded by the dynamic vale Vd 1, x 0 ) since the set of admissible controls is virtally larger in the dynamic case. It amonts to what we refer to as a dynamic compliance effect where the investor follows a niformly bonded risk variance) strategy at each time and ths complies with the adopted reglation rle imposed internally/externally) while the reslting static strategy exhibits an nbonded risk variance). Denoting the stochastic integral martingale) in 3.49) bym t we see that M, M t = t e 2δ2 T s) /e 2δ2 T s) 1) ds as t T. It follows therefore that M t oscillates from to with P t0,x 0 -probability one as t T and hence the same is tre for Xt d whenever δ = 0 for similar behavior arising from the continos-time analoge of a dobling strategy see 9, Example 2.3). We also see from 3.46) and3.49) that nlike in 3.44) we have the strict ineqality lim E t0,x 0 V 1 s t, Xt s ) = lim E t0,x 0 Xt s )<lim E,x 0 Xt d ) = lim E,x 0 V 1 d t, Xt d ) t T t T t T t T 3.60) satisfied for all, x 0 ) 0, T ) R. This shows that the dynamic control d from 3.48) otperforms the static control s from 3.45) in the constrained problem 2.5). Corollary 7 Consider the optimal control problem 2.6) where X solves 2.3) with Xt 0 = x 0 nder P t0,x 0 for, x 0 ) 0, T R given and fixed. Recall that B solves 2.1), S solves 2.2), and we set δ = μ r)/σ for μ R, r R and σ>0. We assme throghot that δ = 0 and r = 0 the cases δ = 0 or r = 0 follow by passage to the limit when the non-zero δ or r tends to 0). A) The statically optimal control is given by s t, x) = δ 1 x 0 e rt t0) x + β x 0 e rt ) ) eδ2 T ) rt t) 3.61) σ x e δ2 T ) 1 if x 0 e rt t0) <βand s t, x) = 0 if x 0e rt t0) β for t, x), T R. The statically optimal controlled process is given by Xt s = x 0 e rt t0) + β x 0 e rt ) ) eδ2 r)t t) e δ2 t ) e δw t W t0 ) δ2 e δ2 T ) 2 t ) 1 ) B t B B t ) 1 δ 2 /r ) δ 2 /r ) δ1/σ +δ σ)/2r) ) δ/σ T B = x 0 + β x T Bt Bt St 0 B t0 B t0 BT ) δ 2 /r B t0 1 B t0 B t0 S t0 3.62) if x 0 e rt t0) <βand Xt s = x 0 e rt t0) if x 0 e rt t0) β for t, T see Fig. 1 below). The static vale fnction Vs 2 := VarX T s ) is given by β Vs 2 x0 t e rt ) ) 2 0, x 0 ) = 3.63) e δ2 T ) 1

16 152 Math Finan Econ 2017) 11: Fig. 1 The dynamically optimal wealth t Xt d and the statically optimal wealth t Xt s in the constrained problem 2.6) of Corollary 7 obtained from the stock price t S t when = 0, x 0 = 1, S 0 = 1, β = 2, r = 0.1, μ = 0.5, σ = 0.4andT = 1. Note that the expected vale of S T eqals e μt 1.64 which is strictly smaller than β if x 0 e rt ) <βand V 2 s, x 0 ) = 0 if x 0 e rt ) β for, x 0 ) 0, T R. B) The dynamically optimal control is given by d t, x) = δ 1 β xe rt t) ) eδ2 r)t t) 3.64) σ x e δ2 T t) 1 if x 0 e rt t0) <βand d t, x) = 0 if x 0e rt t0) β for t, x), T ) R. The dynamically optimal controlled process is given by Xt d = e rt t) β β x 0 e rt ) ) eδ2 T t) 1 e δ2 T ) 1 t e δ2t s) exp δ e δ2 T s) 1 dw s δ2 2 t e 2δ2T s) ) e δ2 T s) 1) ds 2

17 Math Finan Econ 2017) 11: = B t B T β β x 0 B T B t0 + δ3 σ t ) B T B t ) δ 2 /r 1 BT B t0 ) δ 2 /r 1 BT ) δ 2 /r B s Ss ) BT ) δ 2 /r ) log ds 1 2 B s 1 S t0 2 ) 1/2+σ/2δ r/σ δ exp δ σ BT ) δ 2 /r B t0 BT ) δ 2 /r B t BT ) δ 2 /r ) B t 1 BT ) δ 2 /r ) B t0 1 BT ) δ 2 /r B t St ) BT ) log δ 2 /r B t 1 S t0 ) 3.65) with Xt d e rt t) <βfor t, T ) and XT d := lim t T Xt d = β with P t0,x 0 -probability one if x 0 e rt t0) <β, and Xt d = x 0 e rt t0) for t, T if x 0 e rt t0) β see Fig. 1 above). The dynamic vale fnction Vd 2 := VarX T d ) is given by Vd 1, x 0 ) = ) for, x 0 ) 0, T ) R. Proof We assme throghot that the process X solves the stochastic differential eqation 2.3) with Xt 0 = x 0 nder P t0,x 0 for, x 0 ) 0, T R given and fixed where is any admissible control as defined/discssed above. To simplify the notation we will drop the sbscript zero from and x 0 in the first part of the proof below. A): Note that the Lagrangian for the constrained problem 2.6)isdefinedby L t,x, c) = Var t,x XT ) c E t,x XT ) β 3.67) for c > 0. To connect to the reslts of Theorem 3 observe that inf Var t,x X T ) c E t,x XT ) β) = c sp E t,x XT ) 1 c Var t,xxt ) + cβ 3.68) which shows that the control 1/c given in 3.1) is optimal in the nconstrained problem L t,x 1/c, c) := inf L t,x, c) 3.69) for c > 0. Sppose moreover that there exists c = cβ, t, x) >0 sch that E 1/c ) t,x X T = β. 3.70) It then follows that Var t,x XT ) = L t,x 1/c, c) Var t,x XT ) c E t,x XT ) β Var t,x XT ) 3.71) for any admissible control sch that E t,x XT ) β. This shows that the control 1/c from 3.1) with c = cβ, t, x) >0 is statically optimal in 2.6). To realise 3.70) note that taking E t0,x 0 in 3.2) wefindthat E 1/c ) t0,x 0 X T = x0 e rt t0) + c e δ2 T ) ) 2 Setting this expression eqal to β yields c = 2β x 0e rt t0) ). 3.73) e δ2 T ) 1 From 3.73) we see that c > 0 if and only if x 0 e rt t0) <β.ifx 0 e rt t0) β then clearly we can invest all the wealth in B and receive zero variance at T. This shows that the control s t, x) = 0fort, x), T R is statically optimal in this case with Vs 2t 0, x 0 ) = 0

18 154 Math Finan Econ 2017) 11: as claimed. Let s therefore assme that x 0 e rt t0) <βin the seqel. Then by 3.70) and 3.71) we can conclde that the control 1/c is statically optimal in 2.6). Inserting 3.73) into 3.1) and3.2) we obtain 3.61) and3.62) respectively. Inserting 3.73) into3.55) we obtain 3.63) and this completes the first part of the proof. B): Identifying with t and x 0 with x in the statically optimal control s from 3.61)we obtain the control d from 3.64). We claim that this control is dynamically optimal in 2.6) when x 0 e rt t0) <β. For this, take any other admissible control v sch that v, x 0 ) = d, x 0 ) and set w = s.thenw, x 0 ) = d, x 0 ) and 3.33) holds with c from 3.73). Using that E t0,x 0 XT w ) = β by 3.72)and3.73) we see that 3.33) yields Var t0,x 0 XT w )< 1 β E t0,x c 0 XT v ) + c Var,x 0 XT v ) Var t0,x 0 XT v ) 3.74) whenever E t0,x 0 XT v ) β. This shows that the control d from 3.64) is dynamically optimal in 2.6) whenx 0 e rt t0) <βas claimed. If x 0 e rt t0) β then both s t, x) = 0and d t, x) = 0sothatby2.3) we see that X t d := X d t = x 0 e rt t0) for t, T as claimed. Dynamic optimality then follows from the fact singled ot in Remark 9 below) that zero control is the only possible admissible control that can move a given deterministic wealth x 0 at time 0, T ) to another deterministic wealth of zero variance) at time T.Lets therefore assme that x 0 e rt t0) <βin the seqel. Applying Itô s formla to the process Z defined by Z t = β e rt t) X d t 3.75) wherewesetx d := X d and making se of 2.3) we easily find that dz t = δ 2 Z t 1 e dt δ Z t δ2 T t) 1 e dw δ2 T t) t 3.76) with Z t0 = β e rt t0) x 0 nder P t0,x 0. Solving the linear eqation 3.76) explicitly we obtain the closed form expression t δ t Z t = Z t0 exp 1 e dw δ 2 δ2 T s) s 1 e + 1 δ 2 ) ds δ2 T s) 2 1 e δ2 T s) ) ) for t, T ) nder P t0,x 0. Inserting 3.77)into3.75) we easily find that the first identity in 3.65) is satisfied. Integrating by parts and recalling the closed form expressions for B and S stated following 2.2) above we then establish that the second identity in 3.65) also holds. From 3.75)and3.77) we see that Z t = β e rt t) Xt d > 0sothatXt d e rt t) <βfor t, T ) as claimed. Moreover, by the Dambis-Dbins-Schwarz theorem see e.g. 18, p. 181) we know that the continos martingale M defined by M t = δ t e δ2 T s) /e δ2 T s) 1) dw s for t, T ) is a time-changed Brownian motion W in the sense that M t = W M,M t for t, T ) wherewenotethat M, M t = δ 2 t e 2δ2 T s) /e δ2 T s) 1) 2 ds as t T. It follows therefore by the well-known sample path properties of W that M t 2 1 M, M t = W M,M t 2 1 M, M t as t T with P t0,x 0 -probability one. Making se of this fact in 3.65) we see that Xt d β with P t0,x 0 -probability one as t T if x 0 e rt t0) <βas claimed. From the preceding facts we also see that 3.66) holds and the proof is complete. Remark 8 Note from the proof above that Xt d <βwith P t0,x 0 -probability one for all t, T ) if x 0 e rt t0) <βso that X d is not a bridge process bt a time-reversed meander process. The reslt of Corollary 7 shows that it is dynamically optimal to keep the wealth Xt d

19 Math Finan Econ 2017) 11: strictly below β for t, T ) with achieving XT d = β. This behavior is different from the statically optimal wealth Xt s which can go above β on, T ) and end p either above or below β at T see Fig. 1 above). Moreover, it is easily seen from 3.62) thatp t0,x 0 XT s <β)>0 from where we find that E t0,x 0 V 2 s T, X s T ) = > 0 = E t0,x 0 V 2 d T, X d T ) 3.78) if x 0 e rt t0) <βsing 3.63) and3.66) respectively. This shows that the dynamic control d from 3.64) otperforms the static control s from 3.61) in the constrained problem 2.6). Remark 9 Note that no admissible control can move a given deterministic wealth x 0 at time 0, T ) to any other deterministic wealth at time T apart from x 0 e rt t0) in which case eqals zero. This is important since otherwise the optimal control problem 2.6) wold not be well posed. Indeed, this can be seen by a standard martingale measre change d P t0,x 0 = exp δw T W t0 ) δ 2 /2)t )) dp t0,x 0 making W t := W t W t0 +δt ) a standard Brownian motion for t, T. It then follows from 2.3) sing integration by parts that t e rt t0) Xt = x 0 + σ e rs t0) s Xs d W s 3.79) where M t := t σ e rs t0) s Xs d W s is a continos local martingale nder P t0,x 0 for t, T. Moreover, by Hölder s ineqality we see that T ) 1/2 Ẽ t0,x 0 M, M T = E t0,x 0 e δw T W t0 ) δ2 2 T ) σ 2 e 2rt t0) 2 t X t )2 dt E t0,x 0 e 2δW T W t0 ) δ 2 T ) ) 1/2 T ) 1/2 E t0,x0 σ 2 e 2rt t0) 2 t X t )2 dt < 3.80) T since E t0,x t )X t )2 dt < by admissibility of. This shows that M is a martingale nder P t0,x 0. Hence if X T is constant then it follows from 3.79) and the martingale property of M that M t = 0forallt, T. Bt this means that Xt = x 0 e rt t0) for t, T with being eqal to zero as claimed. Remark 10 Note from 3.65)thatE t0,x 0 X d t ) β as t T if x 0e rt ) <β,however,this convergence fails to extend to the variance. Indeed, sing 3.65) it can be verified that Var t0,x 0 Xt d ) = e 2rT t) ) β x 0 e rt t 2 e 0) δ2 T t) 1 e δ2 T ) 1 e δ 2 T ) 1 e δ2 T t) 1 exp e δ2 T ) e δ2 T t) ) 1 e δ2 T t) 1)e δ2 T ) 1) ) ) for t, T ) from where we see that Var t0,x 0 X d t ) as t T if x 0 e rt ) <β.to connect to the comments on the sample path behavior made in Remark 6 note that t X d t is not bonded from below on, T ). Both of these conseqences are de partly to the fact that we allow the wealth process to take both positive and negative vales of nlimited size recall the end of Sect. 1 above). Another reason is that the dynamic optimality by its natre pshes the optimal controls to their limits so that breakdown points are possible.

20 156 Math Finan Econ 2017) 11: Static verss dynamic optimality In this section we address the rationale for introdcing the static and dynamic optimality in the nonlinear optimal control problems nder consideration and explain their relevance for applications of both theoretical and practical interest. We also discss relation of these reslts with the existing approaches to similar problems in the literatre. 1. To simplify the exposition we focs on the nconstrained problem 2.4) and similar argments apply to the constrained problems 2.5) and2.6) as well. Recall that 2.4) represents the optimal portfolio selection problem for an investor who has an initial wealth x 0 R which he wishes to exchange between a risky stock S and a riskless bond B in a self-financing manner dynamically in time so as to maximise his retrn identified with the expectation of his wealth) and minimise his risk identified with the variance of his wealth) at the given terminal time T. De to the qadratic nonlinearity of the variance as a fnction of the expectation) the optimal portfolio strategy 3.1) depends on the initial wealth x 0 in an essential way. This spatial inconsistency not present in the standard/linear optimal control problems) introdces the time inconsistency in the problem becase the investor s wealth process moves from the initial vale x 0 in t nits of time to a new vale x 1 different from x 0 with probability one) which in trn yields a new optimal portfolio strategy that is different from the initial strategy. This time inconsistency repeats itself between any two points in time and the investor may be in dobt which optimal portfolio strategy to se nless already made p his mind. To tackle these inconsistencies we are natrally led to consider two types of investors and conseqently introdce the two notions of optimality as stated in Definitions 1 and 2 respectively. The first investor is a static investor who stays pre-committed to the optimal portfolio strategy evalated initially and does not re-evalate the optimality criterion 2.4) at later times. This investor will determine the optimal portfolio strategy at time and follow it blindly to the terminal time T. The second investor is a dynamic investor who remains non-committed to the optimal portfolio strategy evalated initially as well as sbseqently and continosly re-evalates the optimality criterion 2.4) at each new time. This investor will determine the optimal portfolio strategy at time and contine doing so at each new time ntil the terminal time T. Clearly both the static investor and the dynamic investor embody realistic economic behavior see below for a more detailed accont coming from economics) andtheorem3 discloses their optimal portfolio selection strategies in the nconstrained problem 2.4). Similarly Corollary 5 and Corollary 7 disclose their optimal portfolio selection strategies in the constrained problems 2.5) and2.6). Given that the financial interpretations of these reslts are easy to draw directly and somewhat lengthy to state explicitly we will omit frther details. It needs to be noted that althogh closely related the three problems 2.4) 2.6) are still different and hence it is to be expected that their soltions are also different for some vales of the parameters. Difference between the static and dynamic optimality is best nderstood by analysing each problem on its own first as in this case the complexity of the overall comparison is greatly redced. 2. Apart from the paper 13 where the dynamic optimality was sed in a nonlinear problem of optimal stopping, we are not aware of any other paper on optimal control where nonlinear problems were stdied sing this methodology. The dynamic optimality Definition 2) appears therefore to be original to the present paper in the context of nonlinear problems of optimal control. There are two streams of papers on optimal control however where the static optimality Definition 1) has been sed. Thefirst one belongsto the economics literatre and dates back to the paper by Strotz 21. The second one belongs to