A general necessary and sufficient optimality conditions for singular control problems
aa r X i v : . [ m a t h . P R ] D ec A general necessary and sufficient optimalityconditions for singular control problems
Seid Bahlali ∗ Abstract
We consider a stochastic control problem where the set of strict (classi-cal) controls is not necessarily convex and the the variable control has twocomponents, the first being absolutely continuous and the second singular.The system is governed by a nonlinear stochastic differential equation, inwhich the absolutely continuous component of the control enters both thedrift and the diffusion coefficients. By introducing a new approach, weestablish necessary and sufficient optimality conditions for two models.The first concerns the relaxed-singular controls, who are a pair of pro-cesses whose first component is a measure-valued processes. The secondis a particular case of the first and relates to strict-singular control prob-lems. These results are given in the form of global stochastic maximumprinciple by using only the first order expansion and the associated ad-joint equation. This improves and generalizes all the previous works onthe maximum principle of controlled stochastic differential equations.
Keywords.
Stochastic differential equation, Strict-singular con-trol, Relaxed-singular control, Maximum principle, Adjoint process, Vari-ational inequality.
AMS subject classification.
We study a stochastic control problem where the system is governed by anonlinear stochastic differential equation (SDE for short) of the type ( dx ( v,η ) t = b (cid:16) t, x ( v,η ) t , v t (cid:17) dt + σ (cid:16) t, x ( v,η ) t , v t (cid:17) dW t + G t dη t ,x ( v,η )0 = x, ∗ Laboratory of applied mathematics, University Med Khider, P.O. Box 145, Biskra 07000,Alg´eria. [email protected] b, σ and G are given functions, x is the initial data and W = ( W t ) t ≥ is astandard Brownian motion, defined on a filtered probability space (cid:16) Ω , F , ( F t ) t ≥ , P (cid:17) , satisfying the usual conditions.The control variable, called strict-singular control, is a suitable process ( v, η )where v : [0 , T ] × Ω −→ U ⊂ R k , η : [0 , T ] × Ω −→ U = ([0 , ∞ )) m are B [0 , T ] ⊗F -measurable, ( F t )- adapted, and η is an increasing process (componentwise),continuous on the left with limits on the right with η = 0. We denote by U theclass of all strict-singular controls.The criteria to be minimized, over the set U , has the form J ( v, η ) = E " g (cid:16) x ( v,η ) T (cid:17) + Z T h (cid:16) t, x ( v,η ) t , v t (cid:17) dt + Z T k t dη t , where, g, h and k are given maps and x ( v,η ) t is the trajectory of the systemcontrolled by ( v, η ).A control ( u, ξ ) ∈ U is called optimal if it satisfies J ( u, ξ ) = inf ( v,η ) ∈U J ( v, η ) . This kind of stochastic control problems have been studied by many authors,both by the dynamic programming approach and by the Pontryagin stochasticmaximum principle. The first approach was studied by Ben˘es, Shepp and Wit-senhausen [6] , Chow, Menaldi and Robin [10] , Karatzas and Shreve [21] , Davisand Norman [11] , Haussmann and Suo [17 , , . See [17] for a complete list ofreferences on the subject. It was shown in particular that the value function issolution of a variational inequality, and the optimal state is a reflected diffusionat the free boundary. Note that in [17] , the authors apply the compactificationmethod to show existence of an optimal singular control.In this paper, we are concerned with the second approach, whose objectiveis to establish necessary (as well as sufficient) conditions for optimality of con-trols. The first version of the stochastic maximum principle that covers singularcontrol problems was obtained by Cadenillas and Haussmann [8], in which theyconsider linear dynamics, convex cost criterion and convex state constraints.The method used in [8] is based on the known principle of convex analysis,related to the minimization of convex, continuous and Gˆateaux-differentiablefunctional defined on a convex closed set. Necessary conditions of optimalityfor non linear SDEs with convex control domain, where the coefficients dependexplicitly on the absolutely part of the control, was derived by Bahlali and Chala[1] by applying a convex perturbation on the pair of controls. The result in thenobtained in weak form. Bahlali and Mezerdi [2] generalize the work of [1] to thecase of nonconvex control domain, and derive necessary optimality conditionsby using a strong perturbation (spike variation) on the absolutely continuouscomponent of the control and a convex perturbation on the singular one. ThePeng stochastic maximum principle is then used and te result is given withtwo adjoint equations and a variational inequality of the second order. Version2f stochastic maximum principle for relaxed-singular controls was establishedby Bahlali, Djehiche and Mezerdi [4] in the case of uncontrolled diffusion, byusing the previous works on strict-singular controls, Ekeland’s variational prin-ciple and some stability properties of the trajectories and adjoint processes withrespect to the control variable.In a recent work, Bahlali [5] generalizes and improves all the previous resultson stochastic maximum principle for controlled SDEs, by introducing a newapproach and establish necessary and sufficient optilmality conditions for bothrelaxed and strict controls, by using only the first order expansion and theassociated adjoint equation. The main idea of [5], is to use the property ofconvexity of the set of relaxed controls and treat the problem with the convexperturbation on relaxed controls (instead of the spike variation on strict one).Our aim in this paper, is to follow the method used by [5] and derive neces-sary as well as sufficient conditions of optimality in the form of global stochasticmaximum principle, for both relaxed-singular and strict-singular controls, with-out using the second order expansion. We introduce then a bigger new class R of processes by replacing the U -valued process ( v t ) by a P ( U )-valued process( q t ), where P ( U ) is the space of probability measures on U equipped with thetopology of stable convergence. This new class of processes is called relaxed-singular controls and have a richer structure, for which the control problembecomes solvable.In the relaxed-singular model, the system is governed by the SDE dx ( q,η ) t = Z U b (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) dt + Z U σ (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) dW t + G t dη t ,x ( q,η )0 = x. The functional cost to be minimized, over the class R of relaxed-singularcontrols, is defined by J ( q, η ) = E " g (cid:16) x ( q,η ) T (cid:17) + Z T Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) dt + Z T k t dη t . A relaxed-singular control ( µ, ξ ) is called optimal if it solves J ( µ, ξ ) = inf ( q,η ) ∈R J ( q, η ) . The relaxed-singular control problem finds its interest in two essential points.The first is that an optimal solution exists. Haussmann and Suo [17] haveproved that the relaxed-singular control problem admits an optimal solutionunder general conditions on the coefficients. Indeed, by using a compactificationmethod and under some mild continuity hypotheses on the data, it is shownby purely probabilistic arguments that an optimal solution for the problemexists. Moreover, the value function is shown to be Borel measurable. Thesecond interest is that it is a generalization of the strict-singular control problem.Indeed, if q t ( da ) = δ v t ( da ) is a Dirac measure concentrated at a single point v t U , then we get a strict-singular control problem as a particular case of therelaxed one.To achieve the objective of this paper and establish necessary and sufficientoptimality conditions for these two models, we proceed as follows.Firstly, we give the optimality conditions for relaxed controls. The mainidea is to use the fact that the set of relaxed controls is convex. Then, weestablish necessary optimality conditions by using the classical way of the convexperturbation method. More precisely, if we denote by ( µ, ξ ) an optimal relaxedcontrol and ( q, η ) is an arbitrary element of R , then with a sufficiently small θ > t ∈ [0 , T ], we can define a perturbed control as follows (cid:0) µ θt , ξ θt (cid:1) = ( µ t , ξ t ) + θ [( q t , η t ) − ( µ t , ξ t )] . We derive the variational equation from the state equation, and the varia-tional inequality from the inequality0 ≤ J (cid:0) µ θ , ξ θ (cid:1) − J ( µ, ξ ) . By using the fact that the drift, the diffusion and the running cost coefficientsare linear with respect to the relaxed control variable, necessary optimalityconditions are obtained directly in the global form. The result is given by usingonly the first-order expansion and the associated adjoint equationsTo enclose this part of the paper, we prove under minimal additional hy-potheses, that these necessary optimality conditions for relaxed-singular controlsare also sufficient.The second main result in the paper characterizes the optimality for strict-singular control processes. It is directly derived from the above results by re-stricting from relaxed to strict-singular controls. The main idea is to replacethe relaxed controls by a Dirac measures charging a strict controls. Thus, wereduce the set R of relaxed-singular controls and we minimize the cost J overthe subset δ ( U ) × U = { ( q, η ) ∈ R / q = δ v ; v ∈ U } . Then, we derivenecessary optimality conditions by using only the first order expansion and theassociated adjoint equation. We don’t need anymore the second order expan-sion. Moreover, we show that these necessary optimality conditions for strict-singular controls are also sufficient, without imposing neither the convexity of U nor that of the Hamiltonian H in v .The results of this paper are an important improvement of those of Bahlaliand Mezerdi [2] and an extension of the works by Bahlali [5] to the class ofsingular controls.The paper is organized as follows. In Section 2, we formulate the strict-singular and relaxed-singular control problems and give the various assumptionsused throughout the paper. Section 3 is devoted to study the relaxed-singularcontrol problems and we establish necessary as well as sufficient conditions ofoptimality for relaxed-singular controls. In the last section, we derive directlyfrom the results of Section 3, the optimality conditions for strict-singular con-trols. 4long with this paper, we denote by C some positive constant and for sim-plicity, we need the following matrix notations. We denote by M n × d ( R ) thespace of n × d real matrix and M dn × n ( R ) the linear space of vectors M =( M , ..., M d ) where M i ∈ M n × n ( R ). For any M, N ∈ M dn × n ( R ), L, S ∈M n × d ( R ), Q ∈ M n × n ( R ), α, β ∈ R n and γ ∈ R d , we use the following no-tations αβ = n X i =1 α i β i ∈ R is the product scalar in R n ; LS = d X i =1 L i S i ∈ R , where L i and S i are the i th columns of L and S ; M L = d X i =1 M i L i ∈ R n ; M αγ = d P i =1 ( M i α ) γ i ∈ R n ; M N = d X i =1 M i N i ∈ M n × n ( R ); M QN = d X i =1 M i QN i ∈ M n × n ( R ); M Qγ = d X i =1 M i Qγ i ∈ M n × n ( R ).We denote by L ∗ the transpose of the matrix L and M ∗ = ( M ∗ , ..., M ∗ d ). Let (cid:16) Ω , F , ( F t ) t ≥ , P (cid:17) be a filtered probability space satisfying the usualconditions, on which a d-dimensional Brownian motion W = ( W t ) t ≥ is de-fined. We assume that ( F t ) is the P− augmentation of the natural filtration of( W t ) t ≥ . Let T be a strictly positive real number and consider the following sets U is a non empty subset of R k , U = ([0 , ∞ )) m , U is the class of measurable, adapted processes v : [0 , T ] × Ω −→ U suchthat E " sup t ∈ [0 ,T ] | v t | < ∞ . U is the class of measurable, adapted processes η : [0 , T ] × Ω −→ U suchthat η is nondecreasing (componentwise), left-continuous with right limits, η =0, and E h | η T | i < ∞ . .1 The strict-singular control problem Definition 1
A strict-singular control is a pair of processes ( v, η ) ∈ U × U .We denote by U = U × U the set of all strict-singular controls. For any ( v, η ) ∈ U , we consider the following SDE ( dx ( v,η ) t = b (cid:16) t, x ( v,η ) t , v t (cid:17) dt + σ (cid:16) t, x ( v,η ) t , v t (cid:17) dW t + G t dη t ,x ( v,η )0 = x, (1)where b : [0 , T ] × R n × U −→ R n ,σ : [0 , T ] × R n × U −→ M n × d ( R ) ,G : [0 , T ] −→ M n × m ( R ) . The criteria to be minimized is defined from U into R by J ( v, η ) = E " g (cid:16) x ( v,η ) T (cid:17) + Z T h (cid:16) t, x ( v,η ) t , v t (cid:17) dt + Z T k t dη t , (2)Where g : R d −→ R ,h : [0 , T ] × R d × U −→ R ,k : [0 , T ] −→ ([0 , ∞ )) d . A strict-singular control ( v, η ) is called optimal if it satisfies J ( u, ξ ) = inf ( v,η ) ∈U J ( v, η ) . (3)We assume that b, σ, g and h are continuously differentiable with respect to x .The derivatives b x , σ x , g x and h x , are continuous in ( x, v ) anduniformly bounded. (4) b and σ are bounded by C (1 + | x | + | v | ) .G and k are continuous and G is bounded . Under the above assumptions, for every ( v, η ) ∈ U , equation (1) has anunique strong solution and the functional cost J is well defined from U into R .6 .2 The relaxed-singular model The strict-singular control problem { (1) , (2) , (3) } formulated in the last sub-section may fail to have an optimal solution. Let us begin by a deterministicexamples which shows that even in simple cases, existence of a strict optimalcontrol is not ensured (see Fleming [16] and Yong and Zhou [28] for other ex-amples). Example 1.
The problem is to minimize, over the set U of measurablefunctions v : [0 , T ] → {− , } , the following functional cost J ( v ) = Z T ( x vt ) dt, where x vt denotes the solution of (cid:26) dx vt = v t dt,x v = 0 . We have inf v ∈U J ( v ) = 0 . Indeed, consider the following sequence of controls v nt = ( − k if kn T ≤ t ≤ k + 1 n T , ≤ k ≤ n − . Then clearly (cid:12)(cid:12)(cid:12) x v n t (cid:12)(cid:12)(cid:12) ≤ Tn , | J ( v n ) | ≤ T n . Which implies that inf v ∈U J ( v ) = 0 . There is however no control v such that J ( v ) = 0. If this would have beenthe case, then for every t, x vt = 0. This in turn would imply that v t = 0, whichis impossible. The problem is that the sequence ( v n ) has no limit in the space ofstrict controls. This limit if it exists, will be the natural candidate for optimality.If we identify v nt with the Dirac measure δ v nt ( da ) and set q n ( dt, dv ) = δ v nt ( dv ) dt ,we get a measure on [0 , × U . Then, the sequence ( q n ( dt, dv )) n convergesweakly to 12 dt. [ δ − + δ ] ( da ). Example 2.
Consider the control problem where the system is governed bythe SDE (cid:26) dx t = v t dt + dW t ,x = 0 . J ( v ) = E Z T h x t + (cid:0) − v t (cid:1) i dt.U = [ − ,
1] and x, v, W are one dimensional. The control v (open loop) isa measurable function from [0 , T ] into U .The separation principle applies to this example, the optimal control mini-mizes Z T hb x t + (cid:0) − v t (cid:1) i , where b x t = E [ x t ] satisfies (cid:26) d b x t = v t dt, b x = 0 . This problem has no optimal strict control. A relaxed solution is to let µ t = 12 δ + 12 δ − , where δ a is an Dirac measure concentrated at a single point a. This suggests that the set of strict controls is too narrow and should beembedded into a wider class with a richer topological structure for which thecontrol problem becomes solvable. The idea of relaxed-singular control is toreplace the absolutely continuous part v t of the strict-singular control by a P ( U )-valued process ( q t ), where P ( U ) is the space of probability measures on U equipped with the topology of stable convergence of measures. Definition 2
A relaxed-singular control is a pair ( q, η ) of processes such thati) q is a P ( U ) -valued process progressively measurable with respect to ( F t ) and such that for each t , ]0 ,t ] .q is F t -measurable.ii) η ∈ U .We denote by R = R × U the set of relaxed-singular controls. For more details on relaxed controls, see [3] , [4] , [5] , [15] , [16] , [24] , [25] and[26].For any ( q, η ) ∈ R , we consider the following relaxed-singular SDE dx ( q,η ) t = Z U b (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) dt + Z U σ (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) dW t + G t dη t x ( q,η )0 = x. (5)8he expected cost to be minimized, in the relaxed-singular model, is definedfrom R into R by J ( q, η ) = E " g (cid:16) x ( q,η ) T (cid:17) + Z T Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) dt + Z T k t dη t . (6)A relaxed-singular control ( µ, ξ ) is called optimal if it solves J ( µ, ξ ) = inf ( q,η ) ∈R J ( q, η ) . (7)Haussmann and Suo [17] have proved that the relaxed-singular control prob-lem admits an optimal solution under general conditions on the coefficients.Indeed, by using a compactification method and under some mild continuityhypotheses on the data, it is shown by purely probabilistic arguments that anoptimal solution for the problem exists. Moreover, the value function is shownto be Borel measurable. See Haussmann and Suo [17], Section 3, page 925 topage 934 and essentially Theorem 3.8, page 933. Remark 3
If we put for any ( q, η ) ∈ R b (cid:16) t, x ( q,η ) t , q t (cid:17) = Z U b (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) ,σ (cid:16) t, x ( q,η ) t , q t (cid:17) = Z U σ (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) ,h (cid:16) t, x ( q,η ) t , q t (cid:17) = Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) . Then, equation (5) becomes ( dx ( q,η ) t = b (cid:16) t, x ( q,η ) t , q t (cid:17) dt + σ (cid:16) t, x ( q,η ) t , q t (cid:17) dW t + G t dη t ,x ( q,η ) T = x. With a functional cost given by J ( q, η ) = E " g (cid:16) x ( q,η ) T (cid:17) + Z T h (cid:16) t, x ( q,η ) t , q t (cid:17) dt + Z T k t dη t . Hence, by introducing relaxed-singular controls, we have replaced U by alarger space P ( U ) . We have gained the advantage that P ( U ) is convex. Fur-thermore, the new coefficients of equation (5) and the running cost are linearwith respect to the relaxed control variable. Remark 4
The coefficients b and σ (defined in the above remark) check respec-tively the same assumptions as b and σ . Then, under assumptions (4) , for every ( q, η ) ∈ R , equation (5) has an unique strong solution.On the other hand, It is easy to see that h checks the same assumptions as h . Then, the functional cost J is well defined from R into R . emark 5 If q t = δ v t is an atomic measure concentrated at a single point v t ∈ P ( U ) , then for each t ∈ [0 , T ] we have x ( q,η ) = x ( v,η ) , J ( q, η ) = J ( v, η ) , and we get a strict-singular control problem. So the problem of strict-singularcontrols { (1) , (2) , (3) } is a particular case of relaxed-singular control problem { (5) , (6) , (7) } . In this section, we study the problem { (5) , (6) , (7) } and we establish neces-sary as well as sufficient conditions of optimality for relaxed-singular controls. Since the set of relaxed-singular controls R is convex, a classical way oftreating such a problem is to use the convex perturbation method. More pre-cisely, let ( µ, ξ ) be an optimal relaxed-singular control and x ( µ,ξ ) t the solutionof (5) controlled by ( µ, ξ ). Then, for each t ∈ [0 , T ], we can define a perturbedrelaxed-singular control as follows (cid:0) µ θt , ξ θt (cid:1) = ( µ t , ξ t ) + θ [( q t , η t ) − ( µ t , ξ t )] , where, θ > q, η ) is an arbitrary element of R .Denote by x ( µ θ ,ξ θ ) t the solution of (5) associated with (cid:0) µ θ , ξ θ (cid:1) .From optimality of ( µ, ξ ), the variational inequality will be derived from thefact that 0 ≤ J (cid:0) µ θ , ξ θ (cid:1) − J ( µ, ξ ) . For this end, we need the following classical lemmas.
Lemma 6
Under assumptions (4) , we have lim θ → " sup t ∈ [0 ,T ] E (cid:12)(cid:12)(cid:12)(cid:12) x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (8)10 roof. We have x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t = Z t (cid:20)Z U b (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) µ θs ( da ) − Z U b (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ θs ( da ) (cid:21) ds + Z t (cid:20)Z U b (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ θs ( da ) − Z U b (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) (cid:21) ds + Z t (cid:20)Z U σ (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) µ θs ( da ) − Z U σ (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ θs ( da ) (cid:21) dW s + Z t (cid:20)Z U σ (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ θs ( da ) − Z U σ (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) (cid:21) dW s + Z t G t d (cid:0) ξ θt − ξ t (cid:1) . By using the definition of (cid:0) µ θ , ξ θ (cid:1) and taking expectation, we have E (cid:12)(cid:12)(cid:12)(cid:12) x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C E Z t (cid:12)(cid:12)(cid:12)(cid:12)Z U b (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) µ s ( da ) − Z U b (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds + Cθ E Z t (cid:12)(cid:12)(cid:12)(cid:12)Z U b (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) q s ( da ) − Z U b (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) µ s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds + C E Z t (cid:12)(cid:12)(cid:12)(cid:12)Z U σ (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) µ s ( da ) − Z U σ (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds + Cθ E Z t (cid:12)(cid:12)(cid:12)(cid:12)Z U σ (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) q s ( da ) − Z U σ (cid:18) s, x ( µ θ ,ξ θ ) s , a (cid:19) µ s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds + Cθ E (cid:12)(cid:12)(cid:12)(cid:12)Z t G t d ( η t − ξ t ) (cid:12)(cid:12)(cid:12)(cid:12) . Since b and σ are uniformly Lipschitz with respect to x , and G is bounded,then E (cid:12)(cid:12)(cid:12)(cid:12) x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C E Z t (cid:12)(cid:12)(cid:12)(cid:12) x ( µ θ ,ξ θ ) s − x ( µ,ξ ) s (cid:12)(cid:12)(cid:12)(cid:12) ds + CθE | η T − ξ T | + Cθ . By using Gronwall’s lemma and Buckholder-Davis-Gundy inequality, we ob-tain the desired result. 11 emma 7
Let z be the solution of the linear SDE (called variational equation) dz t = Z U b x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) z t dt + (cid:20)Z U b (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) − Z U b (cid:16) t, x ( µ,ξ ) t , a (cid:17) q t ( da ) (cid:21) dt + Z U σ x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) z t dW t + (cid:20)Z U σ (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) − Z U σ (cid:16) t, x ( µ,ξ ) t , a (cid:17) q t ( da ) (cid:21) dW t + G t d ( η t − ξ t ) ,z = 0 . (9) Then, we have lim θ → E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t θ − z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (10) Proof.
It is easy to see that x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t θ − z t , does not depends on the singular part. Then the result follows immediately bythe same method that in [5 , Lemma 10, page 2086-2088].
Lemma 8
Let ( µ, ξ ) be an optimal relaxed-singular control minimizing the cost J over R and x ( µ,ξ ) t the associated optimal trajectory. Then, for any ( q, η ) ∈ R ,we have ≤ E h g x (cid:16) x ( µ,ξ ) T (cid:17) z T i + E Z T Z U h x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) z t dt (11)+ E Z T (cid:20)Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) q t ( da ) − Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) (cid:21) dt + E Z T k t d ( η t − ξ t ) . roof. Since ( µ, ξ ) minimizes the cost J over R , then0 ≤ J (cid:0) µ θ , ξ θ (cid:1) − J ( µ, ξ ) ≤ E (cid:20) g (cid:18) x ( µ θ ,ξ θ ) T (cid:19) − g (cid:16) x ( µ,ξ ) T (cid:17)(cid:21) + E Z T Z U h (cid:18) t, x ( µ θ ,ξ θ ) t , a (cid:19) µ θt ( da ) dt − E Z T Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ θt ( da ) dt + E Z T Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ θt ( da ) dt − E Z T Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) dt + E Z T k t d (cid:0) ξ θt − ξ t (cid:1) . By using the definition of (cid:0) µ θ , ξ θ (cid:1) , we get0 ≤ E (cid:20) g (cid:18) x ( µ θ ,ξ θ ) T (cid:19) − g (cid:16) x ( µ,ξ ) T (cid:17)(cid:21) + E Z T (cid:20)Z U h (cid:18) t, x ( µ θ ,ξ θ ) t , a (cid:19) µ t ( da ) − Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) (cid:21) dt + θ E Z T (cid:20)Z U h (cid:18) t, x ( µ θ ,ξ θ ) t , a (cid:19) q t ( da ) − Z U h (cid:18) t, x ( µ θ ,ξ θ ) t , a (cid:19) µ t ( da ) (cid:21) dt + θ E Z T k t d ( η t − ξ t ) . Hence,0 ≤ E Z g x (cid:16) x ( µ,ξ ) T + λθ (cid:0) X θT + z T (cid:1)(cid:17) z T dλ (12)+ E Z T Z U Z h x (cid:16) t, x ( µ,ξ ) t + λθ (cid:0) X θt + z t (cid:1) , a (cid:17) µ t ( da ) z t dλdt + E Z T (cid:20)Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) q t ( da ) − Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) (cid:21) dt + E Z T k t d ( η t − ξ t ) + ρ θt , where X θt = x ( µ θ ,ξ θ ) t − x ( µ,ξ ) t θ − z t , and ρ θt is given by ρ θt = E Z g x (cid:16) x ( µ,ξ ) T + λθ (cid:0) X θT + z T (cid:1)(cid:17) X θT dλ + E Z T Z U Z h x (cid:16) t, x ( µ,ξ ) t + λθ (cid:0) X θt + z t (cid:1) , a (cid:17) µ t ( da ) X θt dλdt.
13y (10), we have lim θ → E (cid:12)(cid:12) X θt (cid:12)(cid:12) = 0 . Since g x and h x are continuous and bounded, then by using the Cauchy-Schwartz inequality we get lim θ → ρ θt = 0 , and by letting θ go to 0 in (12), the proof is completed. In this subsection, we introduce the adjoint process. With this process, wederive the variational inequality from (11). The linear terms in (11) may betreated in the following way. Let Φ be the fundamental solution of the linearSDE d Φ t = Z U b x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) Φ t dt + Z U σ x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) Φ t dW t , Φ = I d . This equation is linear with bounded coefficients. Hence, it admits an uniquestrong solution which is invertible, and its inverse Ψ t is the unique solution of d Ψ t = (cid:20)Z U σ x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) Ψ t Z U σ ∗ x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) (cid:21) dt − Z U b x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) Ψ t dt − Z U σ x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) Ψ t dW t , Ψ = I d . Moreover, Φ and Ψ satisfy E " sup t ∈ [0 ,T ] | Φ t | + E " sup t ∈ [0 ,T ] | Ψ t | < ∞ . (13)We introduce the following processes α t = Ψ t z t , (14) X = Φ ∗ T g x ( x ( µ,ξ ) T ) + Z T Z U Φ ∗ t h x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) dt, (15) Y t = E [ X / F t ] − Z t Z U Φ ∗ s h x (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) ds. (16)We remark from (14) , (15) and (16) that E [ α T Y T ] = E h g x (cid:16) x ( µ,ξ ) T (cid:17) z T i . (17)14ince g x and h x are bounded, then by (13), X is square integrable. Hence,the process ( E [ X / F t ]) t ≥ is a square integrable martingale with respect tothe natural filtration of the Brownian motion W . Then, by Itˆo’s representationtheorem we have Y t = E [ X ] + Z t Q s dW s − Z t Z U Φ ∗ s h x (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) ds, where, Q is an adapted process such that E Z T | Q s | ds < ∞ . By applying Itˆo’s formula to α t then with α t Y t and using (17), the variationalinequality (11) becomes0 ≤ E Z T h H (cid:16) t, x ( µ,ξ ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) − H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17)i dt (18)+ E Z T ( k t + G ∗ t p t ) d ( η t − ξ t ) , where, the Hamiltonian H is defined from [0 , T ] × R n × P ( U ) × R n × M n × d ( R )into R by H ( t, x, q, p, P ) = Z U h ( t, x, a ) q ( da )+ Z U b ( t, x, a ) q ( da ) p + Z U σ ( t, x, a ) q ( da ) P, (cid:0) p ( µ,ξ ) , P ( µ,ξ ) (cid:1) is a pair of adapted processes given by p ( µ,ξ ) t = Ψ ∗ t Y t , p ( µ,ξ ) ∈ L ([0 , T ] ; R n ) (19) P ( µ,ξ ) t = Ψ ∗ t Q t − Z U σ ∗ x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) p ( µ,ξ ) t , P ( µ,ξ ) ∈ L (cid:0) [0 , T ] ; R n × d (cid:1) , (20)and the process Q satisfies Z t Q s dW s = E " Φ ∗ T g x ( x ( µ,ξ ) T ) + Z T Φ ∗ t Z U h x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) dt / F t − E " Φ ∗ T g x ( x ( µ,ξ ) T ) + Z T Φ ∗ t Z U h x (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) dt . The process p ( µ,ξ ) is called the adjoint process and from (15) , (16) and (19),it is given explicitly by p ( µ,ξ ) t = E " Ψ ∗ t Φ ∗ T g x ( x ( µ,ξ ) T ) + Ψ ∗ t Z Tt Z U Φ ∗ s h x (cid:16) s, x ( µ,ξ ) s , a (cid:17) µ s ( da ) ds / F t .
15y applying Itˆo’s formula to the adjoint processes p ( µ,ξ ) in (19), we obtainthe adjoint equation, which is a linear backward SDE, given by ( dp ( µ,ξ ) t = −H x (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) dt + P ( µ,ξ ) t dW t ,p ( µ,ξ ) T = g x ( x ( µ,ξ ) T ) . (21) Starting from the variational inequality (18), we can now state the necessaryoptimality conditions, for the relaxed-singular control problem { (5) , (6) , (7) } ,in integral form. Theorem 9 (Necessary optimality conditions for relaxed-singular controls inintegral form). Let ( µ, ξ ) be an optimal relaxed-singular control minimizing thecost J over R and x ( µ,ξ ) denotes the corresponding optimal trajectory. Then,there exists an unique pair of adapted processes (cid:16) p ( µ,ξ ) , P ( µ,ξ ) (cid:17) ∈ L ([0 , T ] ; R n ) × L (cid:0) [0 , T ] ; R n × d (cid:1) , solution of the backward SDE (21) such that, for every ( q, η ) ∈ R ≤ E Z T h H (cid:16) t, x ( µ,ξ ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) − H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17)i dt (22)+ E Z T (cid:16) k t + G ∗ t p ( µ,ξ ) t (cid:17) d ( η t − ξ t ) , Proof.
The result follows immediately from (18).We are ready now state necessary optimality conditions for the relaxed-singular control problem { (5) , (6) , (7) } , in global form. Theorem 10 (Necessary optimality conditions for relaxed-singular controls inglobal form). Let ( µ, ξ ) be an optimal relaxed-singular control minimizing thecost J over R and x ( µ,ξ ) denotes the corresponding optimal trajectory. Then,there exists an unique pair of adapted processes (cid:16) p ( µ,ξ ) , P ( µ,ξ ) (cid:17) ∈ L ([0 , T ] ; R n ) × L (cid:0) [0 , T ] ; R n × d (cid:1) , solution of the backward SDE (21) such that H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) = inf q t ∈ P ( U ) H (cid:16) t, x ( µ,ξ ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) , a.e, a.s, (23) P n ∀ t ∈ [0 , T ] , ∀ i ; (cid:16) k i ( t ) + G ∗ i ( t ) .p ( µ,ξ ) t (cid:17) ≥ o = 1 , (24)16 ( d X i =1 n k i ( t )+ G ∗ i ( t ) p ( µ,ξ ) t ≥ o dξ it = 0 ) = 1 . (25) Proof.
Let ( µ, ξ ) be an optimal solution of problem { (5) , (6) , (7) } . The nec-essary optimality conditions in integral form (22) is valid for every ( q, η ) ∈ R .If we put in (22) η = ξ , then (23) becomes immediately. On the other hand, ifwe choose in (22) q = µ , then we can show (24) and (25), by the same methodthat in [8 , Theorem 4.2] or [4 , Theorem 3.7].
Theorem 11 (Sufficient optimality conditions for relaxed-singular controls).Assume that the functions g and x ( t, x, q, p, P ) are convex. Then, ( µ, ξ ) is an optimal solution of problem { (5) , (6) , (7) } , if it satisfies (23) , (24) and (25) . Proof.
We know that the set of relaxed-singular controls R is convex and theHamiltonian H is linear in q .Let ( µ, ξ ) be an arbitrary element of R (candidate to be optimal). For any( q, η ) ∈ R , we have J ( µ, ξ ) − J ( q, η ) = E h g (cid:16) x ( µ,ξ ) T (cid:17) − g (cid:16) x ( q,η ) T (cid:17)i + E Z T (cid:20)Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) − Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) (cid:21) dt + E Z T k t d ( ξ t − η t ) . Since g is convex, we get g (cid:16) x ( q,η ) T (cid:17) − g (cid:16) x ( µ,ξ ) T (cid:17) ≥ g x (cid:16) x ( µ,ξ ) T (cid:17) (cid:16) x ( q,η ) T − x ( µ,ξ ) T (cid:17) . Thus, g (cid:16) x ( µ,ξ ) T (cid:17) − g (cid:16) x ( q,η ) T (cid:17) ≤ g x (cid:16) x ( µ,ξ ) T (cid:17) (cid:16) x ( µ,ξ ) T − x ( q,η ) T (cid:17) . Hence, J ( µ, ξ ) − J ( q, η ) ≤ E h g x (cid:16) x ( µ,ξ ) T (cid:17) (cid:16) x ( µ,ξ ) T − x ( q,η ) T (cid:17)i + E Z T (cid:20)Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) − Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) (cid:21) dt + E Z T k t d ( ξ t − η t ) .
17e remark that p ( µ,ξ ) T = g x (cid:16) x ( µ,ξ ) T (cid:17) , then we have J ( µ, ξ ) − J ( q, η ) ≤ E h p ( µ,ξ ) T (cid:16) x ( µ,ξ ) T − x ( q,η ) T (cid:17)i + E Z T (cid:20)Z U h (cid:16) t, x ( µ,ξ ) t , a (cid:17) µ t ( da ) − Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) (cid:21) dt + E Z T k t d ( ξ t − η t ) . Applying Itˆo’s formula to p ( µ,ξ ) t (cid:16) x ( µ,ξ ) t − x ( q,η ) t (cid:17) and taking expectation, weobtain J ( µ, ξ ) − J ( q, η ) (26) ≤ E Z T h H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) − H (cid:16) t, x ( q,η ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17)i dt − E Z T H x (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) (cid:16) x ( µ,ξ ) t − x ( q,η ) t (cid:17) dt + E Z T (cid:16) k t + G ∗ t p ( µ,ξ ) t (cid:17) d ( ξ t − η t ) . Since H is convex in x and linear in µ , then by using the Clarke generalizedgradient of H evaluated at ( x t , µ t ) and (23), it follows that H (cid:16) t, x ( q,η ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) − H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) ≥ H x (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) (cid:16) x ( q,η ) t − x ( µ,ξ ) t (cid:17) . Or equivalently0 ≥ H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) − H (cid:16) t, x ( q,η ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) − H x (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) (cid:16) x ( µ,ξ ) t − x ( q,η ) t (cid:17) . By the above inequality and (26), we have J ( µ, ξ ) − J ( q, η ) ≤ E Z T (cid:16) k t + G ∗ t p ( µ,ξ ) t (cid:17) d ( ξ t − η t ) . By (24) and (25), we show that E Z T (cid:16) k t + G ∗ t p ( µ,ξ ) t (cid:17) d ( η t − ξ t ) ≥ . Then, we get J ( µ, ξ ) − J ( q, η ) ≤ . The theorem is proved. 18
Optimality conditions for strict-singular con-trols
In this section, we study the strict-singular control problem { (1) , (2) , (3) } and from the results of section 3, we derive the optimality conditions for strict-singular controls.Throughout this section and in addition to the assumptions (4), we supposethat U is compact, (27) b, σ and h are bounded. (28)Consider the following subset of R δ ( U ) × U = { ( q, η ) ∈ R / q = δ v ; v ∈ U } . Denote by δ ( U ) × U the action set of all relaxed-singular controls in δ ( U ) ×U . If ( q, η ) ∈ δ ( U ) × U , then ( q, η ) = ( δ v , η ) with v ∈ U . In this case we havefor each t , ( q t , η t ) = ( δ v t , η t ) ∈ δ ( U ) × U .We equipped P ( U ) with the topology of stable convergence. Since U iscompact, then with this topology P ( U ) is a compact metrizable space. Thestable convergence is required for bounded measurable functions f ( t, a ) suchthat for each fixed t ∈ [0 , T ], f ( t, . ) is continuous (Instead of functions boundedand continuous with respect to the pair ( t, a ) for the weak topology). Thespace P ( U ) is equipped with its Borel σ -field, which is the smallest σ -field suchthat the mapping q Z f ( s, a ) q ( ds, da ) are measurable for any boundedmeasurable function f , continuous with respect to a. For more details, see Jacodand Memin [20] and El Karoui et al [15].This allows us to summarize some of lemmas that we will be used in thesequel.
Lemma 12 (Chattering Lemma). Let q be a predictable process with values inthe space of probability measures on U . Then there exists a sequence of pre-dictable processes ( u n ) n with values in U such that dtq nt ( da ) = dtδ u nt ( da ) −→ n →∞ dtq t ( da ) stably , P − a.s. (29) where δ u nt is the Dirac measure concentrated at a single point u nt of U . Proof.
See El Karoui et al [15]. 19 emma 13
Let q ∈ R and ( u n ) n ⊂ U such that (29) holds. Then for anybounded measurable function f : [0 , T ] × U → R , such that for each fixed t ∈ [0 , T ] , f ( t, . ) is continuous, we have Z U f ( t, a ) δ u nt ( da ) −→ n →∞ Z U f ( t, a ) q t ( da ) ; dt − a.e (30) Proof.
By (29), and the definition of the stable convergence (see Jacod andMemin [20 , definition 1.1, page 529], we have Z T Z U f ( t, a ) δ u nt ( da ) dt −→ n →∞ Z T Z U f ( t, a ) q t ( da ) dt. Put g ( s, a ) = 1 [0 ,t ] ( s ) f ( s, a ) . It’s clear that g is bounded, measurable and continuous with respect to a. Then, by (29) we get Z T Z U g ( s, a ) δ u ns ( da ) ds −→ n →∞ Z T Z U g ( s, a ) q s ( da ) ds. By replacing g ( s, a ) by its value, we have Z t Z U f ( s, a ) δ u ns ( da ) ds −→ n →∞ Z t Z U f ( s, a ) q s ( da ) ds. The set { ( s, t ) ; 0 ≤ s ≤ t ≤ T } generate B [0 ,T ] . Then for every B ∈ B [0 ,T ] ,we have Z B Z U f ( s, a ) δ u ns ( da ) ds −→ n →∞ Z B Z U f ( s, a ) q s ( da ) ds. This implies that Z U f ( s, a ) δ u ns ( da ) −→ n →∞ Z U f ( s, a ) q s ( da ) , dt − a.e. The lemma is proved.The next lemma gives the stability of the controlled SDE with respect tothe control variable.
Lemma 14
Let ( q, η ) ∈ R be a relaxed-singular control and x ( q,η ) the corre-sponding trajectory. Then there exists a sequence ( u n , η ) n ⊂ U such that lim n →∞ E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) x ( u n ,η ) t − x ( q,η ) t (cid:12)(cid:12)(cid:12) = 0 , (31)lim n →∞ J ( u n , η ) = J ( q, η ) . (32) where x ( u n ,η ) denotes the solution of equation (1) associated with ( u n , η ) . roof. i) Proof of (31). We have E (cid:12)(cid:12)(cid:12) x ( u n ,η ) t − x ( q,η ) t (cid:12)(cid:12)(cid:12) ≤ C Z t E (cid:12)(cid:12)(cid:12) b (cid:16) s, x ( u n ,η ) s , u ns (cid:17) − b (cid:16) s, x ( q,η ) s , u ns (cid:17)(cid:12)(cid:12)(cid:12) ds + C Z t E (cid:12)(cid:12)(cid:12)(cid:12) b (cid:16) s, x ( q,η ) s , u ns (cid:17) − Z U b (cid:16) s, x ( q,η ) s , a (cid:17) q s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds + C Z t E (cid:12)(cid:12)(cid:12) σ (cid:16) s, x ( u n ,η ) s , u ns (cid:17) − σ (cid:16) s, x ( q,η ) s , u ns (cid:17)(cid:12)(cid:12)(cid:12) ds + C Z t E (cid:12)(cid:12)(cid:12)(cid:12) σ (cid:16) s, x ( q,η ) s , u ns (cid:17) − Z U σ (cid:16) s, x ( q,η ) s , a (cid:17) q s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds Since b and σ are uniformly Lipschitz with respect to x , then E (cid:12)(cid:12)(cid:12) x ( u n ,η ) t − x ( q,η ) t (cid:12)(cid:12)(cid:12) (33) ≤ C Z t E (cid:12)(cid:12)(cid:12) x ( u n ,η ) s − x ( q,η ) s (cid:12)(cid:12)(cid:12) ds + C Z t E (cid:12)(cid:12)(cid:12)(cid:12)Z U b (cid:16) s, x ( q,η ) s , a (cid:17) δ u ns ( da ) − Z U b (cid:16) s, x ( q,η ) s , a (cid:17) q s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds + C Z t E (cid:12)(cid:12)(cid:12)(cid:12)Z U σ (cid:16) s, x ( q,η ) s , a (cid:17) δ u ns ( da ) − Z U σ (cid:16) s, x ( q,η ) s , a (cid:17) q s ( da ) (cid:12)(cid:12)(cid:12)(cid:12) ds Since b and σ are bounded, measurable and continuous with respect to a ,then by (30) and the dominated convergence theorem, the second and thirdterms in the right hand side of the above inequality tend to zero as n tends toinfinity. We conclude then by using Gronwall’s lemma and Bukholder-Davis-Gundy inequality.ii) Proof of (32). By using the Cauchy-Schwartz inequality and the fact that g and h are uniformly Lipschitz with respect to x , we get | J ( q n , η ) − J ( q, η ) |≤ C (cid:18) E (cid:12)(cid:12)(cid:12) x ( u n ,η ) T − x ( q,η ) T (cid:12)(cid:12)(cid:12) (cid:19) / + C Z T E (cid:12)(cid:12)(cid:12) x ( u n ,η ) t − x ( q,η ) t (cid:12)(cid:12)(cid:12) ds ! / + E Z T (cid:12)(cid:12)(cid:12)(cid:12)Z U h (cid:16) s, x ( q,η ) s , a (cid:17) δ u ns ( da ) dt − Z U h (cid:16) t, x ( q,η ) t , a (cid:17) q t ( da ) (cid:12)(cid:12)(cid:12)(cid:12) dt ! / . By (31), the first and second terms in the right hand side converge to zero.Moreover, since h is bounded, measurable and continuous in a , then by (30)and the dominated convergence theorem, the third term in the right hand sidetends to zero as n tends to infinity. This prove (32).21 emma 15 As a consequence of (32) , the strict-singular and the relaxed-singularcontrol problems have the same value functions. That is inf ( v,η ) ∈U J ( v, η ) = inf ( q,η ) ∈R J ( q, η ) . (34) Proof.
Let ( u, ξ ) ∈ U and ( µ, ξ ) ∈ R be respectively a strict-singular andrelaxed-singular controls such that J ( u, ξ ) = inf ( v,η ) ∈U J ( v, η ) , (35) J ( µ, ξ ) = inf ( q,η ) ∈R J ( q, η ) . (36)By (36), we have J ( µ, ξ ) ≤ J ( q, η ) , ∀ ( q, η ) ∈ R .Since δ ( U ) × U ⊂ R , then J ( µ, ξ ) ≤ J ( q, η ) , ∀ ( q, η ) ∈ δ ( U ) × U .Since ( q, η ) ∈ δ ( U ) × U , then ( q, η ) = ( δ v , η ), where v ∈ U . Then we get (cid:26) x ( q,η ) = x ( v,η ) , J ( q, η ) = J ( v, η ) . Hence J ( µ, ξ ) ≤ J ( v, η ) , ∀ ( v, η ) ∈ U .The control ( u, ξ ) becomes an element of U , then we get J ( µ, ξ ) ≤ J ( u, ξ ) . (37)On the other hand, by (35) we have J ( u, ξ ) ≤ J ( v, η ) , ∀ ( v, η ) ∈ U . (38)The process µ becomes an element of R , then by the Chattering lemma(Lemma 12), there exists a sequence ( u n ) n ⊂ U such that dtµ nt ( da ) = dtδ u nt ( da ) −→ n →∞ dtµ t ( da ) stably , P − a.s.
Relation (38) holds for every ( v, η ) ∈ U . This is true for ( u n , ξ ) ∈ U , ∀ n ∈ N . We get then J ( u, ξ ) ≤ J ( u n , ξ ) , ∀ n ∈ N ,By using (32) and letting n go to infinity in the above inequality, we get J ( u, ξ ) ≤ J ( µ, ξ ) . (39)22inally, by (37) and (39), we have J ( u, ξ ) = J ( µ, ξ ) . The lemma is proved.To establish necessary optimality conditions for strict-singular controls, weneed the following lemma
Lemma 16
The strict-singular control ( u, ξ ) minimizes J over U if and only ifthe relaxed-singular control ( µ, ξ ) = ( δ u , ξ ) minimizes J over R . Proof.
Suppose that ( u, ξ ) minimizes the cost J over U , then J ( u, ξ ) = inf ( v,η ) ∈U J ( v, η ) .By using (34), we get J ( u, ξ ) = inf ( q,η ) ∈R J ( q, η ) .Since ( µ, ξ ) = ( δ u , ξ ), then (cid:26) x ( µ,ξ ) = x ( u,ξ ) , J ( µ, ξ ) = J ( u, ξ ) , (40)This implies that J ( µ, ξ ) = inf ( q,η ) ∈R J ( q, η ) . Conversely, if ( µ, ξ ) = ( δ u , ξ ) minimize J over R , then J ( µ, ξ ) = inf ( q,η ) ∈R J ( q, η ) . From (34), we get J ( µ, ξ ) = inf ( v,η ) ∈U J ( v, η ) . Since ( µ, ξ ) = ( δ u , ξ ), then relations (40) hold, and we obtain J ( u, ξ ) = inf ( v,η ) ∈U J ( v, η ) . The proof is completed.The following lemma, who will be used to establish sufficient optimalityconditions for strict-singular controls, shows that we get the results of the abovelemma if we replace R by δ ( U ) × U . Lemma 17
The strict-singular control ( u, ξ ) minimizes J over U if and only ifthe relaxed control ( µ, ξ ) = ( δ u , ξ ) minimizes J over δ ( U ) × U . roof. Let ( µ, ξ ) = ( δ u , ξ ) be an optimal relaxed-singular control minimizingthe cost J over δ ( U ) × U , we have then J ( µ, ξ ) ≤ J ( q, η ) , ∀ ( q, η ) ∈ δ ( U ) × U . (41)Since q ∈ δ ( U ), then there exists v ∈ U such that q = δ v . Hence, ( δ v , η ) =( q, η ), and since ( µ, ξ ) = ( δ u , ξ ), it is easy to see that x ( µ,ξ ) = x ( u,ξ ) ,x ( q,η ) = x ( v,η ) , J ( µ, ξ ) = J ( u, ξ ) , J ( q, η ) = J ( v, η ) . (42)By (41) and (42), we get then J ( u, ξ ) ≤ J ( v, η ) , ∀ ( v, η ) ∈ U .Conversely, let ( u, ξ ) be a strict-singular control minimizing the cost J over U . Then J ( u, ξ ) ≤ J ( v, η ) , ∀ ( v, η ) ∈ U .Since the controls u, v ∈ U , then there exist µ, q ∈ δ ( U ) such that µ = δ u and q = δ v . Then ( µ, ξ ) = ( δ u , ξ ) , ( q, η ) = ( δ v , η ) . This implies that relations (42) hold. Consequently, we get J ( µ, ξ ) ≤ J ( q, η ) , ∀ ( q, η ) ∈ δ ( U ) × U . The proof is completed.
Define the Hamiltonian in the strict case from [0 , T ] × R n × U × R n ×M n × d ( R ) into R by H ( t, x, v, p, P ) = h ( t, x, v ) + b ( t, x, v ) p + σ ( t, x, v ) P. Theorem 18 (Necessary optimality conditions for strict-singular controls inglobal form). Suppose that ( u, ξ ) is an optimal strict-singular control minimizingthe cost J over U and x ( u,ξ ) denotes the solution of (1) controlled by ( u, ξ ) . Then,there exists an unique pair of adapted processes (cid:16) p ( u,ξ ) , P ( u,ξ ) (cid:17) ∈ L ([0 , T ] ; R n ) × L (cid:0) [0 , T ] ; R n × d (cid:1) , olution of the backward SDE ( dp ( u,ξ ) t = − H x (cid:16) t, x ( u,ξ ) t , u t , p ( u,ξ ) t , P ( u,ξ ) t (cid:17) dt + P ( u,ξ ) t dW t ,p ( u,ξ ) T = g x ( x ( u,ξ ) T ) , (43) such that H (cid:16) t, x ( u,ξ ) t , u t , p ( u,ξ ) t , P ( u,ξ ) t (cid:17) = inf v t ∈ U H (cid:16) t, x ( u,ξ ) t , v t , p ( u,ξ ) t , P ( u,ξ ) t (cid:17) , a.e, a.s. (44) P n ∀ t ∈ [0 , T ] , ∀ i ; (cid:16) k i ( t ) + G ∗ i ( t ) .p ( u,ξ ) t (cid:17) ≥ o = 1 , (45) P ( d X i =1 n k i ( t )+ G ∗ i ( t ) p ( u,ξ ) t ≥ o dξ it = 0 ) = 1 . (46) Proof.
The optimal strict-singular control ( u, ξ ) is an element of U , then thereexists ( µ, ξ ) ∈ δ ( U ) × U such that( µ, ξ ) = ( δ u , ξ ) . Since ( u, ξ ) minimizes the cost J over U , then by lemma 16, ( µ, ξ ) mini-mizes J over R . Hence, by the necessary optimality conditions for relaxed-singular controls (Theorem 10), there exists an unique pair of adapted processes (cid:0) p ( µ,ξ ) , P ( µ,ξ ) (cid:1) , solution of (21), such that relations (23) , (24) and (25) hold.Since δ ( U ) ⊂ P ( U ), then by (23), we get H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) ≤ H (cid:16) t, x ( µ,ξ ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) , ∀ q t ∈ δ ( U ) , a.e, a.s. (47)Since q ∈ δ ( U ), then there exists v ∈ U such that q = δ v .We note that v is an arbitrary element of U since q is arbitrary.Now, since ( µ, ξ ) = ( δ u , ξ ) and ( q, ξ ) = ( δ v , ξ ), we can easily see that x ( µ,ξ ) = x ( u,ξ ) ,x ( q,ξ ) = x ( v,ξ ) , (cid:0) p ( µ,ξ ) , P ( µ,ξ ) (cid:1) = (cid:0) p ( u,ξ ) , P ( u,ξ ) (cid:1) , H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) = H (cid:16) t, x ( u,ξ ) t , u t , p ( u,ξ ) t , P ( u,ξ ) t (cid:17) , H (cid:16) t, x ( µ,ξ ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) = H (cid:16) t, x ( u,ξ ) t , v t , p ( u,ξ ) t , P ( u,ξ ) t (cid:17) , (48)where (cid:0) p ( u,ξ ) , P ( u,ξ ) (cid:1) is the unique solution of (43).By using (47) and (48), we deduce (44). Relations (45) and (46) followsimmediately from (24) , (25) and (48) . The proof is completed.
Remark 19
Bahlali and Mezerdi [2] , established necessary optimality condi-tions for strict-singular controls of the second-order with two adjoint processes.The result of the above theorem improves that of [2] , in the sense where, we con-sider the same strict-singular control problem, with nonconvex control domainand a general state equation in which the control variable enters both the driftand the diffusion coefficients, and we establish necessary optimality conditionsof the first-order with only one adjoint process. .2 Sufficient optimality conditions for strict-singular con-trols Theorem 20 (Sufficient optimality conditions for strict-singular controls). As-sume that the functions g and x H ( t, x, q, p, P ) are convex. Then, ( u, ξ ) isan optimal solution of problem { (1) , (2) , (3) } if it satisfies (44) , (45) and (46) . Proof.
Let ( u, ξ ) ∈ U be a strict-singular control (candidate to be optimal) and( v, η ) an arbitrary element of U .The controls u, v are elements of U , then there exist µ, q ∈ δ ( U ) such that µ = δ u and q = δ v . Hence, ( µ, ξ ) = ( δ u , ξ ) , ( q, η ) = ( δ v , η ) . This implies that relations (48) hold. Then, by (44) , (45) and (46), wededuce respectively the relaxed relations H (cid:16) t, x ( µ,ξ ) t , µ t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) = inf q t ∈ δ ( U ) H (cid:16) t, x ( µ,ξ ) t , q t , p ( µ,ξ ) t , P ( µ,ξ ) t (cid:17) , a.e, a.s, (49) P n ∀ t ∈ [0 , T ] , ∀ i ; (cid:16) k i ( t ) + G ∗ i ( t ) .p ( µ,ξ ) t (cid:17) ≥ o = 1 , (50) P ( d X i =1 n k i ( t )+ G ∗ i ( t ) p ( µ,ξ ) t ≥ o dξ it = 0 ) = 1 . (51)We remark that the infimum in (49) is taken over δ ( U ) . Now, since H is convex in x , it is easy to see that H is convex in x , and since g is convex, then by using (49) , (50) and (51), and by the same proof that intheorem 11, we show that ( µ, ξ ) minimizes the cost J over δ ( U ) × U . Then,by Lemma 17, we deduce that ( u, ξ ) minimizes the cost J over U . The theoremis proved. Remark 21
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