Draft. Transposition method for BSDEs / BSEEs and applications. Xu Zhang School of Mathematics, Sichuan University zhang

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1 December 13, 2013 Universié Pierre e Marie Curie Transposiion mehod for BSDEs / BSEEs and applicaions Xu Zhang School of Mahemaics, Sichuan Universiy zhang xu@scu.edu.cn

2 Ouline: 1. Backgroud from conrol heory 2. The classical ransposiion mehod in PDEs 3. Moivaion and definiion for ransposiion soluion of BSDEs 4. Well-posedness of BSDEs in he ransposiion sense 5. Transposiion soluion o vecor-valued BSEEs 6. Well-posedness of an operaor-valued BSEE 7. Ponryagin-ype sochasic maximum principle Firs Prev Nex Las Go Back Full Screen Close Qui

3 1. Backgroud from conrol heory Review on LQ problems Given T > 0, consider he following conrol sysem: { ẋ() = Ax() + Bu(), a.e. [0, T ], x(0) = x 0 R n (1), wih a cos funcional J(u( )) = 1 2 T 0 [ Cx(), x() + Du(), u() ]d F x(t ), x(t ), u( ) L2 (0, T ; R m ). Here C = C, D = D and F = F. (2)

4 LQ Problem: Minimize (2) over L 2 (0, T ; R m ). Any u( ) L 2 (0, T ; R m ) saisfying J(u( )) = inf J(u( )) u( ) L 2 (0,T ;R m ) is called an opimal conrol, he corresponding sae rajecory x( ) = x( ; u( )) is called an opimal sae rajecory. How o idenify u( )? For he above LQ, his is easy.

5 Indeed, for any ε R and u( ) L 2 (0, T ; R m ), wrie u ε ( ) = u( ) + εu( ), x ε ( ) = x( ; x 0, u ε ( )). Then, x ε ( ) = x( ) + εr( ), r( ) = By 0 e A( s) Bu(s)ds. J(x ε ( )) { T = J(x( )) + ε [ C x(), r() + Dū(), u() ]d 0 } + F x(t ), r(t ) + O(ε 2 ) J(x( )),

6 We ge Du() = B [ T Wrie ψ() = T ] e A (s ) Cx(s)ds+e A (T ) F x(t ). e A (s ) Cx(s)ds e A (T ) F x(t ) We obain Du() = B ψ(), where ψ( ) solves he following backward ODE: { ψ() = A ψ() + Cx(), ψ(t ) = F x(t ).

7 Review on deerminisic opimal conrol problems Now, consider he following conrolled ODE under sandard assumpions: { ẋ() = b(, x(), u()), a.e. [0, T ], x(0) = x 0 R n (3), wih a cos funcional J(u( )) = T 0 f(, x(), u())d + h(x(t )). (4) Here V[0, T ] = {u( ) : [0, T ] U u( ) measurable} and U R m.

8 Deerminisic opimal conrol problem can be saed as follows. Problem (D): Minimize (4) over V[0, T ]. Any u( ) V[0, T ] saisfying J(u( )) = inf J(u( )) u( ) V[0,T ] is called an opimal conrol, he corresponding sae rajecory x( ) = x( ; u( )) and (x( ), u( )) are called an opimal sae rajecory and opimal pair, respecively.

9 The following resul gives a (firs-order) necessary condiion for opimal pairs. Ponryagin s Maximum Principle. Le (x( ), u( )) be an opimal pair for Problem (D), and le p( ) : [0, T ] R n solve he backward ODE ṗ() = b x (, x(), u()) p() + f x (, x(), u()), a.e. [0, T ], p(t ) = h x (x(t )). (5) Then, for a.e. [0, T ], H(, x(), u(), p()) = max u U H(, x(), u, p()), where H(, x, u, p) = p, b(, x, u) f(, x, u).

10 Key o he proof: Spike Variaion Technique. Tha is, Le (x( ), u( )) be he given opimal pair. Le ε > 0 and E ε [0, T ] be a measurable se whose Lebesgue measure E ε = ε. For any u( ) V[0, T ], define { u(), if [0, T ] \ u ε Eε, () = u(), if E ε. Then u ε ( ) V[0, T ]. Noe ha, U does no necessarily have a linear srucure. Thus, in general, a perurbaion like u() + εu() is meaningless. We refer o u ε ( ) as a spike (or needle) variaion of he conrol u( ).

11 Review on sochasic opimal conrol problems in R n Le (Ω, F, F, P) be a complee filered probabiliy s- pace wih F = {F } [0,T ], on which a 1-dimensional sandard Brownian moion {W ()} [0,T ] is defined. Denoe by W he naural filraion generaed by {W ()} and augmened by all he P-null ses. The filraion {F } [0,T ] plays a crucial role, and i represens he informaion ha one has a each ime. For sochasic differenial equaion (in he Iô sense), one needs o use adaped processes X( ), i.e., for each give, he r.v. X() is a leas F -measurable.

12 We consider he following sochasic conrolled equaion under sandard assumpions: { dx() = b(, x(), u())d + σ(, x(), u())dw (), x(0) = x 0 R n, (6) wih a cos funcional { T } J(u( )) = E f(, x(), u())d+h(x(t )). (7) 0 The conrol u( ) belongs o he following U[0, T ] ={u : [0, T ] Ω U u is {F } 0 -adaped}.

13 The usual opimal conrol problem for (6) can be saed as follows. Problem (S): Minimize (7) over U[0, T ]. Any u( ) U[0, T ] saisfying J(u( )) = inf u( ) U[0,T ] J(u( )), is called an opimal conrol, he corresponding x( ) x( ; u( )) and (x( ), u( )) are called an opimal sae process/rajecory and opimal pair, respecively.

14 To solve Problem (S), one needs o inroduce he following BSDE: { dp() = b x (, x(), u()) p()+σ x (, x(), u()) q() f x (, x(), u()) } d+q()dw (), [0, T ], p(t ) = h x (x(t )). (8) Here he unknown is a pair of {F } 0 -adaped processes (p( ), q( )). Why wo unknowns for a single equaion? Wihou he correced erm q( ), i is impossible o find an adaped soluion o he following simple equaion: { dp() = 0, [0, T ], p(t ) = p T L 2 (Ω, F T, P).

15 Since he conrol variable u( ) appears in he diffusion erm, S. Peng (1990) inroduced an addiional adjoin equaion as follows: [ dp () = b x (, x(), u()) P () + P ()b x (, x(), u()) +σ x (, x(), u()) P ()σ x (, x(), u()) +σ x (, x(), u()) Q() + Q()σ x (, x(), u()) ] +H xx (, x(), u(), p(), q()) d + Q()dW (), P (T ) = h xx (x(t )), where H(, x, u, p, q) = p, b(, x, u) + q, σ(, x, u) f(, x, u). Wrie S n = {A R n n A = A}. Equaion (9) is an S n -valued BSDE. (9) Firs Prev Nex Las Go Back Full Screen Close Qui

16 Define an H-funcion: H(, x, u) 1 2 r [ σ(, x, u) P ()σ(, x, u) ] + p(), b(, x, u) f(, x, u) +r [ q() σ(, x, u) ] r [ σ(, x, u) P ()σ(, x(), u()) ]. Peng s Sochasic Maximum Principle. Assume F = W. Le (x( ), u( )) be an opimal pair of Problem (S). Then here are pairs of processes (p( ), q( )) and (P ( ), Q( )) saisfying he firs-order and secondorder adjoin equaions (8) and (9), respecively, such ha H(, x(), u()) = max u U H(, x(), u), a.e. [0, T ], P-a.s.

17 Sochasic opimal conrol problems in infinie dimensions Consider he following conrolled forward sochasic evoluion equaion dx() = [ Ax() + a(, x(), u()) ] d +b(, x(), u())dw (), (0, T ], x(0) = x 0, (10) where A is an unbounded linear operaor (on a Hilber space H), generaing a C 0 -semigroup. Le U be a meric space. Pu { U[0, T ] u( ) : [0, T ] U } u( ) is F-adaped.

18 Define a cos funcional J ( ) as follows: [ T ] J (u( )) E g(, x(), u())d + h(x(t )). 0 We consider he following opimal conrol problem: Problem (P): Find u( ) U[0, T ] such ha J (ū( )) = inf u( ) U[0,T ] J (u( )). (11) Any u( ) U[0, T ] saisfying (11) is called an opimal conrol, he corresponding x( ) x( ; u( )) and (x( ), u( )) are called an opimal sae process/rajecory and opimal pair, respecively.

19 Our goal is o give a Ponryagin-ype maximum principle for he above sochasic opimal conrol problem. The case when dim H < is now well-undersood, see S. Peng (1990). The case when he conrol does NOT appear in he diffusion erm or he conrol se is convex: A. Bensoussan (1983), Y. Hu and S. Peng (1992), V.V. Anh, W. Grecksch and J. Yong (2010), A. Al-Hussein (2010, 2011), ec. The case when he conrol appears in he diffusion erm and he conrol se is nonconvex: Only wo previous references (i.e., X.Y. Zhou (1993) addressing he linear problem, and S. Tang and X. Li (1994) for he problem wih very special daa).

20 Main difficuly: How o define he soluion o operaor-valued BSEE? When H = R n, an R n n (marix)-bsde can be regarded as an R n2 (vecor)-valued BSDE. When dim H =, L(H) (wih he uniform operaor opology) is sill a Banach space. Neverheless, i is neiher reflexive nor separable even if H iself is separable. There exis no saisfacory sochasic inegraion/evoluion equaion heories in general Banach spaces, say how o define he Iô inegral T 0 Q(s)dW (s) (for an operaor-valued process Q( ))? The exising resul on sochasic inegraion/evoluion equaion in UMD Banach spaces does no fi he presen case because, if a Banach space is UMD, hen i is reflexive. Firs Prev Nex Las Go Back Full Screen Close Qui

21 In his alk, we shall presen a new mehod o solve BSDEs/BSEEs. The main idea comes from he ransposiion mehod for deerminisic non-homogeneous boundary value problems by J.-L. Lions and E. Magenes (1972), and especially he boundary conrollabiliy problem for hyperbolic equaions (e.g., J.-L. Lions (1988)).

22 2. The classical ransposiion mehod in PDEs We now recall he main idea in he classical ransposiion mehod o solve he following wave equaion wih non-homogeneous Dirichel boundary condiions: { y y = 0 in Q (0, T ) G, y = u on Σ (0, T ) Γ, y(0) = y 0, y (0) = y 1 in G, (12) where T > 0, G is a nonempy open bounded domain in R d (d N) wih C 2 boundary Γ, (y 0, y 1 ) L 2 (G) H 1 (G) and u L 2 ((0, T ) Γ).

23 When u 0, one can use Semigroup Theory o show he well-posedness of (12) in he soluion space (y ) C([0, T ]; L 2 (G)) C 1 ([0, T ]; H 1 (G)). When u 0, one needs o use he ransposiion mehod because y Σ = u does NOT make sense by he usual race heorem. For his purpose, for any f L 1 (0, T ; L 2 (G)) and g L 1 (0, T ; H0(G)), 1 consider he following adjoin problem of (12): ζ ζ = f + g, in Q, ζ = 0, on Σ, ζ(t ) = ζ (T ) = 0, in G. (13) Equaion (13) admis a unique soluion ζ C([0, T ]; H0(G)) 1 C 1 ([0, T ]; L 2 (G)), which enjoys he regulariy ζ ν L2 (Σ). Firs Prev Nex Las Go Back Full Screen Close Qui

24 In order o give a reasonable definiion for he soluion o he non-homogenous boundary problem (12) by he ransposiion mehod, we consider firs he case when y is sufficienly smooh. The following resul holds: Assume g C0 (0, T ; H0(G)) 1 and ha y H 2 (Q) saisfies (12). Then, muliplying he firs equaion in (12) by ζ, inegraing i in Q, and using inegraion by pars, we find ha fydxd gy dxd = Q G Q ζ(0)y 1 dx G ζ (0)y 0 dx Σ ζ ν udσ. (14)

25 Noe ha (14) sill makes sense even if he regulariy of y is relaxed as y C([0, T ]; L 2 (G)) C 1 ([0, T ]; H 1 (G)). Definiion 1. We call y C([0, T ]; L 2 (G)) C 1 ([0, T ]; H 1 (G)) a ransposiion soluion o (12), if y(0) = y 0, y (0) = y 1, and for any f L 1 (0, T ; L 2 (G)) and g L 1 (0, T ; H0(G)), 1 i holds ha T fydxd g, y H 1 0 (G),H 1 (G) d Q = ζ(0), y 1 H 1 0 (G),H 1 (G) + 0 Ω ζ (0)y 0 dx where ζ is he unique soluion o (13). Σ ζ ν udσ,

26 One can show he well-posedness of (12) in he sense of ransposiion. The main idea of his mehod is o inerpre he soluion o a forward wave equaion wih non-homogeneous Dirichle boundary condiions in erms of anoher backward wave equaion wih nonhomogeneous source erms. We shall use his idea o inerpre BSDEs/BSEEs in erms of SDEs/SEEs. This enables us To provide a new mehod for solving BS- DEs/BSEEs wih general filraion; To give a new numerical schemes for solving BS- DEs (even wih he naural filraion); To esablish a general Ponryagin-ype sochasic maximum principle in general filraion spaces. The ransposiion mehod is a varian of dualiy mehod. Firs Prev Nex Las Go Back Full Screen Close Qui

27 3. Moivaion and definiion for ransposiion soluion of BSDEs Consider he following semilinear BSDE: { dy() = f(, y(), Y ())d + Y ()dw() in [0, T ], y(t ) = y T L 2 F T (Ω; R n (15) ), where f(,, ) saisfies he global Lipschiz condiion (uniformly w.r.. ) and f(, 0, 0) L 2 F (Ω; L1 (0, T ; R n )).

28 Recall ha, (y( ), Y ( )) L 2 F (Ω; C([0, T ]; Rn )) L 2 F (Ω; L2 (0, T ; R n )) is called a (srong) soluion o he equaion (15) if for any [0, T ], y() = y T T f(s, y(s), Y (s))ds T Y (s)dw(s). The firs sep o esablish he well-posedness of he semilinear BSDE (15) is o sudy he same problem bu for he following linear BSDE wih a nonhomonomous erm f( ) L 2 F (Ω; L1 (0, T ; R n )): { dy() = f()d + Y ()dw (), [0, T ), (16) y(t ) = y T.

29 The case F = W is now well-undersood (J.-M. Bismu (1978) and E. Pardoux & S. Peng (1990)). In his case, he main idea o solve (16) is as follows Define a square inegrable {F }-maringale ( T M() = E y T f(s)ds F ). (17) By MRT, Y ( ) L 2 W (Ω; L2 (0, T ; R n )) so ha Puing M() = M(0) + y() = M() Y (s)dw (s). (18) 0 f(s)ds, (19) one hen find he srong soluion (y( ), Y ( )) for (16).

30 MRT plays a crucial role in he above. In he general case when he filraion F is no equal o he naural one, W migh be a proper sub-class of F, and herefore, he MRT fails. Only a very few works addressing he well-posedness for BSDEs wih he general filraion, say N. El Karoui and S.-J. Huang (1997), and G. Liang, T. Lyons and Z. Qian (2011). The main idea of N. El Karoui and S.-J. Huang (1997) for solving (16) is as follows. Since he filraion F is no equal o he naural one, he following M 2 0,M,F ([0, T ]; Rn ) { g(s)dw (s) 0 } g( ) L2 F(Ω; L 2 (0, T ; R n )) is a proper subspace of M 2 0,F ([0, T ]; Rn ).

31 Then one has he following (unique) orhogonal decomposiion: M( ) M(0) = P ( ) + Q( ), (20) for some P ( ) M 2 0,M,F ([0, T ]; Rn ) and Q( ) ( M 2 0,M,F ([0, T ]; R n ) ). Hence, here is a Y ( ) L 2 F (Ω; L2 (0, T ; R n )) such ha P () = 0 Y (s)dw (s). (21)

32 Sill, we define y( ) as in (19). I is easy o check ha (y( ), Q( ), Y ( )) L 2 F (Ω; D([0, T ]; Rn )) ( M 2 0,M,F ([0, T ]; R n ) ) L 2 F (Ω; L 2 (0, T ; R n )) is he u- nique soluion of he following equaion y() = y T + Q() Q(T ) T f(s)ds [0, T ]. T Y (s)dw(s), (22) This means ha (22) is anoher reasonable modificaion of he linear BSDE (16) (by adding anoher correced erm Q( )).

33 Noe ha he appearance of he exra erm Q( ) makes he rigorous analysis on he properies of y( ) and Y ( ) much more complicaed han he case of naural filraion. Indeed, 1) One needs o use some deep resuls in maringale heory o esablish he dualiy relaionship beween his sor of modified BSDEs and he usual (forward) sochasic differenial equaions alhough i is no difficul o give he desired relaionship formally; 2) Meanwhile, one knows very lile abou he space M 2 0,M,F ([0, T ]; Rn ) (which is acually inroduced o replace he use of MRT), and herefore, i seems very difficul o compue he above Y ( ) in (21).

34 Recenly, by replacing Y ()dw () (in (16)) by dm() (wih M( ) being a square-inegrable maringale), G. Liang, T. Lyons and Z. Qian (2011) developed anoher approach for he well-posedness of BSDEs wih he general filraion. Their main idea o solve he equaion (16) (wih general filraion) is as follows: Alhough formula (18) does no make sense any more, M( ) M 2 F ([0, T ]; Rn ) and y( ) L 2 F (Ω; D([0, T ]; Rn )) are sill well-defined respecively by (17) and (19), and verifies M(0) = y(0), P-a.s. Then, i is easy o check ha he above (y( ), M( )) is he unique soluion of he following equaion y() = y T T f(s)ds + M() M(T ), [0, T ] (23)

35 in he soluion space { Υ = (h( ), N( )) L 2 F(Ω; D([0, T ]; R n )) } M 2 F([0, T ]; R n ) N(0) = h(0). This means ha (23) is a reasonable modificaion of he linear BSDE (16). The advanage of his approach is ha MRT is no required. Bu he cos is ha he adjusing erm Y ( ) in (16) (or more generally, in (15)) is suppressed. Noe ha his erm plays a crucial role in some problems, say he Ponryagin-ype maximum principle for s- ochasic opimal conrol problems. Also, comparison heorem is unclear in his seing because he usual d- ualiy analysis is no available.

36 We now presen a differen approach o rea he wellposedness of BSDEs wih general filraion. The idea is as follows. Fixing [0, T ], we sar from he following linear (forward) sochasic differenial equaion { dz(τ) = u(τ)dτ + v(τ)dw(τ), τ (, T ], (24) z() = η. I is clear ha, for given u( ) L 2 F (Ω; L1 (, T ; R n )), v( ) L 2 F (Ω; L2 (, T ; R n )) and η L 2 F (Ω; R n ), he equaion (24) admis a unique srong soluion z( ) L 2 F (Ω; C([, T ]; Rn )).

37 Now, if he equaion (15) admis a srong soluion (y( ), Y ( )) L 2 F (Ω; C([0, T ]; Rn )) L 2 F (0, T ; L2 (Ω; R n )) (say, when F = W), hen, applying Iô s formula o z(), y(), i follows E z(t ), y T E η, y() = E +E +E T T T z(τ), f(τ, y(τ), Y (τ)) dτ u(τ), y(τ) dτ v(τ), Y (τ) dτ. (25)

38 This inspires us o inroduce he following new noion for he soluion o (15). Definiion 2. We call (y( ), Y ( )) L 2 F (Ω; D([0, T ]; Rn )) L 2 F (Ω; L2 (0, T ; R n )) a ransposiion soluion o (15) if for any [0, T ], u( ) L 2 F (Ω; L1 (, T ; R n )), v( ) L 2 F (Ω; L2 (, T ; R n )) and η L 2 F (Ω; R n ), he ideniy (25) holds. Clearly, any ransposiion soluion of he equaion (15) coincides wih is srong soluion whenever he filraion F is naural.

39 The main advanage of our approach consiss in he fac ha he dualiy analysis is conained in he definiion of soluions, and herefore, one can easily deduce a similar comparison heorem for ransposiion soluions of (15) by using almos he same approach as in N. El Karoui, S. Peng and M. C. Quenez (1997). Also, i is even easier o esablish a Ponryaginype maximum principle for sochasic opimal conrol problems in general filraion spaces han o solve he same problem wih he naural filraion because, again, he desired dualiy analysis is conained in our definiion of soluions.

40 On he oher hand, our definiion can be used as a basis for numerical soluions o BSDEs (even for naural filraion). To see his, we recall he main idea of he classical finie elemen mehod o find he soluion y H0(G)( H 1 2 (G)) o he following ellipic equaion: { y = f, in G, (26) y = 0 on G, where G is a bounded smooh domain in R n, f L 2 (G). A weak (or variaional) formulaion of (26) reads y φdx = fφdx, φ H0(G). 1 (27) G G

41 The key variaional formulaion (27) reminds one o choose a sequence of finie dimensional spaces {H m } H0(G) 1 (ending o H0(G) 1 in some sense), called finie elemen subspaces, and o find approximae soluions {y m } of he equaion (26) so ha y m H m and y m φdx = fφdx, φ H m. (28) G G In P. Wang and X. Zhang, CRAS, 2011, we have used a similar idea o give a numerical scheme solving BSDEs.

42 4. Well-posedness of BSDEs in he ransposiion sense Consider firs he linear case. Theorem 1. (Q. Lü and X. Zhang, JDE, 2013) For any f( ) L 2 F (Ω; L1 (0, T ; R n )) and y T L 2 F T (Ω; R n ), he equaion (16) admis a unique ransposiion soluion (y( ), Y ( )) L 2 F (Ω; D([0, T ]; Rn )) L 2 F (Ω; L2 (0, T ; R n )). Furhermore, (y( ), Y ( )) L 2 F (Ω;D([,T ];R n )) L 2 F (Ω;L2 (,T ;R n )) [ ] C f( ) L 2 F (Ω;L 1 (,T ;R n )) + y T L 2 FT (Ω;R n ), [0, T ]. (29)

43 Skech of he proof: Sep 1. Define a linear funcional l on H L 2 F (Ω; L1 (, T ; R n )) L 2 F (Ω; L2 (, T ; R n )) L 2 F (Ω; R n ) as follows: l ( u( ), v( ), η ) = E z(t ), y T E T ( u( ), v( ), η ) H, z(τ), f(τ) d, where z( ) solves (24). Then, ( ) l u( ), v( ), η [ ] C f( ) L 2 F (Ω;L 1 (,T ;R n )) + y T L 2 FT (Ω;R n ) ( u( ), v( ), η ) H, [0, T ]. (30)

44 From (30), we know l is a bounded linear funcional on H. By a Riesz-ype represenaion heorem in Q. Lü, J. Yong and X. Zhang (JEMS, 2012), here exis y ( ) L 2 F (Ω; L (, T ; R n )), Y ( ) L 2 F (Ω; L2 (, T ; R n )) and ς L 2 F (Ω; R n ) such ha E z(t ), y T E = E T +E η, ς. T z(τ), f(τ) dτ u(τ), y (τ) dτ + E I is clear ha ς T = y T. T v(τ), Y (τ) dτ (31)

45 Sep 2 (The ime consisency). Noe ha he soluion (y ( ), Y ( )) may depend on. I is easy o show ha, for any T, for (τ, ω) [ 1, T ] Ω a.e. ( y 2 (τ, ω), Y 2 (τ, ω) ) = ( y 1 (τ, ω), Y 1 (τ, ω) ). (32) Pu y(, ω) = y 0 (, ω) and Y (, ω) = Y 0 (, ω). Then, E z(t ), y T E η, ς = E +E T T z(τ), f(τ) dτ + E v(τ), Y (τ) dτ. T u(τ), y(τ) dτ (33)

46 Sep 3. We show in his sep ha ς has a càdlàg modificaion. For his, clearly, i suffices o show ha X() ς is an {F }-maringale. 0 f(s)ds, [0, T ] (34) The key observaion o show he above maringale propery is he following: For each [0, T ], ( T ) E y T f(s)ds F = ς, a.s. (35)

47 Sep 4. In his sep, we show ha, for a.e. [0, T ], ς = y() a.s. (36) Indeed, for any fixed any γ L 2 F 2 (Ω; R n ), E γ, ς 2 = 1 E Hence, for 2 [0, T ] a.e., 1 2 γ, f(τ) dτ 1 (τ 2 )γ, f(τ) dτ 2 E γ, y(τ) dτ. E γ, y( 2 ) = E γ, ς 2, γ L 2 F 2 (Ω; R n ).

48 Nex, we consider he case of semilinear BSDEs. By Theorem 1 and Banach fixed poin heorem, we deduce ha Theorem 2. (Q. Lü and X. Zhang, JDE, 2013) For any given y T L 2 F T (Ω), he equaion (15) admis a unique ransposiion soluion (y( ), Y ( )) L 2 F (Ω; D([0, T ]; Rn )) L 2 F (Ω; L2 (0, T ; R n )). Furhermore, (y( ), Y ( )) L 2 F (Ω;D([0,T ];R n )) L 2 F (Ω;L2 (0,T ;R n )) [ ] C f(, 0, 0) L 2 F (Ω;L 1 (0,T ;R n )) + y T L 2 FT (Ω;R n ). (37)

49 5. Transposiion soluion o vecor-valued BSEEs Consider he following BSEE valued in H: { dy = A yd + f(, y, Y )d + Y dw in [0, T ), y(t ) = y T L p F T (Ω; H). (38) Here, p (1, 2]. In order o give he definiion of ransposiion soluion o (38), we inroduce he following forward sochasic differenial equaion: { dz = (Az + v1 )d + v 2 dw in (, T ], (39) z() = η. Here v 1 ( ) L 1 F (, T ; Lq (Ω; H)), v 2 ( ) L 2 F (, T ; Lq (Ω; H)), η L q F (Ω; H), and 1 p + 1 q = 1.

50 Definiion 3. We call (y( ), Y ( )) D F ([0, T ]; L p (Ω; H)) L 2 F (0, T ; Lp (Ω; H)) a ransposiion soluion o (38) if for any [0, T ], v 1 ( ) L 1 F (, T ; Lq (Ω; H)), v 2 ( ) L 2 F (, T ; Lq (Ω; H)) and η L q F (Ω; H), i holds ha E T z(t ), y T H E z(s), f(s, y(s), Y (s)) H = E η, y() H + E T +E T v2 (s), Y (s) H ds. v1 (s), y(s) H ds

51 Theorem 3. (Q. Lü and X. Zhang, 2012) For any y T L p F T (Ω; H), and f(,, ) : [0, T ] H H H saisfying some assumpions, he equaion (38) admis one and only one unique ransposiion soluion (y( ), Y ( )) D F ([0, T ]; L p (Ω; H)) L 2 F (0, T ; Lp (Ω; H)). Furhermore, (y( ), Y ( )) DF ([,T ];L p (Ω;H)) L 2 F (,T ;Lp (Ω;H)) [ ] C f(, 0, 0) L 1 F (,T ;L p (Ω;H)) + y T L p F (Ω;H), T [0, T ]. (40)

52 6. Well-posedness of an operaor-valued BSEE Furher, we consider he following operaor-valued backward sochasic evoluion equaion: dp = (A + J ())P d P (A + J())d K P Kd P (T ) = P T. (K Q + QK)d + F d + Qdw in [0, T ), (41) Here F L 1 F (0, T ; L2 (Ω; L(H))), P T L 2 F T (Ω; L(H)), and J, K L 4 F (0, T ; L (Ω; L(H))). The equaion (41) appeared in he sudy of general sochasic maximum principle in infinie dimensions.

53 In order o define he ransposiion soluion o he e- quaion (41), we inroduce he following wo sochasic differenial equaion: { dx1 = (A + J)x 1 ds + u 1 ds + Kx 1 dw + v 1 dw in (, T ], x 1 () = ξ 1, { dx2 = (A + J)x 2 ds + u 2 ds + Kx 2 dw + v 2 dw in (, T ], x 2 () = ξ 2. (42) (43) Here ξ 1, ξ 2 L 4 F (Ω; H), u 1, u 2 L 2 F (, T ; L4 (Ω; H)) and v 1, v 2 L 4 F (, T ; L4 (Ω; H)).

54 Definiion 4. We call (P ( ), Q( )) D F,w ([0, T ]; L 2 (Ω; L(H))) L 2 F,w (0, T ; L2 (Ω; L(H))) a ransposiion soluion o (41) if for any [0, T ], ξ 1, ξ 2 L 4 F (Ω; H), u 1 ( ), u 2 ( ) L 2 F (, T ; L4 (Ω; H)) and v 1 ( ), v 2 ( ) L 4 F (, T ; L4 (Ω; H)), i holds ha E P T x 1 (T ), x 2 (T ) H E T = E P ()ξ 1, ξ 2 +E +E +E T T T H + E T F (s)x1 (s), x 2 (s) H ds P (s)u1 (s), x 2 (s) H ds P (s)x1 (s), u 2 (s) H ds + E T P (s)k(s)x1 (s), v 2 (s) H ds P (s)v1 (s), Kx 2 (s) H ds + E T P (s)v1 (s), v 2 (s) H ds Q(s)v1 (s), x 2 (s) T H ds + E Q(s)x1 (s), v 2 (s) H ds.

55 Denoe by L 2 (H) he se of he Hilber-Schmid operaors on H. Theorem 4. (Q. Lü and X. Zhang, 2012) Assume ha H is a separable Hilber space and L p F T (Ω) (1 p < ) is a separable Banach space. Then, for any P T L 2 F T (Ω; L 2 (H)), F L 1 F (0, T ; L2 (Ω; L 2 (H))) and J, K L 4 F (0, T ; L (Ω; L(H))), he equaion (41) admis one and only one ransposiion soluion (P, Q) wih he regulariy ( P ( ), Q( ) ) D F ([0, T ]; L 2 (Ω; L 2 (H))) L 2 F (0, T ; L 2(H)). Furhermore, (P, Q) DF ([0,T ];L 2 (Ω;L 2 (H))) L 2 F (0,T ;L 2(H)) [ ] C F L 1 F (0,T ;L 2 (Ω;L 2 (H))) + P T L 2 FT (Ω;L 2 (H)). (44)

56 Theorems 4 indicaes ha, in some sense, he ransposiion soluion inroduced in Definiion 4 is a reasonable noion for he soluion o (41). Unforunaely, we are unable o prove he exisence of ransposiion soluion o (41) in he general case. We shall inroduced below a weaker version of soluion, i.e., relaxed ransposiion soluion (o (41)), which looks awkward bu i suffices o esablish he Ponryagin-ype sochasic maximum principle for Problem (P) in he general seing.

57 ( Definiion 5. We call P ( ), Q ( ), Q ( )) D F,w ([0, T ]; L 4 3(Ω; L(H))) Q[0, T ] a relaxed ransposiion soluion o (41) if for any [0, T ], ξ 1, ξ 2 L 4 F (Ω; H), u 1 ( ), u 2 ( ) L 2 F (, T ; L4 (Ω; H)) and v 1 ( ), v 2 ( ) L 4 F (, T ; L4 (Ω; H)), i holds ha E P T x 1 (T ), x 2 (T ) H E T = E P ()ξ 1, ξ 2 +E +E +E T T T H + E T F (s)x1 (s), x 2 (s) H ds P (s)u1 (s), x 2 (s) H ds P (s)x1 (s), u 2 (s) H ds + E T P (s)v1 (s), Kx 2 (s) H ds + E T P (s)k(s)x1 (s), v 2 (s) H ds P (s)v1 (s), v 2 (s) H ds v1 (s), Q () (ξ 2, u 2, v 2 )(s) T H ds + E Q () (ξ 1, u 1, v 1 )(s), v 2 (s) H ds.

58 I is easy o see ha, if ( P ( ), Q( ) ) is a ransposiion soluion o (41), hen one can find a relaxed ransposiion soluion ( P ( ), Q ( ), Q ( )) o he same equaion (from ( P ( ), Q( ) ) ). Indeed, hey are relaed by Q(s)x 1 (s) = Q () (ξ 1, u 1, v 1 )(s), Q(s) x 2 (s) = Q () (ξ 2, u 2, v 2 )(s). This means ha, we know only he acion of Q(s) (or Q(s) ) on he soluion processes x 1 (s) (or x 2 (s)). However, i is unclear how o obain a ransposiion soluion ( P ( ), Q( ) ) o (41) by means of is relaxed ransposiion soluion ( P ( ), Q ( ), Q ( )). I seems ha his is possible bu we canno do i a his momen.

59 Well-posedness resul for he equaion (41) in he general case Theorem 5. (Q. Lü and X. Zhang, 2012) Assume ha H is a separable Hilber space, and L p F T (Ω; C) (1 p < ) is a separable Banach space. Then, for any P T L 2 F T (Ω; L(H)), F L 1 F (0, T ; L2 (Ω; L(H))) and J, K L 4 F (0, T ; L (Ω; L(H))), he equaion (41) admis one and only one relaxed ransposiion soluion ( P ( ), Q ( ), Q ( )). Furhermore, P L(L 2 F (0,T ;L 4 (Ω;H)), L 2 F (0,T ;L 4 3 (Ω;H))) + sup ( Q (), Q ()) ( ) [0,T ] L(L 4 F (Ω;H) L 2 F (,T ;L4 (Ω;H)) L 2 F (,T ;L4 (Ω;H)), L 2 F (,T ;L (Ω;H)) [ ] C F L 1 F (0,T ; L 2 (Ω;L(H))) + P T L 2 FT (Ω; L(H)). (45)

60 7. Ponryagin-ype sochasic maximum principle Firs Prev Nex Las Go Back Full Screen Close Qui For (, x, u, k 1, k 2 ) [0, T ] H U H H, wrie H(, x, u, k 1, k 2 ) = k 1, a(, x, u) H + k 2, b(, x, u) H g(, x, u). Theorem 6. (Q. Lü and X. Zhang, 2012) Le ( x( ), ū( )) be an opimal pair of Problem (P), and le (y( ), Y ( )) be he ransposiion soluion o (38) wih p = 2, and y T and f(,, ) given by y T = h x ( x(t ) ), f(, y 1, y 2 ) = a x (, x(), ū()) y 1 b x (, x(), ū() ) y2 +g x (, x(), ū() ). (46)

61 Assume b x (, x( ),ū( )) L 4 F (0, T ;L (Ω;L(D(A)))), and (P ( ), Q ( ), Q ( ) ) is he relaxed ransposiion soluion o (41) wih P T, J( ), K( ) and F ( ) given by { PT = h xx ( x(t ) ), J() = ax (, x(), ū()), Then K() = b x (, x(), ū()), F () = H xx (, x(), ū(), y(), Y () ). Re H (, x(), ū(), y(), Y () ) Re H (, x(), u, y(), Y () ) 1 [ P () b (, x(), ū() ) b (, x(), u )], b (, x(), ū() ) b (, x(), u ) 2 H 0, u U, a.e. [0, T ] Ω.

62 Thank You