Singular perturbation control problems: a BSDE approach

Transcription

1 Singular perurbaion conrol problems: a BSDE approach Join work wih Francois Delarue Universié de Nice and Giuseppina Guaeri Poliecnico di Milano Le Mans 8h of Ocober 215 Conference in honour of Vlad Bally

2 We consider a wo scale sysem of conrolled -dimensional SDEs: dx u,v = ( AX u,v + b(x u,v, Q ɛ,u,v, u ) ) d + dw 1 (), X u,v = x, ɛdq ɛ,u,v = (BQ ɛ,u,v +F (X u,v, Q ɛ,u,v )+Gρ(v )) d+ ɛ GdW 2 () Q ɛ = q, he slow variable X akes is values in he Hilber space H he fas variable Q ɛ akes is values in he Hilber space K, ɛ ], 1] is a small parameer. A : D(A) H H and B : D(B) K K are unbounded linear operaors generaing C - semigroups. (W i ), i = 1, 2, are independen cylindrical Wiener processes wih values in H and K respecively, moreover G L(K) u and v are conrols adaped o he filraion generaed by (W 1, W 2 ). They ake values in suiable opological spaces U and V respecively. 1

3 dx u,v = ( AX u,v + b(x u,v, Q ɛ,u,v, u ) ) d + dw 1 (), X u,v = x, ɛdq ɛ,u,v = (BQ ɛ,u,v +F (X u,v, Q ɛ,u,v )+Gρ(v )) d+ ɛ GdW 2 () Q ɛ = q F and b Lipschiz and Gaeaux differeniable w.r.. X and Q b and ρ are bounded he semigroups generaed by A and B are Hilber Schmid and e sa L2 (Ξ,H) + esb L2 (Ξ,K) L (1 s) γ, γ 1/2 B + F is dissipaive wih respec o Q e.g. (q q ), B(q q ) + F (x, q q ) η q q 2 for a suiable η > and all x H, q, q K 2

4 dx u,v = ( AX u,v + b(x u,v, Q ɛ,u,v, u ) ) d + dw 1 (), X u,v = x, ɛdq ɛ,u,v = (BQ ɛ,u,v +F (X u,v, Q ɛ,u,v )+Gρ(v )) d+ ɛ GdW 2 () Q ɛ = q We consider he following opimal conrol problem J ɛ (u, v) = E T ( l1 (X u,v, Q ɛ,u,v, u ) + l 2 (X u,v, Q ɛ,u,v, u ) ) d and he corresponding value funcion V (ɛ) = inf u,v J ɛ (u, v) Our purpose is o sudy he limi of V (ɛ) as ɛ. Idea: if we freeze he slow evoluion hen he conrol problem for he quick one behaves like he opimal sae of an ergodic conrol problem. Thus Ergodic BSDEs mus be involved here! 3

5 Ergodic BSDEs Consider he following sysem in infinie horizon dˇy s = [ Ψ(U s, ˇΞ s ) λ ] ds ˇΞ s dw s, s du s = [LU s + F (U s )]ds + GdW s s U = u Assume ha L + F ( ) is dissipaive and Ψ is Lipschiz w.r.. ˇΞ bounded w.r.. U hen he above sysem admis a unique soluion ((U ), (ˇY ), (ˇΞ ), λ) wih ˇY s C(1 + U s ) where C can be chosen o depend only on he Lipschiz consan of Ψ and on he dissipaiviy consan of L + F ( ). Moreover λ is he value funcion of an ergodic conrol problem (boh in Cesaro and in Abel sense) see [M.Fuhrman, Y. Hu, G.T. 29], [A. Debussche, Y. Hu 211], [Y. Hu, P.Y. Madec, A. Richou 213]] ) 4

6 Recall ha we have BSDE reformulaion of he problem dx u,v = ( AX u,v + b(x u,v, Q ɛ,u,v, u ) ) d + dw 1 (), X u,v = x, ɛdq ɛ,u,v = (BQ ɛ,u,v + F (X u,v, Q ɛ,u,v ) + Gρ(v )) d + ɛ GdW 2 (), Q ɛ = q, hus if ψ(x, q, p, ξ) = inf {pb(x, q, u) + l 1(x, q, u)} + inf {l 2(x, q, v) + ξρ(v)} u U v V hen dx = AX + dw 1, ɛdq ɛ = (BQ ɛ + F (Xɛ, Qɛ ) d + ɛ1/2 GdW 2, dy ɛ = ψ(x ɛ, Qɛ, Zɛ, Ξɛ / ɛ) d Z ɛd W 1 Ξɛ dw 2, X ɛ = x, Q ɛ = q, Y1 ɛ =. V (ɛ) = Y ɛ Proof: Usual eliminaion of conrol by change of P argumen. Noice ha he fas conrolled equaion reeds dq ɛ,u,v =... + ɛ 1/2 G(ɛ 1/2 ρ(v ) + dw 2 ) 5

7 The paramerized ergodic BSDE Fix x H and p H we consider he following version of he fas equaion (noice ha ime has been sreched ha is Q = Q ɛ, Ŵ 2 = e 1/2 W 2 ɛ ) d Q x,q s = B Q x,q s + F (x, Q x,q s ) ds + dŵ 2 s ; Q x,q = q Theorem 1 x H, p H (and Q K),! soluion (Y x,q,p, Ξ x,q,p, λ x,p ) of he infinie horizon ergodic BSDE dˇy x,q,p = [ψ(x, Q x,q, p, ˇΞ x,q,p ) λ(x, p)] d ˇΞ x,q,p dw 2, Moreover ˇY x,q,p c(1 + Q x,q ) where c > only depends on he Lipschiz consans of ψ wih respec o Q and on he dissipaiviy consan of B + F (x, ). 6

8 The corresponding parameyrized - ergodic Conrol Problem λ(x, p) is he value funcion of an ergodic conrol problem wih sae equaion and cos d Q s = B Q v s + F (x, Q v s) ds + Gρ(v s )ds + GdŴ 2 s, Q v = q J(x, p, v) = lim δ E δ e δs ψ 1 (x, Q x,v s, p) + l 2 (x, Q x,v s, v s )) ds where we recall ψ 1 (x, q, p) = inf u U { pb(x, Q x,v s, u) + l 1 (x, q, u) } This implies ha λ is Lipschiz in p and x. 7

9 Limi equaion and main resul We can now inroduce he limi forward-backward sysem: { dȳ = λ(x, Z ) d + Z dw 1, [, 1), Ȳ 1 =, dx = AX d + dw 1, X = x Recall he f.b. sysem for he original, wo scales conrol problem: dx = AX + dw 1, [, 1) ɛdq ɛ = (BQ ɛ + F (Xɛ, Qɛ ) d + ɛ dw 2, [, 1) dy ɛ = ψ(x ɛ, Qɛ, Zɛ, Ξɛ / ɛ) d Z ɛd W 1 Ξɛ dw 2, [, 1) X ɛ = x Q ɛ = q, Y1 ɛ =. Theorem 2 (Main resul) lim Y ɛ Ȳ = ɛ [Alvarez-Bardi 21-27] for he finie dimensional counerpar by viscosiy soluions echniques. Also see [Kabanov-Pergamenshchikov 23],... 8

10 Proof: a freezing/discreizaion argumen The idea is o freeze he slow equaion o give ime o he fas equaion o behave as he opimal ergodic sae. We sar from Y ɛ Ȳ = 1 (ψ(x, Q ɛ, Z ɛ, Ξ ɛ / ɛ) λ(x, Z )) d + 1 (Zɛ Z ) dw Ξɛ dw 2. Adding and subracing he erm: (ψ(x, Q ɛ, Z, Ξ ɛ / ɛ)d ha evenually will be easily reaed by a change of probabiliy we are lef wih 1 (ψ(x, Q ɛ, Z, Ξ ɛ / 1 1 ɛ) λ(x, Z )) d+ (Zɛ Z ) dw 1 + Ξɛ dw 2. 9

11 Le k = k2 N, k =, 1,..., 2 N 1 and define for k < k+1 : X N () = X( k ), Z N () = 2 N k Z s ds. k 1 Fixed k we consider he sysem (wih sreched ime) for k /ɛ: dˇy N,k = [ψ(x k, Q N,k, Z N,k k, ˇΞ N,k ) λ(x k, Z N,k )] d k Ξ N,k dŵ 2, d Q N,k = (B Q N,k + F (X k, Q N,k )) d + dŵ 2, Q N,k k /ɛ = QN,k 1 k /ɛ Recall ha he above sysem admis a unique soluion (Ŷ N,k, Ξ N,k, λ(x N k, Z N k )) such ha Ŷ N,k N,k c(1 + Q ), 1

12 If we se Q N = ˇY N,k k+1 /ɛ ˇY N,k herefore: N,k Q, ˇΞ N = ˇΞ N,k k /ɛ = k+1 /ɛ k /ɛ k+1 /ɛ 2 N k=1 1/ɛ [ + for [ k /ɛ, k+1 /ɛ[ we have [ψ(xɛ N, Q N, ZN ɛ, ˇΞ N ) λ(xn ɛ, ZN ɛ )] d k /ɛ ˇΞ N dŵ 2. (ˇY N,k k /ɛ ˇY N,k k+1 /ɛ ) + k+1 /ɛ k /ɛ ˇΞ N dŵ 2 [ψ(x N ɛ, Q N, Z N ɛ, ˇΞ N ) λ(x N ɛ, Z N ɛ )] d = ] + 11

13 Recall ha we had o esimae (afer change of ime, ha is for: Q ɛ := Qɛ ɛ, Ξ ɛ := Ξɛ ɛ / ɛ ) ɛ 1/ɛ + ɛ (ψ(x ɛ, Q ɛ, Z ɛ, Ξ ɛ ) λ(x ɛ, Z ɛ )) d 1/ɛ (Z ɛ ɛ Z ɛ ) dŵ 1 + 1/ɛ Ξ ɛ dw 2. Adding he erm (in blue) ha we have proved o be null we ge Y ɛ Ȳ = where ɛ 1/ɛ R ɛ,n 1/ɛ +ɛ +ɛ +ɛ 1/ɛ 1/ɛ d + ɛ N (ˇY N,k k=1 (ˇΞ N Ξ ɛ ) dw 2 + ɛ 1 2 k+1 /ɛ ) k /ɛ ˇY N,k 1/ɛ (Z ɛ ɛ Z ɛ ) dw 1 [ψ(x N ɛ, Q N, Z N ɛ, Ξ ɛ ) ψ(x N ɛ, Q N, Z N ɛ, ˇΞ N )] d [ψ(x ɛ, Q ɛ, Z ɛ ɛ, Ξ ɛ ) ψ(x ɛ, Q ɛ, Z ɛ, Ξ ɛ )] d R ɛ,n L( X ɛ ɛ X N ɛ + Q ɛ Q N + Z ɛ Z N ɛ ) (1) 12

14 We ge rid of some erms by Girsanov. Le: and δ 1 () = ψ(x ɛ, Q ɛ, Zɛ ɛ, Ξ ɛ ) ψ(x ɛ, Q ɛ, Z ɛ, Ξ ɛ ) Z ɛ ɛ Z ɛ δ 2 () = ψ(xn ɛ, Q N, ZN ɛ, Ξ ɛ ) ψ(xn ɛ, Q N, ZN ɛ, ˇΞ N ) Ξ ɛ ˇΞ N We se for s [, 1]: W 1 s =: s δ1 (/ɛ) d + W 1 s, W 2 s =: ɛ 1/2 s δ2 (/ɛ) d + W 2 s We denoe by Ẽɛ he expecaion wih respec o he probabiliy Q ɛ under which ( W 1 s, W 2 s ) is a brownian moion (noice ha boh δ 1 and δ 2 are bounded uniformly in ɛ and N). Since he lef hand side is deerminisic, we have Y ɛ Ȳ = Ẽɛ 1 Rɛ,N d + ɛẽɛ N (ˇY N,k k=1 k /ɛ ˇY N,k k+1 /ɛ ) (2) 13

15 We now have o esimae he expecaion on he error in he new probabiliy 1 1 Ẽɛ Rɛ,N /ɛ d Ẽɛ L ( X X N + Q ɛ /ɛ Q N /ɛ + Z Z N 1 Le us sar from Ẽɛ X X N d We noice ha, wih respec o W 1, (X ) saisfies dx() = AX()d δ 1 ()d + d W 1 (), X = x Again by Girsanov since δ 1 ψ ɛ is uniformly bounded Ẽ ɛ 1 X X N d C δ 1E [ 1 X X N 2 d By he coninuiy of rajecories of (X ) sup [,1] X we ge ] 1/2 := C X (N) ) d and inegrabiliy of lim N X (N) = (3) 14

16 Concerning he erm by he same argumen Ẽ ɛ 1 Z Z N d Ẽ ɛ 1 Z Z N d C δ 1E [ 1 Z Z N 2 d and Z (N) = by consrucion of Z N ] 1/2 := C δ 1 Z (N) 15

17 Le us come o he erm: Ẽ ɛ 1 Qɛ Q N /ɛ d = ɛ Ẽɛ 1/ɛ L Q ɛ Q N d Wih respec o W 2 := ɛ 1/2 W ɛ 2 he process ( Q ɛ ) [,1/ɛ] solves d Q ɛ = (B Q ɛ + F (X ɛ, Q ɛ )) d + δ 2 () d + d W 2,, Q ɛ = q, and Q N solves d Q N = (B Q N + F (Xɛ N, Q N )) d + δ2 d + d W 2,, Q = q, hus Q ɛ Q N is a he soluion o: d[ Q ɛ Q N ] = B( Q ɛ Q N ) d + F (X ɛ, Q ɛ ) F (Xɛ N, Q N )) d, Q =. And since B + F (x, ) is dissipaive we sill can say ha, P-a.s. ɛ hus 1/ɛ Q ɛ Q N d ɛ E ɛ 1 1/ɛ X ɛ X N ɛ d = 1 L Qɛ ɛ Q N ɛ d C X (N) X X N d. 16

18 Now we come o he las erm. Recalling ha we ge ˇY s N,k c(1 + Q N s ) and Ẽɛ sup Q N s 2 C s ɛẽɛ N (Ŷ N,k k /ɛ k=1 Ŷ N,k k+1 /ɛ ) ɛ N k=1 Ẽ ɛ (1+ Q N k /ɛ + Q N k+1 /ɛ ) ɛn(1+2 C) A las we sum up all resuls o ge Y ɛ Ȳ Ẽɛ 1 Rɛ,N /ɛ d + ɛ Ẽɛ N (ˇY N,k k=1 k /ɛ ˇY N,k X (N) + Z (N) + ɛn(1 + 2 C) k+1 /ɛ ) So our claim follow leing ɛ end o and hen N o. 17

19 Things o do I) Allow degenerae noise in he slow equaion. In his case he sae equaion reeds dx u,v = ( AX u,v + b(x u,v, Q ɛ,u,v ) ) d + Γr(u )d + ΓdW 1 (), and afer Ghirsanov ransf. he solw equaion sill depends on Q dx = (AX + b(x, Q )) d + ΓdW 1 (), We have o use averaging argumens similar o [Cerrai 99] or [Bréhier 212] akin ino accoun ha he law of he soluion o he fas equaion will converge owards an opimal invarian measure. II?): Guess some informaion on he convergence of he opimal conrols 18

20 Grazie e buon compleanno! 19