Generalized Snell envelope and BSDE With Two general Reflecting Barriers

Transcription

1 1/22 Generalized Snell envelope and BSDE Wih Two general Reflecing Barriers EL HASSAN ESSAKY Cadi ayyad Universiy Poly-disciplinary Faculy Safi Work in progress wih : M. Hassani and Y. Ouknine Iasi, July 2-7, 212 Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 1 / 22 : M

2 2/22 Generalized Snell envelope-noaions Le us firs inroduce he following noaions : (Ω, F, (F ) T, P) is a complee probabiliy space. F = σ(b s, s ) N be a filraion. P is he sigma algebra of F -predicable ses on Ω [, T ]. D is he se of P-measurable and righ coninuous wih lef limis (RCLL for shor) processes (Y ) T wih values in IR. K := {K D : K is nondecreasing and K = }. L 2,d he se of IR d -valued and P-measurable processes (Z ) T such ha Z s 2 ds <, P a.s. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 2 / 22 : M

3 3/22 Snell envelope Le us recall some properies of he Snell envelope. Definiion-Proposiion 1 An F -adaped RCLL process l := (l ) T wih values in IR is called of class D[, T ], if he family (l ν) ν T is uniformly inegrable, where T is he se of all F -sopping imes ν, such ha ν T. 2 Le l := (l ) T be of class D[, T ], hen is Snell envelope S (l) is defined as S (L) = ess sup ν T IE [l ν F ], where T is he se of all sopping imes valued beween and T. I is he smalles RCLL-supermaringale of class D[, T ] which dominaes he process L, i.e., P-a.s, T, l S (l). l Hassan Essaky Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 3 / 22 : M

4 4/22 Generalized Snell envelope-assumpion Quesion : how we can characerize he RCLL local supermaringale which dominaiong he process l? Le L D, ξ L (Ω), l L (Ω [, T ]) and δ K. We assume he following condiion : (A) There exiss a local maringale M = M + κ sdb s such ha P-a.s. L M on [, T [ and l M dδ a.e. on [, T ] and ξ M T. Aim : Characerizing he RCLL local supermaringale which dominaiong he process l as he soluion of some refleced BSDE. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 4 / 22 : M

5 5/22 Generalized Snell envelope Le U = U A + χ sdb s, where A K and χ L 2,d be a RCLL local supermaringale such ha L U, l U dδ a.e. and ξ U T. This laer propery of U will be denoed by (H). Le also (Y, Z, K +, K ) be he minimal soluion of he following RBSDE (i) Y = ξ + dk s + dks Z sdb s, T, (ii) [, T [, L Y U, (iii) on ], T ], l Y, dδ a.e. (iv) L D saisfying < T, L L Y and on ], T ], l L, dδ a.e. we have (Y L )dk + = (U Y )dk =, a.s., (v) Y D, K +, K K, Z L 2,d, (vi) dk + dk. (1) Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 5 / 22 : M

6 6/22 Generalized Snell envelope Then Y is a local supermaringale minimal soluion of he following RBSDE (i) Y = ξ + dk s + Z sdb s, T, (ii) [, T [, L Y, (iii) on ], T ], l Y, dδ a.e. (iv) L D saisfying < T, L L Y and on ], T ], l L, dδ a.e. we have (Y L )dk + =, a.s., (v) Y D, K + K, Z L 2,d. (2) Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 6 / 22 : M

7 7/22 Generalized Snell envelope-resul and definiion Therefore we ge he following resul Theorem-Definiion: Suppose ha (A) hold. Then Y he minimal soluion of (2) is he smalles RCLL local supermaringale saisfying [, T [, L Y, l Y dδ a.e., on [, T ] and ξ Y T. We say ha Y is he generalized Snell envelope associaed o L, l, δ and ξ. We denoe i by S.(L, l, δ, ξ). Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 7 / 22 : M

8 8/22 Generalized Snell envelope and Snell envelope Remark We know ha if L is of class D[, T ] hen L saisfies assumpion (A) (see Dellacherie-Meyer). In his case our generalized snell enveloppe S.(L) = S.(L,,, L T ) coincides wih he usual snell enveloppe esssup τ T IE[L τ F ], where T is he se of all sopping imes valued beween and T, as presened in Dellacherie-Meyer and sudied by several auhors. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 8 / 22 : M

9 9/22 Generalized Snell envelope-example Example If δ = and here exis L D and M a local maringale such ha L l M and ξ M T. Le (Y, Z, K +, K ) be he minimal soluion of he following RBSDE (i) Y = ξ + dk s + Z sdb s, T, (ii) on ], T ], l Y, d a.e (iii) L D saisfying l L Y d a.e. we have (Y L )dk + =, a.s., (v) Y D, K + K, Z L 2,d, Then Y is he smalles local supermaringale such ha l Y, d a.e and ξ Y T. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress wih 9 / 22 : M

10 1/22 Generalized Snell envelope-properies Corollary 1 S.(L, l, δ, ξ) = S.(L, l, δ, ξ), wih l s = l s L s. 2 If L L, dδ dδ, l l, dδ a.e., ξ ξ hen (L, l, δ, ξ ) saisfies condiion (A) and S.(L, l, δ, ξ ) S.(L, l, δ, ξ). 3 S.(L, l, δ, ξ) S(L ξ ) (wih equaliy if l L dδ a.e., on [, T ]) where S(L ξ ) = S.(L,,, ξ) and L ξ = L1 {<T } + ξ1 {=T }. 4 Pu Y = S.(L, l, δ, ξ). If l l Y, dδ a.e., on [, T ] and L L Y, [, T [, and dδ dδ, hen S.(L, l, δ, ξ) = S.(L, l, δ, ξ). 5 Pu Y = S.(L, l, δ, ξ). Then for every L D such ha P a.s., L L Y, [, T [, and l L Y, dδ a.e., on [, T ] and L T = ξ we have S.(L, l, δ, ξ) = S.(L ). Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress1 wih / 22 : M

11 11/22 BSDE wih wo general reflecing barriers-daa We inroduce he following daa : ξ is an F T -measurable one dimensional random variable. L := {L, T } and U := {U, T } are wo barriers which belong o D such ha L U, [, T [. l := {l, T } and u := {u, T } are P measurable processes. A, α and δ are processes in K. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress11 wih / 22 : M

12 12/22 BSDE wih wo general reflecing barriers-definiion Definiion We call (Y, Z, K +, K ) := (Y, Z, K +, K ) T a soluion of he GRBSDE, associaed wih he daa (ξ, L, U, l, u), if he following hold : (i) Y = ξ + dk s + dks Z sdb s, T, (ii) [, T [, L Y U, (iii) on ], T ], Y u, dα a.e. l Y, dδ a.e. (iv) (L, U ) D D saisfying < T, L L Y U U and on ], T ], U u, dα a.e. and l L, dδ a.e. we have (Y L )dk + = (U Y )dk =, a.s., (v) Y D, K +, K K, Z L 2,d, (vi) dk + dk. (3) Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress12 wih / 22 : M

13 13/22 BSDE wih wo general reflecing barriers-assumpion. (H) There exiss a semimaringale S. = S + V. V. + + γ sdb s, wih S IR, V ± K and γ L 2,d, such ha [, T [, L S U and on ], T ], S u, dα a.e. and l S, dδ a.e. We assume, wihou loss of generaliy, ha L T = U T = S T = ξ. Theorem If assumpions (H) hold hen he RBSDE (3) has a maximal (resp. minimal) soluion. l Hassan Essaky Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress13 wih / 22 : M

14 14/22 BSDE wih wo general reflecing barriers-comparison heorem Le (Y, Z, K +, K ) be a soluion for he following RBSDE (i) Y = ξ + dk s + dk s Z s db s, T, (ii) [, T [, L Y U, T (iii) (Y L + )dk = (U Y )dk =, a.s., (iv) Y D, K +, K K, Z L 2,d, (v) dk + dk, (4) where L and U are wo barriers which belong o D such ha L U, [, T [. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress14 wih / 22 : M

15 15/22 BSDE wih wo general reflecing barriers-comparison heorem Assume moreover ha for every [, T ] (a) ξ ξ. (b) Y U, L Y, [, T [. Theorem : Comparison heorem for maximal soluions Assume ha he above assumpions hold hen we have : 1 Y Y, for every [, T ], P a.s. 2 1 {U =U } dk dk and 1 {L =L } dk + dk +. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress15 wih / 22 : M

16 16/22 BSDE wih wo general reflecing barriers-idea of he proof of he exisence resul We proceed by penalizaion mehod. Le (Y n, Z n, K n+, K n ) be he minimal soluion of he following GRBSDE (i) Y n = ξ (dv s + + dvs ) + n T (l s Y n s )+ dα + dk n s Z n s dbs, T, (ii) [, T [, L Y n S, T (iii) (Y n L )dk n+ = (iv) Y n D, K n+, K n K, Z n L 2,d, (v) dk n+ dk n. dk n+ s (S Y n n )dk =, P a.s., (5) Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress16 wih / 22 : M

17 17/22 BSDE wih wo general reflecing barriers-idea of he proof of he exisence resul and (Y n, Z n, K n+, K n ) be he maximal soluion of he following GRBSDE (i) Y n = ξ + (dv s + + dvs ) n (Y n s u s) + dδ s + dk n+ s T dk n s Z n s dbs, T, (ii) [, T [, S Y n U, T (iii) (Y n S )dk n+ = (U Y n )dk n =, P a.s., (iv) Y n D, K n+, K n K, Z n L 2,d, (v) dk n+ dk n. (6) where V +, V are he processes appeared in assumpion (H). We have L Y n Y n+1 S Y n+1 Y n U. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress17 wih / 22 : M

18 18/22 BSDE wih wo general reflecing barriers-idea of he proof of he exisence resul Le Y = inf n Y n and Y = sup Y n, hen our iniial BSDE is equivalen o he following BSDE : n (i) Y = ξ + f (s, Y s, Z s)ds + g(s, Y s, Y s)da s T + dk s + dks Z sdb s, T, (ii) [, T [, Y Y Y, T (iii) (Y Y )dk + = (Y Y )dk =, a.s., (v) Y D, K +, K K, Z L 2,d, (vi) dk + dk. (7) Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress18 wih / 22 : M

19 19/22 BSDE wih wo general reflecing barriers-general case We shall need he following assumpions on he daa : (A.1) There exis wo processes η L (Ω, L 1 ([, T ], ds, IR +)) and C D such ha: (a) (y, z) IR IR d, f (, ω, L (ω) y U (ω), z) η (ω) + C (ω) z 2, dp(dω) a.e. (b) dp(dω) a.e., he funcion (y, z) f (, ω, L (ω) y U (ω), z) is coninuous. (A.2) There exiss β L (Ω, L 1 ([, T ], A(d), IR +)) such ha : (a) A(d)P(dω) a.e., (x, y) IR IR, (b) A(d)P(dω) a.e. he funcion g(, ω, L (ω) x U (ω), L (ω) y U (ω)) β (ω), (x, y) g(, ω, L (ω) x U (ω), L (ω) y U (ω)) is coninuous. (c) P a.s., (, x) ], T ] IR, he funcion y y + g(, ω, L (ω) x U (ω), L (ω) y U (ω)) A is nondecreasing.. (A.3) There exiss a semimaringale S. = S + V. V. + + γ sdb s, wih S IR, V ± K and γ L 2,d, such ha [, T [, L S U and on ], T ], S u, dα a.e. and l S, dδ a.e. We assume, wihou loss of generaliy, ha L T = U T = S T = ξ. Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress19 wih / 22 : M

20 2/22 BSDE wih wo general reflecing barriers-definiion Definiion We call (Y, Z, K +, K ) := (Y, Z, K +, K ) T a soluion of he GRBSDE, associaed wih he daa (ξ, L, U, l, u), if he following hold : (i) Y = ξ + f (s, Y s, Z s)ds + g(s, Y s, Y s)da s T + + dk s + dks Z sdb s, T, (ii) [, T [, L Y U, (iii) on ], T ], Y u, dα a.e. l Y, dδ a.e. (iv) (L, U ) D D saisfying < T, L L Y (8) U U and on ], T ], U u, dα a.e. and l L, dδ a.e. we have (Y L )dk + = (U Y )dk =, a.s., (v) Y D, K +, K K, Z L 2,d, (vi) dk + dk. Quesion : how can we charcerize he Generalized snell envelope as an essenial supremum? Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress2 wih / 22 : M

21 21/22 For Furher Reading E. H. Essaky, M. Hassani, Generalized BSDE wih 2-Reflecing Barriers and Sochasic Quadraic Growh. Applicaion o Dynkin Games. arxiv: v2 [mah.pr] 9 Jul 21. E. H. Essaky, M. Hassani, Y. Ouknine Sochasic Quadraic BSDE Wih Two RCLL Obsacles. arxiv: v1 [mah.pr] 28 Mar 211. Dellacherie, C. and Meyer, P.A. Probabiliies and Poenial B. Theory of Maringales. Norh-Holland Mahemaics Sudies, Amserdam, Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress21 wih / 22 : M

22 22/22 For Furher Reading K. limsiak BSDEs wih monoone generaor and wo irregular reflecing barriers. Bul.Sc. Mahémaiques, in press. S. Peng, M. Xu The smalles g-supermaringale and refleced BSDE wih single and double image L 2 -obsacles Annales de l Insiu Henri Poincare (B) Probabiliy and Saisics, Volume 41, Issue 3, Pages 65-63, 25. L. Sener; J. Zabczyk Srong envelopes of sochasic processes and a penaly mehod Sochasic, Vol 4, pp , Poly-disciplinary Faculy (Cadi Generalized ayyad Universiy Snell envelope Poly-disciplinary and BSDE Faculy Safi Work in progress22 wih / 22 : M