Backward stochastic partial differential equations driven by - PowerPoint PPT Presentation

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Backward stochastic partial differential equations driven by infinite dimensional martingales and applications AbdulRahman Al-Hussein Department of Mathematics, College of Science, Qassim University, P . O. Box 6644, Buraydah, Saudi Arabia E-mail: alhusseinqu@hotmail.com Workshop on Stochastic Control and Finance Roscoff, France March 18-23, 2010

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Outline Aims 1 Notations 2 Example 1 Spaces of solutions 3 Definitions 4 Assumptions 5 Existence & Uniqueness Theorem 6 7 Proof Some advantages 8 An application 9 Maximum principle for controlled stochastic evolution equations An example

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Aims To derive the existence and uniqueness of the solutions to:  − dY ( t ) = ( A ( t ) Y ( t ) + F ( t , Y ( t ) , Z ( t ) Q 1 / 2 ( t )) ) dt  ( BSPDE ) − Z ( t ) dM ( t ) − dN ( t ) , Y ( T ) = ξ.  to provide some applications to the maximum principle for a controlled stochastic evolution system.

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Outline Aims 1 Notations 2 Example 1 Spaces of solutions 3 Definitions 4 Assumptions 5 Existence & Uniqueness Theorem 6 7 Proof Some advantages 8 An application 9 Maximum principle for controlled stochastic evolution equations An example

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application H is a separable Hilbert space. (Ω , F , P ) is a complete probability space equipped with a right continuous filtration {F t } t ≥ 0 . M ∈ M 2 , c [ 0 , T ] ( H ) , i.e M is a continuous square integrable martingale in H . < M > is the predictable quadratic variation of M . ˜ Q M is the predictable process taking values in the space L 1 ( H ) , which is associated with the Dol´ eans measure of M ⊗ M . � t 0 ˜ << M >> t = Q M ( s ) d < M > s . Assume: ∃ a predictable process Q satisfying Q ( t , ω ) is symmetric, positive definite nuclear operator on H and � t << M >> t = Q ( s ) ds . 0 ξ ( ξ ( ω ) ∈ H ) is the terminal value.

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application F : [ 0 , T ] × Ω × H × L 2 ( H ) → H is P ⊗ B ( H ) ⊗ B ( L 2 ( H )) / B ( H ) - measurable. L 2 ( H ) is the space of Hilbert-Schmidt operators on H , inner � � product · , · 2 , norm || · || 2 . A ( t , ω ) is a predictable unbounded linear operator on H . � · The stochastic integral 0 Φ( s ) dM ( s ) is defined for Φ s.t. for Q 1 / 2 Q 1 / 2 (Φ ◦ ˜ M )( t , ω )( H ) ∈ L 2 ( H ) , for every h ∈ H : Φ ◦ ˜ M ( h ) is predictable and � T Q 1 / 2 || Φ ◦ ˜ E [ M || 2 2 d < M > t ] < ∞ . 0 The space of integrands ��� Λ 2 ( H ; P , M ) .

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Example 1 Let m be a 1-dimensional, continuous, square integrable martingale � t with respect to {F t } t s.t. < m > t = 0 h ( s ) ds ∀ 0 ≤ t ≤ T , some cts h : [ 0 , T ] → ( 0 , ∞ ) . � t M ( t ) = β m ( t )(= 0 β dm ( s )) , a fixed element β � = 0 of H . ⇓ ◦ M ∈ M 2 , c ( H ) � t ◦ << M >> t = � 0 h ( s ) ds , where � β ⊗ β β ⊗ β is the identification of β ⊗ β in L 1 ( H ) : ( � � � β ⊗ β )( k ) = β, k β, k ∈ H . ◦ < M > t = | β | 2 � t 0 h ( s ) ds . � ◦ ˜ β ⊗ β Q M = | β | 2 . � t ◦ Let Q ( t ) = � β ⊗ β h ( t ) ⇒ << M >> t = 0 Q ( s ) ds . ◦ Q ( · ) is bounded since Q ( t ) ≤ Q := � β ⊗ β max 0 ≤ t ≤ T h ( t ) . � � β, k β ◦ Q 1 / 2 ( t )( k ) = h 1 / 2 ( t ) . In particular β ∈ Q 1 / 2 ( t )( H ) . | β |

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Outline Aims 1 Notations 2 Example 1 Spaces of solutions 3 Definitions 4 Assumptions 5 Existence & Uniqueness Theorem 6 7 Proof Some advantages 8 An application 9 Maximum principle for controlled stochastic evolution equations An example

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application L 2 F ( 0 , T ; H ) := { φ : [ 0 , T ] × Ω → H , predictable, � T E [ 0 | φ ( t ) | 2 H dt ] < ∞ } . B 2 ( H ) := L 2 F ( 0 , T ; H ) × Λ 2 ( H ; P , M ) . This is a separable Hilbert space, the norm: � � T � � E | φ 1 ( t ) | 2 || ( φ 1 , φ 2 ) || B 2 ( H ) = H dt 0 � � T �� 1 / 2 || φ 2 ( t ) ˜ Q 1 / 2 + E M ( t ) || 2 2 d < M > t . 0 ( V , H , V ′ ) is a rigged Hilbert space: ⊲ V is a separable Hilbert space embedded continuously and densely in H . ⊲ By identifying H with its dual ⇒ get continuous and dense two inclusions: V ⊆ H ⊆ V ′ , V ′ is the dual space of V .

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Outline Aims 1 Notations 2 Example 1 Spaces of solutions 3 Definitions 4 Assumptions 5 Existence & Uniqueness Theorem 6 7 Proof Some advantages 8 An application 9 Maximum principle for controlled stochastic evolution equations An example

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Definition 1 Two elements M and N of M 2 , c [ 0 , T ] ( H ) are very strongly orthogonal (VSO) if E [ M ( u ) ⊗ N ( u )] = E [ M ( 0 ) ⊗ N ( 0 )] , for all [ 0 , T ] - valued stopping times u . In fact: M and N are VSO ⇔ << M , N >> = 0 .

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Definition 2 A solution of the:  − dY ( t ) = ( A ( t ) Y ( t ) + F ( t , Y ( t ) , Z ( t ) Q 1 / 2 ( t )) ) dt  ( BSPDE ) − Z ( t ) dM ( t ) − dN ( t ) , 0 ≤ t ≤ T , Y ( T ) = ξ,  F ( 0 , T ; V ) × Λ 2 ( H ; P , M ) × M 2 , c is a triple ( Y , Z , N ) ∈ L 2 [ 0 , T ] ( H ) s.t. ∀ t ∈ [ 0 , T ] : � T ( A ( s ) Y ( s ) + F ( s , Y ( s ) , Z ( s ) Q 1 / 2 ( s )) ) ds Y ( t ) = ξ + t � T � T − Z ( s ) dM ( s ) − dN ( s ) , t t N ( 0 ) = 0 and N is VSO to M .

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Outline Aims 1 Notations 2 Example 1 Spaces of solutions 3 Definitions 4 Assumptions 5 Existence & Uniqueness Theorem 6 7 Proof Some advantages 8 An application 9 Maximum principle for controlled stochastic evolution equations An example

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application (A1) F : [ 0 , T ] × Ω × H × L 2 ( H ) → H is a mapping such that the following properties are verified. (i) F is P ⊗ B ( H ) ⊗ B ( L 2 ( H )) / B ( H ) - measurable. � T 0 | F ( t , 0 , 0 ) | 2 dt ] < ∞ , where F ( t , 0 , 0 ) = F ( t , ω, 0 , 0 ) . (ii) E [ (iii) ∃ k 1 > 0 such that ∀ y , y ′ ∈ H , ∀ z , z ′ ∈ L 2 ( H ) | F ( t , ω, y , z ) − F ( t , ω, y ′ , z ′ ) | 2 ≤ k 1 ( | y − y ′ | 2 + || z − z ′ || 2 2 ) , uniformly in ( t , ω ) . (A2) ξ ∈ L 2 (Ω , F T , P ; H ) . (A3) There exists a predictable process Q satisfying Q ( t , ω ) is symmetric, positive definite nuclear operator on H and � t << M >> t = Q ( s ) ds . 0

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application (A4) Every square integrable H -valued martingale with respect {F t , 0 ≤ t ≤ T } has a continuous version. (A5) A ( t , ω ) is a linear operator on H , P - measurable, belongs to L ( V ; V ′ ) uniformly in ( t , ω ) and satisfies the following conditions: (i) A ( t , ω ) satisfies the coercivity condition in the sense that 2 [ A ( t , ω ) y , y ] + α | y | 2 V ≤ λ | y | 2 a . e . t ∈ [ 0 , T ] , a . s . ∀ y ∈ V , H for some α, λ > 0 . (ii) A ( t , ω ) is uniformly continuous, i.e. ∃ k 3 ≥ 0 such that for all ( t , ω ) | A ( t , ω ) y | V ′ ≤ k 3 | y | V , for every y ∈ V .

Aims Notations Spaces of solutions Definitions Assumptions Existence & Uniqueness Theorem Proof Some advantages An application Outline Aims 1 Notations 2 Example 1 Spaces of solutions 3 Definitions 4 Assumptions 5 Existence & Uniqueness Theorem 6 7 Proof Some advantages 8 An application 9 Maximum principle for controlled stochastic evolution equations An example