An Extension of the Divergence Operator for Gaussian Processes - PowerPoint PPT Presentation

An Extension of the Divergence Operator for Gaussian Processes Jorge A. León Departamento de Control Automático Cinvestav del IPN Spring School "Stochastic Control in Finance", Roscoff 2010 Jointly with David Nualart and Jaime San Martín Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 1 / 80

Contents Introduction 1 Chaos Decomposition 2 The divergence operator 3 Background 4 Linear Fractional Stochastic Differential Equations 5 Semilinear Fractional Stochastic Differential Equations 6 Semilinear Fractional SPDE’s 7 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 2 / 80

Contents Introduction 1 Chaos Decomposition 2 The divergence operator 3 Background 4 Linear Fractional Stochastic Differential Equations 5 Semilinear Fractional Stochastic Differential Equations 6 Semilinear Fractional SPDE’s 7 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 3 / 80

Equation Consider the linear fractional differential equation of the form � t � t 0 b ( s ) X s dB H X t = η + 0 a ( s ) X s ds + s , t ∈ [ 0 , T ] . Here η ∈ L 2 (Ω) , a , b : [ 0 , T ] → R and B H = { B H t : t ∈ [ 0 , T ] } is a fractional Brownian motion with Hurst parameter H ∈ ( 0 , 1 / 2 ) . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 4 / 80

Equation Consider the linear fractional differential equation of the form � t � t 0 b ( s ) X s dB H X t = η + 0 a ( s ) X s ds + s , t ∈ [ 0 , T ] . Here η ∈ L 2 (Ω) , a , b : [ 0 , T ] → R and B H = { B H t : t ∈ [ 0 , T ] } is a fractional Brownian motion with hurst parameter H ∈ ( 0 , 1 / 2 ) . The stochastic integral is an extension of the divergence operator. Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 5 / 80

Contents Introduction 1 Chaos Decomposition 2 The divergence operator 3 Background 4 Linear Fractional Stochastic Differential Equations 5 Semilinear Fractional Stochastic Differential Equations 6 Semilinear Fractional SPDE’s 7 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 6 / 80

Notation Let H and H 0 be two real separable Hilbert spaces with inner products �· , ·� H and �· , ·� H 0 . W = { W ( h ) : h ∈ H} is a Gaussian process on H such that E ( W ( h ) W ( g )) = � h , g � H , for h , g ∈ H . F is the σ –field generated by W . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 7 / 80

Chaotic representation Let F ∈ L 2 (Ω , F , P ; H 0 ) . Then it has the representation ∞ � f n ∈ H ⊙ n ⊗ H 0 , F = I n ( f n ) , n = 0 where � � E � h , I n ( f n ) � H 0 ( n i 1 )! H n i 1 ( W ( e i 1 )) · · · ( n i k )! H n ik ( W ( e i k ))  if � k ⊗ n i 1 ⊗ n ik  n ! � f n , e ⊗ · · · ⊗ e ⊗ h � H ⊗ n ⊗H 0 , j = 1 n i j = n ,  i 1 i k =   0 , otherwise. Here h ∈ H 0 , { e i : i ∈ N } is an OCS of H and H n ( x ) = ( − 1 ) n e x 2 / 2 d n dx n ( e − x 2 / 2 ) , n ! x ∈ R and n ≥ 0 . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 8 / 80

Contents Introduction 1 Chaos Decomposition 2 The divergence operator 3 Background 4 Linear Fractional Stochastic Differential Equations 5 Semilinear Fractional Stochastic Differential Equations 6 Semilinear Fractional SPDE’s 7 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 9 / 80

Hypotheses Throughout we assume that H is densely and continuously embedded in H 0 and that T : H ⊂ H 0 → H 0 is a linear operator (whose domain D ( T ) is H ) satisfying the following conditions : (H1) |T h | H 0 = | h | H , for all h ∈ H . (H2) T H := { h ∈ H : T h ∈ D ( T ∗ ) } is a dense subset of H . (H3) T H 0 = {T ∗ T h : h ∈ T H } is dense in H 0 . S T is the family of all the smooth random variables of the form F = f ( W ( g 1 ) , . . . , W ( g n )) , p ( R n ) . where { g 1 , . . . , g n } is in D ( T ∗ T ) , f ∈ C ∞ Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 10 / 80

Derivative operator Throughout we assume that H is densely and continuously embedded in H 0 and that T : H ⊂ H 0 → H 0 is a linear operator (whose domain D ( T ) is H ) satisfying the following conditions : (H1) |T h | H 0 = | h | H , for all h ∈ H . (H2) T H := { h ∈ H : T h ∈ D ( T ∗ ) } is a dense subset of H . (H3) T H 0 = {T ∗ T h : h ∈ T H } is dense in H 0 . S T is the family of all the smooth random variables of the form F = f ( W ( g 1 ) , . . . , W ( g n )) , p ( R n ) and where { g 1 , . . . , g n } is in D ( T ∗ T ) , f ∈ C ∞ n � ∂ f DF = ( W ( g 1 ) , . . . , W ( g n )) g i . ∂ x i i = 1 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 11 / 80

Stochastic integral Definition Let u ∈ L 2 (Ω , F , P ; H 0 ) . We say that u belongs to Dom δ ∗ if and only if there exists δ ( u ) ∈ L 2 (Ω) such that E �T ∗ T DF , u � H 0 E � D T F , u � H 0 := = E ( F δ ( u )) , for every F ∈ S T . In this case, the random variable δ ( u ) is called the extended divergence of u . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 12 / 80

Stochastic integral Definition Let u ∈ L 2 (Ω , F , P ; H 0 ) . We say that u belongs to Dom δ ∗ if and only if there exists δ ( u ) ∈ L 2 (Ω) such that E �T ∗ T DF , u � H 0 E � D T F , u � H 0 := = E ( F δ ( u )) , for every F ∈ S T . (1) Remarks. i) If H 0 = H and T = I H , then (1) has the form E � DF , u � H = E ( F δ ( u )) . Thus, δ is equal to the usual divergence operator. ii) Let u ∈ L 2 (Ω , F , P ; H ) . Then � DF , u � H = �T ∗ T DF , u � H 0 . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 13 / 80

Characterization of δ Theorem Assume that (H1)–(H3) hold and that u ∈ L 2 (Ω , F ; H 0 ) has the chaos representation ∞ � f n ∈ H ⊙ n ⊗ H 0 . u = I n ( f n ) , n = 0 Then u ∈ Dom δ ∗ if and only if � f n (the symmetrization of f n as an ) belongs to H ⊙ ( n + 1 ) for all n ≥ 0 , and element of H ⊗ ( n + 1 ) 0 ∞ � n ! | � f n − 1 | 2 H ⊗ n < ∞ . n = 1 In this case δ ( u ) = � ∞ n = 1 I n ( � f n − 1 ) . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 14 / 80

Characterization of δ Proof : Fix n ≥ 1. Let { n 1 , . . . , n k } be a finite sequence of positive integers such that n 1 + · · · + n k = n and { g 1 , . . . , g k } ⊂ T H an orthonormal system on H . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 15 / 80

Characterization of δ Proof : Fix n ≥ 1. Let { n 1 , . . . , n k } be a finite sequence of positive integers such that n 1 + · · · + n k = n and { g 1 , . . . , g k } ⊂ T H an orthonormal system on H . Necessity : We have � � E � u , D T ( n 1 ! H n 1 ( W ( g 1 )) . . . n k ! H n k ( W ( g k ))) � H 0 � k � f n − 1 , ( T ∗ T ) ⊗ ( n − 1 ) ( g ⊗ n 1 = n j ( n − 1 )! 1 j = 1 ⊗ n j − 1 ⊗ ( n j − 1 ) ⊗ · · · ⊗ g ⊗ n k ⊗ · · · ⊗ g ⊗ g n k ) j − 1 j ⊗ T ∗ Tg j � H ⊗ n 0 . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 16 / 80

Characterization of δ Proof : Fix n ≥ 1. Let { n 1 , . . . , n k } be a finite sequence of positive integers such that n 1 + · · · + n k = n and { g 1 , . . . , g k } ⊂ T H an orthonormal system on H . Necessity : We have � � E � u , D T ( n 1 ! H n 1 ( W ( g 1 )) . . . n k ! H n k ( W ( g k ))) � H 0 � k � f n − 1 , ( T ∗ T ) ⊗ ( n − 1 ) ( g ⊗ n 1 = n j ( n − 1 )! 1 j = 1 ⊗ n j − 1 ⊗ ( n j − 1 ) ⊗ · · · ⊗ g ⊗ n k ⊗ · · · ⊗ g ⊗ g n k ) j − 1 j ⊗ T ∗ Tg j � H ⊗ n 0 . Hence, if δ ( u ) has the chaos representation ∞ � v n ∈ H ⊙ n , δ ( u ) = I n ( v n ) , n = 0 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 17 / 80

Characterization of δ Proof : � � E � u , D T ( n 1 ! H n 1 ( W ( g 1 )) . . . n k ! H n k ( W ( g k ))) � H 0 k � n j ( n − 1 )! � f n − 1 , ( T ∗ T ) ⊗ ( n − 1 ) ( g ⊗ n 1 = 1 j = 1 ⊗ n j − 1 ⊗ ( n j − 1 ) ⊗ · · · ⊗ g ⊗ n k ⊗ · · · ⊗ g ⊗ g n k ) j − 1 j ⊗ T ∗ Tg j � H ⊗ n 0 . Hence, if δ ( u ) has the chaos representation ∞ � v n ∈ H ⊙ n , δ ( u ) = I n ( v n ) , n = 0 then the duality relation (1) and (H3) yield that v n = � f n − 1 , and therefore � ∞ n = 1 n ! | � f n − 1 | 2 H ⊗ n < ∞ . Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 18 / 80

Characterization of δ Sufficiency : Let F = f ( W ( g 1 ) , . . . , W ( g k )) be a random variable in S T and K the linear subspace of H generated by { g 1 , . . . , g k } . Then F has the chaos decompostion given by ∞ � k n ∈ K ⊙ n . F = I n ( k n ) , n = 0 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 19 / 80

Characterization of δ Sufficiency : Let F = f ( W ( g 1 ) , . . . , W ( g k )) be a random variable in S T and K the linear subspace of H generated by { g 1 , . . . , g k } . Then F has the chaos decompostion given by ∞ � k n ∈ K ⊙ n . F = I n ( k n ) , n = 0 Consequently, E � u , D T F � H 0 ∞ � ( n + 1 )! � f n , ( T ∗ T ) ⊗ ( n + 1 ) ( k n + 1 ) � H ⊗ ( n + 1 ) = 0 n = 0 ∞ � ( n + 1 )! � � f n , ( T ∗ T ) ⊗ ( n + 1 ) ( k n + 1 ) � H ⊗ ( n + 1 ) = 0 n = 0 Jorge A. León (Cinvestav-IPN) Divergence Operator 2010 20 / 80