Forward Integral and Fractional Stochastic Differential Equations - PowerPoint PPT Presentation

Forward Integral and Fractional Stochastic Differential Equations Jorge A. León Departamento de Control Automático Cinvestav del IPN Spring School "Stochastic Control in Finance", Roscoff 2010 Jointly with Constantin Tudor Jorge A. León (Cinvestav-IPN) Forward Integral 2010 1 / 79

Contents Introduction 1 Preliminaries 2 Forward Integral 3 Semilinear Fractional Stochastic Differential Equations 4 Relation Between Forward and Young Integrals 5 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 2 / 79

Contents Introduction 1 Preliminaries 2 Forward Integral 3 Semilinear Fractional Stochastic Differential Equations 4 Relation Between Forward and Young Integrals 5 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 3 / 79

Equation Consider the semilinear fractional differential equation of the form � t � t 0 σ s X s dB − X t = η + 0 b ( s , X s ) ds + s , t ∈ [ 0 , T ] . Here η : Ω → R , b : Ω × [ 0 , T ] × R → R , σ : Ω × [ 0 , T ] → R and B = { B t : t ∈ [ 0 , T ] } is a fractional Brownian motion with Hurst parameter H ∈ ( 1 / 2 , 1 ) . Jorge A. León (Cinvestav-IPN) Forward Integral 2010 4 / 79

Equation Consider the semilinear fractional differential equation of the form � t � t 0 σ s X s dB − X t = η + 0 b ( s , X s ) ds + s , t ∈ [ 0 , T ] . Here η : Ω → R , b : Ω × [ 0 , T ] × R → R , σ : Ω × [ 0 , T ] → R and B = { B t : t ∈ [ 0 , T ] } is a fractional Brownian motion with Hurst parameter H ∈ ( 1 / 2 , 1 ) . The stochastic integral is the Forward integral introduced by Russo and Vallois. Jorge A. León (Cinvestav-IPN) Forward Integral 2010 5 / 79

Contents Introduction 1 Preliminaries 2 Forward Integral 3 Semilinear Fractional Stochastic Differential Equations 4 Relation Between Forward and Young Integrals 5 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 6 / 79

Fractional Brownian motion { B t } t ∈ [ 0 , T ] is a fractional Brownian motion of Hurst parameter H ∈ ( 1 2 , 1 ) . Jorge A. León (Cinvestav-IPN) Forward Integral 2010 7 / 79

Fractional Brownian motion { B t } t ∈ [ 0 , T ] is a fractional Brownian motion of Hurst parameter H ∈ ( 1 2 , 1 ) . H is the Reproducing Kernel Hilbert Space of the fBm B . That is, H is the closure of the linear space of step functions defined on [ 0 , T ] with respect to the scalar product � t � s 0 | r − u | 2 H − 2 drdu . � � 1 [ 0 , t ] , 1 [ 0 , s ] H = R H ( t , s ) = H ( 2 H − 1 ) 0 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 8 / 79

Fractional Brownian motion { B t } t ∈ [ 0 , T ] is a fractional Brownian motion of Hurst parameter H ∈ ( 1 2 , 1 ) . H is the closure of the linear space of step functions defined on [ 0 , T ] with respect to the scalar product � t � s 0 | r − u | 2 H − 2 drdu . � � 1 [ 0 , t ] , 1 [ 0 , s ] H = R H ( t , s ) = H ( 2 H − 1 ) 0 We consider a subspace of functions included in H via an isometry. This is the space |H| of all measurable functions ϕ : [ 0 , T ] → R such that � T � T 0 | ϕ r | | ϕ s | | r − s | 2 H − 2 drds < ∞ . || ϕ || 2 |H| = H ( 2 H − 1 ) 0 The space ( |H| , || · || |H| ) is a Banach one and the class of all the step functions defined on [ 0 , T ] is dense in it. Jorge A. León (Cinvestav-IPN) Forward Integral 2010 9 / 79

Fractional Brownian motion The space |H| is the setof all measurable functions ϕ : [ 0 , T ] → R such that � T � T 0 | ϕ r | | ϕ s | | r − s | 2 H − 2 drds < ∞ || ϕ || 2 |H| = H ( 2 H − 1 ) 0 and the Banach space |H| ⊗ |H| is the class of all the measurable functions ϕ : [ 0 , T ] 2 → R such that || ϕ || 2 |H|⊗|H| � [ H ( 2 H − 1 )] 2 [ 0 , T ] 4 | ϕ r ,θ || ϕ u ,η || r − u | 2 H − 2 | θ − η | 2 H − 2 drdud θ d η = < ∞ . Jorge A. León (Cinvestav-IPN) Forward Integral 2010 10 / 79

Derivative operator Let V be a Hilbert space and S V the family of V –valued smooth random variables of the form n � F = F i v i , F i ∈ S and v i ∈ V . i = 1 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 11 / 79

Derivative operator Let V be a Hilbert space and S V the family of V –valued smooth random variables of the form n � F = F i v i , F i ∈ S and v i ∈ V . i = 1 Set D k F = � n i = 1 D k F i ⊗ v i . We define the space D k , p ( V ) as the completion of S V with respect to the norm k || F || p k , p , V = E ( || F || p E ( || D i F || p � V ) + H ⊗ i ⊗ V ) . i = 1 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 12 / 79

Gradient operator For p > 1, D 1 , p ( |H| ) ⊆ D 1 , p ( H ) is the family of all the elements u ∈ |H| a.s. such that ( Du ) ∈ |H| ⊗ |H| a.s., and || u || p D 1 , p ( |H| ) = E ( || u || p |H| ) + E ( || Du || p |H|⊗|H| ) < ∞ . Jorge A. León (Cinvestav-IPN) Forward Integral 2010 13 / 79

Gradient operator D 1 , p ( |H| ) ⊆ D 1 , p ( H ) is the family of all the elements u ∈ |H| a.s. such that ( Du ) ∈ |H| ⊗ |H| a.s., and || u || p D 1 , p ( |H| ) = E ( || u || p |H| ) + E ( || Du || p |H|⊗|H| ) < ∞ . D 1 , p ( |H| ) ⊂ Dom δ . 1 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 14 / 79

Gradient operator D 1 , p ( |H| ) ⊆ D 1 , p ( H ) is the family of all the elements u ∈ |H| a.s. such that ( Du ) ∈ |H| ⊗ |H| a.s., and || u || p D 1 , p ( |H| ) = E ( || u || p |H| ) + E ( || Du || p |H|⊗|H| ) < ∞ . D 1 , p ( |H| ) ⊂ Dom δ . 1 E ( | δ ( u ) | 2 ) ≤ || u || 2 D 1 , 2 ( |H| ) . 2 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 15 / 79

Gradient operator D 1 , p ( |H| ) ⊆ D 1 , p ( H ) is the family of all the elements u ∈ |H| a.s. such that ( Du ) ∈ |H| ⊗ |H| a.s., and || u || p D 1 , p ( |H| ) = E ( || u || p |H| ) + E ( || Du || p |H|⊗|H| ) < ∞ . D 1 , p ( |H| ) ⊂ Dom δ . 1 E ( | δ ( u ) | 2 ) ≤ || u || 2 D 1 , 2 ( |H| ) . 2 A process u ∈ D 1 , p ( |H| ) belongs to L 1 , p if 3 H || u || p H = E ( || u || p H ([ 0 , T ]) ) + E ( || Du || p H ([ 0 , T ] 2 ) ) < ∞ . L 1 , p 1 1 L L Jorge A. León (Cinvestav-IPN) Forward Integral 2010 16 / 79

Gradient operator D 1 , p ( |H| ) ⊆ D 1 , p ( H ) is the family of all the elements u ∈ |H| a.s. such that ( Du ) ∈ |H| ⊗ |H| a.s., and || u || p D 1 , p ( |H| ) = E ( || u || p |H| ) + E ( || Du || p |H|⊗|H| ) < ∞ . D 1 , p ( |H| ) ⊂ Dom δ . 1 E ( | δ ( u ) | 2 ) ≤ || u || 2 D 1 , 2 ( |H| ) . 2 A process u ∈ D 1 , p ( |H| ) belongs to L 1 , p if 3 H || u || p H = E ( || u || p H ([ 0 , T ]) ) + E ( || Du || p H ([ 0 , T ] 2 ) ) < ∞ . L 1 , p 1 1 L L Then, || u || p D 1 , p ( |H| ) ≤ b H || u || p H . L 1 , p Jorge A. León (Cinvestav-IPN) Forward Integral 2010 17 / 79

Gradient operator D 1 , p ( |H| ) ⊆ D 1 , p ( H ) is the family of all the elements u ∈ |H| a.s. such that ( Du ) ∈ |H| ⊗ |H| a.s., and || u || p D 1 , p ( |H| ) = E ( || u || p |H| ) + E ( || Du || p |H|⊗|H| ) < ∞ . D 1 , p ( |H| ) ⊂ Dom δ . 1 E ( | δ ( u ) | 2 ) ≤ || u || 2 D 1 , 2 ( |H| ) . 2 A process u ∈ D 1 , p ( |H| ) belongs to L 1 , p if 3 H || u || p H = E ( || u || p H ([ 0 , T ]) ) + E ( || Du || p H ([ 0 , T ] 2 ) ) < ∞ . L 1 , p 1 1 L L Then, || u || p D 1 , p ( |H| ) ≤ b H || u || p H . L 1 , p L 1 , p H ⊂ Dom δ. 4 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 18 / 79

Gradient operator Theorem (Alòs and Nualart) Let { u t } t ∈ [ 0 , T ] be a process in L 1 , 2 H − ε for some 0 < ε < H − 1 2 . Then � t � 2 � � � � � E sup 0 u s δ B s � � � � 0 ≤ t ≤ T  � 2 ( H − ε ) �� T 1  H − ε ds ≤ C 0 | E ( u s ) |  2 ( H − ε )  H   � T �� T � H − ε  1  H dr + E 0 | D s u r | ds  ,   0  where C = C ( ε, H , T ) . Jorge A. León (Cinvestav-IPN) Forward Integral 2010 19 / 79

Contents Introduction 1 Preliminaries 2 Forward Integral 3 Semilinear Fractional Stochastic Differential Equations 4 Relation Between Forward and Young Integrals 5 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 20 / 79

Forward integral Definition (Russo and Vallois) Let { u t } t ∈ [ 0 , T ] be a process with integrable paths. We say that u is forward integrable with respect to B (or u ∈ Dom δ − ) if the stochastic process � t � � � � ε − 1 0 u s B ( s + ε ) ∧ T − B s ds t ∈ [ 0 , T ] converges uniformly on [ 0 , T ] in probability as ε → 0. The limit is � · 0 u s dB − denoted by s and it is called the forward integral of u with respect to B . Jorge A. León (Cinvestav-IPN) Forward Integral 2010 21 / 79

Forward integral Proposition Assume that u ∈ L 1 , 2 H − ρ , for some 0 < ρ < H − 1 2 , and that the trace condition � T � T 0 | D s u t | | t − s | 2 H − 2 dsdt < ∞ a.s. 0 holds. Then u ∈ Dom δ − and for every t ∈ [ 0 , T ] , � t � t � t � T 0 D s u r | r − s | 2 H − 2 dsdr . 0 u s dB − s = 0 u s δ B s + H ( 2 H − 1 ) 0 Jorge A. León (Cinvestav-IPN) Forward Integral 2010 22 / 79