Existence and smoothness of the density of the solution to fractional stochastic integral Volterra equations
aa r X i v : . [ m a t h . P R ] A p r EXISTENCE AND SMOOTHNESS OF THE DENSITY OF THESOLUTION TO FRACTIONAL STOCHASTIC INTEGRAL VOLTERRAEQUATIONS
MIREIA BESALÚ, DAVID MÁRQUEZ-CARRERAS AND EULALIA NUALART
Abstract.
We consider stochastic Volterra integral equations driven by a fractionalBrownian motion with Hurst parameter
H > . We first derive supremum norm esti-mates for the solution and its Malliavin derivative. We then show existence and smooth-ness of the density under suitable nondegeneracy conditions. This extends the results in[11] and [15] where stochastic differential equations driven by fractional Brownian motionare considered. The proof uses a priori estimates for deterministic differential equationsdriven by a function in a suitable Sobolev space. Introduction
We consider the stochastic integral Volterra equation on R d X t = X + Z t b ( t, s, X s ) ds + Z t σ ( t, s, X s ) dW Hs , t ∈ (0 , T ] , (1.1)where σ = ( σ i,j ) d × m : [0 , T ] × R d → R d × R m and b = ( b i ) d × : [0 , T ] × R d → R d aremeasurable functions, W H = { W H,jt , t ∈ [0 , T ] , j = 1 , . . . , m } are independent fractionalBrownian motions (fBm) with Hurst parameter H > defined in a complete probabilityspace (Ω , F , P ) , and X is a d-dimensional random variable.As H > , the integral with respect to W H can be defined as a pathwise Riemann-Stieltjes integral using the results by Young [19]. Moreover, Zälhe [20] introduced ageneralized Stieltjes integral using the techniques of fractional calculus. In particular, sheobtained a formula for the Riemann-Stieltjes integral using fractional derivatives (see (2.2)below). Using this formula, Nualart and Rascanu [14] proved a general result on existence,uniqueness and finite moments of the solution to a class of general differential equationsincluded in (1.1). These results were extended by Besalú and Rovira [5] for the Volterraequation (1.1). The proof of these results uses a priori estimates for a deterministicdifferential equation driven by a function in a suitable Sobolev space.The first aim of this paper it is to obtain supremum norm estimates of the solution to(1.1). We first consider the case where σ is bounded since, in this case, the estimates areof polynomial type, while in the general case are of exponential type. In the case where σ is bounded, we also obtain estimates for the Malliavin derivative of the solution andshow existence and smoothness of the density. To obtain these results, we first derivea priori estimates for some deterministic equations. Finally, in the case where σ is not Date : April 8, 2020.M. Besalú and D. Márquez-Carreras are supported by the grant MTM 2015-65092-P from MINECO,Spain. E. Nualart is supported from the Spanish Government grants PGC2018-101643-B-I00, SEV-2015-0563, and Ayudas Fundación BBVA a Equipos de Investigación Cientifica 2017. necessarily bounded, we also show existence of the density by first showing the Fréchetdifferentiability of the solution to the corresponding deterministic equation.These results provide extensions of the works by Hu and Nualart [11] and Nualartand Saussereau [15], where stochastic differential equations driven by fBm are considered.In particular, we provide a corrected proof of [11, Theorem 7], as there is a problemin their argument. The techniques used to obtain the a priori estimates in the presentpaper are much more involved than those in [11] and [15] due to the time-dependence ofthe coefficients. As in those papers, our nondegeneracy assumption is an ellipticity-typecondition, see Baudoin and Hairer [2] for the existence and smoothness of the densityunder Hörmander’s condition for stochastic differential equations driven by a fBm withHurst parameter
H > .Volterra equations driven by general Itô processes or semimartingales are widely stud-ied, see for instance [1, 3, 4, 17]. Concerning Volterra equations driven by fBm, the mainreferences are the papers of Deya and Tindel [7, 8], where existence and uniqueness isstudied separately for the case H > and H > , using an algebraic integration settingand the Young integral, respectively. For the case H > / and using the Young inte-gral, existence and uniqueness of the solution to equation (1.1) with an extra term drivenby an independent Wiener process is proved in [18]. See also [16, 9] for the existenceand uniqueness of fBm driven Volterra equations in a Hilbert space. In [21], a class offractional stochastic Volterra equations of convolution type driven by infinite dimensionalfBm with Hurst index H ∈ (0 , is considered, and existence and regularity results ofthe stochastic convolution process are established. Last but not least, existence of thedensity of the solution to equation (1.1) in the one dimensional case is obtained in [10]as a consequence of a Bismut type formula. However, supremum norm estimates andexistence and smoothness of the density in the multidimensional case do not seem to bestudied yet in the literature for this kind of equations.The structure of this paper is as follows: in the next section we introduce all the spaces,norms and operators used through the paper. In Section 3, we obtain a priori estimatesfor the solution of some systems of equations in a deterministic framework and study theFréchet differentiability of one of them. Section 4 is devoted to apply the results obtainedin Section 3 to the Volterra equation (1.1) and derive the existence and smoothness of thedensity. Notation:
For any integer k ≥ , we denote by C kb the class of real-valued functionson R d which are k times continuously differentiable with bounded partial derivatives upto the k th order. We denote by C ∞ b the the class of real-valued functions on R d which areinfinitely differentiable and bounded together with all their derivatives.Throughout all the paper, C α , C α,β , c α,T , etc. will denote generic constants that maychange from line to line. RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 3 Preliminaries
For any α ∈ (0 , ) , we denote by W α (0 , T ; R d ) the space of measurable functions f :[0 , T ] → R d such that k f k α, := sup t ∈ [0 ,T ] (cid:18) | f ( t ) | + Z t | f ( t ) − f ( s ) || t − s | α +1 ds (cid:19) < ∞ . For any α ∈ (0 , ) , we denote by W − α (0 , T ; R m ) the space of measurable functions g : [0 , T ] → R m such that k g k − α, := sup ≤ s
Ω = C ([0 , T ]; R m ) be the Banach space of continuous functions, null at time 0,equipped with the supremum norm. Let P be the unique probability measure on Ω suchthat the canonical process { W Ht , t ∈ [0 , T ] } is an m -dimensional fractional Brownianmotion with Hurst parameter H > . M. BESALÚ, D. MÁRQUEZ-CARRERAS AND E. NUALART
We denote by E the space of step functions on [0 , T ] with values in R m . Let H be theHilbert space defined as the closure of E with respect to the scalar product h ( [0 ,t ] , . . . , [0 ,t m ] ) , ( [0 ,s ] , . . . , [0 ,s m ] ) i H = m X i =1 R H ( t i , s i ) , where R H ( t, s ) = Z t ∧ s K H ( t, r ) K H ( s, r ) dr, and K H ( t, s ) is the square integrable kernel defined by K H ( t, s ) = c H s / − H Z ts ( u − s ) H − / u H − / du, (2.3)where c H = q H (2 H − β (2 − H,H − / , β denotes the Beta function and t > s . For t ≤ s , we set K H ( t, s ) = 0 .The mapping ( [0 ,t ] , . . . , [0 ,t m ] ) → P mi =1 W H,it i can be extended to an isometry between H and the Gaussian space H associated to W H . We denote this isometry by ϕ → W H ( ϕ ) .Consider the operator K ∗ H from E to L (0 , T ; R m ) defined by ( K ∗ H ϕ ) i ( s ) = Z Ts ϕ i ( t ) ∂ t K H ( t, s ) dt. From (2.3), we get ∂ t K H ( t, s ) = c H (cid:18) ts (cid:19) H − / ( t − s ) H − / . Notice that K ∗ H ( [0 ,t ] , . . . , [0 ,t m ] ) = ( K H ( t , · ) , . . . , K H ( t m , · )) . For any ϕ, ψ ∈ E , h ϕ, ψ i H = h K ∗ H ϕ, K ∗ H ψ i L (0 ,T ; R m ) = E( W H ( ϕ ) W H ( ψ )) and K ∗ H provides an isometry between the Hilbert space H and a closed subspace of L (0 , T ; R m ) .Following [15], we consider the fractional version of the Cameron-Martin space H H := K H ( L (0 , T ; R m )) , where for h ∈ L (0 , T ; R m ) , ( K H h )( t ) := Z t K H ( t, s ) h s ds. We finally denote by R H = K H ◦ K ∗ H : H → H H the operator R H ϕ = Z · K H ( · , s )( K ∗ H h )( s ) ds. We remark that for any ϕ ∈ H , R H ϕ is Hölder continuous of order H . Therefore, for any − H < α < / , H H ⊂ C H (0 , T ; R m ) ⊂ W − α (0 , T ; R m ) . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 5
Notice that R H [0 ,t ] = R H ( t, · ) , and, as a consequence, H H is the Reproducing KernelHilbert Space associated with the Gaussian process W H . The injection R H : H → Ω embeds H densely into Ω and for any ϕ ∈ Ω ∗ ⊂ H , E (cid:16) e iW H ( ϕ ) (cid:17) = exp (cid:18) − k ϕ k H (cid:19) . As a consequence, (Ω , H , P) is an abstract Wiener space in the sense of Gross.3. Deterministic differential equations
Fix < α < . Consider the deterministic differential equation on R d x t = x + Z t b ( t, s, x s ) ds + Z t σ ( t, s, x s ) dg s , t ∈ [0 , T ] , (3.1)where g ∈ W − α (0 , T ; R m ) , x ∈ R d , and b and σ are as in (1.1).Consider the following hypotheses on b and σ : (H1) σ : [0 , T ] × R d → R d × R m is a measurable function such that the derivatives ∂ x σ ( t, s, x ) , ∂ t σ ( t, s, x ) and ∂ x,t σ ( t, s, x ) exist. Moreover, there exist some con-stants < β, µ, δ ≤ and for every N ≥ there exists K N > such that thefollowing properties hold:(1) | σ ( t, s, x ) − σ ( t, s, y ) | + | ∂ t σ ( t, s, x ) − ∂ t σ ( t, s, y ) | ≤ K | x − y | , ∀ x, y ∈ R d , ∀ s, t ∈ [0 , T ] ,(2) | ∂ x i σ ( t, s, x ) − ∂ y i σ ( t, s, y ) | + (cid:12)(cid:12) ∂ x i ,t σ ( t, s, x ) − ∂ y i ,t σ ( t, s, y ) (cid:12)(cid:12) ≤ K N | x − y | δ , ∀| x | , | y | ≤ N, ∀ s, t ∈ [0 , T ] , i = 1 . . . d ,(3) | σ ( t , s, x ) − σ ( t , s, x ) | + | ∂ x i σ ( t , s, x ) − ∂ x i σ ( t , s, x ) | ≤ K | t − t | µ , ∀ x ∈ R d , ∀ t , t , s ∈ [0 , T ] , i = 1 . . . d ,(4) | σ ( t, s , x ) − σ ( t, s , x ) | + | ∂ t σ ( t, s , x ) − ∂ t σ ( t, s , x ) | ≤ K | s − s | β , ∀ x ∈ R d , ∀ s , s , t ∈ [0 , T ] ,(5) (cid:12)(cid:12) ∂ x i ,t σ ( t, s , x ) − ∂ x i ,t σ ( t, s , x ) (cid:12)(cid:12) + | ∂ x i σ ( t, s , x ) − ∂ x i σ ( t, s , x ) | ≤ K | s − s | β , ∀ x ∈ R d , ∀ s , s , t ∈ [0 , T ] , i = 1 , . . . , d . (H2) b : [0 , T ] × R d → R d is a measurable function such that there exists b ∈ L ρ ([0 , T ] ; R d ) with ρ ≥ , < µ ≤ and ∀ N ≥ there exists L N > suchthat:(1) | b ( t, s, x ) − b ( t, s, y ) | ≤ L N | x − y | , ∀| x | , | y | ≤ N, ∀ s, t ∈ [0 , T ] ,(2) | b ( t , s, x ) − b ( t , s, x ) | ≤ L | t − t | µ , ∀ x ∈ R d , ∀ s, t , t ∈ [0 , T ] ,(3) | b ( t, s, x ) | ≤ L | x | + b ( t, s ) , ∀ x ∈ R d , ∀ s, t ∈ [0 , T ] ,(4) | b ( t , s, x ) − b ( t , s, x ) − b ( t , s, x ) + b ( t , s, x ) | ≤ L N | t − t || x − x | , ∀| x | , | x | ≤ N, ∀ t , t , s ∈ [0 , T ] . Remark . Actually, we can consider σ and b defined only in the set D × R d with D = { ( t, s ) ∈ [0 , T ] ; s ≤ t } .The following existence and uniqueness result holds. M. BESALÚ, D. MÁRQUEZ-CARRERAS AND E. NUALART
Theorem 3.2. [5, Theorem 4.1]
Assume that σ and b satisfy hypotheses ( H1 ) and ( H2 ) with ρ = 1 /α , min { β, δ + δ } > − µ and < − µ < α < α := min (cid:26) , β, δ δ (cid:27) . Then, equation (3.1) has a unique solution x ∈ C − α (0 , T ; R d ) . The first aim of this section is to obtain estimates for the supremum norm of thesolution to (3.1). We first consider the case where σ is bounded and the bound on b doesnot depend on x . Theorem 3.3.
Assume the hypotheses of Theorem 3.2 with µ = 1 and (H2) (3) replacedby | b ( t, s, x ) | ≤ L + b ( t, s ) , ∀ x ∈ R d , ∀ s, t ∈ [0 , T ] . (3.2) Assume that σ is bounded. Then, there exists a constant C α,β > such that k x k ∞ ≤ | x | + 1 + T (cid:18)(cid:16) K (1) T,α + K (2) T,α,β k g k − α (cid:17) / (1 − α ) ∨ ∨ T (cid:19) , (3.3) where K (1) T,α = 4( L ( T ∨
1) + L + B ,α ) and K (2) T,α,β = C α,β ( T + 1 + k σ k ∞ ) , L, L are theconstants in Hypothesis (H2) , and B ,α := sup t ∈ [0 ,T ] (cid:16)R t | b ( t, u ) | /α du (cid:17) α .Remark . The techniques used in the proof do not seem to extend to the case < µ < ,thus it is left open for future work. More specifically, if µ < , the first term in equation(3.14) is of order i ∆ µ +1 − α . Then, when dividing by ( t − s ) − α we obtain a term of order i ∆ µ which cannot be bounded by T . Proof.
We divide the interval [0 , T ] into n = [ T / ∆] + 1 subintervals, where [ a ] denotes thelargest integer strictly bounded by a and ∆ ≤ will be chosen below. Step 1.
We start studying k x k , ∆ , − α . For s, t ∈ [0 , ∆] , s < t , | x t − x s | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z s ( b ( t, r, x r ) − b ( s, r, x r )) dr (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ts b ( t, r, x r ) dr (cid:12)(cid:12)(cid:12)(cid:12) (3.4) + (cid:12)(cid:12)(cid:12)(cid:12)Z s ( σ ( t, r, x r ) − σ ( s, r, x r )) dg r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ts σ ( t, r, x r ) dg r (cid:12)(cid:12)(cid:12)(cid:12) = A + B + C + D. Using the Hypothesis (H2) (2), the term A is easy to bound A ≤ Ls ( t − s ) . (3.5)For the second term we use (3.2) to obtain B ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z ts ( L + b ( s, r )) dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ L ( t − s ) + B ,α ( t − s ) − α . (3.6)For the next term, we use [5, Lemma A.2] to get (cid:12)(cid:12) D α [ σ ( t, · , x · ) − σ ( s, · , x · )] ( r ) (cid:12)(cid:12) ≤ K ( t − s )Γ(1 − α ) (cid:18) r α + α Z r ( r − u ) β + | x r − x u | ( r − u ) α +1 du (cid:19) ≤ C α,β ( t − s ) (cid:0) r − α + r β − α + k x k ,s, − α r − α (cid:1) . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 7
Putting together the previous estimate, equation (2.2) and the estimate in [11, (3.5)] weconclude that C ≤ C α,β k g k − α ( t − s ) (cid:12)(cid:12)(cid:12)(cid:12)Z s (cid:0) r − α + r β − α + k x k ,s, − α r − α (cid:1) dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α,β k g k − α ( t − s ) (cid:0) s − α + s β − α + s − α k x k ,s, − α (cid:1) . (3.7)For term D, we obtain, proceeding similarly as for term C , (cid:12)(cid:12) D αs + [ σ ( t, · , x · )] ( r ) (cid:12)(cid:12) ≤ − α ) (cid:0) k σ k ∞ ( r − s ) − α + αK Z rs ( r − u ) β + | x r − x u | ( r − u ) α +1 du (cid:19) ≤ C α,β (cid:0) k σ k ∞ ( r − s ) − α + ( r − s ) β − α + k x k s,t, − α ( r − s ) − α (cid:1) . Therefore, D ≤ C α,β k g k − α Z ts (cid:0) k σ k ∞ ( r − s ) − α + ( r − s ) β − α + k x k s,t, − α ( r − s ) − α (cid:1) dr ≤ C α,β k g k − α ( t − s ) − α (cid:0) k σ k ∞ + ( t − s ) β + k x k s,t, − α ( t − s ) − α (cid:1) . (3.8)Next, introducing (3.5), (3.6), (3.7) and (3.8) into (3.4), we obtain | x t − x s | ( t − s ) − α ≤ Ls ( t − s ) α + L ( t − s ) α + B ,α + C α,β k g k − α (cid:0) ( t − s ) α (cid:0) s − α + s β − α + s − α k x k ,s, − α (cid:1) + k σ k ∞ + ( t − s ) β + k x k s,t, − α ( t − s ) − α (cid:1) . Thus, k x k , ∆ , − α ≤ L ∆ α + L ∆ α + B ,α + C α,β k g k − α (cid:0) ∆ + ∆ β + ∆ β + k σ k ∞ + k x k , ∆ , − α (∆ − α + ∆ − α ) (cid:1) ≤ L + L + B ,α + C α,β k g k − α (cid:0) k σ k ∞ + k x k , ∆ , − α ∆ − α (cid:1) , as ∆ ≤ . Choosing ∆ such that ∆ − α ≤ C α,β k g k − α , (3.9)we obtain that k x k , ∆ , − α ≤ L + L + B ,α + C α,β k g k − α (1 + k σ k ∞ )) . (3.10)Therefore, k x k , ∆ , ∞ ≤ | x | + k x k , ∆ , − α ∆ − α ≤ | x | + 12 , (3.11)if ∆ is such that ∆ − α ≤
14 ( L + L + B ,α + C α,β k g k − α (1 + k σ k ∞ )) . (3.12) M. BESALÚ, D. MÁRQUEZ-CARRERAS AND E. NUALART
Step 2.
We next study k x k s,t, − α for s, t ∈ [ i ∆ , ( i + 1)∆] , s < t . We write | x t − x s | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z s ( b ( t, r, x r ) − b ( s, r, x r )) dr (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ts b ( t, r, x r ) dr (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z i ∆0 ( σ ( t, r, x r ) − σ ( s, r, x r )) dg r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z si ∆ ( σ ( t, r, x r ) − σ ( s, r, x r )) dg r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ts σ ( t, r, x r ) dg r (cid:12)(cid:12)(cid:12)(cid:12) = A + B + C i + C i + D. (3.13)The terms A , B , and D can be bounded exactly as in Step 1. Thus, it suffices to boundthe terms C i and C i . We start with C i . We write C i ≤ i X ℓ =1 (cid:12)(cid:12)(cid:12)(cid:12)Z ℓ ∆( ℓ − ( σ ( t, r, x r ) − σ ( s, r, x r )) dg r (cid:12)(cid:12)(cid:12)(cid:12) . Using [5, Lemma A.2], we get (cid:12)(cid:12) D α ( ℓ − [ σ ( t, · , x · ) − σ ( s, · , x · )] ( r ) |≤ K ( t − s )Γ(1 − α ) r − ( ℓ − α + α Z r ( ℓ − (cid:0) ( r − u ) β + | x r − x u | (cid:1) ( r − u ) α +1 du ! ≤ C α,β ( t − s )( r − ( ℓ − α (cid:0) r − ( ℓ − β + ( r − ( ℓ − − α k x k ( ℓ − ,ℓ ∆ , − α (cid:1) . Then, by the estimate in [11, (3.5)], we obtain C i ≤ C α,β k g k − α ( t − s ) i X ℓ =1 (cid:0) ∆ − α + ∆ β − α + ∆ − α k x k ( ℓ − ,ℓ ∆ , − α (cid:1) . (3.14)Similarly, for the term C i we obtain C i ≤ C α,β k g k − α ( t − s ) Z si ∆ r − i ∆) α (cid:0) r − i ∆) β + ( r − i ∆) − α k x k i ∆ ,s, − α (cid:1) dr ≤ C α,β k g k − α ( t − s ) (cid:0) ( s − i ∆) − α + ( s − i ∆) β − α + ( s − i ∆) − α k x k i ∆ ,s, − α (cid:1) . (3.15) RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 9
Hence, from (3.5), (3.6), (3.8), (3.14) and (3.15), and using the fact that ∆ ≤ , t − s ≤ ∆ and i ∆ ≤ T , we obtain | x t − x s | ( t − s ) − α ≤ Ls ( t − s ) α + L ( t − s ) α + B ,α + C α,β k g k − α " ( t − s ) α i X ℓ =1 (cid:0) ∆ − α + ∆ β − α + ∆ − α k x k ( ℓ − ,ℓ ∆ , − α (cid:1) + ( t − s ) α (cid:0) ( s − i ∆) − α + ( s − i ∆) β − α + ( s − i ∆) − α k x k i ∆ ,s, − α (cid:1) + ( t − s ) β + k σ k ∞ + k x k s,t, − α ( t − s ) − α ≤ LT + L + B ,α + C α,β k g k − α (cid:20) T + 1 + k σ k ∞ + ∆ − α i X ℓ =1 k x k ( ℓ − ,ℓ ∆ , − α + ∆ − α k x k i ∆ , ( i +1)∆ , − α (cid:21) . Choosing ∆ such that ∆ − α ≤ C α,β k g k − α , (3.16)we obtain that k x k i ∆ , ( i +1)∆ , − α ≤ A + A ∆ − α i X ℓ =1 k x k ( ℓ − ,ℓ ∆ , − α , (3.17)where A = 2( LT + L + B ,α + C α,β k g k − α ( T + 1 + k σ k ∞ )) ,A = 2 C α,β k g k − α . Step 3.
We now use an induction argument in order to show that for all i ≥ , ∆ − α k x k i ∆ , ( i +1)∆ , − α ≤ . For i = 0 it is proved in Step 1. Assuming that it is true up to i − and using (3.17), weget that ∆ − α k x k i ∆ , ( i +1)∆ , − α ≤ A ∆ − α + A ∆ − α i X ℓ =1 k x k ( ℓ − ,ℓ ∆ , − α ≤ ∆ − α ( A + A T ) . Finally, it suffices to choose ∆ such that ∆ − α ≤ A + A T , (3.18)to conclude the desired claim.Therefore, we have that k x k i ∆ , ( i +1)∆ , ∞ ≤ | x i ∆ | + ∆ − α k x k i ∆ , ( i +1)∆ , − α ≤ | x i ∆ | + 1 . (3.19) Applying this inequality recursively, we conclude that sup ≤ t ≤ T | x t | ≤ sup ≤ t ≤ ( n − | x t | + 1 ≤ · · · ≤ | x | + n, and the desired bound follows choosing ∆ such that ∆ = 1(4( L ( T ∨
1) + L + B ,α + C α,β k g k − α ( T + 1 + k σ k ∞ ))) / (1 − α ) ∧ ∧ T, where C α,β is such that (3.9), (3.12), (3.16) and (3.18) hold. (cid:3) The next result is an exponential bound for the supremum norm of the solution to (3.1)under more general hypotheses than the previous theorem.
Theorem 3.5.
Assume the hypotheses of Theorem 3.2 with µ = 1 . Then, there exists aconstant C α,β > such that k x k ∞ ≤ ( | x | + 1) exp (cid:18) T (cid:18)(cid:16) K (3) T,α + K (4) T,α,β k g k − α (cid:17) / (1 − α ) ∨ ∨ T (cid:19)(cid:19) , where K (3) T,α = 6( L + L ( T + 1) + B ,α ) , K (4) T,α,β = C α,β ( T + 1) , and L, L , and B ,α are asin Theorem 3.3.Proof. The proof follows similarly as the proof of Theorem 3.3. We divide the interval [0 , T ] into n = [ T / ∆] + 1 subintervals, where ∆ ≤ will be chosen below. Step 1.
We start bounding k x k , ∆ , − α . We can use the same bound for | x t − x s | obtainedin (3.4). Then, terms A and C can be bounded as in (3.5) and (3.7) respectively. Forterm B , using (H2) (3), we get that B ≤ L ( t − s ) k x k s,t, ∞ + B ,α ( t − s ) − α . (3.20)For term D , we obtain D ≤ C α,β k g k − α ( t − s ) − α (cid:2) k x k s,t, ∞ + ( t − s ) β + ( t − s ) − α k x k s,t, − α (cid:3) . (3.21)Thus, we get that | x t − x s | ( t − s ) − α ≤ Ls ( t − s ) α + B ,α + k x k s,t, ∞ [ L ( t − s ) α + C α,β k g k − α ]+ C α,β k g k − α ( t − s ) α (cid:2) s − α + s β − α + s − α k x k ,s, − α (cid:3) + C α,β k g k − α (cid:2) ( t − s ) β + ( t − s ) − α k x k s,t, − α (cid:3) . Hence, as ∆ ≤ , k x k , ∆ , − α ≤ B + B k x k , ∆ , ∞ + B k x k , ∆ , − α , where B = L + B ,α + C α,β k g k − α ,B = L + C α,β k g k − α ,B = ∆ − α C α,β k g k − α . Thus, k x k , ∆ , − α ≤ B (1 − B ) − + B (1 − B ) − k x k , ∆ , ∞ . (3.22) RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 11
Therefore, using the fact that sup t ∈ [0 , ∆] | x t | ≤ | x | + k x k , ∆ , − α ∆ − α , we conclude that sup t ∈ [0 , ∆] | x t | ≤ B − | x | + B − B (1 − B ) − ∆ − α , (3.23)where B = 1 − B (1 − B ) − ∆ − α . Step 2.
We next study k x k i ∆ , ( i +1)∆ ,β , for i ≥ . For s, t ∈ [ i ∆ , ( i + 1)∆] , s < t , | x t − x s | can be bounded as in (3.13). Then using (3.5), (3.14), (3.15), (3.20), and (3.21), we getthat | x t − x s | ( t − s ) − α ≤ Ls ( t − s ) α + L ( t − s ) α k x k s,t, ∞ + B ,α + C α,β k g k − α " ( t − s ) α i X ℓ =1 (cid:0) ∆ − α + ∆ β − α + ∆ − α k x k ( ℓ − ,ℓ ∆ , − α (cid:1) + ( t − s ) α (cid:0) ( s − i ∆) − α + ( s − i ∆) β − α + ( s − i ∆) − α k x k i ∆ ,s, − α (cid:1) + ( t − s ) β + k x k s,t, ∞ + k x k s,t, − α ( t − s ) − α ≤ LT + L k x k i ∆ , ( i +1)∆ , ∞ + B ,α + C α,β k g k − α (cid:20) T + 1 + k x k i ∆ , ( i +1)∆ , ∞ + ∆ − α i X ℓ =1 k x k ( ℓ − ,ℓ ∆ , − α + ∆ − α k x k i ∆ , ( i +1)∆ , − α (cid:21) . Therefore, we obtain that k x k i ∆ , ( i +1)∆ , − α ≤ C − " C + C k x k i ∆ , ( i +1)∆ , ∞ + C ∆ − α i X ℓ =1 k x k ( ℓ − ,ℓ ∆ , − α , (3.24)where C = 1 − C α,β k g k − α ∆ − α ,C = LT + B ,α + C α,β k g k − α [ T + 1] ,C = L + C α,β k g k − α ,C = C α,β k g k − α . Thus, k x k i ∆ , ( i +1)∆ , ∞ ≤ C − | x i ∆ | + C − C − ∆ − α (cid:0) C + C ∆ − α i X ℓ =1 k x k ( ℓ − ,ℓ ∆ , − α (cid:1) , (3.25)where C = 1 − C − C ∆ − α .We next show by induction that for all i ≥ , ∆ − α k x k i ∆ , ( i +1)∆ , − α ≤ k x k , ( i +1)∆ , ∞ . For i = 0 it is proved in (3.22) that k x k , ∆ , − α ≤ B (1 − B ) − + B (1 − B ) − k x k , ∆ , ∞ . Then, it suffices to choose ∆ such that B ≤ and ∆ − α ≤ (cid:18) B ∧ B (cid:19) , to conclude the claim for i = 0 .Assuming that it is true up to i − and using (3.24), we get that k x k i ∆ , ( i +1)∆ , − α ≤ C − (cid:2) C + C T + k x k , ( i +1)∆ , ∞ ( C + C T ) (cid:3) . Finally, it suffices to choose ∆ such that C ≥ and ∆ − α ≤ C + C T ∧ C + C T , to conclude the desired claim.By (3.25), we conclude that k x k i ∆ , ( i +1)∆ , ∞ ≤ C − | x i ∆ | + C − C − ∆ − α ( C + T C (1 + k x k ,i ∆ , ∞ )) . (3.26) Step 3.
Using (3.26), we get that sup ≤ t ≤ ( i +1)∆ | x t | ≤ sup ≤ t ≤ i ∆ | x t | + sup i ∆ ≤ t ≤ ( i +1)∆ | x t |≤ sup ≤ t ≤ i ∆ | x t | + C − | x i ∆ | + C − C − ∆ − α ( C + T C (1 + k x k ,i ∆ , ∞ )) ≤ K sup ≤ t ≤ i ∆ | x t | + K , where K = 1 + C − (1 + T C − C ∆ − α ) ,K = C − C − ( C + T C )∆ − α . Iterating, we obtain that sup ≤ t ≤ T | x t | ≤ K sup ≤ t ≤ ( n − | x t | + K ≤ · · · ≤ K n − sup ≤ t ≤ ∆ | x t | + K n − X i =0 K i . We next choose ∆ such that C ∆ − α ≤ and C − ≤ . Then, C − ≤ . Moreover, wechoose ∆ such that T C C − ∆ − α ≤ . This implies that K ≤ . Thus, n − X i =0 K i ≤ n − X i =0 (cid:18) (cid:19) i = 37 (cid:18) (cid:19) n − ≤ e n − . In order to bound K , it suffices to choose ∆ such that C ∆ − α ≤ . Then, we easilyobtain that K ≤ . We finally bound sup ≤ t ≤ ∆ | x t | using (3.23). Again we choose ∆ such that (1 − B ) − ≤ and B ∆ − α ≤ so that B − ≤ . We also choose ∆ such that ∆ − α B ≤ so that sup ≤ t ≤ ∆ | x t | ≤ | x | + 2 · ·
14 = 2 | x | + 34 < | x | + 1 . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 13
Finally, we conclude that sup t ∈ [0 ,T ] | x t | ≤ (2 | x | + 1) e n − ≤ ( | x | + 1) e [ T ∆ ] , which implies the desired estimate choosing ∆ such that ∆ = 1( C α,β k g k − α (1 + T ) + 6( L + L (1 + T ) + B ,α )) / (1 − α ) ∧ ∧ T. (cid:3) The next result provides a supremum norm estimate of the solution z t of the followingsystem of equations x t = x + Z t b ( t, r, x r ) dr + Z t σ ( t, r, x r ) dg r z t = w t + Z t h ( t, r, x r ) z r dr + Z t f ( t, r, x r ) z r dg r , (3.27)where g belongs to W − α (0 , T ; R m ) , w belongs to C − α (0 , T ; R d ) , b : [0 , T ] × R d → R d , σ : [0 , T ] × R d → R d × R m , h : [0 , T ] × R d → R d × R d , and f : [0 , T ] × R d → R d × R m are measurable functions, and x ∈ R d .We will use the following hypotheses on h, f and w : (H3) h is Lipschitz continuous with respect to t and bounded .f is bounded and satisfies (H1) .w is Lipschitz continuous and bounded . Theorem 3.6.
Assume that b and σ satisfy the hypotheses of Theorem 3.3 and that h, f and w satisfy hypothesis (H3) . Then there exists a unique solution z ∈ C − α (0 , T ; R d ) toequation (3.27) . Moreover, there exists a constant C α,β > such that k z k ∞ ≤ k w k ∞ ) exp (cid:18) T (cid:16) K (5) T,α + K (6) T,α,β k g k − α (cid:17) / (1 − α ) ∨ ∨ T (cid:19) , where K (5) T,α = 16 ( K + k h k ∞ + L + L + B ,α ) e T ( T + 1) and K (6) T,α,β = C α,β ( k f k ∞ + k σ k ∞ + 1) e T ( T + 1) .Proof. The existence and uniqueness of the solution follows similarly as [14, Theorem 5.1].We next prove the estimate of the supremum norm of the solution. We divide the interval [0 , T ] into n = [ T / ˜∆] + 1 subintervals, where ˜∆ ≤ will be chosen below. Step 1.
We first estimate k z k , ˜∆ , ∞ . Let t, t ′ ∈ [0 , ˜∆] with t < t ′ . We write | z t ′ − z t | ≤ | w t ′ − w t | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t ′ t h ( t ′ , r, x r ) z r dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t ( h ( t ′ , r, x r ) − h ( t, r, x r )) z r dr (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t ′ t f ( t ′ , r, x r ) z r dg r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t ( f ( t ′ , r, x r ) − f ( t, r, x r )) z r dg r (cid:12)(cid:12)(cid:12)(cid:12) = E + F + G + H + I. The first three terms are easily bounded as E ≤ K ( t ′ − t ) ,F ≤ k h k ∞ ( t ′ − t ) k z k t,t ′ , ∞ , and G ≤ L ( t ′ − t ) t k z k t,t ′ , ∞ . We next bound H and I . Using (2.2) and the estimate in [11, (3.5)], we get H ≤ K k g k − α Z t ′ t (cid:12)(cid:12) D αt + [ f ( t ′ , · , x · ) z · ] ( r ) (cid:12)(cid:12) dr, where | D αt + [ f ( t ′ , · , x · ) z · ] ( r ) |≤ − α ) (cid:18) | f ( t ′ , r, x r ) z r | ( r − t ) α + α Z rt | f ( t ′ , r, x r ) z r − f ( t ′ , u, x u ) z u | ( r − u ) α +1 du (cid:19) ≤ C α ( H + H ) ,H ≤ C α k f k ∞ k z k t,t ′ , ∞ ( r − t ) − α , and H ≤ C α,β k z k t,t ′ , ∞ ( r − t ) β − α + C α,β k z k t,t ′ , ∞ k x k t,t ′ , − α ( r − t ) − α + C α,β k f k ∞ k z k t,t ′ , − α ( r − t ) − α . Therefore, we obtain H ≤ C α,β ( t ′ − t ) − α k g k − α (cid:2) k z k t,t ′ , ∞ (cid:0) k f k ∞ + ( t ′ − t ) β + k x k t,t ′ , − α ( t ′ − t ) − α (cid:1) + k f k ∞ k z k t,t ′ , − α k ( t ′ − t ) − α (cid:3) . Similarly, I ≤ C α,β k g k − α ( t ′ − t ) t − α (cid:2) k z k t,t ′ , ∞ (cid:0) t β + k x k t,t ′ , − α t − α (cid:1) + k z k t,t ′ , − α t − α (cid:3) . Hence, we conclude that | z t ′ − z t | ( t ′ − t ) − α ≤ K + D k z k t,t ′ , ∞ + D k z k t,t ′ , − α , where D = k h k ∞ + L + C α,β k g k − α (cid:16) k f k ∞ + 1 + k x k , ˜∆ , − α ˜∆ − α (cid:17) ,D = C α,β k g k − α ( k f k ∞ + 1) ˜∆ − α . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 15
Thus, k z k , ˜∆ , − α ≤ (1 − D ) − ( K + D k z k , ˜∆ , ∞ ) . Moreover, k z k , ˜∆ , ∞ ≤ k w k ∞ + ˜∆ − α (cid:2) (1 − D ) − ( K + D k z k , ˜∆ , ∞ ) (cid:3) . Choosing ˜∆ satisfying (3.12), we obtain by (3.11) that k x k , ˜∆ , − α ˜∆ − α ≤ ≤ . We nextchoose ˜∆ such that ˜∆ − α K ≤ , ˜∆ − α D ≤ , and D ≤ . Then, we obtain that k z k , ˜∆ , ∞ ≤ k w k ∞ + 1 . (3.28) Step 2.
We next estimate k z k i ˜∆ , ( i +1) ˜∆ , ∞ for i = 1 , . . . , n . Fix t, t ′ ∈ [ i ˜∆ , ( i + 1) ˜∆] with t < t ′ . Similar bounds can be obtained for the corresponding terms E, F, G and H as inStep 1. Thus, we just need to bound the term I i := I , that is, I i ≤ i X ℓ =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ℓ ˜∆( ℓ −
1) ˜∆ ( f ( t ′ , r, x ) − f ( t, r, x r )) z r dg r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ti ˜∆ ( f ( t ′ , r, x ) − f ( t, r, x r )) z r dg r (cid:12)(cid:12)(cid:12)(cid:12) . Following the same computations as for I , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ℓ ˜∆( ℓ −
1) ˜∆ ( f ( t ′ , r, x ) − f ( t, r, x r )) z r dg r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C α,β k g k − α ( t ′ − t ) ˜∆ h k z k ℓ −
1) ˜∆ ,ℓ ˜∆ , − α ˜∆ − α i . + k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , ∞ (cid:16) − α + k x k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , − α ˜∆ − α (cid:17)i . Therefore, the term I i is bounded by C α,β k g k − α ( t ′ − t ) ˜∆ − α (cid:20) k z k i ˜∆ , ( i +1) ˜∆ , ∞ (cid:16) k x k i ˜∆ , ( i +1) ˜∆ , − α ˜∆ − α (cid:17) + k z k i ˜∆ , ( i +1) ˜∆ , − α ˜∆ − α + i X ℓ =1 h k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , ∞ (cid:16) k x k ( l −
1) ˜∆ ,ℓ ˜∆ , − α ˜∆ − α (cid:17) + k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , − α ˜∆ − α i(cid:21) . Hence, we obtain that k z k i ˜∆ , ( i +1) ˜∆ , − α ≤ K + E k z k i ˜∆ , ( i +1) ˜∆ , − α + E i k z k i ˜∆ , ( i +1) ˜∆ , ∞ + i X ℓ =1 h E ℓ k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , ∞ ) + E k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , − α i . where E = C α k g k − α ( k f k ∞ + 1) ˜∆ − α ,E i = k h k ∞ + L + C α,β k g k − α (cid:16) k f k ∞ + 1 + k x k i ˜∆ , ( i +1) ˜∆ , − α ˜∆ − α (cid:17) ,E ℓ = C α,β k g k − α ˜∆ (cid:16) k x k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , − α ˜∆ − α (cid:17) ,E = C α,β k g k − α ˜∆ − α . Choosing ˜∆ such that E ≤ , we obtain that k z k i ˜∆ , ( i +1) ˜∆ , − α ≤ K + 2 E i k z k i ˜∆ , ( i +1) ˜∆ , ∞ + i X ℓ =1 h E ℓ k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , ∞ ) + 2 E k z k ( ℓ −
1) ˜∆ ,ℓ ˜∆ , − α i . (3.29)Choosing ˜∆ satisfying (3.18), we obtain by the Step 3 in Theorem 3.3 that for all ℓ =1 , . . . , i , k x k ℓ ˜∆ , ( ℓ +1) ˜∆ , − α ˜∆ − α ≤ . Thus, E i ≤ E := k h k ∞ + L + C α,β k g k − α ( k f k ∞ + 1) E ℓ ≤ E := C α,β k g k − α ˜∆ . Applying expression (3.29) recurrently we obtain that k z k i ˜∆ , ( i +1) ˜∆ , − α ≤ K (1 + 2 E ) i − + 2 E k z k i ˜∆ , ( i +1) ˜∆ , ∞ + (2 E + 4 E E ) i − X ℓ =1 (1 + 2 E ) ℓ − k z k ( i − ℓ ) ˜∆ , ( i − ( ℓ − , ∞ . This implies that k z k i ˜∆ , ( i +1) ˜∆ , ∞ ≤ | z i ˜∆ | + k z k i ˜∆ , ( i +1) ˜∆ , − α ˜∆ − α ≤ E − | z i ˜∆ | + K i + E − (2 E + 4 E E )(1 + 2 E ) i − i k z k ,i ˜∆ , ∞ ˜∆ − α , where E = 1 − E ˜∆ − α and K i = E − E (1 + 2 E ) i − ˜∆ − α . Step 3.
Using the result of Step 2 yields that sup t ∈ [0 , ( i +1) ˜∆] | z t | ≤ L i sup t ∈ [0 ,i ˜∆] | z t | + K i , (3.30)where L i = E − (cid:16) − α (2 E + 4 E E )(1 + 2 E ) i − i (cid:17) .We finally bound L i and K i . We choose ˜∆ such that E ˜∆ − α ≤ , so that E − ≤ .We also choose ˜∆ such that E ≤ ˜∆ so that (1 + 2 E ) i − ≤ (1 + ˜∆) i − ≤ (1 + ˜∆) n − ≤ (1 + ˜∆) T/ ˜∆ ≤ e T . Hence, choosing ˜∆ such that E e T ˜∆ − α ≤ we conclude that K i ≤ . Moreover, as i ˜∆ ≤ T , we have that L i ≤ (cid:16) (cid:16) ˜∆ − α C α,β k g k − α + 2 ˜∆ − α E (cid:17) e T T (cid:17) . We finally choose ˜∆ such that that ˜∆ − α C α,β k g k − α e T T ≤ and − α E e T T ≤ , sothat L i ≤ e .Iterating (3.30) and using (3.28), we conclude that sup ≤ t ≤ T | z t | ≤ e sup ≤ t ≤ ( n −
1) ˜∆ | z t | + 1 ≤ · · · ≤ e n − sup ≤ t ≤ ˜∆ | z t | + n − X i =0 e i ≤ e [ T/ ˜∆] ( k w k ∞ + 1) , which implies the desired result. (cid:3) RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 17
We end this section by showing the Fréchet differentiability of the solution to thedeterministic equation (3.1), which extends [15, Lemma 3 and Proposition 4].
Lemma 3.7.
Assume the hypotheses of Theorem . Assume that b ( t, s, · ) , σ ( t, s, · ) be-long to C b for all s, t ∈ [0 , T ] and that the partial derivatives of b and σ satisfy ( H2 ) and ( H1 ) , respectively. Then the mapping F : W − α (0 , T ; R m ) × W α (0 , T ; R d ) → W α (0 , T ; R d ) defined by ( h, x ) → F ( h, x ) := x − x − Z · b ( · , s, x s ) ds − Z · σ ( · , s, x s ) d ( g s + h s ) (3.31) is Fréchet differentiable. Moreover, for any ( h, x ) ∈ W − α (0 , T ; R m ) × W α (0 , T ; R d ) , k ∈ W − α (0 , T ; R m ) , v ∈ W α (0 , T ; R d ) , and i = 1 , . . . , d , the Fréchet derivatives withrespect to h and x are given respectively by D F ( h, x )( k ) it = − m X j =1 Z t σ i,j ( t, s, x s ) dk js , (3.32) D F ( h, x )( v ) it = v it − d X k =1 Z t ∂ x k b i ( t, s, x s ) v ks ds − d X k =1 m X j =1 Z t ∂ x k σ i,j ( t, s, x s ) v ks d ( g js + h js ) . (3.33) Proof.
For ( h, x ) and (˜ h, ˜ x ) in W − α (0 , T ; R m ) × W α (0 , T ; R d ) we have F ( h, x ) t − F (˜ h, ˜ x ) t = x t − ˜ x t − Z t ( b ( t, s, x s ) − b ( t, s, ˜ x s )) ds − Z t ( σ ( t, s, x s ) − σ ( t, s, ˜ x s )) d ( g s + h s ) − Z t σ ( t, s, ˜ x s ) d ( h s − ˜ h s ) . Using [5, Proposition 2.2(2)], we get that (cid:13)(cid:13)(cid:13)(cid:13) x − ˜ x − Z · ( b ( · , s, x s ) − b ( · , s, ˜ x s )) ds (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,T k x − ˜ x k α, . From [5, Proposition 3.2(2)], we obtain (cid:13)(cid:13)(cid:13)(cid:13) Z · ( σ ( · , s, x s ) − σ ( · , s, ˜ x s )) d ( g s + h s ) (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,T k x − ˜ x k α, k g + h k − α, (1 + ∆( x ) + ∆(˜ x )) , where ∆( x ) := sup u ∈ [0 ,T ] Z u | x u − x s | δ ( u − s ) α +1 ds ≤ c α,δ,T k x k δ − α and similarly ∆(˜ x ) ≤ c α,δ,T k ˜ x k δ − α . Finally, [5, Proposition 3.2(1)] yields to (cid:13)(cid:13)(cid:13)(cid:13) Z · σ ( · , s, ˜ x s ) d ( h s − ˜ h s ) (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,T (1 + k ˜ x k α, ) k h − ˜ h k − α, . Therefore, F is continuous in both variables ( h, x ) . We next show the Fréchet differentia-bility. Let v, w ∈ W α (0 , T ; R d ) . By [5, Proposition 2.2(2) and 3.2(2)], we have that k D F ( h, x )( v ) − D F ( h, x )( w ) k α, ≤ c α,T k v − w k α, (1 + k g + h k − α, ) . Thus, D F ( h, x ) is a bounded linear operator. Moreover, F ( h, x + v ) t − F ( h, x ) t − D F ( h, x )( v ) t = Z t ( b ( t, s, x s ) − b ( t, s, x s + v s ) + ∂ x b ( t, s, x s ) v s ) ds + Z t ( σ ( t, s, x s ) − σ ( t, s, x s + v s ) + ∂ x σ ( t, s, x s ) v s ) d ( g s + h s ) . By the mean value theorem and [5, Proposition 2.2(2)], (cid:13)(cid:13)(cid:13)(cid:13) Z · ( b ( · , s, x s ) − b ( · , s, x s + v s ) + ∂ x b ( · , s, x s ) v s ) ds (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,T k v k α, . Similarly, using [5, Proposition 3.2(2)], we obtain (cid:13)(cid:13)(cid:13)(cid:13) Z · ( σ ( · , s, x s ) − σ ( · , s, x s + v s ) + ∂ x σ ( · , s, x s ) v s ) d ( g s + h s ) (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,δ,T k v k α, k g + h k − α, . This shows that D F is the Fréchet derivative with respect to x of F ( h, x ) . Similarly,we show that it is Fréchet differentiable with respect to h and the derivative is given by(3.32). (cid:3) Proposition 3.8.
Assume the hypotheses of Lemma . Then the mapping g ∈ W − α (0 , T ; R m ) → x ( g ) ∈ W α (0 , T ; R d ) is Fréchet differentiable and for any h ∈ W − α (0 , T ; R m ) the derivative in the direction h is given by D h x it = m X j =1 Z t Φ ijt ( s ) dh js , where for i = 1 , . . . , d , j = 1 , . . . , m , ≤ s ≤ t Φ ijt ( s ) = σ ij ( t, s, x s ) + d X k =1 m X ℓ =1 Z ts ∂ x k σ i,ℓ ( t, u, x u )Φ kju ( s ) dg ℓu + d X k =1 Z ts ∂ x k b i ( t, u, x u )Φ kju ( s ) du, (3.34) and Φ ijt ( s ) = 0 if s > t .Proof. The proof follows similarly as the proof of [15, Proposition 4] once we have extended[15, Proposition 2 and 9]. We proceed with both extensions below. (cid:3)
The next propositions are the extensions of [15, Proposition 2 and 9], respectively.
RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 19
Proposition 3.9.
Assume the hypotheses of Lemma . Fix g ∈ W − α (0 , T ; R m ) andconsider the linear equation v t = w t + Z t ∂ x b ( t, s, x s ) v s ds + Z t ∂ x σ ( t, s, x s ) v s dg s . where w ∈ C − α (0 , T ; R d ) . Then there exists a unique solution v ∈ C − α (0 , T ; R d ) suchthat k v k α, ≤ c (1) α,T k w k α, exp (cid:16) c (2) α,T k g k / (1 − α )1 − α, (cid:17) , (3.35) for some positive constants c (1) α,T and c (2) α,T .Proof. Existence and uniqueness follows from [5] and the estimate (3.35) follows from [5,Proposition 4.2] with γ = 1 . (cid:3) Proposition 3.10.
Assume the hypotheses of Lemma . Then the solution to thelinear equation (3.34) is Hölder continuous of order − α in t , uniformly in s and Höldercontinuous of order β ∧ (1 − α ) in s , uniformly in t .Proof. By the estimates in [5], we get sup s ∈ [0 ,T ] k Φ · ( s ) k − α ≤ c α,T (1 + (1 + k g k − α, ) sup s ∈ [0 ,T ] k Φ · ( s ) k α, ) . which is bounded by Proposition 3.9. Therefore, Φ t ( s ) is Hölder continuous of order − α in t , uniformly in s . On the other hand, appealing again to Proposition 3.9, for s ′ ≤ s ≤ t ,we have k Φ · ( s ) − Φ · ( s ′ ) k α, ≤ c (1) α,T k w · ( s, s ′ ) k α, exp (cid:16) c (2) α,T k g k / (1 − α )1 − α, (cid:17) , where w t ( s, s ′ ) = σ ( t, s, x s ) − σ ( t, s ′ , x s ′ ) + Z ss ′ ∂ x σ ( t, u, x u )Φ u ( s ′ ) dg u + Z ss ′ ∂ x b ( t, u, x u )Φ u ( s ′ ) du. We next bound the k · k α, -norm of w · ( s, s ′ ) . For the first term, by the definition of the k · k α, -norm, we have k σ ( · , s, x s ) − σ ( · , s ′ , x s ′ ) k α, ≤ c α,T k σ ( · , s, x s ) − σ ( · , s ′ , x s ′ ) k − α ≤ c α,β,T ( s − s ′ ) β ∧ (1 − α ) , where we have used [5, Lemma A.2] in the last inequality.For the second term, as ∂ x σ is bounded, we obtain (cid:13)(cid:13)(cid:13)(cid:13) Z ss ′ ∂ x σ ( · , u, x u )Φ u ( s ′ ) dg u (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,T (cid:12)(cid:12)(cid:12)(cid:12) Z ss ′ Φ u ( s ′ ) dg u (cid:12)(cid:12)(cid:12)(cid:12) ≤ c α,T ( s − s ′ ) − α k g k − α, sup s ∈ [0 ,T ] k Φ · ( s ) k α, , where the last inequality follows from [14, Proposition 4.1]. Finally, for the last term, as ∂ x b is bounded, we get (cid:13)(cid:13)(cid:13)(cid:13) Z ss ′ ∂ x b ( t, u, x u )Φ u ( s ′ ) du (cid:13)(cid:13)(cid:13)(cid:13) α, ≤ c α,T (cid:12)(cid:12)(cid:12)(cid:12) Z ss ′ Φ u ( s ′ ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ c α,T ( s − s ′ ) sup s ∈ [0 ,T ] k Φ · ( s ) k α, . Therefore, we conclude that k Φ · ( s ) − Φ · ( s ′ ) k α, ≤ c α,β,T ( s − s ′ ) β ∧ (1 − α ) exp (cid:16) c (2) α,T k g k / (1 − α )1 − α, (cid:17) , which implies that Φ t ( s ) is Hölder continuous of order β ∧ (1 − α ) in s uniformly in t . (cid:3) Stochastic Volterra equations driven by fBm
In this section we apply the results obtained in Section 3 to the Volterra equation (1.1).Recall that W H = { W Ht , t ∈ [0 , T ] } is an m -dimensional fractional Brownian motion withHurst parameter H > . That is, a centered Gaussian process with covariance function E ( W H,it W H,jt ) = R H ( t, s ) = 12 (cid:0) t H + s H − | t − s | H (cid:1) δ ij . Fix α ∈ (1 − H, ) . As the trajectories of W H are (1 − α + ǫ ) -Hölder continuous for all ǫ < H + α − , by the first inclusion in (2.1), we can apply the framework of Section 3.In particular, under the assumptions of Theorem 3.5, there exists a unique solution toequation (1.1) satisfying sup ≤ t ≤ T | X t | ≤ ( | X | + 1) exp (cid:18) T (cid:18)(cid:16) K (3) T,α + K (4) T,α,β k W H k − α (cid:17) / (1 − α ) ∨ ∨ T (cid:19)(cid:19) . Moreover, under the further assumptions of Theorem 3.3, we have the estimate sup ≤ t ≤ T | X t | ≤ | X | + 1 + T (cid:18)(cid:16) K (1) T,α + K (2) T,α,β k W H k − α (cid:17) / (1 − α ) ∨ ∨ T (cid:19) . As a consequence of these estimates we can establish the following integrability prop-erties of the solution to (1.1).
Theorem 4.1.
Assume that E ( | X | p ) < ∞ for all p ≥ and that σ and b satisfy thehypotheses of Theorem 3.5. Then for all p ≥ E (cid:18) sup ≤ t ≤ T | X t | p (cid:19) < ∞ . Moreover, if for any λ > and γ < H , E (exp( λ | X | γ )) < ∞ , then under the assumptionsof Theorem 3.3, we have E (cid:18) exp (cid:18) λ (cid:18) sup ≤ t ≤ T | X t | γ (cid:19)(cid:19)(cid:19) < ∞ , for any λ > and γ < H . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 21
We next proceed with the study of the existence and smoothness of the density of thesolution to (1.1). From now on we assume that the initial condition is constant, that is, X = x ∈ R d . We start by extending the results in [15] in order to show the existenceof the density of the solution to the Volterra equation (1.1) when σ is not necessarilybounded. We first derive the (local) Malliavin differentiability of the solution. Theorem 4.2.
Assume the hypotheses of Lemma 3.7. Then the solution to (1.1) isalmost surely differentiable in the directions of the Cameron-Martin space. Moreover,for any t > , X it belongs to the space D , loc and the derivative satisfies for i = 1 , . . . , d , j = 1 , . . . , m , D js X it = σ ij ( t, s, X s ) + d X k =1 m X ℓ =1 Z ts ∂ x k σ iℓ ( t, r, X r ) D js X kr dW H,ℓr + d X k =1 Z ts ∂ x k b i ( t, r, X r ) D js X kr dr, (4.1) if s ≤ t and 0 if s > t .Proof. By Proposition 3.8, the mapping ω ∈ W − α (0 , T ; R m ) → X ( ω ) ∈ W α (0 , T ; R d ) is Fréchet differentiable and for all ϕ ∈ H and i = 1 , . . . , d , the Fréchet derivative D R H ϕ X it = ddǫ X it ( ω + ǫ R H ϕ ) | ǫ =0 exists, which proves the first statement of the theorem. Moreover, by [12, Proposition4.1.3.], this implies that for any t > X it belongs to the space D , loc .The derivative D R H ϕ X it coincides with h DX it , ϕ i H , where D is the usual Malliavinderivative. Furthermore, by Proposition 3.8, for any ϕ ∈ H and i = 1 , . . . , d , D R H ϕ X it = m X j =1 Z t Φ ijt ( s ) d ( R H ϕ ) j ( s )= m X j =1 Z t Φ ijt ( s ) (cid:18)Z s ∂ s K H ( s, u )( K ∗ H ϕ ) j ( u ) du (cid:19) ds = m X j =1 Z T ( K ∗ H Φ it ) j ( s )( K ∗ H ϕ ) j ( s ) ds = h Φ it , ϕ i H and equation (4.1) follows from (3.34). This concludes the proof. (cid:3) We next derive the existence of the density.
Theorem 4.3.
Assume the hypotheses of Lemma 3.7. Assume also the following nonde-generacy condition on σ : for all s, t ∈ [0 , T ] , the vector space spanned by { ( σ j ( t, s, x ) , . . . , σ dj ( t, s, x )) , ≤ j ≤ m } is R d . Then, for any t > the law of the random vector X t is absolutely continuous withrespect to the Lebesgue measure on R d .Proof. By Theorem 4.2 and [12, Theorem 2.1.2] it suffices to show that the Malliavinmatrix Γ t of X t defined by Γ ijt = h DX it , DX jt i H is invertible a.s., which follows along the same lines as in the proof of [15, Theorem 8]. (cid:3) We finally consider the case that σ is bounded and show the existence and smoothnessof the density. As before, we first study the Malliavin differentiability of the solution. Theorem 4.4.
Assume the hypotheses of Theorem 3.3, that b i ( t, s, · ) , σ i,j ( t, s, · ) belongto C ∞ b for all s, t ∈ [0 , T ] and that the partial derivatives of all orders of b and σ satisfy ( H2 ) and ( H1 ) respectively. Then for any t > , X it belongs to the space D ∞ and the n thiterated derivative satisfies the following equation for i = 1 , . . . , d , j , . . . , j n ∈ { , . . . , m } , D j s · · · D j n s n X it = n X q =1 D j s · · · ˇ D j q s q · · · D j n s n σ ij ℓ ( t, s ℓ , X s ℓ )+ m X ℓ =1 Z ts ∨···∨ s n D j s · · · D j n s n σ iℓ ( t, r, X r ) dW H,ℓr + Z ts ∨···∨ s n D j s · · · D j n s n b i ( t, r, X r ) dr, (4.2) if s ∨ · · · ∨ s n ≤ t and 0 otherwise. The notation ˇ D j q s q means that the factor D j q s q is omittedin the sum. When n = 1 this equation coincides with (4.1) .Proof. By Theorem 4.2, for any t > X it belongs to D , loc and the Malliavin derivativesatisfies (4.1). Applying Theorem 3.6 to the system formed by equations (1.1) and (4.1)we obtain that a.s. sup s,t ∈ [0 ,T ] | D js X it | ≤ k σ k ∞ + 1) exp (cid:18) T (cid:16) K (5) T,α + K (6) T,α,β k W H k − α (cid:17) / (1 − α ) ∨ ∨ T (cid:19) , (4.3)which implies that for all p ≥ , sup t ∈ [0 ,T ] E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =1 Z t Z t D js X it D jr X it | r − s | H − dsdr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! < ∞ . This and [12, Lemma 4.1.2] show that the random variable X it belongs to the Sobolevspace D ,p for all p ≥ . Similarly, it can be proved that X it belongs to the Sobolev space D k,p for all p, k ≥ . For the sake of conciseness, we only sketch the main steps. First, byinduction, following exactly along the same lines as in the proofs of [15, Proposition 5 andLemma 10] and Proposition 3.8, it can be shown that the deterministic mapping x definedin Section 3 is infinitely differentiable. Second, by a similar argument as in the proof ofTheorem 4.2, we have that for all t > , X it is almost surely infinitely differentiable in thedirections of the Cameron-Martin space and it belongs to the space D k,p loc for all p, k ≥ . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 23
Finally, using equation (4.2), the estimate for linear equations obtained in Theorem 3.6and an induction argument, we obtain that for all k, p ≥ , sup t ∈ [0 ,T ] E (cid:0) k D ( k ) X t k p H ⊗ k (cid:1) < ∞ , where D ( k ) denotes the k th iterated derivative. This concludes the desired claim. (cid:3) The next theorem extends and corrects the proof of [11, Theorem 7] as there is a mistakein the last step of the proof.
Theorem 4.5.
Assume the hypotheses of Theorem 4.4 and that σ ( t, s, · ) is uniformlyelliptic, that is, for all s, t ∈ [0 , T ] , x, ξ ∈ R d with | ξ | = 1 , m X j =1 (cid:0) d X i =1 σ ij ( t, s, x ) ξ i (cid:1) ≥ ρ > , for some ρ > . Then for any t > the probability law of X t has an C ∞ density.Proof. By [13, Theorem 7.2.6] it suffices to show that
E(( det (Γ t )) − p ) < ∞ for all p > .We write det (Γ t ) ≥ inf | ξ | =1 ( ξ T Γ t ξ ) d . Fix ξ ∈ R d with k ξ k = 1 and ǫ ∈ (0 , . Then ξ T Γ t ξ = k d X i =1 DX it ξ i k H = k d X i =1 K ∗ H ( DX it ) ξ i k L (0 ,t ; R m ) ≥ k d X i =1 K ∗ H ( DX it ) ξ i k L ( t − ǫ,t ; R m ) ≥ A − B, where A := m X j =1 Z tt − ǫ d X i,k =1 Z ts Z ts σ ij ( t, u, X u ) σ kj ( t, v, X v ) ∂ u K H ( u, s ) ∂ v K H ( v, s ) ξ i ξ k dudvds,B := m X j =1 Z tt − ǫ (cid:18) d X i =1 Z ts (cid:18) d X k =1 m X ℓ =1 Z tu ∂ x k σ iℓ ( t, r, X r ) D ju X kr dW H,ℓr + d X k =1 Z tu ∂ x k b i ( t, r, X r ) D ju X kr dr (cid:19) ∂ u K H ( u, s ) ξ i du (cid:19) ds. We next we add and substract the term σ ij ( t, u, X u ) σ kj ( t, u, X u ) inside A to obtain that A = A + A , where A := m X j =1 Z tt − ǫ Z ts Z ts d X i =1 σ ij ( t, u, X u ) ξ i ! ∂ u K H ( u, s ) ∂ v K H ( v, s ) dudvds,A := m X j =1 Z tt − ǫ d X i,k =1 Z ts Z ts σ ij ( t, u, X u ) (cid:0) σ kj ( t, v, X v ) − σ kj ( t, u, X u ) (cid:1) × ∂ u K H ( u, s ) ∂ v K H ( v, s ) ξ i ξ k dudvds. By the uniform ellipticity property, we get that A ≥ ρ Z tt − ǫ (cid:18)Z ts ∂ u K H ( u, s ) du (cid:19) ds = c H Z tt − ǫ (cid:18)Z ts ( us ) H − / ( u − s ) H − du (cid:19) ds ≥ c H Z tt − ǫ (cid:18)Z ts ( u − s ) H − du (cid:19) ds = c H ǫ H . Moreover, since σ is bounded, using Hölder’s inequality and hypothesis (H1) , for any q ≥ , we get that E | A | q ≤ C T,q ǫ q − Z tt − ǫ Z tt − ǫ Z tt − ǫ (cid:0) E( | X u − X v | q ) + E( | X u | q ) | u − v | βq + E( | X u | q | X u − X v | δq ) (cid:1) ( ∂ u K H ( u, s ) ∂ v K H ( v, s )) q dudvds ≤ C T,q ǫ q − Z tt − ǫ Z tt − ǫ Z tt − ǫ (cid:0) | u − v | (1 − α ) q + | u − v | βq + | u − v | δq (1 − α ) (cid:1) × ( ∂ u K H ( u, s ) ∂ v K H ( v, s )) q dudvds ≤ C T,q ǫ q (2 H +min { − α,β,δ (1 − α ) } . We are left to bound E | B | q . Since ∂ x b ( t, s, x ) is bounded, using Hölder’s inequality and(4.3), we obtain that for all q ≥ , E (cid:12)(cid:12)(cid:12)(cid:12) m X j =1 Z tt − ǫ (cid:18) d X i,k =1 Z ts Z tu ∂ x k b i ( t, r, X r ) D ju X kr ∂ u K H ( u, s ) ξ i drdu (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12) q ≤ C α,T,q ǫ q − Z tt − ǫ Z tt − ǫ ( ∂ u K H ( u, s )) q duds ≤ C α,T,q ǫ q (2 H +2) . Similarly, for all q ≥ , we have that E (cid:12)(cid:12)(cid:12)(cid:12) m X j =1 Z tt − ǫ (cid:18) d X i,k =1 m X ℓ =1 Z ts (cid:18)Z tu ∂ x k σ iℓ ( t, r, X r ) D ju X kr dW H,ℓr (cid:19) ∂ u K H ( u, s ) ξ i du (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12) q ≤ C α,T,q ǫ q − − α )2 q Z tt − ǫ Z tt − ǫ ( ∂ u K H ( u, s )) q duds ≤ C α,T,q ǫ q (2 H +2(1 − α )) . RACTIONAL STOCHASTIC INTEGRAL VOLTERRA EQUATIONS 25
Appealing to [6, Proposition 3.5] we conclude the desired result. (cid:3)
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Mireia Besalú, Dep. Genètica, Microbiologia i Estadística, Universitat de Barcelona.Diagonal, 645, 08028 Barcelona
E-mail address : [email protected] David Márquez-Carreras, Facultat de Matemàtiques i Informàtica, Universitat de Barcelona.Gran Via de les Corts Catalanes, 585, 08007 Barcelona
E-mail address : [email protected] Eulalia Nualart, Universitat Pompeu Fabra and Barcelona Graduate School of Eco-nomics, Ramón Trias Fargas 25-27, 08005 Barcelona, Spain.
E-mail address ::