Stochastic Volterra equations driven by cylindrical Wiener process

0.18 MB·English

by  Anna Karczewska

Stochastic Volterra equations driven by
7 cylindrical Wiener process
0
0
2
Anna Karczewska and Carlos Lizama
n
a
J
Department of Mathematics, University of Zielona G´ora
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1
ul. Szafrana 4a, 65-246 Zielona G´ora, Poland, e-mail: [email protected]

]
R
Universidad de Santiago de Chile, Departamento de Matem´atica, Facultad de Ciencias
P
. Casilla 307-Correo 2, Santiago, Chile, e-mail: [email protected]
h
t
a
m Abstract
[

In this paper, stochastic Volterra equations driven by cylindrical Wiener pro-
2
v cess in Hilbert space are investigated. Sufficient conditions for existence of strong
1 solutions are given. The key role is played by convergence of α-times resolvent
4
2 families.
0
1
6
0 1 Introduction
/
h
t
a Let H be a separable Hilbert space with a norm |·| and A be a closed linear unbounded
m H
operator with dense domain D(A) ⊂ H equipped with the graph norm | · | . The
: D(A)
v
purpose of this paper is to study the existence of strong solutions for a class of stochastic
i
X
Volterra equations of the form
r
a
t t
X(t) = X + a(t−τ)AX(τ)dτ + Ψ(τ)dW(τ), t ≥ 0, (1)
0
Z0 Z0
tα−1
wherea(t) = , α > 0,andW,Ψareappropriatestochasticprocesses. Itiswellknown
Γ(α)
that there are several situations that can be modeled by stochastic Volterra equations (see
e.g. [7, Section 3.4 ] and references therein). We note that stochastic Volterra equations
driven by white noise have been studied in [3] among other authors. A similar equation
and very related to our case appears first studied in [2]. Here we are interested in the
study of strong solutions when equation (1) is driven by a cylindrical Wiener process W.
2000 Mathematics Subject Classification: primary: 60H20; secondary: 60H05, 45D05.
Keywords and phrases: stochasticVolterraequation,α-timesresolventfamily,strongsolution,stochastic
convolution, convergence of resolvent families.
1
Whena(t)isacompletely positivefunction, sufficient conditionsforexistence ofstrong
solutions for (1) were obtained in [9]. This was done using a method which involves the
use of a resolvent family associated to the deterministic version of equation (1):
t
u(t) = a(t−τ)Au(τ)dτ +f(t), t ≥ 0, (2)
Z0
where f is an H-valued function.
However, there are two kinds of problems that arise when we study (1). On the
one hand, the kernels tα−1 are α-regular and απ -sectorial but not completely positive
Γ(α) 2
functions for α > 1, so e.g. the results in [9] cannot be used directly for α > 1. On the
other hand, for α ∈ (0,1), we have a singularity of the kernel in t = 0. This fact strongly
suggests the use of α-times resolvent families associated to equation (2). This new tool
appears carefully studied in [1] as well as their relationship with fractional derivatives.
For convenience of the reader, we provide below the main results on α-times resolvent
families to be used in this paper.
Our second main ingredient to obtain strong solutions of (1) relies on approximation
of α-times resolvent families. This kind of result was very recently formulated by Li
and Zheng [10]. It enables us to prove a key result on convergence of α-times resolvent
families (see Theorem 2 below). Then we canfollowthe methods employed in [9]to obtain
existence of strong solution for the stochastic equation (1) (see Theorem 4).
Our plan for the paper is the following. In section 2 we formulate the deterministic
results which will play the key role for the paper. Section 3 is devoted to weak and mild
solutions while in section 4 we provide strong solution to (1). More precisely, we give
sufficient condition for a stochastic convolution to be a strong solution to (1).
2 Convergence of α-times resolvent families
In this section we formulate the main deterministic results on convergence of resolvents.
We denote
tα−1
g (t) = , α > 0, t > 0,
α
Γ(α)
where Γ is the gamma function.
By S (t), t ≥ 0, we denote the family of α-times resolvent families corresponding to
α
the Volterra equation (2), if it exists, and defined as follows.
Definition 1 (see [1])
A family (S (t)) of bounded linear operators in a Banach space B is called α-times
α t≥0
resolvent family for (2) if the following conditions are satisfied:
1. S (t) is strongly continuous on R and S (0) = I;
α + α
2
2. S (t) commutes with the operator A, that is, S (t)(D(A)) ⊂ D(A) and AS (t)x =
α α α
S (t)Ax for all x ∈ D(A) and t ≥ 0;
α
3. the following resolvent equation holds
t
S (t)x = x+ g (t−τ)AS (τ)xdτ (3)
α α α
Z0
for all x ∈ D(A), t ≥ 0.
Necessary and sufficient conditions for existence of the α-times resolvent family have
been studied in [1]. Observe that the α-times resolvent family corresponds to a C -
0
semigroup in case α = 1 and a cosine family in case α = 2. In consequence, when
1 < α < 2 such resolvent families interpolate C -semigroups and cosine functions. In
0
particular, for A = ∆, the integrodifferential equation corresponding to such resolvent
family interpolates the heat equation and the wave equation (see [6]).
Definition 2 An α-times resolvent family (S (t)) is called exponentially bounded
α t≥0
if there are constants M ≥ 1 and ω ≥ 0 such that
kS (t)k ≤ Meωt, t ≥ 0. (4)
α
If there is the α-times resolvent family (S (t)) for A and satisfying (4), we write
α t≥0
A ∈ Cα(M,ω). Also, set Cα(ω) := ∪ Cα(M,ω) and Cα := ∪ Cα(ω).
M≥1 ω≥0
Remark 1 It was proved by Bazhlekova [1, Theorem 2.6] that if A ∈ Cα for some α > 2,
then A is bounded.
The following subordination principle is very important in the theory of α-times re-
solvent families (see [1, Theorem 3.1]).
Theorem 1 Let 0 < α < β ≤ 2,γ = α/β,ω ≥ 0. If A ∈ Cβ(ω) then A ∈ Cα(ω1/γ) and
the following representation holds
∞
S (t)x = ϕ (s)S (s)xds, t > 0, (5)
α t,γ β
Z0
where ϕ (s) := t−γΦ (st−γ) and Φ (z) is the Wright function defined as
t,γ γ γ
∞
(−z)n
Φ (z) := , 0 < γ < 1. (6)
γ
n!Γ(−γn+1−γ)
n=0
X
3
Remark 2 (i)WerecallthattheLaplacetransformoftheWrightfunctioncorrespondsto
E (−z) where E denotes theMittag-Leffler function. Inparticular, Φ (z) isa probability
γ γ γ
density function.
(ii) Also we recall from [1, (2.9)] that the continuity in t ≥ 0 of the Mittag-Leffler
function together with the asymptotic behavior of it, imply that for ω ≥ 0 there exists a
constant C > 0 such that
E (ωtα) ≤ Ceω1/αt, t ≥ 0, α ∈ (0,2). (7)
α
As we have already written, in this paper the results concerning convergence of α-
times resolvent families in a Banach space B will play the key role. Using a very recent
result due to Li and Zheng [10] we are able to prove the following theorem.
Theorem 2 Let A be the generator of a C -semigroup (T(t)) in a Banach space B
0 t≥0
such that
kT(t)k ≤ Meωt, t ≥ 0. (8)
Then, for each 0 < α < 1 we have A ∈ Cα(M,ω1/α). Moreover, there exist bounded opera-
tors A and α-times resolvent families S (t) for A satisfying ||S (t)|| ≤ MCe(2ω)1/αt,
n α,n n α,n
for all t ≥ 0, n ∈ N, and
S (t)x → S (t)x as n → +∞ (9)
α,n α
for all x ∈ B, t ≥ 0. Moreover, the convergence is uniform in t on every compact subset
of R .
+
Proof Since A is the generator of a C semigroup satisfying (8), we have A ∈ C1(ω).
0
Hence, the first assertion follows directly from Theorem 1, that is, for each 0 < α < 1
there is an α-times resolvent family (S (t)) for A given by
α t≥0
∞
S (t)x = ϕ (s)T(s)xds, t > 0. (10)
α t,α
Z0
Since A generates a C -semigroup, the resolvent set ρ(A) of A contains the ray [w,∞)
0
and
M
||R(λ,A)k|| ≤ for λ > w, k ∈ N.
(λ−w)k
Define
A := nAR(n,A) = n2R(n,A)−nI, n > w, (11)
n
the Yosida approximation of A.
Then
∞
n2ktk
||etAn|| = e−nt||en2R(n,A)t|| ≤ e−nt ||R(n,A)k||
k!
k=0
X
2
≤ Me(−n+nn−w)t = Menn−wwt .
4
Hence, for n > 2w we obtain
||eAnt|| ≤ Me2wt. (12)
Next, since each A is bounded, it follows also from Theorem 1 that for each 0 < α < 1
n
there exists an α-times resolvent family (S (t)) for A given as
α,n t≥0 n
∞
S (t) = ϕ (s)esAnds, t > 0. (13)
α,n t,α
Z0
By (12) and Remark 2(i) it follows that
∞
kS (t)k ≤ ϕ (s)kesAnkds
α,n t,α
Z0
∞ ∞
≤ M ϕ (s)e2ωsds = M Φ (τ)e2ωtατdτ = ME (2ωtα), t ≥ 0.
t,α α α
Z0 Z0
This together with Remark 2(ii), gives
kS (t)k ≤ MCe(2ω)1/αt, t ≥ 0. (14)
α,n
Now, we recall the fact that R(λ,A )x → R(λ,A)x as n → ∞ for all λ sufficiently
n
large (see e.g. [11, Lemma 7.3]), so we can conclude from [10, Theorem 4.2] that
S (t)x → S (t)x as n → +∞ (15)
α,n α
for all x ∈ B, uniformly for t on every compact subset of R .
+
An analogous result can be proved in the case when A is the generator of a strongly
continuous cosine family.
Theorem 3 Let A be the generator of a C -cosine family (T(t)) in a Banach space B.
0 t≥0
Then, for each 0 < α < 2 we have A ∈ Cα(M,ω2/α). Moreover, there exist bounded opera-
tors A and α-times resolvent families S (t) for A satisfying ||S (t)|| ≤ MCe(2ω)1/αt,
n α,n n α,n
for all t ≥ 0, n ∈ N, and
S (t)x → S (t)x as n → +∞
α,n α
for all x ∈ B, t ≥ 0. Moreover, the convergence is uniform in t on every compact subset
of R .
+
In the following, we denote by Σ (ω) the open sector with vertex ω ∈ R and opening
θ
angle 2θ in the complex plane which is symmetric with respect to the real positive axis,
i.e.
Σ (ω) := {λ ∈ C : |arg(λ−ω)| < θ}.
θ
5
We recall from [1, Definition 2.13] that an α-times resolvent family S (t) is called
α
analytic if S (t) admits an analytic extension to a sector Σ for some θ ∈ (0,π/2]. An
α θ0 0
α-times analytic resolvent family is said to be of analyticity type (θ ,ω ) if for each
0 0
θ < θ and ω > ω there is M = M(θ,ω) such that
0 0
kS (t)k ≤ MeωRet, t ∈ Σ .
α θ
The set of all operators A ∈ Cα generating α-times analytic resolvent families S (t) of
α
type (θ ,ω ) is denoted by Aα(θ ,ω ). In addition, denote Aα(θ ) := {Aα(θ ,ω );ω ∈
0 0 0 0 0 0 0 0
R }, Aα := {Aα(θ );θ ∈ (0,π/2]}. For α = 1 we obtain the set of all generators of
+ 0 0 S
analytic semigroups.
S
Remark 3 We note that the spatial regularity condition R(S (t)) ⊂ D(A) for all t > 0
α
is satisfied by α-times resolvent families whose generator A belongs to the set Aα(θ ,ω )
0 0
where 0 < α < 2 (see [1, Proposition 2.15]). In particular, setting ω = 0 we have that
0
A ∈ Aα(θ ,0) if and only if −A is a positive operator with spectral angle less or equal to
0
π −α(π/2+θ). Note that such condition is also equivalent to the following
Σ ⊂ ρ(A) and kλ(λI −A)−1k ≤ M, λ ∈ Σ . (16)
α(π/2+θ) α(π/2+θ)
The above considerations give us the following remarkable corollary.
Corollary 1 Suppose A generates an analytic semigroup of angle π/2 and α ∈ (0,1).
Then A generates an α-times analytic resolvent family.
Proof Since A generates an analytic semigroup of angle π/2 we have
kλ(λI −A)−1k ≤ M, λ ∈ Σ .
π−ǫ
Thenthecondition(16)(seealso[1,Corollary2.16])impliesA ∈ Aα(min{2−απ, 1π},0),
2α 2
α ∈ (0,2), that is A generates an α-times analytic resolvent family.
In the sequel we will use the following assumptions concerning Volterra equations:
(A1) A is the generator of C -semigroup and α ∈ (0,1); or
0
(A2) A is the generator of a strongly continuous cosine family and α ∈ (0,2).
Observe that (A2) implies (A1) but not vice versa.
6
3 Weak vs. mild solutions
Assume that H and U are separable Hilbert spaces. Let the cylindrical Wiener process W
be defined on a stochastic basis (Ω,F,(F) ,P), with the positive symmetric covariance
t≥0
operator Q ∈ L(U). This is known that the process W takes values in some superspace
of U. (For more details concerning cylindrical Wiener process we refer to [4] or [8].)
We define the subspace U := Q1/2(U) of the space U, endowed with the inner product
0
hu,vi := hQ−1/2u,Q−1/2vi . The set L0 := L (U ,H) of all Hilbert-Schmidt operators
U0 U 2 2 0
fromU intoH,equippedwiththenorm|C| := ( +∞|Cf |2 )1/2,where{f } ⊂ U
0 L2(U0,H) k=1 k H k 0
is an orthonormal basis of U , is a separable Hilbert space. We assume that Ψ belongs to
0 P
the class of measurable L0-valued processes.
2
By N2(0,T;L0) we denote a Hilbert space of all L0-predictable processes Ψ such that
2 2
||Ψ|| < +∞, where
T
1
T 2
||Ψ|| := E |Ψ(τ)|2 dτ
T L02
(cid:26) (cid:18)Z0 (cid:19)(cid:27)
1
T 2
= E Tr(Ψ(τ)Q21)(Ψ(τ)Q12)∗ dτ .
(cid:26) Z0 h i (cid:27)
We shall use the following Probability Assumptions (abbr. (PA)):
1. X is an H-valued, F -measurable random variable;
0 0
2. Ψ ∈ N2(0,T;L0) and the interval [0,T] is fixed.
2
Definition 3 Assume that (PA) hold. An H-valued predictable process X(t), t ∈ [0,T],
is said to be a strong solution to (1), if X takes values in D(A), P-a.s.,
t
for any t ∈ [0,T], |g (t−τ)AX(τ)| dτ < +∞, P −a.s., α > 0, (17)
α H
Z0
and for any t ∈ [0,T] the equation (1) holds P-a.s.
Let A∗ denote the adjoint of A with a dense domain D(A∗) ⊂ H and the graph norm
|·|D(A∗).
Definition 4 Let (PA) hold. An H-valued predictable process X(t), t ∈ [0,T], is said to
t
be a weak solution to (1), if P( |g (t−τ)X(τ)| dτ < +∞) = 1, α > 0, and if for all
0 α H
ξ ∈ D(A∗) and all t ∈ [0,T] the following equation holds
R
t t
hX(t),ξi = hX ,ξi +h g (t−τ)X(τ)dτ,A∗ξi +h Ψ(τ)dW(τ),ξi , P−a.s.
H 0 H α H H
Z0 Z0
7
Definition 5 Assume that X is F -measurable random variable. An H-valued pre-
0 0
dictable process X(t), t ∈ [0,T], is said to be a mild solution to the stochastic Volterra
equation (1), if E( t|S (t−τ)Ψ(τ)|2 dτ) < +∞, α > 0, for t ≤ T and, for arbitrary
0 α L02
t ∈ [0,T],
R
t
X(t) = S (t)X + S (t−τ)Ψ(τ)dW(τ), P −a.s. (18)
α 0 α
Z0
where S (t) is the α-times resolvent family.
α
We will use the following result.
Proposition 1 (see, e.g. [4, Proposition 4.15])
Assume that A is a closed linear unbounded operator with the dense domain D(A) ⊂ H.
Let Φ(t),t ∈ [0,T], be an L (U ,H)-predictable process. If Φ(t) ∈ D(A), P − a.s. for
2 0
all t ∈ [0,T] and
T T
P |Φ(s)|2 ds < ∞ = 1, P |AΦ(s)|2 ds < ∞ = 1,
L02 L02
(cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19)
T
then P Φ(s)dW(s) ∈ D(A) = 1 and
(cid:18)Z0 (cid:19)
T T
A Φ(s)dW(s) = AΦ(s)dW(s), P −a.s.
Z0 Z0
We define the stochastic convolution
t
WΨ(t) := S (t−τ)Ψ(τ)dW(τ), (19)
α α
Z0
where Ψ ∈ N2(0,T;L0). Because α-times resolvent families S (t), t ≥ 0, are bounded,
2 α
then S (t−·)Ψ(·) ∈ N2(0,T;L0), too.
α 2
Analogously like in [8], we can formulate the following result.
Proposition 2 Assume that S (t),t ≥ 0, are the resolvent operators to (2). Then, for
α
any process Ψ ∈ N2(0,T;L0), the convolution WΨ(t), t ≥ 0, α > 0, given by (19) has a
2 α
predictable version. Additionally, the process WΨ(t), t ≥ 0, α > 0, has square integrable
α
trajectories.
Under some conditions a mild solution to Volterra equations is a weak solution and
vice versa, see [8, Propositions 4 and 5].
Now, we can prove that a mild solution to the equation (1) is a weak solution to (1).
8
Proposition 3 If Ψ ∈ N2(0,T;L0) and Ψ(·,·)(U ) ⊂ D(A), P-a.s., then the stochastic
2 0
convolution WΨ(t), t ≥ 0, α > 0, given by (19), fulfills the equation
α
t t
hWΨ(t),ξi = hg (t−τ)WΨ(τ),A∗ξi + hξ,Ψ(τ)dW(τ)i , α ∈ (0,2), (20)
α H α α H H
Z0 Z0
for any t ∈ [0,T] and ξ ∈ D(A∗).
Proof Let us notice that the process WΨ has integrable trajectories. For any ξ ∈ D(A∗)
α
we have
t
hg (t−τ)WΨ(τ),A∗ξi dτ ≡ (from (19))
α α H
Z0
t τ
≡ hg (t−τ) S (τ −σ)Ψ(σ)dW(σ),A∗ξi dτ =
α α H
Z0 Z0
(from Dirichlet’s formula and stochastic Fubini’s theorem)
t t
= h g (t−τ)S (τ −σ)dτ Ψ(σ)dW(σ),A∗ξi
α α H
Z0 (cid:20)Zσ (cid:21)
t t−σ
= h g (t−σ −z)S (z)dz Ψ(σ)dW(σ),A∗ξi
α α H
Z0 (cid:20)Z0 (cid:21)
(where z := τ −σ and from definition of convolution)
t
= h A[(g ⋆S )(t−σ)]Ψ(σ)dW(σ),ξi =
α α H
Z0
(from the resolvent equation (3) because A(g ⋆S )(t−σ)x = (S (t−σ)−I)x,
α α α
where x ∈ D(A))
t
= h [S (t−σ)−I]Ψ(σ)dW(σ),ξi =
α H
Z0
t t
= h S (t−σ)Ψ(σ)dW(σ),ξi − h Ψ(σ)dW(σ),ξi .
α H H
Z0 Z0
Hence, we obtained the following equation
t t
hWΨ(t),ξi = hg (t−τ)WΨ(τ),A∗ξi dτ + hξ,Ψ(τ)dW(τ)i
α H α α H H
Z0 Z0
for any ξ ∈ D(A∗).
Immediately from the equation (20) we deduce the following result.
Corollary 2 If A is a bounded operator and Ψ ∈ N2(0,T;L0), then the following equality
2
holds
t t
WΨ(t) = g (t−τ)AWΨ(τ)dτ + Ψ(τ)dW(τ), (21)
α α α
Z0 Z0
for t ∈ [0,T], α > 0.
Remark 4 The formula (21) says that the convolution WΨ(t), t ≥ 0, α > 0, is a strong
α
solution to (1) with X ≡ 0 if the operator A is bounded.
0
9
4 Strong solutions
In this section we provide sufficient conditions under which the stochastic convolution
WΨ(t), t ≥ 0, α > 0, defined by (19) is a strong solution to the equation (1).
α
Lemma 1 Let A be a closed linear unbounded operator with dense domain D(A) equipped
with the graph norm |·| . Assume that (A1) or (A2) holds. If Ψ and AΨ belong to
D(A)
N2(0,T;L0) and in addition Ψ(·,·)(U ) ⊂ D(A), P-a.s., then (21) holds.
2 0
Proof Because formula (21) holds for any bounded operator, then it holds for the Yosida
approximation A of the operator A, too, that is
n
t t
WΨ (t) = g (t−τ)A WΨ (τ)dτ + Ψ(τ)dW(τ),
α,n α n α,n
Z0 Z0
where
t
WΨ (t) := S (t−τ)Ψ(τ)dW(τ).
α,n α,n
Z0
By Proposition 1, we have
t
A WΨ (t) = A S (t−τ)Ψ(τ)dW(τ).
n α,n n α,n
Z0
By assumption Ψ ∈ N2(0,T;L0). Because the operators S (t) are deterministic and
2 α,n
bounded for any t ∈ [0,T], α > 0, n ∈ N, then the operators S (t −·)Ψ(·) belong to
α,n
N2(0,T;L0), too. In consequence, the difference
2
Φ (t−·) := S (t−·)Ψ(·)−S (t−·)Ψ(·) (22)
α,n α,n α
belongs to N2(0,T;L0) for any t ∈ [0,T], α > 0 and n ∈ N. This means that
2
t
E |Φ (t−τ)|2 dτ < +∞ (23)
α,n L02
(cid:18)Z0 (cid:19)
for any t ∈ [0,T].
Let us recall that the cylindrical Wiener process W(t), t ≥ 0, can be written in the
form
+∞
W(t) = f β (t), (24)
j j
j=1
X
where {f } is an orthonormal basis of U and β (t) are independent real Wiener processes.
j 0 j
From (24) we have
t +∞ t
Φ (t−τ)dW(τ) = Φ (t−τ)f dβ (τ). (25)
α,n α,n j j
Z0 j=1 Z0
X
10