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Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Time regularity of the solution to a
stochastic Volterra equation by dilation
theorem
Szymon Peszat
Institute of Mathematics, Jagiellonian University, Kraków
Stochastic Processes and Differential Equations in Infinite
Dimensional Spaces, King’s College, London 2014
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
The talk is based on the joint work: Szymon Peszat and Jerzy
Zabczyk, Time regularity for stochastic Volterra equations by
the dilation theorem, J. Math. Anal. Appl. 409 (2014),
676–683.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Consider stochastic Volterra equation
Z t
X (t) = X (0) +
v (t − s)AX (s)ds + L(t),
X (0) = x0 ,
0
where (A, D(A)) is a closed densely defined linera operator on
a Hilbert space H, v : [0, +∞) 7→ R is a locally integrable
function, and L is a semimartingale in H.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Our aim is to study path regularity of X (càdlàg, continuity)
as well as maximal inequalities which enables us to study
non-linear problem where L(t) = F (X (t)) + σ(X (t))Z (t).
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v ≡ 1 and A generates a C0 -semigroup S, then
Z t
X (t) = X (0) +
AX (s)ds + L(t)
0
dX = AX dt + dL,
X (0) = x0 .
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v ≡ 1 and A generates a C0 -semigroup S, then
Z t
X (t) = X (0) +
AX (s)ds + L(t)
0
dX = AX dt + dL,
X (0) = x0 .
The weak solution
Z t
hX (t), hiH = hx0 , hiH + hX (s), A∗ hiH +hL(t), hiH , h ∈ D(A∗ ),
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v ≡ 1 and A generates a C0 -semigroup S, then
Z t
X (t) = X (0) +
AX (s)ds + L(t)
0
dX = AX dt + dL,
X (0) = x0 .
The weak solution
Z t
hX (t), hiH = hx0 , hiH + hX (s), A∗ hiH +hL(t), hiH , h ∈ D(A∗ ),
0
is given by
Z
t
S(t − s)dL(s).
X (t) = S(t)x0 +
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
X is usually not a semimartingale. The fact that t appears in
the integrant plays an important role.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
X is usually not a semimartingale. The fact that t appears in
the integrant plays an important role.
1)
0 such that
R t There is f ∈ C ([0, +∞)), f 6≡
∞
f (t − s)dB(s), t ≥ 0, has C , trajectories.
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
X is usually not a semimartingale. The fact that t appears in
the integrant plays an important role.
1)
0 such that
R t There is f ∈ C ([0, +∞)), f 6≡
∞
f (t − s)dB(s), t ≥ 0, has C , trajectories.
0
Rt
2) For a “typical” f ∈ C (S 1 ), 0 f (t − s)dB(s), t ≥ 0, has
unbounded trajectories on each interval,
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
X is usually not a semimartingale. The fact that t appears in
the integrant plays an important role.
1)
0 such that
R t There is f ∈ C ([0, +∞)), f 6≡
∞
f (t − s)dB(s), t ≥ 0, has C , trajectories.
0
Rt
2) For a “typical” f ∈ C (S 1 ), 0 f (t − s)dB(s), t ≥ 0, has
unbounded trajectories on each interval, Z Brzezniak, S.P.,
and J. Zabczyk 2002.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If S is a C0 -group, then
Z
X (t) = S(t)x0 + S(t)
S(−s)dL(s),
0
càdlàg (cont. ) follows.
t
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
The group assumption can be relaxed. On can assume that S
is a generalised contraction semigroup kS(t)k ≤ e ωt , ω ∈ R,
see P. Kotelenez 1987, or E. Hausenblas and J. Seidler 2001.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
The group assumption can be relaxed. On can assume that S
is a generalised contraction semigroup kS(t)k ≤ e ωt , ω ∈ R,
see P. Kotelenez 1987, or E. Hausenblas and J. Seidler 2001.
By the Riesz and Nagy dilation theorem S(t) = PT (t), where
T is a C0 -group on a Hilbert space H such that H ,→ H, and
P is an orthogonal projection.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
The group assumption can be relaxed. On can assume that S
is a generalised contraction semigroup kS(t)k ≤ e ωt , ω ∈ R,
see P. Kotelenez 1987, or E. Hausenblas and J. Seidler 2001.
By the Riesz and Nagy dilation theorem S(t) = PT (t), where
T is a C0 -group on a Hilbert space H such that H ,→ H, and
P is an orthogonal projection. Thus
Z t
X (t) = PT (t)
T (−s)dL(s).
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
The case of general semigroup is open except the case where L
is a Wiener process. Then one can apply Da Prato, Kwapien,
Zabczyk factorisation, or Kolmogorov test.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Da Prato–Kwapień–Zabczyk Factorisation
Z
t
S(t − s)dL(s) = πIA,α (Yα )(t),
0
where IA,α is the Liouville–Riemann operator
Z t
1
(t − s)α−1 S(t − s)ψ(s)ds,
IA,α ψ(t) =
Γ(α) 0
and
1
Yα (t) :=
Γ(1 − α)
Z
t
(t − s)−α S(t − s)dL(s).
0
Then IA,α maps Lq (0, T ; H) into C ([0, T ]; H) if 1/q < α. It is
enough to show that Yα has trajectories in Lq (0, T ; H).
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
One can use Kolmogorov test (for continuity) or N.N.
Chentsov (for càdlàg) modification.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
One can use Kolmogorov test (for continuity) or N.N.
Chentsov (for càdlàg) modification.
E |X (t) − X (t − h)|pH |X (t) − X (t + h)|pH ≤ Kh1+r .
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
One can use Kolmogorov test (for continuity) or N.N.
Chentsov (for càdlàg) modification.
E |X (t) − X (t − h)|pH |X (t) − X (t + h)|pH ≤ Kh1+r .
The existence of càdlàg modification to linear SPDE with
cylindrical Lévy noise, see: S.P. and J. Zabczyk 2013, Y. Liu,
Y. and J. Zhai 2012, and Z. Brzezniak, B. Goldys, P. Imkeller,
S. Peszat, E. Priola, and J. Zabczyk 2010.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
For Volterra
Z
t
v (t − s)AX (s)ds + L(t)
X (t) = x0 +
0
do we have mild formulation?
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
For Volterra
Z
t
v (t − s)AX (s)ds + L(t)
X (t) = x0 +
0
do we have mild formulation?
Z
t
R(t − s)dL(s).
X (t) = R(t)x0 +
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Definition
A family R(t), t ≥ 0, of bounded linear operators on H is
called resolvent to the equation
Z t
y (t) = f (t) +
v (t − s)Ay (s)ds,
t ∈ [0, T ],
0
if the following conditions are satisfied
(i) R is strongly continuous on R+ = [0, +∞) and R(0)
equals the identity operator I ,
(ii) R(t)D(A) ⊂ D(A) and AR(t)x = R(t)Ax for all t ≥ 0
and x ∈ D(A),
(iii) for all x ∈ D(A),
Z t
R(t)x = x +
v (t − s)AR(s)ds,
t ≥ 0.
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Proposition
If L is a semimartingale with càdlàg trajectories in H, then the
weak solution exists and is given by the formula
Z t
X (t) = R(t)x0 +
R(t − s)dL(s),
t ∈ [0, T ].
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
We would like to know if R(t) = ΠT (t) where T is a C0 -group
on H ←- H and Π : H 7→ H is a continuous projection.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
We would like to know if R(t) = ΠT (t) where T is a C0 -group
on H ←- H and Π : H 7→ H is a continuous projection. Then
Z t
X (t) = R(t)x0 + ΠT (t)
T (−s)dL(s).
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Definition
We say that a family R(t), t ∈ R, of bounded linear operators
on a Hilbert space (H, h·, ·iH ) is positive definite if for any
finite sequences (tj ) in R, and (ψj ) in H,
X
hR(tj − tk )ψj , ψk iH ≥ 0.
j,k
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Definition
We say that a family R(t), t ∈ R, of bounded linear operators
on a Hilbert space (H, h·, ·iH ) is positive definite if for any
finite sequences (tj ) in R, and (ψj ) in H,
X
hR(tj − tk )ψj , ψk iH ≥ 0.
j,k
We say that the family R is strongly continuous if for any
ψ ∈ H and t ∈ R,
lim |R(s)ψ − R(t)ψ|H = 0.
s→t
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Nagy delation theorem
Assume that R is a strongly continuous positive definite family
of bounded operators on a Hilbert space H, such that R(0)
equals the identity operator I . Then there exist: a Hilbert
space H containing isometrically H and a strongly continuous
unitary group T (t), t ∈ R, on H, such that
R(t)ψ = ΠT (t)ψ,
t ≥ 0, ψ ∈ H,
where Π is the orthogonal projection of H onto H.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Any C0 -semigroup S of contractions is positive definite.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Any C0 -semigroup S of contractions is positive definite. Thus
any generalised contraction C0 semigroup S can be written as
S(t) = Πe ωt T (t),
T is a C0 -unitary group.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Any C0 -semigroup S of contractions is positive definite. Thus
any generalised contraction C0 semigroup S can be written as
S(t) = Πe ωt T (t),
T is a C0 -unitary group.
Under which condition e −ωt R(t), t ≥ 0, is positive definite?
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Any C0 -semigroup S of contractions is positive definite. Thus
any generalised contraction C0 semigroup S can be written as
S(t) = Πe ωt T (t),
T is a C0 -unitary group.
Under which condition e −ωt R(t), t ≥ 0, is positive definite?
Then
R(t) = eωt ΠT (t),
t ≥ 0.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
S.P. and J. Zabczyk
Assume that A is a self-adjoint negative definite operator on a
Hilbert space H and that v is non-increasing then the
resolvent R exists and e−ωt R(t) is positive definite for some ω.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
S.P. and J. Zabczyk
Assume that A is a self-adjoint negative definite operator on a
Hilbert space H and that v is non-increasing then the
resolvent R exists and e−ωt R(t) is positive definite for some ω.
Consequently, if L is càdlàg (or continuous) semimartingale in
H, then the weak solution to VE has a càdlàg (resp.
continuous) modification.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Z
R(t) = s(t; −A) =
s(t; −µ)E (d µ)
σ(A)
E is the spectral measure,
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Z
R(t) = s(t; −A) =
s(t; −µ)E (d µ)
σ(A)
E is the spectral measure, s(t; µ), t ≥ 0, is scalar resolvent
Z t
s(t; µ) + µ
v (t − τ )s(τ ; µ)d τ = 1,
t ≥ 0.
0
t 7→ e −ωt R(|t|) is positive definite if t 7→ e −ω|t| s(|t|; µ) is
positive definite for µ ∈ σ(A).
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Z
R(t) = s(t; −A) =
s(t; −µ)E (d µ)
σ(A)
E is the spectral measure, s(t; µ), t ≥ 0, is scalar resolvent
Z t
s(t; µ) + µ
v (t − τ )s(τ ; µ)d τ = 1,
t ≥ 0.
0
t 7→ e −ωt R(|t|) is positive definite if t 7→ e −ω|t| s(|t|; µ) is
positive definite for µ ∈ σ(A). The later holds if
Z +∞
ω+µ
e −ωt (ω cos βt + β sin βt) v (t)dt ≥ 0,
∀ β ≥ 0.
0
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v is non-increasing then the condition is satisfied.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v (0) > 0 and t 7→ v 0 (|t|) is positive definite then the
condition is satisfied.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v (0) > 0 and t 7→ v 0 (|t|) is positive definite then the
condition is satisfied. In particular v 0 (t) ≥ 0, v 00 (t) ≤ 0 and
v 000 (t) ≥ 0.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v (0) > 0 and t 7→ v 0 (|t|) is positive definite then the
condition is satisfied. In particular v 0 (t) ≥ 0, v 00 (t) ≤ 0 and
v 000 (t) ≥ 0.
Example
Let v (t) = t, t ≥ 0. Then v 0 ≡ 1 is positive definite. This is
√
the classical case leading to s(t; µ) = cos( µt).
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
If v (0) > 0 and t 7→ v 0 (|t|) is positive definite then the
condition is satisfied. In particular v 0 (t) ≥ 0, v 00 (t) ≤ 0 and
v 000 (t) ≥ 0.
Example
Let v (t) = t, t ≥ 0. Then v 0 ≡ 1 is positive definite. This is
√
the classical case leading to s(t; µ) = cos( µt).
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Example
Let a, b > 0, and let
(
a − ba t, t ∈ [0, b]
v 0 (t) =
0,
t ≥ b.
Then v 0 (t) ≥ 0 is non-increasing and concave. Note that
(
a 2
v (0) + at − 2b
t , t ∈ [0, b]
v (t) =
ab
t ≥ b.
v (0) + 2 ,
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
Example
Let α ∈ (0, 1), and let
v (t) = t −α ,
t > 0.
Then v is locally integrable, strictly decreasing and
positive.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
References
References
Z. Brzezniak, S. Peszat, J. Zabczyk, Continuity of
stochastic convolutions, Czechoslovak Math. J. 51 (2001),
679–684.
Z. Brzeźniak, B. Goldys, P. Imkeller, S. Peszat, E. Priola,
and J. Zabczyk, Time irregularity of generalized
Ornstein–Uhlenbeck processes, C. R. Math. Acad. Sci.
Paris 348 (2010), 273–276.
G. Da Prato, S. Kwapien, and J. Zabczyk, Regularity of
solutions of linear stochastic equations in Hilbert spaces,
Stochastics 23 (1987), 1–23.
E. Hausenblas, and J. Seidler, A note on maximal
inequality for stochastic convolutions, Czechoslovak Math.
Time regularity of the solution to a stochastic Volterra equation by dilation theorem
References
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Time regularity of the solution to a stochastic Volterra equation by dilation theorem
References
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Time regularity of the solution to a stochastic Volterra equation by dilation theorem
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