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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 I. Iscoe, M. B. Marcus, D. McDonald, M. Talagrand, and J. Zinn, Continuity of l 2 -Valued Ornstein-Uhlenbeck Processes, Ann. Probab. 18 (1990), 68–84. A. Karczewska, Regularity of solutions to stochastic Volterra equations of convolution type, Integral Transforms Spec. Funct. 20 (2009), 171–176. A. Karczewska, Convolution Type Stochastic Volterra Equations, Lecture Notes in Nonlinear Analysis, 10. Juliusz Schauder Center for Nonlinear Studies, Toruń, 2007. A. Karczewska, and J. Zabczyk, Regularity of solutions to stochastic Volterra equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), 141–154. Time regularity of the solution to a stochastic Volterra equation by dilation theorem References P. Kotelenez, A maximal inequality for stochastic convolution integrals on Hilbert space and space-time regularity of linear stochastic partial differential equations, Stochastics 21 (1987), 345–458. Y. Liu, and J. Zhai, A note on time regularity of generalized Ornstein–Uhlenbeck process with cylindrical stable noise, C. R. Acad. Sci. Paris 350 (2012), 97–100. S. Nagy, and C. Foias, Harmonic Analysis of Operators on Hilbert Spaces, 1970, North Holland, Amsterdam. S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, 2007, Cambridge University Press, Cambridge. Time regularity of the solution to a stochastic Volterra equation by dilation theorem References S. Peszat and J. Zabczyk, Time regularity of solutions to linear equations with Lévy noise in infinite dimensions, Stochastic Processes Appl. 123 (2013), 719–751. S. Peszat, J. Zabczyk, Time regularity for stochastic Volterra equations by the dilation theorem, J. Math. Anal. Appl. 409 (2014), 676–683. J. Prüss, Evolutionary Integral Equations and Applications, 1993, Birkhaüser Verlag, Basel.