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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
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A New Inequality for Stochastic ConvolutionIntegrals
Erfan Salavati
(in a joint work with Bijan Z. Zangeneh)Sharif University of Technology
Tehran, Iran
7th International Conference on Stochastic Analysis, Seoul,2014
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Outline
...1 Motivation: Stochastic Evolution Equations
...2 Maximal Inequalities
...3 Itö Type Inequalities
...4 Applications
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Semilinear Stochastic Evolution Equations
Let H be a Hilbert space and St : H → H a C0-semigroupof contractions with generator A.
dX (t) = AX (t)dt + f (t ,X (t))dt + g(X−t )dMt
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Semilinear Stochastic Evolution Equations
Let H be a Hilbert space and St : H → H a C0-semigroupof contractions with generator A.
dX (t) = AX (t)dt + f (t ,X (t))dt + g(X−t )dMt
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Example
Consider this semilinear SPDE on a region in D ⊂ Rd
∂tu(t , x) = ∆u(t , x) + f (x ,u) + g(x ,u)Wt
H = L2(D) A = ∆ S(t) = et∆ D(A) = H2(D)
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Example
Consider this semilinear SPDE on a region in D ⊂ Rd
∂tu(t , x) = ∆u(t , x) + f (x ,u) + g(x ,u)Wt
H = L2(D) A = ∆ S(t) = et∆ D(A) = H2(D)
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Stochastic Convolution Integrals
.Definition..
......
By a mild solution of equation (*) we mean a cadlag adaptedprocess X (t) in H which satisfies
X (t) = StX0 +
∫ t
0St−sf (X (s))ds
+
∫ t
0St−sg(X (s−))dMs
Integrals of the form∫ t
0 St−sdMs are called StochasticConvolution Integrals.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Lipschitz case
.Theorem (Existence and uniqueness, Lipschitz case)..
......If f and g are Lipschitz operators then equation (*) has a uniquemild solution with initial value X0.
See Kotelenez, Da Prato and Zabczyk,...
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Method of proof
X (t)− Y (t) =
∫ t
0St−s(f (X (s))− f (Y (s)))ds
+
∫ t
0St−s(g(X (s−))− g(Y (s−)))dMs
E sup0≤t≤T
∥X (t)− Y (t)∥2 ≤ C∫ T
0E∥f (X (s))− f (Y (s))∥2ds
+ C∫ T
0E∥g(X (s))− g(Y (s))∥2ds
≤ C′∫ T
0E∥X (s)− Y (s)∥2ds
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Maximal Inequalities for Stochastic Convolutions
.Theorem (Kotelenez (’84), Ichikawa (’84))..
......
Let M(t) be an H-valued cadlag locally square integrablemartingale. Then
E sup0≤t≤T
∥∫ t
0S(t − s)dM(s)∥2 ≤ CE[M](T )
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Maximal Inequalities for Stochastic Convolutions
.Theorem (Burkholder Type Inequality, Zangeneh (’95))..
......
Let p ≥ 2 and T > 0. Let St be a contraction semigroup on Hand Mt be an H-valued square integrable càdlàg martingale.Then
E sup0≤t≤T
∥∫ t
0St−sdMs∥p ≤ CpE([M]
p2T )
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Itö Type Inequalities: Motivation
Many interesting operators on Hilbert spaces are notLipschitz, e.g the evaluation operators on function spaces,
F (u)(x) = f (u(x))
.A more general class of operators are semi-monotoneoperators.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Itö Type Inequalities: Motivation
Many interesting operators on Hilbert spaces are notLipschitz, e.g the evaluation operators on function spaces,
F (u)(x) = f (u(x))
.A more general class of operators are semi-monotoneoperators.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.Definition..
......
f : D(f ) ⊂ H → H (not necessarily linear) is said to bemonotone if
∀x , y ∈ D(f ), ⟨f (x)− f (y), x − y⟩ ≤ 0
and is said to be semi-monotone if for a real constant M,
∀x , y ∈ D(f ), ⟨f (x)− f (y), x − y⟩ ≤ M∥x − y∥2.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Stochastic Equations with monotone non-linearity
dX (t) = AX (t)dt + f (t ,X (t))dt + g(t ,X (t))dM(t)
E∥f (X (t))− f (Y (t))∥2 ≤?
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Stochastic Equations with monotone non-linearity
dX (t) = AX (t)dt + f (t ,X (t))dt + g(t ,X (t))dM(t)
E∥f (X (t))− f (Y (t))∥2 ≤?
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Literature
Deterministic equations: Browder (’64), Kato (’64)Stochastic equations using the variational approach:Krylov and Rozovski (’81), Pardoux (’75) for Wiener noiseand Gyongy and Krylov (’82) for general martingale noiseStochastic equations using the semigroup approach:Zangeneh (’95) for Wiener noise, Marinelli and Röckner(’10) for Poisson noise.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.Theorem (Browder (’64), Kato (’64))..
......
Let f (t , x) : H → H be demi-continuous and semi-monotone,then the equation
dx(t)dt
= Ax(t) + f (x(t))
with initial condition x0 has a unique mild solution in H.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.Theorem (Zangeneh (’95))..
......
Let f : H → H be demi-continuous and semi-monotone and gbe Lipschitz, then the equation
dX (t) = AX (t)dt + f (t ,X (t))dt + g(t ,X (t))dW (t)
with initial condition X0 has a unique mild solution in H.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Itô Type Inequality for second power
.Theorem (Zangeneh (’95))..
......
Let M(t) be an H-valued square integrable martingale and
X (t) =∫ t
0S(t − s)dM(s)
then
∥X (t)∥2 ≤∫ t
0⟨X (s−),dM(s)⟩+ [M](t)
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Itô Type Inequality for pth power
.Theorem (S., Zangeneh (’13))..
......
Let p ≥ 2 and M(t) be an H-valued square integrable
semi-martingale where E[M]p2T < ∞. Let
X (t) =∫ t
0 S(t − s)dM(s), then
∥X (t)∥p ≤ p∫ t
0∥X (s−)∥p−2⟨X (s−),dM(s)⟩
+12
p(p − 1)∫ t
0∥X (s−)∥p−2d [M]c(s)
+∑
0≤s≤t
(∥X (s)∥p − ∥X (s−)∥p − p∥X (s−)∥p−2⟨X (s−),∆X (s)⟩
)
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Stochastic Evolution Equations with Lévy Noise
Let H be a Hilbert space. We consider the problem
dX (t) = AX (t)dt + f (X (t))dt + g(X (t))dW (t)
+
∫E
k(ξ,X (t−))N(dt ,dξ)
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Poisson Random Measure
To every Lévy process on a Banach space U, correspondsa Poisson random measure (Prm)
∀A ⊂ U, N(t ,A) = number of jumps in [0, t ] with value in A
Define the compensated Poisson random measure (cPrm)as N(t ,A) = N(t ,A)− tν(A).
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Existence and Uniqueness of the Mild Solution
.Theorem (Existence and Uniqueness, S.-Zangeneh (’13))..
......
Let f : H → H be demi-continuous and semi-monotone and gbe Lipschitz and k satisfies∫
E∥k(ξ, x)− k(ξ, y)∥2ν(dξ) ≤ C∥x − y∥2,
∫E∥k(ξ, x)− k(ξ, y)∥pν(dξ) ≤ C∥x − y∥p,
then the equation (*) for initial value X0 has a unique mildsolution with E sup0≤t≤T ∥Xt∥p < ∞ and the solution dependscontinuously in pth norm on initial condition and coefficients.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Example
∂2u∂t2 = ∆u − 3
√∂u∂t +u(t−, x)∂Z
∂t on [0,∞)×Du = 0 on [0,∞)× ∂Du(0, x) = u0(x) on D.∂u∂t (0, x) = 0 on D.
(1)
− 3√
x can be replaced by any continuous and decreasingfunction R → R with linear growth.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Example
∂2u∂t2 = ∆u − 3
√∂u∂t +u(t−, x)∂Z
∂t on [0,∞)×Du = 0 on [0,∞)× ∂Du(0, x) = u0(x) on D.∂u∂t (0, x) = 0 on D.
(1)
− 3√
x can be replaced by any continuous and decreasingfunction R → R with linear growth.
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
.. Example
Let v = dudt .
H = H1(D)× L2(D)
f (u, v) =(
0− 3√
v(x)
), k(ξ, u, v) =
(0
u(x)ξ
)
A =
(0 I∆ 0
)
Erfan Salavati A New Inequality for Stochastic Convolution Integrals
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Motivation: Stochastic Evolution EquationsMaximal InequalitiesItö Type Inequalities
Applications
Thank you for your attention!
Erfan Salavati A New Inequality for Stochastic Convolution Integrals