a new inequality for stochastic convolution integralspkim/7icsaa/slides/erfan_salavati.pdf ·...

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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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Applications

.. Outline

...1 Motivation: Stochastic Evolution Equations

...2 Maximal Inequalities

...3 Itö Type Inequalities

...4 Applications


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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


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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)


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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.


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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,...


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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


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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 )


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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 )


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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.


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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.


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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 ≤?


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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.


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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.


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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.


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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)


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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)⟩

)


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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)


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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).


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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.


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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.


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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

)


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Applications

Thank you for your attention!


a new inequality for stochastic convolution integralspkim/7icsaa/slides/erfan_salavati.pdf ·...

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