Kolmogorov equations and weak order analysis for SPDEs with - - PowerPoint PPT Presentation

Kolmogorov equations and weak order analysis for SPDEs with multiplicative noise

Charles-Edouard Bréhier Joint work with Arnaud Debussche (ENS Rennes) CNRS & Université Lyon 1, Institut Camille Jordan

C-E Bréhier Multiplicative noise SPDEs: weak rates 1 / 21

Plan of the talk

1

Model, assumptions and results SPDEs with non constant diffusion coefficient Numerical scheme Weak convergence result

2

Study of the Kolmogorov equation The Kolmogorov equation Basic regularity estimates Improved estimates: strategy

C-E Bréhier Multiplicative noise SPDEs: weak rates 2 / 21

Model and assumptions

Parabolic, semilinear, SPDE: t ≥ 0, z ∈ (0, 1) ∂x(t, z) ∂t = ∂2x(t, z) ∂z2 +f1

• x(t, z)
• + ∂
• f2(x(t, z))
• ∂z

+σ

• x(t, z)
• ξ(t, z)

driven by space-time white noise ξ(t, z).

Boundary conditions: x(t, 0) = x(t, 1) = 0 (Dirichlet). Initial condition: x(0, ·) = x0.

Coefficients f1, f2 and σ are smooth, bounded and Lipschitz continuous.

In the drift: Burgers type nonlinearity. Multiplicative noise: diffusion coefficient is not constant

C-E Bréhier Multiplicative noise SPDEs: weak rates 3 / 21

Stochastic evolution equation formulation

Formulation: in the Da Prato-Zabczyk framework, X(t) = X(t, ·) ∈ H = L2(0, 1) dX(t) = AX(t)dt + G(X(t))dt + σ(X(t))dW (t), with

A : D(A) → H unbounded, self-adjoint linear operator Aen = −λnen, where en(·) = √ 2 sin(nπ·) and λn = −n2π2; G = F1 + BF2 and σ : H → L(H) are Nemytskii coefficients. W is a cylindrical Wiener process: W (t) =

• n∈N

βn(t)en. where βn are independent standard Wiener processes, for n ∈ N.

C-E Bréhier Multiplicative noise SPDEs: weak rates 4 / 21

Well-posedness

The equation dX(t) = AX(t)dt + G(X(t))dt + σ(X(t))dW (t) admits a unique, global, mild solution: for all t ≥ 0

X(t) = etAx0 + t e(t−s)A F1(X(s) + BF2(X(s))

• ds +

t e(t−s)Aσ(X(s))dW (s)

where etAx =

n∈N e−tλnx, enen is the semi-group associated with A.

C-E Bréhier Multiplicative noise SPDEs: weak rates 5 / 21

Well-posedness

The equation dX(t) = AX(t)dt + G(X(t))dt + σ(X(t))dW (t) admits a unique, global, mild solution: for all t ≥ 0

X(t) = etAx0 + t e(t−s)A F1(X(s) + BF2(X(s))

• ds +

t e(t−s)Aσ(X(s))dW (s)

where etAx =

n∈N e−tλnx, enen is the semi-group associated with A.

Regularization by the semi-group: for every α ∈ [0, 1)

• etAh
• H ≤ Cαt−α|h|−α.

with |h|2

−α = ∞ n=1 λ−2α n

|h, en|2.

C-E Bréhier Multiplicative noise SPDEs: weak rates 5 / 21

Temporal discretization

Discretization of dX(t) = AX(t)dt + G(X(t))dt + σ(X(t))dW (t) : linear-implicit Euler scheme X ∆t

n+1 = X ∆t n

+ ∆tAX ∆t

n+1 + ∆tG(X ∆t n ) + σ(X ∆t n )∆W ∆t n

= S∆tX ∆t

n

+ ∆tS∆tG(X ∆t

n ) + S∆tσ(X ∆t n )∆W ∆t n

with S∆t = (I − ∆tA)−1, T = N∆t, X ∆t = x0, ∆W ∆t

n

= W

• (n + 1)∆t
• − W (n∆t).

Strong order of convergence is 1

4:

E

• X(N∆t) − X ∆t

N

• ≤ Cκ(T, x0)∆t

1 4 −κ.

where Cκ(T, x0) ∈ (0, ∞), for all κ ∈ (0, 1

4).

C-E Bréhier Multiplicative noise SPDEs: weak rates 6 / 21

Main result: weak order 1

2 Consider real-valued test functions ϕ, with C3 regularity, and derivatives controlled in the following sense: for some p, q ∈ [2, ∞)

• Dnϕ(x).
• h1, . . . , hn
• ≤ C(1 + |x|Lp)K|h1|Lq . . . |hn|Lq.

Example: ϕ(x) = 1

0 φ(x(z))dz.

Theorem (B.-Debussche)

For such test functions ϕ, for every T ∈ (0, ∞), and every κ ∈ (0, 1

2), there exists Cκ(T, ϕ) ∈ (0, ∞) such that

• Eϕ
• X(N∆t)
• − Eϕ
• X ∆t

N

• ≤ Cκ(T, ϕ)P(x0)∆t

1 2 −κ.

Preprint 2017 Kolmogorov Equations and Weak Order Analysis for

SPDES with Nonlinear Diffusion Coefficient.

C-E Bréhier Multiplicative noise SPDEs: weak rates 7 / 21

Comparison with previous results

Approaches in the literature (not exhaustive): Kolmogorov equation: SDEs: Talay, Tubaro, Milstein, Bally, Tretyakov, Kloeden, Platen, . . . SPDEs: Debussche, Larsson, Kovacs, Andersson, Printems, Wang, . . . mild Itô calculus: Jentzen, Conus, Kurniawan, Cox, . . . Malliavin calculus: SDEs: Clément, Kohatsu-Higa, Lamberton SPDEs: Andersson, Larsson, Kruse; Lindner

C-E Bréhier Multiplicative noise SPDEs: weak rates 8 / 21

Comparison with previous results

Approaches in the literature (not exhaustive): Kolmogorov equation: SDEs: Talay, Tubaro, Milstein, Bally, Tretyakov, Kloeden, Platen, . . . SPDEs: Debussche, Larsson, Kovacs, Andersson, Printems, Wang, . . . mild Itô calculus: Jentzen, Conus, Kurniawan, Cox, . . . Malliavin calculus: SDEs: Clément, Kohatsu-Higa, Lamberton SPDEs: Andersson, Larsson, Kruse; Lindner Novelties of our work:

1 extension of the Kolmogorov equation approach when the

diffusion coefficient σ is not constant.

2 treatment of Burgers type nonlinearities. C-E Bréhier Multiplicative noise SPDEs: weak rates 8 / 21

Plan of the talk

1

Model, assumptions and results SPDEs with non constant diffusion coefficient Numerical scheme Weak convergence result

2

Study of the Kolmogorov equation The Kolmogorov equation Basic regularity estimates Improved estimates: strategy

C-E Bréhier Multiplicative noise SPDEs: weak rates 9 / 21

Simplified problem

Evolution equation: dY (t) = AY (t)dt + G(Y (t))dt + σ(Y (t))dW (t), Y (0) = y. where the coefficients G : H → H and σ : H → L(H) are of class C3

b.

Discretization: spectral Galerkin approximation dYN(t) = AYN(t)dt + PNG(YN(t))dt + PNσ(YN(t))dW (t). where PNy = N

n=1y, enen.

Test function: ϕ : H → R is of class C3

b.

Weak error estimate:

• Eϕ(YN(T)) − Eϕ(Y (T))
• ≤ Cκ(ϕ, T)λ

− 1

2+κ

N+1 .

C-E Bréhier Multiplicative noise SPDEs: weak rates 10 / 21

The Kolmogorov equation

The function (t, y) → u(t, y) = E

• ϕ(Y (t))
• Y (0) = y
• is solution of

the Kolmogorov equation ∂u(t, y) ∂t = Lu(t, y) = Ay + G(y), Du(t, y) + 1 2Tr

• σ(y)σ(y)⋆D2u(t, y)
• .

C-E Bréhier Multiplicative noise SPDEs: weak rates 11 / 21

The Kolmogorov equation

The function (t, y) → u(t, y) = E

• ϕ(Y (t))
• Y (0) = y
• is solution of

the Kolmogorov equation ∂u(t, y) ∂t = Lu(t, y) = Ay + G(y), Du(t, y) + 1 2Tr

• σ(y)σ(y)⋆D2u(t, y)
• .

Some regularity properties are needed: Du(t, y).h ≤ Cα(t)|h|−α , D2u(t, y).(h, k) ≤ Cβ,γ(t)|h|−β|k|−γ.

C-E Bréhier Multiplicative noise SPDEs: weak rates 11 / 21

The Kolmogorov equation

The function (t, y) → u(t, y) = E

• ϕ(Y (t))
• Y (0) = y
• is solution of

the Kolmogorov equation ∂u(t, y) ∂t = Lu(t, y) = Ay + G(y), Du(t, y) + 1 2Tr

• σ(y)σ(y)⋆D2u(t, y)
• .

Some regularity properties are needed: Du(t, y).h ≤ Cα(t)|h|−α , D2u(t, y).(h, k) ≤ Cβ,γ(t)|h|−β|k|−γ. Basic estimates: α ∈ [0, 1

2) and β, γ ∈ [0, 1 2) with β + γ < 1 2.

Improved estimates: α ∈ [0, 1) and β, γ ∈ [0, 1

2).

C-E Bréhier Multiplicative noise SPDEs: weak rates 11 / 21

Decomposition of the weak error

Eϕ(Y (T)) − Eϕ(YN(T)) = u(T, y) − Eu(0, YN(T)) = u(T, y) − u(T, PNy) + u(T, PNy) − Eu(0, YN(T)) and using Itô’s formula and the Kolmogorov equation u(T, PNy) − Eu(0, YN(T)) = E T

• LN − ∂

∂t

• u
• T − t, YN(t)
• dt

= E T

• LN − L
• u
• T − t, YN(t)
• dt

C-E Bréhier Multiplicative noise SPDEs: weak rates 12 / 21

Required regularity results for numerical analysis

First-order derivative:

• (PN − I)G(y), Du
• T − t, Y (t)
• ≤ Cα(T − t)
• (PN − I)G(y)
• −α ≤ Cα(T − t)

λα

N+1

.

So we need α ∈ [0, 1

2). Burgers: G(y) = F1(y) + BF2(y): α ∈ [0, 1).

C-E Bréhier Multiplicative noise SPDEs: weak rates 13 / 21

Required regularity results for numerical analysis

First-order derivative:

• (PN − I)G(y), Du
• T − t, Y (t)
• ≤ Cα(T − t)
• (PN − I)G(y)
• −α ≤ Cα(T − t)

λα

N+1

.

So we need α ∈ [0, 1

2). Burgers: G(y) = F1(y) + BF2(y): α ∈ [0, 1).

Second-order derivative:

Tr

• (PN − I)σσ⋆(y)(PN − I)D2u
• T − t, Y (t))
• ≤ C Tr
• (−A)−1/2−κ/2

λ1/2−κ

N+1

• (−A)1/2−κD2u
• T − t, Y (t)
• (−A)1/2−κ
• L(H)

we need δ = β = γ = 1

2 − κ ∈ [0, 1 2).

C-E Bréhier Multiplicative noise SPDEs: weak rates 13 / 21

Basic regularity estimates: 1st order derivative

Du(t, y).h = E

• Dϕ(Y (t, y)
• .ηh

t

• where ηh

t is the solution of the first variation equation

dηh

t = Aηh t dt + G ′

Y (t, y)

• .ηh

t dt + σ′

Y (t, y)

• .ηh

t dW (t),

ηh

0 = h.

C-E Bréhier Multiplicative noise SPDEs: weak rates 14 / 21

Basic regularity estimates: 1st order derivative

Du(t, y).h = E

• Dϕ(Y (t, y)
• .ηh

t

• where ηh

t is the solution of the first variation equation

dηh

t = Aηh t dt + G ′

Y (t, y)

• .ηh

t dt + σ′

Y (t, y)

• .ηh

t dW (t),

ηh

0 = h.

Explicit expression:

ηh

t = etAh +

t e(t−s)AG

′

Y (s, y)

• .ηh

s ds +

t e(t−s)Aσ′ Y (s, y)

• .ηh

s dW (s).

Using Itô’s formula: E|ηh

t |2 ≤ Ct−2α|h|2 −α + C

t

• 1 +

1 (t − s)1/2+κ

• E|ηh

s |2ds,

and Gronwall’s Lemma: E|ηh

t | ≤ Ct−α|h|−α for α < 1/2.

C-E Bréhier Multiplicative noise SPDEs: weak rates 14 / 21

Basic regularity estimates: 2nd order derivative

D2u(t, y).(h, k) = E

• Dϕ
• Y (t, y)
• .ζh,k

t

• +E
• D2ϕ
• X(t, y)
• .
• ηh

t , ηk t

• ,

where ζh,k

t

is solution of the second variation equation

dζh,k

t

= Aζh,k

t

dt + G

′

Y (t, y)

• .ζh,k

t

dt + σ′ Y (t, y)

• .ζh,k

t

dW (t) + G

′′

Y (t, y)

• .
• ηh

t , ηk t

• dt + σ′′

Y (t, y)

• .
• ηh

t , ηk t

• dW (t).

C-E Bréhier Multiplicative noise SPDEs: weak rates 15 / 21

Basic regularity estimates: 2nd order derivative

D2u(t, y).(h, k) = E

• Dϕ
• Y (t, y)
• .ζh,k

t

• +E
• D2ϕ
• X(t, y)
• .
• ηh

t , ηk t

• ,

where ζh,k

t

is solution of the second variation equation

dζh,k

t

= Aζh,k

t

dt + G

′

Y (t, y)

• .ζh,k

t

dt + σ′ Y (t, y)

• .ζh,k

t

dW (t) + G

′′

Y (t, y)

• .
• ηh

t , ηk t

• dt + σ′′

Y (t, y)

• .
• ηh

t , ηk t

• dW (t).

Restriction β + γ < 1

2 comes from

E

• t

e(t−s)Aσ′′ Y (s, y)

• .
• ηh

s , ηk s

• dW (s)
• 2

≤ C t 1 (t − s)1/2+κ E|ηh

s |2|ηk s |2ds

≤ C t 1 (t − s)1/2+κ 1 s2β+2γ |h|2

−β|k|2 −γds.

C-E Bréhier Multiplicative noise SPDEs: weak rates 15 / 21

Is there a hope to improve these estimates?

C-E Bréhier Multiplicative noise SPDEs: weak rates 16 / 21

Is there a hope to improve these estimates?

The estimates given above are sharp, when σ is not constant: E|ηh

t | ≤ Cα(t)|h|−α with Cα(t) < ∞ if and only if α < 1 2.

E|ζh,k

t

| ≤ Cβ,γ(t)|h|−β|k|−γ with Cβ(t) < ∞ if and only if β + γ < 1

2.

References:

On the differentiability of solutions of stochastic evolution equations with respect to their initial values by A. Andersson, A. Jentzen, R. Kurniawan and T. Welti. Preprint 2016. Counterexamples to regularities for the derivative processes associated to stochastic evolution equations by M. Hefter, A. Jentzen and R. Kurniawan. Preprint 2017.

C-E Bréhier Multiplicative noise SPDEs: weak rates 16 / 21

What else can we do now?

C-E Bréhier Multiplicative noise SPDEs: weak rates 17 / 21

What else can we do now?

Fundamental ideas: estimates on E

• Dϕ(Y (t, y)
• .ηh

t

• and E
• Dϕ
• Y (t, y)
• .ζh,k

t

• .

avoid the use of Gronwall’s Lemma. avoid the use of Itô’s formula.

C-E Bréhier Multiplicative noise SPDEs: weak rates 17 / 21

What else can we do now?

Fundamental ideas: estimates on E

• Dϕ(Y (t, y)
• .ηh

t

• and E
• Dϕ
• Y (t, y)
• .ζh,k

t

• .

avoid the use of Gronwall’s Lemma. avoid the use of Itô’s formula. Ingredients:

• riginal “explicit” expressions for ηh

t and ζh,k t

Malliavin integration by parts. Rigorous justification: with a discrete-time approximation. In this talk: formal computations.

C-E Bréhier Multiplicative noise SPDEs: weak rates 17 / 21

How can we get new expressions?

Define ˜ ηh

t = ηh t − etAh. Then

d ˜ ηh

t = A˜

ηh

t dt + G ′

Y (t, y)

• .˜

ηh

t dt + σ′

Y (t, y)

• .˜

ηh

t dW (t)

+ G

′

Y (t, y)

• .etAhdt + σ′

Y (t, y)

• .etAhdW (t).

Similarly,

dζh,k

t

= Aζh,k

t

dt + G

′

Y (t, y)

• .ζh,k

t

dt + σ′ Y (t, y)

• .ζh,k

t

dW (t) + G

′′

Y (t, y)

• .
• ηh

t , ηk t

• dt + σ′′

Y (t, y)

• .
• ηh

t , ηk t

• dW (t).

C-E Bréhier Multiplicative noise SPDEs: weak rates 18 / 21

How can we get new expressions?

Define ˜ ηh

t = ηh t − etAh. Then

d ˜ ηh

t = A˜

ηh

t dt + G ′

Y (t, y)

• .˜

ηh

t dt + σ′

Y (t, y)

• .˜

ηh

t dW (t)

+ G

′

Y (t, y)

• .etAhdt + σ′

Y (t, y)

• .etAhdW (t).

Similarly,

dζh,k

t

= Aζh,k

t

dt + G

′

Y (t, y)

• .ζh,k

t

dt + σ′ Y (t, y)

• .ζh,k

t

dW (t) + G

′′

Y (t, y)

• .
• ηh

t , ηk t

• dt + σ′′

Y (t, y)

• .
• ηh

t , ηk t

• dW (t).

We introduce random linear operators Π(t, s) such that θt,s = Π(t, s)θ solves the linear SPDE dθt,s = Aθtdt + G

′

Y (t, y)

• .θtdt + σ′

Y (t, y)

• .θtdW (t)

with t > s and θs,s = θ.

C-E Bréhier Multiplicative noise SPDEs: weak rates 18 / 21

New formulas

Natural expressions (Duhamel):

˜ ηh

t =

t Π(t, s)G

′

Y (s, y)

• .esAhds +

t Π(t, s)σ′ Y (s, y)

• .esAhdW (s)

and

ζh,k

t

= t Π(t, s)G

′′

Y (s, y)

• .
• ηh

s , ηk s

• ds+

t Π(t, s)σ′′ Y (s, y)

• .
• ηh

s , ηk s

• dW (s)

C-E Bréhier Multiplicative noise SPDEs: weak rates 19 / 21

New formulas

Natural expressions (Duhamel):

˜ ηh

t =

t Π(t, s)G

′

Y (s, y)

• .esAhds +

t Π(t, s)σ′ Y (s, y)

• .esAhdW (s)

and

ζh,k

t

= t Π(t, s)G

′′

Y (s, y)

• .
• ηh

s , ηk s

• ds+

t Π(t, s)σ′′ Y (s, y)

• .
• ηh

s , ηk s

• dW (s)

Anticipating, two-sided stochastic integrals!

C-E Bréhier Multiplicative noise SPDEs: weak rates 19 / 21

New formulas

Natural expressions (Duhamel):

˜ ηh

t =

t Π(t, s)G

′

Y (s, y)

• .esAhds +

t Π(t, s)σ′ Y (s, y)

• .esAhdW (s)

and

ζh,k

t

= t Π(t, s)G

′′

Y (s, y)

• .
• ηh

s , ηk s

• ds+

t Π(t, s)σ′′ Y (s, y)

• .
• ηh

s , ηk s

• dW (s)

Anticipating, two-sided stochastic integrals! No reference in the literature. Some related works: Pardoux-Protter 87, A two-sided stochastic integral and its calculus Leon-Nualart 98, Stochastic evolution equations with random generators Only formal computations. Justified in a discrete-time setting.

C-E Bréhier Multiplicative noise SPDEs: weak rates 19 / 21

Auxiliary time-discretization

We use the linear implicit Euler-scheme (∆t > 0): Y τ,∆t

n+1 = S∆tY τ,∆t n

+ ∆tS∆tG(Y τ,∆t

n

) + eτAS∆tσ(Y τ,∆t

n

)∆Wn. and an additional approximation, τ > 0.

C-E Bréhier Multiplicative noise SPDEs: weak rates 20 / 21

Auxiliary time-discretization

We use the linear implicit Euler-scheme (∆t > 0): Y τ,∆t

n+1 = S∆tY τ,∆t n

+ ∆tS∆tG(Y τ,∆t

n

) + eτAS∆tσ(Y τ,∆t

n

)∆Wn. and an additional approximation, τ > 0. Define uτ,∆t(n∆t, y) = E

• ϕ(Y τ,∆t

n

)

• .

With the strategy outlined above, and many nontrivial arguments,

Duτ,∆t(T, y).h ≤ Cα,κ τ κT α |h|−α , D2uτ,∆t(T, y).(h, k) ≤ Cβ,γ,κ τ κT β+γ |h|−β|k|−γ

for α ∈ [0, 1) and β, γ ∈ [0, 1

2), κ > 0. Finally ∆t → 0, then τ → 0.

C-E Bréhier Multiplicative noise SPDEs: weak rates 20 / 21

Conclusion

New results, concerning SPDEs with nonlinear diffusion coefficient: new regularity results for solutions of Kolmogorov equations application: weak convergence rates for time-discretization. Perspectives theoretical study: the stochastic two-sided integrals numerical analysis: discretization in space (Finite Elements)

• ther models: with less restrictive assumptions

Thanks for your attention.

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