Numerical approximations for nonlinear stochastic partial ... · stochastic partial differential...

Numerical approximations for nonlinearstochastic partial differential equations

Arnulf Jentzen (ETH Zurich, Switzerland)Joint works with

Sonja Cox (University of Amsterdam, the Netherlands),Arnaud Debussche (ENS Rennes, France),

Giuseppe Da Prato (Scuola Normale Superiore di Pisa, Italy),Martin Hairer (University of Warwick, UK),

Mario Hefter (University of Kaiserslautern, Germany),Martin Hutzenthaler (University of Duisburg-Essen, Germany),

Ladislas Jacobe de Naurois (ETH Zurich, Switzerland),Thomas Müller-Gronbach (University of Passau, Germany),

Ryan Kurniawan (ETH Zurich, Switzerland),Michael Röckner (Bielefeld University, Germany),

Timo Welti (ETH Zurich, Switzerland), andLarisa Yaroslavtseva (University of Passau, Germany)

Workshop on Numerics for Stochastic Partial DifferentialEquations and their Applications, Linz, Austria

Monday, 10 December 2016

Some examples of SDEs

Heston model Consider α, γ ∈ R, β, δ, X(1)0 , X

(2)0 > 0, ρ ∈ [−1, 1] and

∂∂t X

(1)t = α X

(1)t +

(2)t X

∂∂t W

∂∂t X

(2)t = δ − γX

(2)t + β

(ρ ∂∂t W

(1)t +

√1− ρ2 ∂

∂t W(2)t

)for t ∈ [0, T ], where (Wt)t∈[0,T ] = ((W

(1)t ,W

(2)t ))t∈[0,T ] is a two-dim. BM.

Nonlinear stochastic heat equation (Parabolic Anderson model)

∂∂t Xt(x) = ∂2

∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where (Wt)t∈[0,T ] is acylindrical IdL2((0,1);R)-Wiener process and b : R→ R is regular.Nonlinear stochastic Wave equation (Hyperbolic Anderson model)

∂t2 Xt(x) = ∂2

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].Stochastic Burgers equation

∂∂t Xt(x) = ∂2

∂x2 Xt(x)− Xt(x) · ∂∂x Xt(x) + ∂∂t Wt(x)

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].

(2)0 > 0, ρ ∈ [−1, 1] and

∂∂t X

(1)t = α X

(1)t +

(2)t X

∂∂t W

∂∂t X

(2)t = δ − γX

(2)t + β

(ρ ∂∂t W

(1)t +

√1− ρ2 ∂

∂t W(2)t

(1)t ,W

∂∂t Xt(x) = ∂2

∂t2 Xt(x) = ∂2

∂∂t Xt(x) = ∂2

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].

(2)0 > 0, ρ ∈ [−1, 1] and

∂∂t X

(1)t = α X

(1)t +

(2)t X

∂∂t W

∂∂t X

(2)t = δ − γX

(2)t + β

(ρ ∂∂t W

(1)t +

√1− ρ2 ∂

∂t W(2)t

(1)t ,W

∂∂t Xt(x) = ∂2

∂t2 Xt(x) = ∂2

∂∂t Xt(x) = ∂2

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].

(2)0 > 0, ρ ∈ [−1, 1] and

∂∂t X

(1)t = α X

(1)t +

(2)t X

∂∂t W

∂∂t X

(2)t = δ − γX

(2)t + β

(ρ ∂∂t W

(1)t +

√1− ρ2 ∂

∂t W(2)t

(1)t ,W

∂∂t Xt(x) = ∂2

∂t2 Xt(x) = ∂2

∂∂t Xt(x) = ∂2

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].

(2)0 > 0, ρ ∈ [−1, 1] and

∂∂t X

(1)t = α X

(1)t +

(2)t X

∂∂t W

∂∂t X

(2)t = δ − γX

(2)t + β

(ρ ∂∂t W

(1)t +

√1− ρ2 ∂

∂t W(2)t

(1)t ,W

∂∂t Xt(x) = ∂2

∂t2 Xt(x) = ∂2

∂∂t Xt(x) = ∂2

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].

Some difficulties in the numerical approximations of SDEs

Theorem (Hairer, Hutzenthaler & J 2015 AOP)

Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of

dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,

and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

0 = X0 and

Y Nn+1 = Y N

n + µ(Y Nn ) T

N + σ(Y Nn )(

W (n+1)TN

−W nTN

limN→∞

(Nα∥∥E[XT

]− E

]∥∥) =

0 : α = 0

∞ : α > 0.

Plot of∥∥E[XT

]− E

]∥∥ for T = 2 and N ∈ 21, 22, . . . , 230.

10−12

10−10

10−8

10−6

10−4

10−2

Number N of time discretizations

Approximation error of the mean

A function with order 0

Order line 1/2

Order line 1

Theorem (Gerencsér, J, & Salimova 2016)

Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of

dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,

and every N ∈ N we have

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS

Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)

Adaptive approximations and d ≥ 4: Yaroslavtseva 2016

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

Theorem (Hefter & J 2016)

Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of

dXt = (δ − γXt) dt + β√

Xt dWt , t ∈ [0, T ], X0 = ξ.

Then there exists a c ∈ (0,∞) such that for all N ∈ N we have

infu : RN→Rmeasurable

E[∣∣XT − u(W T

N,W 2T

N, . . . ,WT )

∣∣] ≥ c · N−min1, 2δβ2 .

Xt dWt , t ∈ [0, T ], X0 = ξ.

N,W 2T

N, . . . ,WT )

∣∣] ≥ c · N−min1, 2δβ2 .

Xt dWt , t ∈ [0, T ], X0 = ξ.

N,W 2T

N, . . . ,WT )

∣∣] ≥ c · N−min1, 2δβ2 .

Xt dWt , t ∈ [0, T ], X0 = ξ.

N,W 2T

N, . . . ,WT )

∣∣] ≥ c · N−min1, 2δβ2 .

Xt dWt , t ∈ [0, T ], X0 = ξ.

N,W 2T

N, . . . ,WT )

∣∣] ≥ c · N−min1, 2δβ2 .

Xt dWt , t ∈ [0, T ], X0 = ξ.

N,W 2T

N, . . . ,WT )

∣∣] ≥ c · N−min1, 2δβ2 .

0 5 10 15 20 250

absolute

quantity

ofstocksin

theS&P500

2δ/β2 in the Heston model

The S&P 500 (the Standard & Poor’s 500) is a stock market index.In Hutzenthaler, J & Noll 2016 we calibrate 498 stocks from the S&P 500 within theHeston model: 359 stocks satisfy 2δ

β2 ≤ 25, 162 stocks (≈ 32%) satisfy 2δβ2 < 1.

More than 100 stocks (= 20%) satisfy 2δβ2 ≤ 1

0 5 10 15 20 250

absolute

quantity

ofstocksin

theS&P500

0 5 10 15 20 250

absolute

quantity

ofstocksin

theS&P500

0 5 10 15 20 250

absolute

quantity

ofstocksin

theS&P500

0 5 10 15 20 250

absolute

quantity

ofstocksin

theS&P500

Convergence results

Weak convergence results for SPDEs in the literature, e.g.,Hausenblas 2003 Progr. Probab.

Shardlow 2003 BIT

De Bouard & Debussche 2006 Appl. Math. Optim.

Debussche & Printemps 2007 arXiv (2009 Math. Comp.)Debussche 2008 arXiv (2011 Math. Comp.)Geissert, Kovacs & Larsson 2009 BIT

Lindner & Schilling 2009 arXiv (2013 Potential Anal.)Hausenblas 2011 J. Comput. Appl. Math.

Dörsek 2011 arXiv (2012 SIAM J. Numer. Anal.)Dörsek 2011 PhD thesis, TU Wien

Kovacs, Larsson & Lindgren 2012 BIT

Kovacs, Larsson & Lindgren 2012 arXiv (2013 BIT)Wang & Gan 2012 J. Math. Anal. Appl.

Bréhier 2012 arXiv (2014 Potential Anal.)Kruse 2012 PhD thesis, Bielefeld University (2014 Lecture Notes in Mathematics)Lindgren 2012 PhD thesis, Chalmers University of Technology and University of Gothenburg

Andersson & Larsson 2012 arXiv

Andersson, Kruse & Larsson 2013 arXiv ( 2016 SPDEs: Anal. and Comp.)Bréhier & Kopec 2013 arXiv

Wang 2013 arXiv . . .

use for Xt (x) = ∂2

∂x2 Xt (x) + b(Xt (x))Wt (x) or Xt (x) = ∂2

∂x2 Xt (x) + b(Xt (x))Wt (x) that b

is constant.

Shardlow 2003 BIT

is constant.

Shardlow 2003 BIT

is constant.

Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)

Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of

Xt(x) = ∂2

∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)

with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Xt(x) = ∂2

∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

N )]∣∣ ≤ C · N(ε−1/2)

∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N

N ‖2H

])1/2 ≤ C · N(ε−1/4) .

∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )

]− E

[ϕ(Y N

T )]∣∣ ≥ C · N−1/2.

Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)

Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1

0 ((0, 1);R)× H,f ∈ C3

b ((0, 1)× R,R), for every n ∈ N let en(·) =√

2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =

∑Nn=1 〈en, ·〉H en ∈ L(H) and let

X N : [0, T ]× Ω→ PN(H) be a mild solution of

Xt(x) = ∂2

∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)

with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

Rate can essentially not be improved.

Similar results for time discretizations (Cox, J & Welti 2016).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

0 ((0, 1);R)× H,f ∈ C3

Xt(x) = ∂2

]− E

[ϕ(X N

T )]∣∣ ≤ C · N(ε−1).

Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)

Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let

en(·) =√

2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =

∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a

continuous mild solution of

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

en(·) =√

X Nt (x) = ∂2

∂x2 X Nt (x) + PN

(κ · X N

t (x) · ∂∂x X Nt (x)

)+ PN Wt(x)

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

c · N−1 ≤∣∣E[ϕ(X∞T )

]− E

[ϕ(X N

T )]∣∣ ≤ C · N−1.

Cahn-Hilliard Cook equation driven by a standard Wiener process:

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:

P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]

∥∥Xt − Y ht

∥∥ ≤ Cα,ε hα

(semi strong convergence rate) and hence

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

∂∂t Xt(x) =

[− ∂4

x Xt(x) + ∂2x

(Xt(x)3 − Xt(x)

)]dt + ∂

∂t Wt(x)

∥∥Xt − Y ht

t∈[0,T ]

∥∥Xt − Y ht

∥∥2

∥∥Xt − Y Nt

∥∥∥∥∥∥∥Lp(Ω;R)

≤ Cα,p N−α.

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:

P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

infs1,...,sN∈[0,T ]

infu : RN→Rd

measurable

E[∥∥XT − u

(Ws1 , . . . ,WsN

)∥∥] ≥ aN .

∥∥Xt − Y Nt

∥∥ ≤ Cα,ε N−α

Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :

supt∈[0,T ]

Plot of∥∥E[XT

]− E

]∥∥ for T = 2 and N ∈ 21, 22, . . . , 230.

10−12

10−10

10−8

10−6

10−4

10−2

Number N of time discretizations

Approximation error of the mean

A function with order 0

Order line 1/2

Order line 1

Methods of proof

Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))

Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that

ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +

ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds

ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs

∑u∈U

ϕ′′(eA(t−s)Xs)(

eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)

Mild Itô formula shows

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

t∫0E[ϕ′′(eA(t−s)Xs)

(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

t∫0E[ϕ′′(eA(t−s)PN(Xs))

(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

E[ϕ(Xt)

[ϕ(eAt X0)

∑u∈U

E[ϕ(PN(Xt))

[ϕ(eAt PN(X0)

∑u∈U

Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))

Consider

a separable Hilbert space H with H ⊆ H continuously and densely,

strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t

0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Consider

∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3

Xt = S0,t X0 +

0Ss,t Ys ds +

0Ss,t Zs dWs.

Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))

Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that

ϕ(Xt) = ϕ(St0,t Xt0 ) +

ϕ′(Ss,t Xs) Ss,t Ys ds +

ϕ′(Ss,t Xs) Ss,t Zs dWs

∑u∈U

ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that∫ t

‖ϕ′(Ss,t Xs)Ss,t Ys‖V + ‖ϕ′(Ss,t Xs)Ss,t Zs‖2HS(U,V) ds <∞,

‖ϕ′′(Ss,t Xs)‖L(2)(H,V) ‖Ss,t Zs‖2HS(U,H) ds <∞

ϕ(Xt) = ϕ(St0,t Xt0 ) +

∑u∈U

Thanks for your attention!

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