Skorohod and Stratonovich in the plane Samy Tindel Universit de - - PowerPoint PPT Presentation

▶
skorohod and stratonovich in the plane
SMART_READER_LITE
LIVE PREVIEW

Skorohod and Stratonovich in the plane Samy Tindel Universit de - - PowerPoint PPT Presentation

Feb 27, 2023 254 likes •571 views

Skorohod and Stratonovich in the plane Samy Tindel Universit de Lorraine Workshop on Random Dynamical Systems - Tianjin 2013 Ongoing joint work with Khalil Chouk (Paris Dauphine) Samy T. (Lorraine) Skorohod and Stratonovich in the plane

slide-1
SLIDE 1

Skorohod and Stratonovich in the plane

Samy Tindel

Université de Lorraine

Workshop on Random Dynamical Systems - Tianjin 2013 Ongoing joint work with Khalil Chouk (Paris Dauphine)

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 1 / 30

slide-2
SLIDE 2

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 2 / 30

slide-3
SLIDE 3

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 3 / 30

slide-4
SLIDE 4

Overview of relations between integrals

Case of Brownian motion W : Stratonovich:

t

s zu dWu Brackets

← → Itô:

t

s zu d⋄Wu

Case of fractional Brownian motion B: Strato, Rough paths:

t

s zu dBu Trace terms

← → Skorohod:

t

s zu d⋄Bu

Case of fractional Brownian sheet x indexed by [0, 1]2: Strato, Rough paths: (??)

??

← → Skorohod:

s2

s1

t2

t1

zu d⋄Wu

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 4 / 30

slide-5
SLIDE 5

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 5 / 30

slide-6
SLIDE 6

Rough paths assumptions

Context: Consider a Hölder path x and For n ≥ 1, x n ≡ linearization of x with mesh 1/n ֒ → x n piecewise linear. For 0 ≤ s < t ≤ 1, set x1,n,i

st

≡ x i

t − x j s,

x2,n,i,j

st

≡

  • s<u<v<t dx n,i

u dx n,j v

Main rough paths assumption: x is a Cγ function with γ > 1/3. The process x2,n converges to a process x2 as n → ∞ ֒ → in a C2γ space. Notation: X ≡ (x1, x2) called rough path above x

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 6 / 30

slide-7
SLIDE 7

Guiding example: fractional Brownian motion

FBm definition: B = (B1, . . . , Bd) Bj centered Gaussian process, independence of coordinates Variance of the increments: E[|Bj

t − Bj s|2] = |t − s|2H

H− ≡ Hölder-continuity exponent of B If H = 1/2, B = Brownian motion If H = 1/2 natural generalization of BM Remarks:

1

FBm widely used in applications

2

Main rough path assumption verified for fBm

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 7 / 30

slide-8
SLIDE 8

Rough paths definition of

  • ϕ(x) dx

Consider: x satisfying the rough paths assumptions ϕ : Rd → Rd of class C 3

b

Then: limn→∞

d

i=1

t

s ϕi(x n u ) dx i,n u

exists We call the limit

d

i=1

t

s ϕi(xu) dx i u

Convergence of Riemann sums along partition πst = (sj):

t

s ϕi(xu) dx i u =

lim

|πst|→0

  • sj∈πst
  • ϕi1(xsj) x1,i1

sjsj+1 + ∂i2ϕi1(xsj) x2,i1i2 sjsj+1

  • Stratonovich type formula holds true

Proposition 1.

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 8 / 30

slide-9
SLIDE 9

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 9 / 30

slide-10
SLIDE 10

Stratonovich and Skorohod in 1-d

Notation: πn typical partition whose mesh goes to 0 Typical pathwise (Stratonovich, rough paths) integral:

t

s zu dxu = lim n→∞

  • πn

zsi(xsi+1 − xsi) + Higher order terms Wick products, crash course: Hn = nth Hermite polynomial, G1, G2 independent Gaussian G⋄,n1

1

⋄ G⋄,n2

2

≡ Hn1(G1)Hn2(G2) Fundamental property: E[G⋄,n1

1

⋄ G⋄,n2

2

] = 0 Typical Skorohod integral: for a Gaussian process x

t

s zu d⋄xu = lim n→∞

  • πn

zsi ⋄ (xsi+1 − xsi) + Higher order terms

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 10 / 30

slide-11
SLIDE 11

Stratonovich and Skorohod in 1-d (2)

Other way to look at Skorohod: for test r.v L on Wiener space, E

t

s zu d⋄xu

  • L
  • = E [z, DxLH]

֒ → Divergence on Wiener space Assumptions for integration: For Stratonovich: Regularity and rough paths hypothesis on z, x For Skorohod: Regularity on the Wiener space

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 11 / 30

slide-12
SLIDE 12

Comparison of 1-d Stratonovich and Skorohod

Summary of a paper by Hu-Jolis-T (AOP 2013): Consider a rather general multidimensional Gaussian process x Including fractional Brownian motion for H ∈ (0, 1) Assume that x gives raise to a rough path Then a Stratonovich type change of variable holds for x A Malliavin-Skorohod type change of variable is available for x Corrections between both integrals are computed Conclusion: Existence of rough path = ⇒ Skorohod and Strato changes of variables Comparison between Skorohod and Strato integrals Remark: Similar ideas in preprint by Kruk-Russo

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 12 / 30

slide-13
SLIDE 13

From Stratonovich to Skorohod

Stratonovich/Skorohod corrections: for a sequence of partitions πn

  • Start from some convergent Riemann-Stratonovich sums involving:

∂k

i1...ikf (xtq) xk,i1,...,ik tq,tq+1

  • By Wick calculus, compute the corrections between

∂k

ik...i1f (xtq) xk,i1,...,ik tq,tq+1

and ∂k

ik...i1f (xtq) ⋄ xk,⋄,i1,...,ik tq,tq+1

֒ → Nice combinatorial formula

  • Analyze limits as |πn| → 0

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 13 / 30

slide-14
SLIDE 14

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 14 / 30

slide-15
SLIDE 15

2d-Itô type formulas: history

Martingale case (70-80’s): Wong-Zakai, Cairoli-Walsh, Nualart Itô type formulas Stochastic, mixed bracket/stochastic and pure bracket terms Fractional Brownian sheet case (03-06): Tudor-Viens Itô-Skorohod formulas from Malliavin calculus Mixed and bracket terms involve covariance of the fBs Rough sheet case (ongoing): Chouk-Gubinelli Definition of a planar rough path Related Stratonovich differential calculus Study of RRDS: widely open!

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 15 / 30

slide-16
SLIDE 16

Aim of the talk

For a process x indexed by the plane: Get some handy Strato-Skorohod formulas Make a link between both Strato and Skorohod worlds Additional difficulties with respect to 1d-case: Clumsy formulas for 2d-indexed processes Terrible boundary terms Lack of Riemann sums representations ֒ → They were at the heart of strategy in 1-d Restricted framework: Rather general Gaussian process in the Young case Fractional Brownian sheet with H1, H2 > 1/3

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 16 / 30

slide-17
SLIDE 17

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 17 / 30

slide-18
SLIDE 18

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 18 / 30

slide-19
SLIDE 19

Notations

Notations for integration in the plane: x: generic plane-indexed function, here real valued Directions of integration: 1 and 2 y ≡ ϕ(x), y j ≡ ϕ(j)(x) Planar differentials: d12x ≡ ∂2

stx and dˆ 1ˆ 2x ≡ ∂sx ∂tx

Planar increment: δxs1s2;t1t2 ≡ xs2;t2 − xs2;t1 − xs1;t2 + xs1;t1 Change of variables, smooth x: [δϕ(x)]s1s2;t1t2 =

  • [s1,s2]×[t1,t2] ϕ(1)(xu;v) duvxu;v

+

  • [s1,s2]×[t1,t2] ϕ(2)(xu;v) duxu;vdvxu;v,

Change of variables, shorthand: δy =

  • 1
  • 2 y 1 d12x +
  • 1
  • 2 y 2 dˆ

1ˆ 2x := z1 + z2.

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 19 / 30

slide-20
SLIDE 20

Notations 2

Index conventions: 1; 2 ≡ integration of type d12, ˆ 1; ˆ 2 ≡ integration of type dˆ

1;ˆ 2 = d1 × d2

1; 0 ≡ integration of type d1 (1-dimensional) First terms of the rough path: x1;2 =

  • 1
  • 2 d12x = δx,

and x

ˆ 1;ˆ 2 =

  • 1
  • 2 d1x d2x =
  • 1
  • 2 dˆ

1ˆ 2x

Increments z1 and z2: z1 =

  • 1
  • 2 y 1 d12x,

and z2 =

  • 1
  • 2 y 2 dˆ

1ˆ 2x

Young type assumptions: x is (γ1, γ2)-Hölder, with γ1, γ2 > 1/2

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 20 / 30

slide-21
SLIDE 21

Young type change of variable

Under Young assumptions ֒ → z1 and z2 are well defined in the 2d-Young sense. Moreover: (i) Both z1 and z2 can be decomposed as: z1 = y 1 x1;2 + ρ1, and z2 = y 2 x

ˆ 1;ˆ 2 + ρ2,

where ρ1, ρ2 are increments with double regularity (2γ1, 2γ2) (ii) Nice continuity properties when x n → x and x n smooth Theorem 2.

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 21 / 30

slide-22
SLIDE 22

Young type change of variable (2)

Under Young assumptions: (iii) Riemann sums convergences: if limn→∞(|π1

n| + |π2 n|) = 0,

lim

n→∞

  • π1

n,π2 n

y 1

σi;τj x1;2 σiσi+1;τjτj+1

= z1

s1s2;t1t2

lim

n→∞

  • π1

n,π2 n

y 2

σi;τj x1;0 σiσi+1;τj x0;2 σi;τjτj+1

= z2

s1s2;t1t2.

(iv) The change of variables formula δy =

  • 1
  • 2 y 1 d12x +
  • 1
  • 2 y 2 dˆ

1ˆ 2x := z1 + z2.

still holds true with integrals ≡ Young integrals. Theorem 3 (following last slide).

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 22 / 30

slide-23
SLIDE 23

Notations for Skorohod integration

Skorohod differentials: d⋄

12x and d⋄ ˆ 1ˆ 2x

֒ → Only formal, since Skorohod integral is defined by duality Increments z1,⋄ and z2,⋄: z1,⋄ =

  • 1
  • 2 y 1 d⋄

12x,

and z2,⋄ =

  • 1
  • 2 y 2 d⋄

ˆ 1ˆ 2x

Gaussian assumptions: x is a centered Gaussian process and E[xs1;t1xs2;t2] = Rs1s2;t1t2 = R1

s1s2R2 t1t2

R1, R2 cov. functions on [0, 1] such that R1, R2 ∈ C1-var([0, 1]2)

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 23 / 30

slide-24
SLIDE 24

Skorohod change of variables, Young case

Under Young plus Gaussian assumptions ֒ → z1,⋄ and z2,⋄ are well defined in the Skorohod-Malliavin sense. Moreover: (i) Explicit corrections between z1, z2 and z1,⋄, z2,⋄ (ii) Riemann-Wick sums convergences: lim

n→∞

  • π1

n,π2 n

y 1

σi;τj ⋄ x1;2 σiσi+1;τjτj+1

= z1,⋄

s1s2;t1t2

lim

n→∞

  • π1

n,π2 n

y 2

σi;τj ⋄ x1;0 σiσi+1;τj ⋄ x0;2 σi;τjτj+1

= z2,⋄

s1s2;t1t2.

Theorem 4.

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 24 / 30

slide-25
SLIDE 25

Skorohod type change of variable (2)

Under Young assumptions: (iii) The change of variables formula for y = ϕ(x) becomes δys;t = z1,⋄ + z2,⋄+1 2

  • 1
  • 2 y 2

u;v d1R1 u d2R2 v

+1 2

  • 1
  • 2 y 3

u;v R1 u d2R2 v d⋄ 1xu;v + 1

2

  • 1
  • 2 y 3

u;v R2 v d1R1 u d⋄ 2xu;v

+1 4

  • 1
  • 2 y 4

u;v R1 u R2 v d1R1 u d2R2 v .

Theorem 5 (following last slide).

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 25 / 30

slide-26
SLIDE 26

Sketch

1

Introduction Rough paths integrals Skorohod and Stratonovich in 1-d 2d program

2

2d Stratonovich and Skorohod formulas Young case Rough case

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 26 / 30

slide-27
SLIDE 27

A part of the level 2 rough path

Hypothesis 1: x is (γ1, γ2)-Hölder, with γ1, γ2 > 1/3 Hypothesis 2: The following rough path X can be constructed out of x. Increment Interpretation Regularity Increment Interpretation x1;2

  • 1
  • 2 d12x

(γ1, γ2) xˆ

1;ˆ 2

  • 1
  • 2 dˆ

1ˆ 2x

x11;02

  • 1 d1x
  • 2 d12x

(2γ1, γ2) x1ˆ

1;0ˆ 2

  • 1 d1x
  • 2 dˆ

1ˆ 2x

x01;22

  • 2 d2x
  • 1 d12x

(γ1, 2γ2) x0ˆ

1;2ˆ 2

  • 2 d2x
  • 1 dˆ

1ˆ 2x

x11;22

  • 1
  • 2 d12xd12x

(2γ1, 2γ2) x1ˆ

1;2ˆ 2

  • 1
  • 2 d12xdˆ

1ˆ 2x

xˆ

11;ˆ 22

  • 1
  • 2 dˆ

1ˆ 2xd12x

(2γ1, 2γ2) xˆ

1ˆ 1;ˆ 2ˆ 2

  • 1
  • 2 dˆ

1ˆ 2xdˆ 1ˆ 2x

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 27 / 30

slide-28
SLIDE 28

Stratonovich formula in the rough case

If rough path X above x is well defined ֒ → z1 and z2 are well defined in the rough path sense. Moreover: (i) Increment z1 can be decomposed as: z1 = y x1;2 + y 1 x11;02 + y 1 x01;22 + y 1 x11;22 + y 2 x

ˆ 11;ˆ 22 + ρ1,

where ρ1 is increment with triple regularity (3γ1, 3γ2) ֒ → Similar expression for z2 (ii) Nice continuity properties when x n → x and x n smooth Theorem 6.

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 28 / 30

slide-29
SLIDE 29

Stratonovich formula in the rough case (2)

If rough path above x is well defined: (iii) The change of variables formula δy =

  • 1
  • 2 y 1 d12x +
  • 1
  • 2 y 2 dˆ

1ˆ 2x := z1 + z2.

still holds true when integrals are understood in the rough sense. Theorem 7. Remark: No related Riemann sums in this case!

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 29 / 30

slide-30
SLIDE 30

Skorohod formula for fractional Brownian sheet

Fractional Brownian sheet: centered Gaussian process such that E[xs1;t1xs2;t2] = Rs1s2;t1t2 = R1

s1s2R2 t1t2

Rj

u1u2 = 1 2 (|u1|2γj + |u2|2γj − |u1 − u2|2γj), with γj > 1/3

Skorohod formula, rough statement: When x ≡ fBs ֒ → One can take limits in the Young-Skorohod formula and δys;t = z1,⋄ + z2,⋄+1 2

  • 1
  • 2 y 2

u;v d1R1 u d2R2 v

+1 2

  • 1
  • 2 y 3

u;v R1 u d2R2 v d⋄ 1xu;v + 1

2

  • 1
  • 2 y 3

u;v R2 v d1R1 u d⋄ 2xu;v

+1 4

  • 1
  • 2 y 4

u;v R1 u R2 v d1R1 u d2R2 v .

Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 30 / 30