Rough paths methods 3: Second order structures Samy Tindel - - PowerPoint PPT Presentation

Rough paths methods 3: Second order structures

Samy Tindel

University of Lorraine at Nancy

KU - Probability Seminar - 2013

Samy T. (Nancy) Rough Paths 3 KU 2013 1 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 2 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 3 / 46

Examples of fBm paths

H = 0.3 H = 0.5 H = 0.7

Samy T. (Nancy) Rough Paths 3 KU 2013 4 / 46

General strategy

Aim: Define and solve an equation of the type: yt = a +

t

0 σ(ys) dBs, where B is fBm.

Properties of fBm: Generally speaking, take advantage of two aspects of fBm: Gaussianity Regularity Remark: For 1/3 < H < 1/2, Young integral isn’t suficient Levy area: We shall see that the following exists: B2,i,j

st

=

t

s dBi u

u

s dBj v ∈ C2γ 2

for γ < H Strategy: Given B and B2 solve the equation in a pathwise manner

Samy T. (Nancy) Rough Paths 3 KU 2013 5 / 46

Pathwise strategy

Aim: For x ∈ Cγ

1 con 1/3 < γ < 1/2, define and solve an equation of

the type: yt = a +

t

0 σ(yu) dxu

(1) Main steps: Define an integral

zs dxs for z: function whose increments are

controlled by those of x Solve (1) by fixed point arguments in the class of controlled processes Remark: Like in the previous chapters, we treat a real case and b ≡ 0 for notational sake. Caution: d-dimensional case really different here, because of x2

Samy T. (Nancy) Rough Paths 3 KU 2013 6 / 46

Heuristics (1)

Hypothesis: Solution yt exists in a space Cγ

1 ([0, T])

A priori decomposition for y: δyst ≡ yt − ys =

t

s σ(yv)dxv

= σ(ys) δxst +

t

s [σ(yv) − σ(ys)]dxv

= ζs δxst + rst Expected coefficients regularity: ζ = σ(y): bounded, γ-Hölder, r: 2γ-Hölder

Samy T. (Nancy) Rough Paths 3 KU 2013 7 / 46

Heuristics (2)

Start from controlled structure: Let z such that δzst = ζs δxst + rst, with ζ ∈ Cγ, r ∈ C2γ (2) Formally:

t

s zvdxv

= zs δxst +

t

s δzsv dxv

= zs δxst + ζs

t

s δxsv dxv +

t

s rsv dxv

= zs δxst + ζs x2

st +

t

s rsv dxv

Samy T. (Nancy) Rough Paths 3 KU 2013 8 / 46

Heuristics (3)

Formally, we have seen: z satisfies

t

s zvdxv = zs δxst + ζsx2 st +

t

s rsvdxv

Integral definition: zs δxst trivially defined ζsx2

st well defined, if Levy area x2 provided

t

s rsvdBv defined through operator Λ if r ∈ C2γ 2 , x ∈ Cγ 1 and

3γ > 1 Remark:

• We shall define

t

s zv dxv more rigorously

• Equation (1) solved within class of proc. with decomposition (2)

Samy T. (Nancy) Rough Paths 3 KU 2013 9 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 10 / 46

Let z ∈ Cκ

1 with 1/3 < κ ≤ γ.

We say that z is a process controlled by x, if z0 = a ∈ R, and δz = ζδx + r, i.e. δzst = ζs δxst + rst, s, t ∈ [0, T], (3) with ζ ∈ Cκ

1 , and r is a remainder such that r ∈ C2κ 2 .

Controlled processes space denoted by Qκ,a, and a controlled process z ∈ Qκ,a should be considered as a couple (z, ζ). Natural semi-norm on Qκ,a: N[z; Qκ,a] = N[z; Cκ

1 ] + N[ζ; Cb 1] + N[ζ; Cκ 1 ] + N[r; C2κ 2 ]

with N[g; Cκ

1 ] = gκ and N[ζ; Cb 1(V )] = sup0≤s≤T |ζs|V.

Definition 1.

Samy T. (Nancy) Rough Paths 3 KU 2013 11 / 46

Operations on controlled processes

In order to solve equations, two preliminary steps:

1

Study of transformation z → ϕ(z) for

◮ Controlled process z ◮ Smooth function ϕ 2

Integrate controlled processes with respect to x

Samy T. (Nancy) Rough Paths 3 KU 2013 12 / 46

Composition of controlled processes

Consider z ∈ Qκ,a, ϕ ∈ C 2

• b. Define

ˆ z = ϕ(z), ˆ a = ϕ(a). Then ˆ z ∈ Qκ,ˆ

a, and

δˆ z = ˆ ζδx + ˆ r, with ˆ ζ = ∇ϕ(z)ζ and ˆ r = ∇ϕ(z)r + [δ(ϕ(z)) − ∇ϕ(z)δz] . Furthermore, N[ˆ z; Qκ,ˆ

a] ≤ cϕ,T (1 + N 2[z; Qκ,a]).

Proposition 2.

Samy T. (Nancy) Rough Paths 3 KU 2013 13 / 46

Composition of controlled processes (2)

Remark: In previous proposition Quadratic bound instead of linear as in the Young case Due to Taylor expansions of order 2 Next step: Define J (z dx) for a controlled process z: Start with smooth x, z Try to recast J (z dx) with expressions making sense for a controlled process z ∈ Cκ

1

Samy T. (Nancy) Rough Paths 3 KU 2013 14 / 46

Integration of smooth controlled processes

Hypothesis: x, ζ smooth functions, r smooth increment Smooth controlled process z ∈ Q1,a, namely δzst = ζs δxst + rst Expression of the integral: J (zdx) defined as Riemann integral and

t

s zudxu = zs[xt − xs] +

t

s [zu − zs]dxu

Otherwise stated: J (z dx) = z δx + J (δz dx).

Samy T. (Nancy) Rough Paths 3 KU 2013 15 / 46

Integration of smooth controlled processes (2)

Levy area shows up: if δzst = ζs δxst + rst, J (z dx) = z δx + J (ζδx dx) + J (r dx). (4) Transformation of J (ζδx dx): Jst(ζδx dx) =

t

s ζs [(δx)sudxu] = ζsx2 st

Plugging in (4) we get J (z dx) = z δx + ζ x2 + J (r dx) Multidimensional case:

t

s ζs [δxsu dxu] ←

→

t

s ζij s

• δx i

su dx j u

• = ζij

s x2,ij st

Samy T. (Nancy) Rough Paths 3 KU 2013 16 / 46

Levy area

Recall: J (z dx) = z δx + ζ x2 + J (r dx) ֒ → For γ < 1/2, x2 enters as an additional data Hypothesis: Path x is γ-Hölder with γ > 1/3, and admits a Levy area, i.e x2 ∈ C2γ

2 (Rd,d),

formally defined as x2 = ”J (dxdx)”, and satisfying: δx2 = δx ⊗ δx, i.e. δx2,ij = δx i

su δx j ut,

for any s, u, t ∈ [0, T] and i, j ∈ {1, . . . , d}. Remark:

• If x is a regular path, Levy area in the Riemann sense.
• fBm also admits a Levy area in the Stratonovich sense.

Samy T. (Nancy) Rough Paths 3 KU 2013 17 / 46

Integration of smooth controlled processes (3)

Analysis of J (r dx): we have seen J (r dx) = J (z dx) − z δx − ζ x2 Apply δ on each side of the identity: [δ(J (r dx))]sut = δzsu δxut + δζsu x2

ut − ζs δx2 sut

= ζs δxsu δxut + rsu δxut + δζsu x2

ut − ζs δxsu δxut

= rsu δxut + δζsu x2

ut.

Expression with Λ: If r ∈ C2κ

2 , δx ∈ Cγ 2 , δζ ∈ Cκ 2 , x2 ∈ C2γ 2 ,

with κ + 2γ > 2κ + γ > 1, then: δ(J (r dx)) = r δx + δζ x2 ⇒ J (r dx) = Λ(r δx + δζ x2)

Samy T. (Nancy) Rough Paths 3 KU 2013 18 / 46

Integration of smooth controlled processes (4)

Conclusion: We have seen: J (z dx) = z δx + ζ x2 + J (r dx) J (r dx) = Λ(r δx + δζ x2) Thus, if m, x are smooth paths: J (z dx) = z δx + ζ x2 + Λ(r δx + δζ x2) Substantial gain: This expression can be extended to irregular paths!

Samy T. (Nancy) Rough Paths 3 KU 2013 19 / 46

Let x ∈ Cγ

1 , with 1/3 < κ < γ, and Levy area x2. Let z ∈ Qκ,b,

such that δzst = ζs(δx)st + rst, with ζ ∈ Cκ

1 , r ∈ C2κ 2

Define ℓ by z0 = a ∈ R, and δℓ ≡ J (z dx) = z δx + ζ · x2 − Λ(r δx + δζ · x2). Then ℓ is an element of Qκ,a and:

1

The semi-norm of ℓ in Qκ,a satisfies: N[ℓ; Qκ,a] ≤ cx

• 1 + T γ−κN[z; Qκ,b]
• 2

We have Jst(z dx) = lim

|πst|→0 n

• i=0
• zti(δx)ti,ti+1 + ζti · x2

ti,ti+1

• Proposition 3.

Samy T. (Nancy) Rough Paths 3 KU 2013 20 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 21 / 46

Pathwise strategy

Hypothesis: x is a function of Cγ

1 with 1/3 < γ ≤ 1/2.

It x admits a Levy area x2 Aim: We wish to define and solve an equation of the form: yt = a +

t

0 σ(ys) dxs

(5) Meaning of the equation: y ∈ Qa,κ, and δy = J (σ(y) dx)

Samy T. (Nancy) Rough Paths 3 KU 2013 22 / 46

Fixed point: strategy

A map on a small interval: Consider an interval [0, τ], with τ to be determined later Consider κ such that 1/2 < κ < γ < 1 In this interval, consider Γ : Qa,κ([0, τ]) → Qa,κ([0, τ]) defined by: Γ(z) = ˆ z, with ˆ z0 = a, and for s, t ∈ [0, τ]: δˆ zst =

t

s σ(zr)dxr = Jst(σ(z) dx)

Aim: See that for a small enough τ, the map Γ is a contraction ֒ → our equation admits a unique solution in Cκ

1 ([0, τ])

Remark: Same kind of computations as in the Young case ֒ → but requires more work (quadratic estimates, patching)!

Samy T. (Nancy) Rough Paths 3 KU 2013 23 / 46

Existence-uniqueness theorem

Let x ∈ Cγ

1 , with 1/3 < κ < γ and Levy area x2.

Let σ : R → R be a C 3

b function. Then

1

Equation δy = J (σ(y) dx) admits a unique solution y in Qκ,a for any 1/3 < κ < γ.

2

Application (a, x, x2) → y is continuous from R × Cγ

1 × C2γ 2

to Qκ,a. Theorem 4.

Samy T. (Nancy) Rough Paths 3 KU 2013 24 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 25 / 46

Levy area of fBm

Let B be a d-dimensional fBm, with H > 1/3, and 1/3 < γ <

• H. Almost surely, the paths of B:

1

Lye into Cγ

1

2

Admit a Levy area B2 ∈ C2γ

2

such that δB2 = δB ⊗ δB, i.e. B2,ij

sut = δBi su δBj ut

Proposition 5. Proof: B ∈ Cγ

1 almost surely: already seen (Kolmogorov criterion)

Samy T. (Nancy) Rough Paths 3 KU 2013 26 / 46

Geometric and weakly geometric Levy area

Remark: The stack B2 as defined in Proposition is called a weakly geometric second order rough path above X ֒ → allows a reasonable differential calculus When there exists a family Bε such that

◮ Bε is smooth ◮ B2,ε is the iterated Riemann integral of Bε ◮ B2 = limε→0 B2,ε

then one has a so-called geometric rough path above B ֒ → easier physical interpretation

Samy T. (Nancy) Rough Paths 3 KU 2013 27 / 46

Levy area construction for fBm: history

Situation 1: H > 1/4 ֒ → 3 possible geometric rough paths constructions for B. Malliavin calculus tools (Ferreiro-Utzet) Regularization or linearization of the fBm path (Coutin-Qian) Analytic approximation (Unterberger) Situation 2: d = 1 ֒ → Then one can take B2

st = (Bt−Bs)2 2

Situation 3: H ≤ 1/4, d > 1 The constructions by approximation diverge Existence result by dyadic approximation (Lyons-Victoir) Recent advances (Unterberger, Nualart-T) for weakly geometric Levy area construction

Samy T. (Nancy) Rough Paths 3 KU 2013 28 / 46

fBm kernel

Recall: B is a d-dimensional fBm, with Bi

t =

• R Kt(u) dW i

u,

t ≥ 0, where W is a d-dimensional Wiener process and Kt(u) ≈ (t − u)H− 1

2 1{0<u<t}

∂tKt(u) ≈ (t − u)H− 3

2 1{0<u<t}. Samy T. (Nancy) Rough Paths 3 KU 2013 29 / 46

Heuristics: fBm differential

Formal differential: we have Bj

v =

v

0 Kv(u) dW j u and thus formally for H > 1/2

˙ Bj

v =

v

0 ∂vKv(u) dW j u

Formal definition of the area: Consider Bi. Then formally

1

0 Bi v dBj v

=

1

0 Bi v

v

0 ∂vKv(u) dW j u

• dv

=

1 1

u ∂vKv(u) Bi v dv

• dW j

u

This works for H > 1/2 since H − 3/2 > −1.

Samy T. (Nancy) Rough Paths 3 KU 2013 30 / 46

Heuristics: fBm differential for H < 1/2

Formal definition of the area for H < 1/2: Use the regularity of Bi and write

1

0 Bi v dBj v =

1 1

u ∂vKv(u) Bi v dv

• dW j

u

=

1 1

u ∂vKv(u) δBi uv dv

• dW j

u

+

1

0 K1(u) Bi u dW j u.

Control of singularity: ∂vKv(u) δBi

uv ≈ (v − u)H−3/2+H

֒ → Definition works for 2H − 3/2 > −1, i.e. H > 1/4! Hypothesis:

1

0 Bi v dBj v well defined as stochastic integral

Samy T. (Nancy) Rough Paths 3 KU 2013 31 / 46

Levy area construction

Admitted: for 0 ≤ s < t ≤ T, one can define the stochastic integral B2

st =

t

s dBu ⊗

u

s dBv,

• i. e.

B2,ij

st =

t

s dBi u

u

s dBj v,

If i = j: B2

st(i, i) = 1 2(Bt − Bs)2

If i = j: B2,ij

st , with Bi considered as deterministic path, is a Wiener

integral w.r.t Bj Algebraic relation: δB2 = δB ⊗ δB, trivial

Samy T. (Nancy) Rough Paths 3 KU 2013 32 / 46

Regularity criterion in C2

Let g ∈ C2. Then, for any γ > 0 and p ≥ 1 we have gγ ≤ c (Uγ;p(g) + δgγ) , with Uγ;p(g) =

T T

|gst|p |t − s|γp+2ds dt

1/p

. Lemma 6.

Samy T. (Nancy) Rough Paths 3 KU 2013 33 / 46

Levy area of fBm: regularity

Strategy: Apply our regularity criterion to g = B2 Term 2: We have seen: δB2 = δB ⊗ δB B ∈ Cγ

1

⇒ δB ⊗ δB ∈ C2γ

3

Term 1: For p ≥ 1 we shall control E

• Uγ;p(B2)
• p

=

T T

E

• |B2

st|p

|t − s|γp ds dt

Samy T. (Nancy) Rough Paths 3 KU 2013 34 / 46

Moments of B2

Aim: Control of E

• |B2

st|p

Scaling and stationarity arguments: E

• |B2

st(i, j)|p

= E

• t

s dBi u

u

s dBj v

• p

= |t − s|2pH E

• 1

0 dBi u

u

0 dBj v

• p

Stochastic analysis arguments: Since

1

0 dBi u

u

0 dBj v is element of the second chaos of fBm:

E

• 1

0 dBi u

u

0 dBj v

• p

≤ cp,1 E

• 1

0 dBi u

u

0 dBj v

• 2

≤ cp,2

Samy T. (Nancy) Rough Paths 3 KU 2013 35 / 46

Levy area of fBm: regularity (2)

Recall: B2γ ≤ c

• Uγ;p(B2) + δB2γ
• Computations for Uγ;p(B2):

Let γ < 2H, and p such that γ + 2/p < 2H. Then: E

• Uγ;p(B2)
• p

=

T T

E

• |B2

st|p

|t − s|γp+2ds dt ≤ cp

T T

|t − s|2pH |t − s|p(γ+2/p)ds dt ≤ cp Conclusion:

• B2 ∈ C2γ

2

for any γ < H

• One can solve equation (5) driven by fBm with H > 1/3!

Samy T. (Nancy) Rough Paths 3 KU 2013 36 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 37 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 38 / 46

Rough path assumptions

Regularity of X: X ∈ Cγ(Rd) with γ > 0. Iterated integrals: X allows to define Xn

st(i1, . . . , in) =

• s≤u1<···<un≤t dXu1(i1) dXu2(i2) · · · dXun(in),

for 0 ≤ s < t ≤ T, n ≤ ⌊1/γ⌋ and i1, . . . , in ∈ {1, . . . , d}. Regularity of the iterated integrals: Xn ∈ Cnγ

2 (Rdn), where

N[g; Cκ

2 ] ≡

sup

0≤s<t≤T

|gst| |t − s|κ

Samy T. (Nancy) Rough Paths 3 KU 2013 39 / 46

Main rough paths result

Theorem (loose formulation): Under the assumption of the previous slide, plus regularity assumptions on σ, one can

1

Obtain change of variables formula of Itô’s type

2

Solve equations of the form dYt = σ(Yt)dXt Moreover, the application F : Rn × Cγ

2 (Rd) × · · · × Cnγ 2 (Rdn)

− → Cγ(Rm) (a, x1, . . . , xn) → Y is a continuous map

Rough paths theory

dx, dxdx

Smooth V0, . . . , Vd

Vj(x) dx j

dy = Vj(y)dx j

Samy T. (Nancy) Rough Paths 3 KU 2013 40 / 46

Meaning of the nth iterated integral

Definition: The nth order iterated integral associated to X is an element {Xn

st(i1, . . . , in); s ≤ t, 1 ≤ i1, . . . , in ≤ d} satisfying:

(i) The regularity condition Xn ∈ Cnγ

2 (Rdn).

(ii) The multiplicative property: δXn

sut(i1, . . . , in) = n−1

• n1=1

Xn1

su(i1, . . . , in1)Xn−n1 ut

(in1+1, . . . , in). (iii) The geometric relation: Xn

st(i1, . . . , in) Xm st(j1, . . . , jm)

can be expressed in terms of higher order integrals Remark: The notion of controlled process is also more complicated for higher order rough paths.

Samy T. (Nancy) Rough Paths 3 KU 2013 41 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 42 / 46

Geometrical structures

Lie algebra: In general (1, X1, . . . , Xn) ∈ R ⊕ Rd ⊕ (Rd)n ֒ → Lie algebra structure and associated Lie group: Gn(Rd) ֒ → Structures introduced by Chen in the ’50s Rough path: γ-Hölder function with values in Gn(Rd) Two important relations:

• (1, X1, . . . , Xn) determines all the iterated integrals if n ≥ ⌊1/γ⌋
• Any element of Gn(Rd) can be realized as iterated integrals of a

smooth function Solving equations: Two possibilities

• Show that (y, x) is a single rough path
• Approximations, due to the second important relation above

Samy T. (Nancy) Rough Paths 3 KU 2013 43 / 46

Lyons theory vs. algebraic integration

Advantages of Lyons’ approach: Elegant formalism (mixing geometry, analysis, probability) Approximation in Gn(Rd) yields powerful estimates:

◮ Moments of solution to RDEs ◮ Differential of RDEs

Advantages of algebraic integration: Simpler formalism Notion of controlled process can be adapted easily to many situations:

◮ Evolution equations ◮ Volterra equations ◮ Delay equations ◮ Integration in the plane

Some results are hard to express without controlled processes: ֒ → Norris type lemma, application to Burgers (Hairer)

Samy T. (Nancy) Rough Paths 3 KU 2013 44 / 46

Sketch

1

Heuristics

2

Controlled processes

3

Differential equations

4

Aplication to fBm

5

Final remarks Higher order structures Lyons theory Some projects

Samy T. (Nancy) Rough Paths 3 KU 2013 45 / 46

Current research directions

Non exhaustive list: Hörmander type theorem for SDEs driven by Gaussian processes Further study of the law of Gaussian SDEs: Capacity, Gaussian bounds Statistical aspects of rough differential equations Rough paths in the plane: Comparison between Skorohod and pathwise integrations Viscosity solutions for rough PDEs

Samy T. (Nancy) Rough Paths 3 KU 2013 46 / 46