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Rough paths methods 1: Introduction Samy Tindel University of Lorraine at Nancy KU - Probability Seminar - 2013 Samy T. (Nancy) Rough Paths 1 KU 2013 1 / 44 Sketch Introduction 1 Motivations for rough paths techniques Summary of rough

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Rough paths methods 1: Introduction

Samy Tindel

University of Lorraine at Nancy

KU - Probability Seminar - 2013

Samy T. (Nancy) Rough Paths 1 KU 2013 1 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 2 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 3 / 44

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Nancy 1

04/09/13 05:45 Nancy, France - Google Maps

Address Nancy

Samy T. (Nancy) Rough Paths 1 KU 2013 4 / 44

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Nancy 2: Stanislas square

Samy T. (Nancy) Rough Paths 1 KU 2013 5 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 6 / 44

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Equation under consideration

Equation: Standard differential equation driven by fBm, Rn-valued Yt = a +

t

0 V0(Ys) ds + d

  • j=1

t

0 Vj(Ys) dBj s,

(1) with t ∈ [0, 1]. Vector fields V0, . . . , Vd in C∞

b .

A d-dimensional fBm B with 1/3 < H < 1. Note: some results will be extended to H > 1/4.

Samy T. (Nancy) Rough Paths 1 KU 2013 7 / 44

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Fractional Brownian motion

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 Remark: FBm widely used in applications

Samy T. (Nancy) Rough Paths 1 KU 2013 8 / 44

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Examples of fBm paths

H = 0.3 H = 0.5 H = 0.7

Samy T. (Nancy) Rough Paths 1 KU 2013 9 / 44

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Paths for a linear SDE driven by fBm

dYt = −0.5Ytdt + 2YtdBt, Y0 = 1 H = 0.5 H = 0.7 Blue: (Bt)t∈[0,1] Red: (Yt)t∈[0,1]

Samy T. (Nancy) Rough Paths 1 KU 2013 10 / 44

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Some applications of fBm driven systems

Biophysics, fluctuations of a protein: New experiments at molecule scale ֒ → Anomalous fluctuations recorded Model: Volterra equation driven by fBm ֒ → Samuel Kou Statistical estimation needed Finance: Stochastic volatility driven by fBm (Sun et al. 2008) Captures long range dependences between transactions

Samy T. (Nancy) Rough Paths 1 KU 2013 11 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 12 / 44

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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 x2,n,i,j

st

≡

  • s<u<v<t dx n,i

u dx n,j v

Rough paths assumption 1: x is a Cγ function with γ > 1/3. The process x2,n converges to a process x2 as n → ∞ ֒ → in a C2γ space. Rough paths assumption 2: Vector fields V0, . . . , Vj in C∞

b .

Samy T. (Nancy) Rough Paths 1 KU 2013 13 / 44

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Brief summary of rough paths theory

Main rough paths theorem (Lyons): Under previous assumptions ֒ → Consider y n solution to equation y n

t = a +

t

0 V0(y n u ) du + d

  • j=1

t

0 Vj(y n u ) dx n,j u .

Then y n converges to a function Y in Cγ. Y can be seen as solution to ֒ → Yt = a +

t

0 V0(Yu) du + d j=1

t

0 Vj(Yu) dx j u.

Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44

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Brief summary of rough paths theory

Main rough paths theorem (Lyons): Under previous assumptions ֒ → Consider y n solution to equation y n

t = a +

t

0 V0(y n u ) du + d

  • j=1

t

0 Vj(y n u ) dx n,j u .

Then y n converges to a function Y in Cγ. Y can be seen as solution to ֒ → Yt = a +

t

0 V0(Yu) du + d j=1

t

0 Vj(Yu) dx j u. Rough paths theory

Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44

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Brief summary of rough paths theory

Main rough paths theorem (Lyons): Under previous assumptions ֒ → Consider y n solution to equation y n

t = a +

t

0 V0(y n u ) du + d

  • j=1

t

0 Vj(y n u ) dx n,j u .

Then y n converges to a function Y in Cγ. Y can be seen as solution to ֒ → Yt = a +

t

0 V0(Yu) du + d j=1

t

0 Vj(Yu) dx j u. Rough paths theory

dx, dxdx

Smooth V0, . . . , Vd

Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44

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Brief summary of rough paths theory

Main rough paths theorem (Lyons): Under previous assumptions ֒ → Consider y n solution to equation y n

t = a +

t

0 V0(y n u ) du + d

  • j=1

t

0 Vj(y n u ) dx n,j u .

Then y n converges to a function Y in Cγ. Y can be seen as solution to ֒ → Yt = a +

t

0 V0(Yu) du + d j=1

t

0 Vj(Yu) dx j u. Rough paths theory

dx, dxdx

Smooth V0, . . . , Vd

Vj(x) dx j

dy = Vj(y)dx j

Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44

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Iterated integrals and fBm

Nice situation: H > 1/4 ֒ → 2 possible constructions for geometric iterated integrals of B. Malliavin calculus tools ֒ → Ferreiro-Utzet Regularization or linearization of the fBm path ֒ → Coutin-Qian, Friz-Gess-Gulisashvili-Riedel Conclusion: for H > 1/4, one can solve equation dYt = V0(Yt) dt + Vj(Yt) dBj

t,

in the rough paths sense. Remark: Recent extensions to H ≤ 1/4 (Unterberger, Nualart-T).

Samy T. (Nancy) Rough Paths 1 KU 2013 15 / 44

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Study of equations driven by fBm

Basic properties:

1

Moments of the solution

2

Continuity w.r.t initial condition, noise More advanced natural problems:

1

Density estimates ֒ → Hu-Nualart + Lots of people

2

Numerical schemes ֒ → Neuenkirch-T, Friz-Riedel

3

Invariant measures, ergodicity ֒ → Hairer-Pillai, Cohen-Panloup-T

4

Statistical estimation (H, coeff. Vj) ֒ → Berzin-León, Hu-Nualart, Neuenkirch-T

−10 −5 5 10 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Samy T. (Nancy) Rough Paths 1 KU 2013 16 / 44

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Extensions of the rough paths formalism

Stochastic PDEs: Equation: ∂tYt(ξ) = ∆Yt(ξ) + σ(Yt(ξ)) ˙ xt(ξ) (t, ξ) ∈ [0, 1] × Rd Easiest case: x finite-dimensional noise Methods: ֒ → viscosity solutions or adaptation of rough paths methods KPZ equation: Equation: ∂tYt(ξ) = ∆Yt(ξ) + (∂ξYt(ξ))2 + ˙ xt(ξ) − ∞ (t, ξ) ∈ [0, 1] × R ˙ x ≡ space-time white noise Methods:

◮ Extension of rough paths to define (∂xYt(ξ))2 ◮ Renormalization techniques to remove ∞ Samy T. (Nancy) Rough Paths 1 KU 2013 17 / 44

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Aim

1

Define rigorously an equation driven by fBm or general Gaussian processes

2

Solve this kind of equation

3

Investigate some properties of the solution

◮ Law of Xt ◮ Statistical issues ◮ Other systems (delayed equations, stochastic PDEs) Samy T. (Nancy) Rough Paths 1 KU 2013 18 / 44

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General strategy

1

In order to reach the general case, we shall go through the following steps:

◮ Itô integral for usual Brownian motion ◮ Young integral (2 versions) for H > 1/2 ◮ Case 1/3 < H < 1/2, with a semi-pathwise method 2

For each case, 2 main steps:

◮ Definition of a stochastic integral

usdBs

for a reasonable class of processes u

◮ Resolution of the equation by means of a fixed point method Samy T. (Nancy) Rough Paths 1 KU 2013 19 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 20 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 21 / 44

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Definition of Brownian motion

Complete probability space: (Ω, F, (Ft)t≥0, P) One-dimensional Brownian motion: continuous adapted (Bt ∈ Ft) process such that B0 = 0 and: If 0 ≤ s < t, Bt − Bs independent of Fs (Bt − Bs) ∼ N(0, t − s) Definition 1. d-dimensional Brownian motion: B = (B1, . . . , Bd), where Bi independent 1-d Brownian motions

Samy T. (Nancy) Rough Paths 1 KU 2013 22 / 44

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Kolmogorov criterion

Notation: If f : [0, T] → Rd is a function, we shall denote: δfst = ft − fs, and f µ = sup

s,t∈[0,T]

|δfst| |t − s|µ Let X = {Xt; t ∈ [0, T]} be a process defined on (Ω, F, P), such that E [|δXst|α] ≤ c|t − s|1+β, for s, t ∈ [0, T], c, α, β > 0 Then there exists a modification ˆ X of X such that almost surely ˆ X ∈ Cγ

1 for any γ < β/α, i.e P(ω; ˆ

X(ω)γ < ∞) = 1. Theorem 2.

Samy T. (Nancy) Rough Paths 1 KU 2013 23 / 44

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Brownian pathwise regularity

The Brownian motion B is γ-Hölder continuous for any γ < 1/2, up to modification. Proposition 3. Proof: We have δBst ∼ N(0, t − s). Hence, for n ≥ 1, E

  • |δBst|2n

= cn|t − s|n i.e E

  • |δBst|2n

= cn|t − s|1+(n−1) Kolmogorov: B is γ-Hölder for γ < (n − 1)/2n = 1/2 − 1/(2n). Taking limits n → ∞, the proof is finished.

Samy T. (Nancy) Rough Paths 1 KU 2013 24 / 44

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Examples of fBm paths

H = 0.3 H = 0.5 H = 0.7

Samy T. (Nancy) Rough Paths 1 KU 2013 25 / 44

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Brownian irregularity

Recall our strategy in order to solve SDEs: Definition of a stochastic integral

usdBs

for a reasonable class of processes u Resolution by fixed point method For all ω ∈ Ω almost surely, the path {Bt(ω); t ∈ [0, T]} is not differentiable in t for all t ∈ [0, T]. Theorem 4. Consequence for the definition of integrals:

  • ne cannot define

usdBs as a Riemann integral: the sums

  • si∈π

usi δB(ω)sisi+1 are not convergent in general as |π| → 0.

Samy T. (Nancy) Rough Paths 1 KU 2013 26 / 44

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Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 27 / 44

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Simple processes

A process u is called simple if it can be decomposed as: ut = ξ01{0}(t) +

N−1

  • i=1

ξi 1(ti,ti+1](t), with N ≥ 1, (t1, . . . , tN) partition of [0, T] with t1 = 0, tN = T ξi ∈ Fti and |ξi| ≤ c with c > 0 Definition 5. Remark:

  • For notational sake, most of our computations in dimension 1.
  • We shall mention when the d-dimensional extension is non trivial.

Samy T. (Nancy) Rough Paths 1 KU 2013 28 / 44

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Integral of a simple process

Let u be a simple process. Let s, t ∈ [0, T] such that s ≤ t and: tm ≤ s < tm+1, and tn ≤ t < tn+1 We define Jst(u dB) = ”

t

s uwdBw” as:

Jst(u dB) = ξm δBstm +

n−1

  • i=m

ξi δBtiti+1 + ξn δBtnt Definition 6. Remark: For simple processes, stochastic integral coincides with usual one.

Samy T. (Nancy) Rough Paths 1 KU 2013 29 / 44

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Integral of a simple process: properties

Let u be a simple process, Jst(u dB) its stochastic integral. Then:

1

Jtt(u dB) = 0

2

Jst((αu + βv) dB) = αJst(u dB) + βJst(v dB) for α, β ∈ R

3

E[Jst(u dB)|Fs] = 0, i.e martingale property

4

E[(Jst(u dB))2] =

t

s E[u2 τ] dτ

5

E[(Jst(u dB))2|Fs] =

t

s E[u2 τ|Fs] dτ

Proposition 7.

Samy T. (Nancy) Rough Paths 1 KU 2013 30 / 44

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Proof of claim 5

If i < j, then (independence of increments of B) E

  • ξi(δB)titi+1 ξj(δB)tjtj+1
  • Ftj
  • = ξiξj (δB)titi+1 E
  • (δB)tjtj+1
  • Ftj
  • = ξiξj (δB)titi+1 E
  • (δB)tjtj+1
  • = 0

Samy T. (Nancy) Rough Paths 1 KU 2013 31 / 44

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Proof of claim 5 (2)

Hence E[(Jst(u dB))2|Fs] = E

 

  • ξm−1(δB)stm +

n−1

  • i=m

ξi(δB)titi+1 + ξn(δB)tnt

2

  • Fs

 

= E

  • ξ2

m−1(δB)2 stm + n−1

  • i=m

ξ2

i (δB)2 titi+1 + ξ2 n(δB)2 tnt

  • Fs
  • = E[ξ2

m−1|Fs](tm − s) + n−1

  • i=m

E[ξ2

i |Fs](ti+1 − ti) + E[ξ2 n|Fs](t − tn)

=

t

s E[u2 τ|Fs] dτ

Samy T. (Nancy) Rough Paths 1 KU 2013 32 / 44

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L2

a space The set of left continuous square integrable processes u such that ut ∈ Ft is called L2

  • a. The norm on L2

a is defined by:

u2

L2

a ≡

T

0 E

  • u2

s

  • ds

Definition 8. Let u ∈ L2

  • a. There exists a sequence (un)n≥0 of simple processes

such that lim

n→∞ u − unL2

a = 0

Proposition 9.

Samy T. (Nancy) Rough Paths 1 KU 2013 33 / 44

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Extension of the stochastic integral

Let u ∈ L2

a, limit of simple processes un. Then for s < t

The sequence (Jst(un dB))n≥0 converges in L2(Ω) Its limit does not depend on the sequence (un)n≥0 Proposition 10. Proof: We know that: E

  • (Jst(un − um))2

=

t

s E

  • (un

w − um w )2

dw. Therefore, (Zn)n≥0 ≡ (Jst(un))n≥0 is a Cauchy sequence in L2(Ω).

Samy T. (Nancy) Rough Paths 1 KU 2013 34 / 44

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Extension of the stochastic integral (2)

Let u ∈ L2

  • a. The stochastic integral of u with respect to B is

the process J (u dB) such that, for all s < t, Jst(u dB) = L2(Ω) − lim

n→∞ Jst(un dB),

where (un)n≥0 is any sequence of simple processes converging to u. Definition 11. Remark: Properties (1)–(5) established for simple processes are still satisfied for processes in L2

a.

Samy T. (Nancy) Rough Paths 1 KU 2013 35 / 44

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Integral of a process in L2

a: properties Let u be a process in L2

a, Jst(u dB) its stochastic integral. Then:

1

Jtt(u dB) = 0

2

Jst((αu + βv) dB) = αJst(u dB) + βJst(v dB) for α, β ∈ R

3

E[Jst(u dB)|Fs] = 0, i.e martingale property

4

E[(Jst(u dB))2] =

t

s E[u2 τ] dτ

5

E[(Jst(u dB))2|Fs] =

t

s E[u2 τ|Fs] dτ

Proposition 12. Remark: For this construction, essential use of Independence of increments of B Probabilistic convergence of L2 type

Samy T. (Nancy) Rough Paths 1 KU 2013 36 / 44

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SLIDE 40

Sketch

1

Introduction Motivations for rough paths techniques Summary of rough paths theory

2

Usual Brownian case Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations

Samy T. (Nancy) Rough Paths 1 KU 2013 37 / 44

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SLIDE 41

Equation under consideration

Xt = a +

t

0 σ(Xs)dBs +

t

0 b(Xs)ds,

t ∈ [0, T] (2) a ∈ Rn initial condition b, σ coefficients in C 1

b

B = (B1, . . . , Bd) d-dimensional Brownian motion Bi iid Brownian motions

Samy T. (Nancy) Rough Paths 1 KU 2013 38 / 44

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Notational simplification

Simplified setting: In order to ease notations, we shall consider: Real-valued solution and Brownian motion: n = d = 1. However, we shall use d-dimensional methods b ≡ 0 Simplified equation: we end up with Xt = a +

t

0 σ(Xs)dBs,

t ∈ [0, T] (3) a ∈ R, σ ∈ C 1

b(R)

B is a 1-d Brownian motion

Samy T. (Nancy) Rough Paths 1 KU 2013 39 / 44

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Fixed point: strategy

A map on a small interval: Consider an interval [0, τ], with τ to be determined later In this interval, consider Γ : L2

a([0, τ]) → L2 a([0, τ]) defined by:

Γ(Y ) = ˆ Y , with ˆ Y0 = a, and for all s, t ∈ [0, τ]: δ ˆ Yst =

t

s σ(Yu)dBu = Jst(σ(Y ) dB)

⇔ ˆ Yt = a +

t

0 σ(Yu)dBu

Aim: See that for a small enough τ, the map Γ is a contraction ֒ → our equation admits a unique solution in L2

a([0, τ])

Samy T. (Nancy) Rough Paths 1 KU 2013 40 / 44

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Contraction argument in [0, τ]

Definition of 2 processes: Let Y 1, Y 2 ∈ L2

a([0, τ]). Set ˆ

Y i = Γ(Y i). Then ˆ Y 1

t − ˆ

Y 2

t =

t

  • σ(Y 1

s ) − σ(Y 2 s )

  • dBs

Evaluation of the difference: With cσ = σ′∞, we have: ˆ Y 1 − ˆ Y 22

L2

a([0,τ]) =

τ

0 E

  • ( ˆ

Y 1

t − ˆ

Y 2

t )2

dt =

τ

0 dt

t

0 E

  • (σ(Y 1

s ) − σ(Y 2 s ))2

ds ≤ c2

σ

τ

0 dt

t

0 E

  • (Y 1

s − Y 2 s )2

ds ≤ c2

σY 1 − Y 22 L2

a([0,τ]) τ Samy T. (Nancy) Rough Paths 1 KU 2013 41 / 44

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SLIDE 45

Contraction argument in [0, τ] (2)

Contraction: Choose τ so that c2

στ = 1

4, i.e τ = 1 4c2

σ

We have obtained: Γ(Y 1) − Γ(Y 2)L2

a([0,τ]) ≤ 1

2Y 1 − Y 2L2

a([0,τ])

Conclusion: Classical fixed point argument ֒ → Unique solution for equation (3) in [0, τ]. Let τ =

1 4c2

σ , with cσ = σ′∞. There exists a unique solution

X to equation (3) in L2

a([0, τ]).

Lemma 13.

Samy T. (Nancy) Rough Paths 1 KU 2013 42 / 44

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SLIDE 46

From [0, τ] to [τ, 2τ]

New map Γ: In [τ, 2τ], consider the map Γ : L2

a([τ, 2τ]) → L2 a([τ, 2τ])

defined by: Γ(Y ) = ˆ Y , with ˆ Yτ = Xτ, and for s, t ∈ [τ, 2τ]: (δ ˆ Y )st =

t

s σ(Yu)dBu = Jst(σ(Y ) dB)

⇔ ˆ Yt = Xτ +

t

τ σ(Ys)dBs = Xτ +

t

τ σ(Ys)dBτ s ,

with Bτ

s = Bs − Bτ, independent of Fτ.

New fixed point argument: the same fixed point arguments yield a unique solution X to (3) in L2

a([τ, 2τ]).

Samy T. (Nancy) Rough Paths 1 KU 2013 43 / 44

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SLIDE 47

Existence and uniqueness result

Iterating the fixed point arguments, we get: Let σ ∈ C 1

b(R), T > 0 and a ∈ R.

Then equation (3) admits a unique solution in L2

a([0, T]).

Theorem 14. Remark: possible simple improvements: d-dimensional case Drift coefficient b Lipschitz coefficients σ, b

Samy T. (Nancy) Rough Paths 1 KU 2013 44 / 44