Rough paths methods 1: IntroductionSamy.Tindel/some-slides/ku-rough-paths1.… · Samy T. (Nancy) Rough Paths 1 KU 2013 8 / 44. ExamplesoffBmpaths H = 0.3 H = 0.5 H = 0.7 Samy T

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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1 IntroductionMotivations for rough paths techniquesSummary of rough paths theory

2 Usual Brownian caseBasic properties of Brownian motionItô’s stochastic integralStochastic differential equations


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Nancy 104/09/13 05:45Nancy, France - Google Maps

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Address Nancy


Nancy 2: Stanislas square


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Equation under consideration

Equation:Standard differential equation driven by fBm, Rn-valued

Yt = a +∫ t

0V0(Ys) ds +

d∑j=1

∫ t

0Vj(Ys) dB j

s , (1)

witht ∈ [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.


Fractional Brownian motion

B = (B1, . . . ,Bd)

B j centered Gaussian process, independence of coordinatesVariance of the increments:

E[|B jt − B j

s |2] = |t − s|2H

H− ≡ Hölder-continuity exponent of BIf H = 1/2, B = Brownian motionIf H 6= 1/2 natural generalization of BM

Remark: FBm widely used in applications


Examples of fBm paths

H = 0.3 H = 0.5 H = 0.7Samy T. (Nancy) Rough Paths 1 KU 2013 9 / 44

Paths for a linear SDE driven by fBmdYt = −0.5Ytdt + 2YtdBt , Y0 = 1

H = 0.5 H = 0.7

Blue: (Bt)t∈[0,1] Red: (Yt)t∈[0,1]


Some applications of fBm driven systems

Biophysics, fluctuations of a protein:New experiments at molecule scale→ Anomalous fluctuations recordedModel: Volterra equation driven by fBm→ Samuel KouStatistical estimation needed

Finance:Stochastic volatility driven by fBm (Sun et al. 2008)Captures long range dependences between transactions


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Rough paths assumptionsContext: Consider a Hölder path x and

For n ≥ 1, xn ≡ linearization of x with mesh 1/n→ xn piecewise linear.For 0 ≤ s < t ≤ 1, set

x2,n,i ,jst ≡∫

s<u<v<tdxn,i

u dxn,jv

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

Brief summary of rough paths theoryMain rough paths theorem (Lyons): Under previous assumptions→ Consider yn solution to equation

ynt = a +

∫ t

0V0(yn

u ) du +d∑

j=1

∫ t

0Vj(yn

u ) dxn,ju .

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

∫ t0 V0(Yu) du +

∑dj=1

∫ t0 Vj(Yu) dx j

u.

Rough paths theoryRough paths theory

∫dx ,

∫ ∫dxdx

Smooth V0, . . . ,VdRough paths theory

∫dx ,

∫ ∫dxdx

Smooth V0, . . . ,Vd

∫Vj(x) dx j

dy = Vj(y)dx j



ynt = a +

∫ t

0V0(yn

u ) du +d∑

j=1

∫ t

0Vj(yn

u ) dxn,ju .


∫ t0 V0(Yu) du +

∑dj=1

∫ t0 Vj(Yu) dx j

u.

Rough paths theory

Rough paths theory

∫dx ,

∫ ∫dxdx

Smooth V0, . . . ,VdRough paths theory

∫dx ,

∫ ∫dxdx


∫Vj(x) dx j

dy = Vj(y)dx j



ynt = a +

∫ t

0V0(yn

u ) du +d∑

j=1

∫ t

0Vj(yn

u ) dxn,ju .


∫ t0 V0(Yu) du +

∑dj=1

∫ t0 Vj(Yu) dx j

u.

Rough paths theory

Rough paths theory

∫dx ,

∫ ∫dxdx


Rough paths theory

∫dx ,

∫ ∫dxdx


∫Vj(x) dx j

dy = Vj(y)dx j



ynt = a +

∫ t

0V0(yn

u ) du +d∑

j=1

∫ t

0Vj(yn

u ) dxn,ju .


∫ t0 V0(Yu) du +

∑dj=1

∫ t0 Vj(Yu) dx j

u.

Rough paths theoryRough paths theory

∫dx ,

∫ ∫dxdx


Rough paths theory

∫dx ,

∫ ∫dxdx


∫Vj(x) dx j

dy = Vj(y)dx j


Iterated integrals and fBm

Nice situation: H > 1/4→ 2 possible constructions for geometric iterated integrals of B.

Malliavin calculus tools→ Ferreiro-UtzetRegularization 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) dB jt ,

in the rough paths sense.

Remark: Recent extensions to H ≤ 1/4 (Unterberger, Nualart-T).


Study of equations driven by fBmBasic properties:

1 Moments of the solution2 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 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


Extensions of the rough paths formalismStochastic PDEs:

Equation: ∂tYt(ξ) = ∆Yt(ξ) + σ(Yt(ξ)) xt(ξ)

(t, ξ) ∈ [0, 1]× Rd

Easiest case: x finite-dimensional noiseMethods:→ viscosity solutions or adaptation of rough paths methods

KPZ equation:Equation: ∂tYt(ξ) = ∆Yt(ξ) + (∂ξYt(ξ))2 + xt(ξ)−∞(t, ξ) ∈ [0, 1]× Rx ≡ space-time white noiseMethods:

I Extension of rough paths to define (∂xYt(ξ))2

I Renormalization techniques to remove ∞


Aim

1 Define rigorously an equation driven by fBm or general Gaussianprocesses

2 Solve this kind of equation3 Investigate some properties of the solution

I Law of XtI Statistical issuesI Other systems (delayed equations, stochastic PDEs)


General strategy

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

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

2 For each case, 2 main steps:I Definition of a stochastic integral

∫usdBs

for a reasonable class of processes uI Resolution of the equation by means of a fixed point method


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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 B i independent 1-d Brownian motions


Kolmogorov criterionNotation: If f : [0,T ]→ Rd is a function, we shall denote:

δfst = ft − fs , and ‖f ‖µ = sups,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 X of X such that almost surelyX ∈ Cγ1 for any γ < β/α, i.e P(ω; ‖ X (ω)‖γ <∞) = 1.

Theorem 2.


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.


Examples of fBm paths

H = 0.3 H = 0.5 H = 0.7Samy T. (Nancy) Rough Paths 1 KU 2013 25 / 44

Brownian irregularityRecall our strategy in order to solve SDEs:

Definition of a stochastic integral∫

usdBsfor a reasonable class of processes uResolution by fixed point method

For all ω ∈ Ω almost surely, the path Bt(ω); t ∈ [0,T ] is notdifferentiable in t for all t ∈ [0,T ].

Theorem 4.

Consequence for the definition of integrals:one cannot define

∫usdBs as a Riemann integral: the sums∑

si∈πusi δB(ω)si si+1

are not convergent in general as |π| → 0.Samy T. (Nancy) Rough Paths 1 KU 2013 26 / 44

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Simple processes

A process u is called simple if it can be decomposed as:

ut = ξ010(t) +N−1∑i=1

ξi 1(ti ,ti+1](t),

withN ≥ 1, (t1, . . . , tN) partition of [0,T ] with t1 = 0, tN = Tξi ∈ 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.


Integral of a simple process

Let u be a simple process. Let s, t ∈ [0,T ] such that s ≤ tand:

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 δBti ti+1 + ξn δBtnt

Definition 6.

Remark:For simple processes, stochastic integral coincides with usual one.


Integral of a simple process: properties

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

1 Jtt(u dB) = 02 Jst((αu + βv) dB) = αJst(u dB) + βJst(v dB) forα, β ∈ R

3 E[Jst(u dB)|Fs ] = 0, i.e martingale property4 E[(Jst(u dB))2] =

∫ ts E[u2

τ ] dτ5 E[(Jst(u dB))2|Fs ] =

∫ ts E[u2

τ |Fs ] dτ

Proposition 7.


Proof of claim 5

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

E[ξi(δB)ti ti+1 ξj(δB)tj tj+1

∣∣∣∣Ftj

]= ξiξj (δB)ti ti+1 E

[(δB)tj tj+1

∣∣∣∣Ftj

]= ξiξj (δB)ti ti+1 E

[(δB)tj tj+1

]= 0


Proof of claim 5 (2)

Hence

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

= E(ξm−1(δB)stm +

n−1∑i=m

ξi(δB)ti ti+1 + ξn(δB)tnt

)2 ∣∣∣∣Fs

= E

[ξ2

m−1(δB)2stm +

n−1∑i=m

ξ2i (δB)2

ti ti+1 + ξ2n(δB)2

tnt

∣∣∣∣Fs

]

= E[ξ2m−1|Fs ](tm − s) +

n−1∑i=m

E[ξ2i |Fs ](ti+1 − ti) + E[ξ2

n|Fs ](t − tn)

=∫ t

sE[u2

τ |Fs ] dτ


L2a space

The set of left continuous square integrable processes u suchthat ut ∈ Ft is called L2

a. The norm on L2a is defined by:

‖u‖2L2

a≡∫ T

0E[u2

s

]ds

Definition 8.

Let u ∈ L2a. There exists a sequence (un)n≥0 of simple processes

such thatlim

n→∞‖u − un‖L2

a = 0

Proposition 9.


Extension of the stochastic integral

Let u ∈ L2a, 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

sE[(un

w − umw )2

]dw .

Therefore,(Zn)n≥0 ≡ (Jst(un))n≥0

is a Cauchy sequence in L2(Ω).


Extension of the stochastic integral (2)

Let u ∈ L2a. 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)− limn→∞Jst(un dB),

where (un)n≥0 is any sequence of simple processes convergingto u.

Definition 11.

Remark: Properties (1)–(5) established for simple processes are stillsatisfied for processes in L2

a.


Integral of a process in L2a: properties

Let u be a process in L2a, Jst(u dB) its stochastic integral. Then:

1 Jtt(u dB) = 02 Jst((αu + βv) dB) = αJst(u dB) + βJst(v dB) forα, β ∈ R

3 E[Jst(u dB)|Fs ] = 0, i.e martingale property4 E[(Jst(u dB))2] =

∫ ts E[u2

τ ] dτ5 E[(Jst(u dB))2|Fs ] =

∫ ts E[u2

τ |Fs ] dτ

Proposition 12.

Remark: For this construction, essential use ofIndependence of increments of BProbabilistic convergence of L2 typeSamy T. (Nancy) Rough Paths 1 KU 2013 36 / 44

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Equation under consideration

Xt = a +∫ t

0σ(Xs)dBs +

∫ t

0b(Xs)ds, t ∈ [0,T ] (2)

a ∈ Rn initial conditionb, σ coefficients in C 1

b

B = (B1, . . . ,Bd) d-dimensional Brownian motionB i iid Brownian motions


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 methodsb ≡ 0

Simplified equation: we end up with

Xt = a +∫ t

0σ(Xs)dBs , t ∈ [0,T ] (3)

a ∈ R, σ ∈ C 1b (R)

B is a 1-d Brownian motion


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 : L2a([0, τ ])→ L2

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

δYst =∫ t

sσ(Yu)dBu = Jst(σ(Y ) dB)

⇔Yt = a +∫ t

0σ(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 ])


Contraction argument in [0, τ ]Definition of 2 processes:Let Y 1,Y 2 ∈ L2

a([0, τ ]). Set Y i = Γ(Y i). Then

Y 1t − Y 2

t =∫ t

0

[σ(Y 1

s )− σ(Y 2s )]

dBs

Evaluation of the difference:With cσ = ‖σ′‖∞, we have:

‖Y 1 − Y 2‖2L2

a([0,τ ]) =∫ τ

0E[(Y 1

t − Y 2t )2

]dt

=∫ τ

0dt∫ t

0E[(σ(Y 1

s )− σ(Y 2s ))2

]ds

≤ c2σ

∫ τ

0dt∫ t

0E[(Y 1

s − Y 2s )2

]ds

≤ c2σ‖Y 1 − Y 2‖2

L2a([0,τ ]) τ


Contraction argument in [0, τ ] (2)Contraction: Choose τ so that

c2στ =

14 , i.e τ =

14c2σ

We have obtained:

‖Γ(Y 1)− Γ(Y 2)‖L2a([0,τ ]) ≤

12‖Y

1 − Y 2‖L2a([0,τ ])

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

Let τ = 14c2

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

X to equation (3) in L2a([0, τ ]).

Lemma 13.


From [0, τ ] to [τ, 2τ ]

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

Γ : L2a([τ, 2τ ])→ L2

a([τ, 2τ ])

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

(δY )st =∫ t

sσ(Yu)dBu = Jst(σ(Y ) dB)

⇔Yt = Xτ +∫ t

τσ(Ys)dBs = Xτ +

∫ t

τσ(Ys)dBτ

s ,

with Bτs = Bs − Bτ , independent of Fτ .

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

a([τ, 2τ ]).


Existence and uniqueness result

Iterating the fixed point arguments, we get:

Let σ ∈ C 1b (R), T > 0 and a ∈ R.

Then equation (3) admits a unique solution in L2a([0,T ]).

Theorem 14.

Remark: possible simple improvements:d-dimensional caseDrift coefficient bLipschitz coefficients σ, b


Documents

Rough paths methods 1: IntroductionSamy.Tindel/some-slides/ku-rough-paths1.… · Samy T. (Nancy) Rough Paths 1 KU 2013 8 / 44. ExamplesoffBmpaths H = 0.3 H = 0.5 H = 0.7 Samy T