yamada-watanabe theorem in the fractional case · yamada-watanabe theorem in the fractional case...

$: Yamada-Watanabe theorem in the fractional case · Yamada-Watanabe theorem in the fractional case Antoine Bordas June 17, 2019 ENS Paris-Saclay & CIMAT 1$
Yamada-Watanabe theorem in the fractionalcase

Antoine BordasJune 17, 2019

ENS Paris-Saclay & CIMAT

1

Introduction

dXt = g(Xt)dBHt h(Xt) + h(Xt)d(BH

t )Tg(Xt) + b(Xt)dt

where

• g, h, b : R → R

• X0 ∈ Sp

• Xt is an Sp-valued process

• BH is a matrix fractional Brownian motion with H >12

Existence? Uniqueness?

2

Table of contents

1. Yamada-Watanabe theorem

2. Fractional Brownian Motion

3. Fractional Wishart process

4. Outcome

3

Yamada-Watanabe theorem

The original one

dxt = σ(xt)dBt + b(xt)dt

with Bt a Brownian motion,|σ(x) − σ(y)|2 ≤ ρ(|x − y|),∫ +∞

0ρ−1(x)dx = ∞ and b

Lipschtiz continuous

Pathwise uniqueness

4

Multidimensional version

Theorem (Graczyk, Malecki)Denote by Bt a p × p Brownian matrix and consider the matrix SDE onSp

dXt = g(Xt)dBth(Xt) + h(Xt)dBTt g(Xt) + b(Xt)dt

where g, h, b : R → R and X0 ∈ Sp. Suppose that

|g(x)h(x) − g(y)h(y)|2 ≤ ρ(|x − y|)

where ρ is a function such that∫ ∞

0ρ−1(x)dx = ∞, b is locally Lipschitz

and G(x, y) = g2(x)h2(y) + g2(y)h2(x) is locally Lipschitz and strictlypositive on {x = y}.Then the pathwise uniqueness holds, up to the collision time.1Graczyk, P., & Małecki, J. (2013). Multidimensional Yamada-Watanabe theorem

and its applications to particle systems. Journal of Mathematical Physics, 54(2),021503.

5

Sketch of proof

• Derive a SDE on the eigenvalues λi(t) of Xt

• Show that starting from λ1(0) < ... < λp(0) the first collision time isinfinite a.s.

• Prove that the SDEs on the eigenvalues and eigenvectors have aunique strong solution

Non colliding starting point

6

Particle systems

X = (x1, ...xp)• Symmetric polynomials of

particles:en(X) =

∑i1<...<in

xi1 ...xin

• Symmetric polynomials ofsquare of differences:Vn = en(A) with A ={(xi − xj)2 : 1 ≤ i < j ≤ p}

7

Fractional Brownian Motion

Definition

DefinitionA centered Gaussian process B = (Bt)t≥0 is called a fractionalBrownian motion (fBm) of Hurst parameter H ∈]0, 1[ if it has thecovariance function RH(t, s) = E(BtBs) = 1

2(s2H + t2H − |t − s|2H)

.

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Main properties

• Stationary increments• Non independent increments

Theorem

The fractional Brownian motion is a semi-martingale if and only if H = 12 .

Ito’s calculus non usable

2Nualart, D. (2006). Fractional Brownian motion: Stochastic, calculus andapplications. Proceedings oh the International Congress of Mathematicians, Vol. 3,2006-01-01, ISBN 978-3-03719-022-7, pags. 1541-1562. 3.

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Stochastic integration

In the case H >12 :

• Pathwise approach: Young integral for processes with γ−Holdertrajectories where γ > 1 − H

• Malliavin calculus: Skorohod integral

3Nualart, D. (2006). The Malliavin calculus and related topics (Vol. 1995). Berlin:Springer.

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Ito’s formula

TheoremFor a N-dimensional fractional Brownian motion BH(t) and for a functionF ∈ C2 (

RN), we have:

F(BH(t)) = F(0) +N∑

i=1

∫ t

0

∂F∂xi

(BH(s))δBHi (s)

+ HN∑

i=1

∫ t

0

∂2F∂x2

i(BH(s))s2H−1ds.

where∫ t

0

∂F∂xi

(BH(s))δBHi (s) stands for the Skorohod integral.

4Pardo, J. C., Pérez, J. L., & Pérez-Abreu, V. (2017). On the non-commutativefractional Wishart process. Journal of Functional Analysis, 272(1), 339-362.

11

Fractional Wishart process

A matrix process

DefinitionLet (BH(t))t≥0 be the matrix fractional Brownian motion with parameterH. We define the fractional Wishart process of order n and parameterH the process (X(t))t≥0 satisfying X(t) =

(BH(t)

)T BH(t) for all t ≥ 0.

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Hadamard’s formula

TheoremLet A be a self-adjoint matrix which depends smoothly on a parameter t,that has simple spectrum. We denote by λj the eigenvalues and vj theeigenvectors. Then we have the evolution equations:

λk = v∗kAvk

vk =∑j =k

v∗j Avk

λk − λjvj + ckvk

λk = v∗kAvk +

∑j =k

|v∗kAvj|2

λk − λj

4Tao, T. (2012). Topics in random matrix theory (Vol. 132). AmericanMathematical Soc..

13

Spectral dynamic

Theorem

Let X be the fractional Wishart process of order n and parameter H >12 ,

λ1, ..., λn its eigenvalues. We denote by ϕi the functions such thatλi(t) = ϕi(BH(t)).Then for any i and t > 0, we have:

λi(t) = λi(0) +p∑

k=1

n∑h=1

∫ t

0

∂ϕi∂bkh

(BH(s))δbkh(s)

+ 2H∫ t

0

p +∑i=j

λi(s) + λj(s)λi(s) − λj(s)

s2H−1ds

5Pardo, J. C., Pérez, J. L., & Pérez-Abreu, V. (2017). On the non-commutativefractional Wishart process. Journal of Functional Analysis, 272(1), 339-362.

14

Matricial diffusion

Theorem

The fractional Wishart process of order n and parameter H >12 ,

(WH(t))t≥0 satisfies the stochastic differential equation:

dWH(t) =√

WH(t)δBH(t) + δ(BH(t))T√

WH(t) + 2Hnt2H−1Idt

Generalization to a non-integer order α ∈ R.

dWH(t) =√

WH(t)δBH(t) + δ(BH(t))T√

WH(t) + 2Hαt2H−1Idt

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Outcome

Directions to go

• Fractional Wishart process with non-integer orderEigenvalues dynamicUniqueness

• Yamada-Watanabe type theoremEigenvalues diffusionExistence and uniqueness

16

Thank You

Bibliography

[1] Graczyk, P., & Małecki, J. (2013). MultidimensionalYamada-Watanabe theorem and its applications to particle systems.Journal of Mathematical Physics, 54(2), 021503.

[2] Nualart, D. (2006). Fractional Brownian motion: Stochastic, calculusand applications. Proceedings oh the International Congress ofMathematicians, Vol. 3, 2006-01-01, ISBN 978-3-03719-022-7, pags.1541-1562. 3.

[3] Nualart, D. (2006). The Malliavin calculus and related topics (Vol.1995). Berlin: Springer.

[4] Tao, T. (2012). Topics in random matrix theory (Vol. 132)

[5] Pardo, J. C., Pérez, J. L., & Pérez-Abreu, V. (2017). On thenon-commutative fractional Wishart process. Journal of FunctionalAnalysis, 272(1), 339-362.

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