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
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
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
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
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
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
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)
.
8
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.
9
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.
9
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.
9
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.
10
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.
12
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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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
15
Outcome
Directions to go
• Fractional Wishart process with non-integer orderEigenvalues dynamicUniqueness
• Yamada-Watanabe type theoremEigenvalues diffusionExistence and uniqueness
16
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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