Malliavin Calculus

How far can you go with integration by parts? L. Gross

L.Decreusefond Singular Greeks SDE’s Optimal Γ calculus (Condi- transport 2 tional) densities Dirichlet calculus Malliavin Stein- Antici- Malliavin- pative calculus Dirichlet calculus method Inde- pendent rand. var. Ind. increments Lévy

Spatial Poisson

Brownian fractional motion Brownian motion Martin- gales Sample-paths Itô calculus Rough paths

SDE’s

Stochastic integral

2 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Invariance yields IBP

Invariance with respect to translation Z Z d f (x + τ)g(x + τ) dx = f (x)g(x) dx0 dτ R τ=0 R

Integration by parts Z Z f 0(x)g(x) dx + f (x)g0(x) dx = 0 R R

3 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Brownian motion

Definition The Brownian motion is the unique centered Gaussian process such that 1 E [B(t)B(s)] = min(t, s) = (t + s − |t − s|) 2

Independent and stationary increments Hölder (1/2 − ε) sample-paths Finite quadratic variation

4 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Construction

1.2 0.0 0.4 1.0 0.2 0.1 − 0.8 0.0 0.2 − 0.6 0.2 − 0.3 0.4 − 0.4 −

0.6 0.4 0.2 − −

0.8 − 0.5 0.0 − 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Donsker Karhunen-Loève Poisson [ ] √ sin(π(n+ 1 )t) ( )−λ √1 P nt P 2 Nλ √t t j=1 Xj 2 Yn 1 n π(n+ 2 ) λ

5 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Standard Wiener measure

Distribution of B supported by

C0([0, 1]; R) Hol(1/2 − ε)

Wη,p for 0 < η − 1/p < 1/2

ZZ |f (t) − f (s)|p kf kp = ds dt Wη,p 1+ηp [0,1]2 |t − s|

How to characterize a Gaussian measure on these Banach spaces ?

6 / 98 June 2018 Institut Mines-Télécom Malliavin calculus RKHS

Definition (Reproducing Kernel Hilbert Space) n o H = span t ∧ ., t ∈ [0, 1]

for the norm induced by the scalar product

ht ∧ ., s ∧ .iH = t ∧ s

7 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Identification Since Z s 1 1 2 t ∧ s = 1[0,t](u) du = I0+ (1[0,t])(s) ⇒ t ∧ . ∈ I0+ L ([0, 1]) 0 Z 1 and t ∧ s = 1[0,t](u)1[0,s](u) du = h1[0,t], 1[0,s]iL2 0

H as a Besov-Liouville space n o 1 2 H = span t ∧ ., t ∈ [0, 1] ' I0+ L ([0, 1]) with 1 ˙ ˙ kI0+ (h)kH = khkL2([0,1])

8 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Abstract Wiener space

Theorem

For any choice of W among C, Hol(1/2 − ε) or Wη,p, the triplet emb : H −→ W is an AWS: H is dense in W , emb is radonifying and the standard Gaussian measure is characterized by

1 ∗ 2 E [exp(−hζ, ωi ∗ )] = exp(− k emb ζk ) W ,W 2 H

where ∗ emb emb W ∗ −−−→H∗ 'H −−→ W

9 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Wiener integral

∗ emb emb W ∗ −−−→H∗ 'H −−→ W

Definition The Wiener integral is the isometric extension of the map

∗ ∗ 2 δ : emb (W ) ⊆ H −→ L (P1/2) ∗ emb (ζ) 7−→ hζ, ωiW ∗,W .

10 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Identification

∗ ∗ emb ∗ emb W −−−→H 'H −−→ W and H = I1,2 ∗ ht = emb εt must satisfy for ω ∈ H ⊂ W and differentiable

Z 1 ∗ ˙ ω(t) = hεt, emb ωiW ∗,W = hemb εt, ωiH = ht(u)ω ˙ (u) du 0 ˙ ⇒ ht(u) = 1[0,t](u) ⇒ ht(u) = t ∧ u

Consequence

∗ ∗ 2 ∗ δ : emb (W ) ⊆ I1,2 −→ L (P1/2) δ(t ∧ .) = δ(emb (ζ)) = B(t) ∗ emb (ζ) 7−→ hζ, ωiW ∗,W

11 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Fractional Brownian motion

Definition

For any H in (0, 1), {BH (t); t ≥ 0} is the centered Gaussian process whose covariance kernel is given by

VH 2 2 2 R (s, t) = E [B (s)B (t)] = s H + t H − |t − s| H H H H 2 where Γ(2 − 2H) cos(πH) V = . H πH(1 − 2H)

stationary increments Hölder (H − ε) sample-paths Finite 1/H-variation

12 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Some realizations

H = 0.2 H = 0.8 0.10 2.0

1.5 0.05

1.0 0.00

0.5 0.05 −

0.0 0.10 − 0.5 − 0.15 − 1.0 − 0.20 − 1.5 − 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

H = 0.5

0.25

0.00

0.25 −

0.50 −

0.75 −

1.00 −

1.25 −

0.0 0.2 0.4 0.6 0.8 1.0

Figure: Sample-path example for H = 0.2, H = 0.5 (middle) and H = 0.8. 13 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Besov-Liouville spaces

Definition (Riemann-Liouville fractional spaces) For α > 0, for f ∈ Lp([0, 1]),

Z t α 1 α−1 I0+ f (t) = (t − s) f (s) ds. Γ(α) 0

α p The space Iα,p is the set I0+ (L [0, 1]) equipped with the scalar product Z 1 α α p hI0+ f , I0+ giIα,p = hf , giL = f (s)g(s) ds. 0

14 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Sobolev embeddings

For a > b > c and p, c − 1/p > 0

Wa,p ⊂ Ib,p ⊂ Wc,p ⊂ Hol(c − 1/p)

15 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Key lemma

Square root of RH

There exists KH such that Z 1 RH (t, s) = KH (t, r)KH (s, r) dr 0

2 6 7

5 6

1.5

4 5

4 1 3

3 2

0.5

2 1

0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

K0.25(1,s) K0.5(1,s) K0.75(1,s)

16 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Properties

Properties

s K (t, s) = tH−1/2 K (1, ) H H t KH (t, s) = 0 if s > t

K1/2(t, s) = 1[0,t](s) −|H−1/2| H−1/2 ∞ KH (t, s) = r (t − s) × C −function

Warning !

Singularity of KH increases as H ↑

17 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Regularity

p KH : L −→ IH+1/2,p Z t f 7−→ KH f (t) = KH (t, s)f (s) ds 0 is continuous. Essential remarks

Z 1 RH (t, s) = KH (t, r)KH (s, r) dr ⇒ RH (t, s) = KH KH (t,.) (s) 0

1 K1/2 = I0+

18 / 98 June 2018 Institut Mines-Télécom Malliavin calculus RKHS

Identification

RKHS(BH ) = span {RH (t,.), t ∈ [0, 1]} 2 = KH (L ) = IH+1/2,2

with scalar product Z 1 h ˙, ˙ i 2 = ˙( ) ˙ ( ) KH h KH g KH (L ) h s g s ds 0

19 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Identification

Abstract Wiener space for fBm ∗ ∗ emb ∗ emb W −−−→ IH+1/2,2 ' IH+1/2,2 −−→ W

˙ For h = KH (h) ∈ IH+1/2,2 ⊂ W

∗ hε , i ∗ = h ε , i = ( ) t emb h W ,W emb t h IH+1/2,2 h t Z 1 = ( , )˙( ) = h ( ,.), ˙i 2 = h ( ( ,.)), i KH t s h s ds KH t h L KH KH t h IH+1/2,2 0 hence ∗ emb (εt) = KH (KH (t,.)) = RH (t,.)

20 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Consequences

Definition The Wiener integral is the isometric extension of the map

∗ ∗ 2 δ : emb (W ) ⊆ I1,2 −→ L (µW ) ∗ emb (ζ) 7−→ hζ, ωiW ∗,W

Lemma

δ(RH (t,.)) = BH (t)

(δ(KH (1[0,t])), t ∈ [0, 1]) is an ordinary BM

2 2 2 E δ(K (1 )) = kK (1 ))k = k1 k 2 = t H [0,t] H [0,t] IH+1/2,2 [0,t] L

21 / 98 June 2018 Institut Mines-Télécom Malliavin calculus In brief

Embeddings and identifications ∗ ∗ emb ∗ ∗ W H = (IH+1/2,2)

2 KH emb L H = IH+1/2,2 W

for W = C, or Wη,p with 0 < η − 1/p < H 1 ∗ 2 E [exp(−hζ, B i ∗, )] = exp − k emb ζk H W W 2 H

22 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Quasi invariance

Theorem (Cameron-Martin) For any h ∈ H, h i 1 E F B + emb(h) = E F(B ) exp δh − khk2 H H 2 H

23 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Girsanov theorem

Theorem

For u = KH u˙ adapted,

Z t Z t dQ 1 2 = exp u˙ (s) dB(s) − u˙ (s) ds P 2 d H Ft 0 0 then Z t ( ) − ( , ) ˙ ( ) = ( ) LawQ BH t KH t s u s ds LawPH BH 0

24 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Differentiation on W

Obstacles Why not consider the Fréchet derivative

1 0 lim F(ω + εω ) − F(ω) ? ε→0 ε The Itô map is not continuous on W , so not Fréchet differentiable A random variable F is defined up to a negligeable set. We must ensure that for any admissible direction of differentiation h

F = G a.s. ⇒ F(BH + h) = G(BH + h) a.s.

The Cameron-Martin theorem ensures that this is true for h ∈ H

25 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Cylindrical functionals

Definition A function F is said to be cylindrical if there exists f ∈ S(Rn), n (h1, ··· , hn) ∈ H such that

F(ω) = f (δh1, ··· , δhn).

The set of such functionals is denoted by Cyl.

26 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Gross-Sobolev derivative

Definition

For F(ω) = f (δh1, ··· , δhn). Set

n X ∇F = ∂j F(δh1, ··· , δhn) hj , j=1

so that n X h∇F, hiH = ∂j F(δh1, ··· , δhn) hhj , hiH. j=1

−1 h∇F, hiH = lim ε (F(BH + εh) − F(BH )) ε→0

27 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Examples

∇(BH (t))(s) = ∇ δ RH (t,.) (s) = RH (t, s) ˙ −1 ∇s BH (t) := KH (∇(BH (t)))(s) = KH (t, s)

Recall For H = 1/2

2 −→ H = −→ L IH+1/2,2 W ∇s B(t) = t ∧ s ˙ ˙ −1 ∇s B(t) = 1[0,t](s) ∀h ∈ H, h = KH h ˙ −1 ∇ = KH ∇

28 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Classical formulas

Derivation and chain rule

∇(FG) = F∇G + G∇F ∇ϕ(F) = ϕ0(F)∇F

29 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Extension to a larger set

Closability Fn ∈ Cyl −→ 0 and ∇Fn −→ ξ ⇒ ξ = 0 ?

Theorem (Integration by parts)

E [Gh∇F, hiH] = E [FG δh] − E [F h∇G, hiH]

E [F(ω + τh)G(ω)] = E [F ◦ Th G ◦ T−h ◦ Th] 1 2 = E FG ◦ T− exp δh − khk h 2 H 1τ 2 = E F(ω)G(ω − τh) exp τδh(ω) − khk2 2 H

30 / 98 June 2018 Institut Mines-Télécom Malliavin calculus where ThG(ω) = G(ω + h) Gross-Sobolev spaces

Definition D2,1 is the closure of Cyl for the norm

2 2 2 kFk2,1 = E F + E k∇FkH Z 1 2 ˙ 2 = E F + E (∇s F) ds 0 ˙ −1 where ∇ = KH ∇.

31 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Support of the derivative

Theorem ˙ F ∈ Ft ⇐⇒ ∇s F = 0 for s > t

h i h i E f B(t1), ··· , B(tn + s) | Ftn = E f B(t1), ··· , B(tn) + N (0, s) and n X ∇˙ ( ( ), ··· , ( ))) = ∂ (...) ( ) = 0 > r f B t1 B tn j f 1[0,tj ] s for s tn j=1

32 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Iteration

An example 2 s 7−→ ∇s BH (t) = 2BH (t) ∇s BH (t) = 2BH (t) RH (t, .s) ∈ H

2 ∇r ∇s BH (t) = 2RH (t, r) RH (t, s) meaning that

(2) 2 ∇r,s BH (t) = 2RH (t, r) RH (t, s) ∈ H ⊗ H

33 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Higher order derivative

Definition

D2,k is the closure of Cyl for the norm

k 2 X h (j) 2 i kFk2,k = E k∇ FkH⊗j j=0 k "Z # 2 X ˙ (j) 2 = E F + E ∇s1,··· ,sj F ds1 ... dsj [0,1]j j=1

34 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Divergence

Theorem 2 2 ∇ : D2,1 ⊂ L (W ; PH ) 7−→ L (W ⊗ H) is continuous.

Consequence ∗ ∗ 2 ∗ 2 ∗ ∇ : dom ∇ ⊂ L (W ⊗ H) −→ L (W ; PH ) by identification of the Hilbert spaces and their dual

35 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Divergence

Definition

dom δ = U(ω, s) ∈ L2(W ⊗ H), ∃c > 0, ∀F ∈ Cyl, | [h∇ , i ]| ≤ k k 2 E F U H c F L (PH )

Then, ∗ E [h∇F, UiH] = E [F ∇ U]

If U is deterministic,

∗ E [h∇F, UiH] = E [F δU] ⇒ ∇ = δ on H

36 / 98 June 2018 Institut Mines-Télécom Malliavin calculus A key formula

Theorem a a random variable, U a process

δ(aU) = aδU − h∇a, UiH

E [δ(aU) ψ] = E [ahU, ∇ψiH]

= E [hU, ∇(aψ) − ψ∇aiH]

= E [aδU ψ] − E [h∇a, UiHψ]

37 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Divergence extends Itô integral

Corollary (H = 1/2) If U ∈ dom δ and U˙ is adapted, Z 1 δU = U˙ (s) dB(s) 0

X X 1 ˙ ( ) = ( ) ⇐⇒ ( ) = + ( )( ) U s Uti 1(ti ,ti+1] s U t Uti I0 1(ti ,ti+1] t Z 1 1 δ( + ( )) = δ ∧ . − ∧ . − ∇˙ ( ) ( ) Uti I0 1(ti ,ti+1] Uti ti+1 ti s U ti 1(ti ,ti+1] s ds 0

= Uti B(ti+1 − B(ti )) − 0

38 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Extension of the Itô isometry formula

Theorem Z 1 ZZ 1 2 ˙ 2 ˙ ˙ ˙ ˙ E δU = E U(s) ds + E ∇r Us ∇s Ur ds dr 0 0

39 / 98 June 2018 Institut Mines-Télécom Malliavin calculus For the sake of notations

Recall that δ(KH 1[0,t]) is a B.M. Recall that δ extends Itô integral on adapted processes

Z 1 ˙ ˙ U(s) δB(s) = δ(KH U) 0

Z 1 B(t) = 1[0,t](s) δB(s) 0 Z 1 BH (t) = KH (t, s) δB(s) 0

40 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Inversion formula

α β α+β H+1/2 Theorem (I ◦ I = I and KH ' I )

−1 continuous K = KH ◦ K1/2 : I1/2,p −−−−−−→ IH,p bijective ˙ −1 Formally, BH = KH (B) = KH ◦ K1/2(B) Theorem For any H, we have −1 B = K (BH ), PH − a.s. Remark

−1 t 7→ BH (t) ∈ IH,p ⇒ t 7→ K (BH )(t) ∈ I1/2,p ⊂ Hol(1/2 − 1/p)

41 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Proof

If for any U˙ ∈ L2(W × [0, 1])

Z 1 Z 1 ˙ −1 ˙ E B(t) U(t) dt = E K (BH )(t) U(t) dt 0 0

then −1 B(t) = K (BH )(t) PH ⊗ dt a.s. Continuity argument and the proof holds. It suffices to consider U˙ (ω, s) = ψ(ω)g(s). Integration by parts + adjunction + Fubini

42 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Itô formula

Theorem (LD) 2 For f ∈ Cb,

Z t ∗ 0 f (BH (t)) = f (0) + Kt f ◦ BH (s) δB(s) 0 Z 1 00 2H−1 + HVH f BH (s) s ds. 0

∗ 2 where K1 is the adjoint of K in L ([0, 1]) and

∗ ∗ Kt f = K1(f 1[0,t])

43 / 98 June 2018 Institut Mines-Télécom Malliavin calculus The usual way for H = 1/2

X f (B(t)) − f (0) = f (B(ti+1)) − f (B(ti )) X 0 = f (B(ti ))(B(ti+1) − B(ti )) 2 1 X 00 + f (B(s )) B(t +1) − B(t ) 2 i i i

First term: Riemann approximations converge in probability to stochastic integrals 00 Second term: Converge in probability to the integral of f (Bs ) with respect to the square bracket of the BM

44 / 98 June 2018 Institut Mines-Télécom Malliavin calculus By parts, integrate you shall

Lemma

f (BH (t)) − f (BH (s)) n X 2−2j B (t) + B (s) = (B (t) − B (s))2j+1 f (2j+1) H H (2j + 1)! H H 2 j=0 2(n+1) BH (t) − BH (s) Z 1 2n+1 (2(n+1)) + λ f λBH (t)+(1−λ)BH (s) dλ 2 0

h 2(n+1)i 2(n+1)H E |BH (t) − BH (s)| ≤ c|t − s| ⇒ 2(n + 1)H > 1

45 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Fundamental theorem of calculus

Z 1 d E ψ f BH (t) − E ψ f BH (0) = E ψ f BH (t + ε) dε 0 dε

If H > 1/2

E ψ f BH (t + ε) − E ψ f BH (t) 0 BH (t) + BH (t + ε) 2 = E B (t + ε) − B (t) f ψ + O(ε H ) H H 2

46 / 98 June 2018 Institut Mines-Télécom Malliavin calculus First order term

0 BH (t) + BH (t + ε) E B (t + ε) − B (t) f ψ H H 2 Z 1 0 BH (t) + BH (t + ε) = E KH (t + ε, s) − KH (t, s) δB(s) f ψ 0 2 Z 1 ˙ 0 BH (t) + BH (t + ε) = E KH (t + ε, s) − KH (t, s) ∇s f ψ ds 0 2

47 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Proof (cont’d)

Z 1 0 BH (t) + BH (t + ε) ˙ A0 = E f KH (t + ε, s) − KH (t, s) ∇s ψ ds 2 0 B (t) + B (t + ε) + E ψ f 00 H H 2 Z 1 × KH (t + ε, s) − KH (t, s) KH (t + ε, s) + KH (t, s) ds 0 = B1 + B2.

48 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Proof (cont’d)

−1 ε→0 h 0 ˙ i ε B1 −−−→ E f (BH (t)) K∇ψ(t) .

−1 ε→0 00 d ε B2 −−−→ E ψ f (B (t)) R (t, t) H dt H 2H−1 00 = HVH t E ψ f (BH (t))

49 / 98 June 2018 Institut Mines-Télécom Malliavin calculus The final step: Fundamental theorem of calculus

Z t 0 ˙ E ψ f BH (t) − E ψ f BH (0) = E f BH (s) K∇ψ(s) ds 0 Z t 00 2H−1 + HVH E ψ f BH (s) s ds . 0

+ Integration by parts

Z t Z 1 0 ˙ 0 ˙ E f BH (s) K∇ψ(s) ds = E f BH (s) 1[0,t](s) K∇ψ(s) ds 0 0 Z 1 Z 1 ∗ 0 ˙ ∗ 0 = E K1 f ◦ BH 1[0,t] ∇s ψ ds = E ψ K1 f ◦ BH 1[0,t] (s) δB(s) . 0 0

50 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Stochastic integrals: Discretize BH

Stratonovitch type integrals

X RSπ(u) = u(ti ) BH (ti+1) − BH (ti ) or ti ∈π X 1 Z ti+1 SSπ(u) = u(s) ds BH (ti+1) − BH (ti ) θi t ti ∈π i

51 / 98 June 2018 Institut Mines-Télécom Malliavin calculus ˙ Discretize B and use BH = KHB

π 1 B (t) = B(ti ) + ∆Bi (t − ti ) for t ∈ [ti , ti+1), θi

Z ti+1 π X 1 BH (t) = K(t, s) ds ∆Bi θi t ti ∈π i X 1 = ( )( )∆ K 1[ti ,ti+1] t Bi θi ti ∈π

Riemann-Stratonovitch integral X 1 Z T d π ( ) := ( ) ( )( ) ∆ RT u u t K 1[ti ,ti+1] t dt Bi θi 0 dt ti ∈π

52 / 98 June 2018 Institut Mines-Télécom Malliavin calculus aδU = δ(aU) + h∇a, Ui

Z ti+1 π X 1 ∗ RT (u) = δ KT u(t) dt θi t ti ∈π i ZZ X 1 ∗ ˙ + KT (∇r u)(t) dt dr θi ti ∈π 2 [ti ,ti+1]

Theorem Under technical conditions on u,

Z t π ∗ ˙ ∗ RT (u) −→ δ(KT u) + ∇(KT u)(s) ds 0

53 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Back to BH discretization

Z t ∗ ˙ SSπ(u) −→ δ(KT u) + (K∇)s u(s) ds 0

Why is H < 1/2 so hard ?

H−1/2 KH behaves as I

54 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Ornstein-Uhlenbeck semi-group

Definition Z 1 −t p −2t ∀F ∈ L (PH ), PtF(ω) = F(e ω + 1 − e ζ) dPH (ζ) W Theorem Invariance of Gaussian measures with respect to rotations imply

Pt+s = Pt ◦ Ps

(ω) −−−→t→∞ R PtF W F dPH PH is the stationary measure of Pt 1 T∞ If F ∈ L , PtF ∈ k=1 D2,k for any t > 0

55 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Generator

Theorem (A consequence of IBP) 1 For any F ∈ L (PH ) ∩ Lip(W ), d P F = LP F dt t t where ∞ X (2) ∗ LF(ω) = −h∇F, ωiW ,W + h∇ F, hj ⊗ hj iI1,2 j=1 = −δ∇F

with (hj , j ≥ 1) a CONB of I1,2

56 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Stein-Malliavin representation formula

Z Z Z Z r∞ d Pr F(ω) F dPH − Ps0F(ω) dQ(ω) = LPtF(ω) dt dQ(ω) W W W s0 dt

57 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Stein method in one picture

m = T ∗ν

∗ Pt m

µ ∗ µ Pt µ = µ

58 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Dirichlet-Malliavin structure

= LF dPt F dt t=0 L = −∇∗∇

LF/ G s.t. E( , )= F G tL E(F,H)= d Pt =e hP F,Gi 2 dt t L (P ) R hG,Hi 2 , ∀H H P F→ F dP L (P ) at t = 0 t H H

D s.t. E(F,G)= E h∇F,∇GiH

E(F,G) =

hLF,Gi 2 L (PH ) PH , L R Γ = LF dPH = 0 F L(F 2) − 2FLF

59 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Malliavin calculus for Poisson point processes

60 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Configuration space

Definition A configuration is a locally finite set of points ofaset E. NE = the set of configurations of E.

Definition Let µ σ-finite Radon measure on a Polish space E. The Poisson process with intensity µ is such that for any function f : E → R+, Z Z E exp(− f dN) = exp − (1 − e−f (s)) dµ(s) . E R P where f dN = x∈N f (x)

61 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Campbell-Mecke formula

Theorem Z Z E F(N ⊕ x, x) dµ(x) = E F(N, x) dN(x) E E

Z d E exp(− (f + t 1B) dN) dt E t=0 Z d −( ( )+ ( )) = exp − (1 − e f s t 1B s ) dµ(s) dt E t=0 R yields CM formula for F(N, x) = exp(− f dN) g(x)

62 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Gradient

Definition (Discrete gradient)

Z 2 dom D = F : NE −→ R, E |F(N ⊕ x) − F(N)| dµ(x) < ∞ E

For F ∈ dom D, we set

Dx F(N) = F(N ⊕ x) − F(N).

63 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Divergence

Definition (Poisson divergence) We denote by dom δ, the set of vector fields such that (

dom δ = U : NE × E → R,

"Z 2# ) E U(N x, x)( dN(x) − dµ(x)) < ∞ E

Then, Z δU(N) = U(N x, x)( dN(x) − dµ(x)). E

64 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Campbell-Mecke formula is equivalent to IBP

Theorem (Integration by parts for Poisson process)

For F ∈ dom D and any U ∈ dom δ, Z E Dx F(N) U(N, x) dµ(x) = E [F(N) δU(N)] . E

Corollary (Skorohod isometry) For any U ∈ dom δ, Z 2 2 E δU = E U(N, x) dµ(x) E Z Z + E Dx U(N, y) Dy U(N, x) dµ(x) dµ(y) . E E

65 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Glauber process

Construction

N(0) = η ∈ NE , Each atom of η has a life duration, independent of that of the other atoms, exponentially distributed with parameter 1. Atoms are born at moments following a Poisson process with intensity µ(Λ). On its appearance, each atom is localised independently from all the others according to µ/µ(Λ). It is also assigned in an independent manner, a life duration exponentially distributed with parameter 1.

66 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Glauber process E N(t)

N(0)

Time

67 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Generator

Theorem The infinitesimal generator of N is given by

− LNF(N) = (δD)F(N) Z dµ(x) = µ(E) (F(N ⊕ x) − F(N)) dµ(x) E µ(E) Z 1 + N(E) (F(N − δ ) − F(N)) dN(x) x N(E)

for F bounded from NE into R. Definition

PtF(N) = E [F(N(t)) | N(0) = N]

68 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Similarities and difference Brownian Poisson 2 2 kδ k 2 = k k + [ (∇ ◦ ∇ )] kδ k 2 = k k + [ ( ◦ )] U L U L2 E tr U U U L U L2 E tr DU DU P n (n) P n (n) F = E [F] + n≥1 δ (E[∇ F]) F = E [F] + n≥1 δ (E[D F]) ( ) ≤ k∇ k2 ( ) ≤ k k2 var F F L2(W ;H) var F DF L2(W ×E)

∇(FG) = F ∇G + G ∇F D(FG) = F DG + G DF + DF DG PtF ∈ ∩k≥1D2,k No regularizing property ( ) ≤ h∇ , ∇ i ( ) ≤ [R ( −1| |2, Ent F E F log F H Ent F E E min F Dx F Dx FDx log F) dµ(x)]

69 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Independent random variables

Definition (LD, H. Halconruy)

For XA = (Xa, a ∈ A) independent random variables, Xa ∈ Ea a Polish space.

DaF(XA) = F(XA) − E [F | Xb, b 6= a] X δU = DaUa a∈A

Then, " # X E DaFUa = E [F δU] a∈A

70 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Similarities and difference Brownian Independent r.v. 2 kδ k 2 = k k + [ (∇ ◦ ∇ )] kδ k 2 = [ ( ◦ )] U L U L2 E tr U U U L E tr DU DU P n (n) F = E [F] + n≥1 δ (E[∇ F]) No chaos decomposition 2 2 ( ) ≤ k∇ k var(F) ≤ kDFk 2 var F F L2(W ;H) L (W ×E)

∇(FG) = F ∇G + G ∇F Da(FG) = ... + DaFDaG − E [FG|Ga] + E [F|Ga] E [F|Ga] PtF ∈ ∩k≥1D2,k No regularizing property P 2 Ent(F) ≤ E h∇F, ∇ log FiH Ent(F) ≤ a∈AE Dk G /E [G|Gk ]

71 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Clark representation formula

Brownian motion Independent random variables

∞ EN = ×j=1{1, ··· , j} 1 Pt(k) = if k 6= j j t + j − 1 t = for k = j t + j − 1 ∞ O P = Pj j=1

Xj = the i-th coordinate of law Pj

73 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Construction

Start with σ = (1) If X2 = 2 then σ = (1, 2) else σ = (2, 1) If Xk = k then concatenate k to the current permutations If Xk = j, then σ ← (k, j) ◦ σ Example: 1 2 3 4 σ = 1 33 4 2

If X5 = 3 1 2 3 4 5 σ = 1 5 4 2 3

74 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Mallows distribution

Theorem (Kerov et al.)

For any σ ∈ SN,

tcyc(σ) Pt({σ}) = , (t + 1)(t + 2) × · · · × (t + N − 1)

where cyc(σ) is the number of cycles of σ. t = 1: uniform distribution t ⇒ ∞: σ close to identity t ⇒ 0: almost a unique cycle

75 / 98 June 2018 Institut Mines-Télécom Malliavin calculus A new representation of the number of fixed points

Number of cycles N X = C1 1(Ik =k)1(Im6=k, m∈{k+1,··· ,N}) k=1

Theorem (LD, HH)

N N t − 1 X t Y C1 = t 1 − + (1( = ) − ) 1( 6= ) N + t − 1 Il l t + l − 1 Im l l=1 m=l+1 N−1 l−1 N X t X 1 Y − 1( = ) − 1( 6= ). t + l − 2 Il k t + l − 1 Im k l=2 k=1 m=l+1 76 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Variance of C1

Theorem (LD, HH)

For any t ∈ R, we get

2 N ! ˜ Nt t 2t X 1 var[C1] = + 1 − · t + N − 1 t + N − 1 N t + k − 1 k=1

77 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Dirichlet forms

N UN F (EN, ⊗j=1µj ) −−→ (W , P) −→ R E ( ) = h ( ◦ ), ( ◦ )i 2 N F E DN F UN DN F UN L (EN ) is a family of Dirichlet forms on W

78 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Convergence of Dirichlet forms

Definition

Let (En, n ≥ 1) be a sequence of Dirichlet forms on W . It converges in the Dirichlet sense to the Dirichlet form E if for F ∈ dom E ∩ Lip ,

n→∞ En(F, F) = E [h∇nF, ∇nFi] −−−→E(F, F) = E [h∇F, ∇Fi]

79 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Poisson point process as limit of discrete random variables

80 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Brownian motion as a limit of a random walk

√ Z t N N N ek (t) = N 1[(k−1)/N, k/N)(t) and hk (t) = ek (s) ds. 0 N N X N ω (t) = Mk hk (t), for all t ∈ [0, 1], k=1 where (Mk , k = 1, ··· , N) are iid standard Gaussian r.v. Theorem (LD, HH)

N h i2 UN X N 0 N 0 N E (F) = E F(ω ) − E F(ω(k) + Mk hk ) , k=1 ωN = ωN − N 0 where (k) Mk hk and Mk is an independent copy of Mk . For F ∈ Lip(H) U N→∞ E N (F) −−−−→E(F) = E [h∇F, ∇FiH] 81 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Process convergence

Barbour’90 Nn − n log n sup E F( √ ) − E [F(B)] ≤ c √ k k ≤1 n n F C3( ; R) b D

82 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Common space

Brownian motion 1/2−ε0 ∞ 1/2−ε0 2 (t 7→ B(t)) ∈ Hol(1/2 − ε) ⊂ I0+ (L ) ⊂ I0+ (L )

Poisson process X 1/2−ε ( ) = ( ) ∈ ( 2), ∀ > 0 N t 1[Tj , 1] t I0+ L j

Common space 1/2−ε 2 I0+ (L ) for arbitrary ε > 0

83 / 98 June 2018 Institut Mines-Télécom Malliavin calculus β 2 Wiener measure on I0+ (L ) Itô-Nisio theorem X 1 B(t) := Xn I0+ (en)(t) a.s. n≥1 2 with (en, n ≥ 1) CONB of L and (Xn, n ≥ 1) independent N (0, 1)

2 β 2 Convergence in L (Ω; I0+ (L )) X 1 2 X 1−β 2 1−β k + k = k k 2 = k k < ∞ I0 en Iβ, 2 I0+ en L I0+ HS n≥1 n≥1 implies β < 1/2

84 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Wiener measure

  X Z 1 ( hη, ωi ) = ( ( 1−β ◦ −β)η( ) ( ) ) Eµβ exp i Iβ, 2 EP1/2 exp i I1− I0+ s en s ds Xn  0 n≥1 Z 1 2 1 X 1−β −β = exp(− (I1− ◦ I0+ )η(s) en(s) ds ) 2 0 n≥1 1 1−β −β 2 = exp(− k(I − ◦ I + )ηk 2 ) 2 1 0 L ([0, 1]) Z 1 1 1−β 1−β = exp(− (I0+ ◦ I1− )η ˙(s)η ˙(s) ds), 2 0

Conclusion 1 Eµ exp(ihη, ωi ) = exp(− hVβη, ηi ) β Iβ, 2 2 Iβ, 2 85 / 98 June 2018 Institut Mines-Télécom Malliavin calculus where β 1−β 1−β −β Vβ = I0+ ◦ I0+ ◦ I1− ◦ I0+ . Gaussian structure on l2(N)

Hilbert spaces isometry

β 2 2 Jβ : I0+ (L ) −→ l (N) −β f 7−→ hen, I0+ f iL2([0, 1]), n ≥ 1

Commutative diagram

β 2 Jβ 2 I0+ (L ) −−−−→ l (N)     :=J ◦ ◦J−1 Vβ y ySβ β Vβ β β 2 Jβ 2 I0+ (L ) −−−−→ l (N)

86 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Gaussian measure on l 2(N) ∗ 2 Then Jβµβ = mβ, where mβ is the Gaussian measure on l (N) such that for any v ∈ l2(N), Z 1 exp(i v.u) dmβ(u) = exp(− Sβv.v) l2(N) 2

Expression of Sβ

2 2 Sβ : l (N) −→ l (N) X u = (un, n ∈ N) 7−→ ( hhn, hj iL2 uj , n ∈ N) j≥1

1−β where hn = I0+ (en)

87 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Stein equation

Ornstein-Uhlenbeck process Z β −t p −2t Pt F(x) = F(e x + 1 − e v) dmβ(v) l2(N)

Infinitesimal generator

d β β P F(x) = AβP F(x) where dt f t β (2) 2 A F(x) = hx, ∇F(x)il2 − trace(Sβ ◦ ∇ F(x)), x ∈ l (N)

Stein equation Z Z ∞ β β F(x) dmβ(x) − F(x) = A Pt F(x) dt, m-a.s. l2(N) 0

88 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Embedding of the Poisson process

Poisson process

λ 1 N (t) = √ (Nλ(t) − λt) λ Z 1 λ 1 JβN = √ hn(s)( dNλ(s) − λ ds), n ∈ N λ 0 where 1−β hn = I0+ (en) Integration by parts

Z 1 Z 1 λ λ E F(N ) g(τ)( dNλ(τ) − λ dτ) = λ E Dτ F(N ) g(τ) dτ 0 0

89 / 98 June 2018 Institut Mines-Télécom Malliavin calculus β 2 Convergence in I0+ (L )

Theorem Z h λ i cβ sup E F(N ) − F dmβ ≤ √ k k λ F C3(l2(N); R)≤1 b

90 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Proof I

−1/2 X 1−β −1/2 Hλ = λ hn ⊗ xn = λ H1 n≥1

1 X h λ 1−β i E [JβNλ.G(JβNλ)] = √ E δ (h )F(JβNλ) xn.x λ n n≥1 Z 1 1 X 1−β = √ E hn (τ)Dτ F(JβNλ)λ dτ xn.x λ 0 n≥1 Z 1 = E Dτ F(JβNλ).Hλ(τ) dτ . 0

91 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Proof II

According to the Taylor formula,

Dτ F(JβNλ) = F(JβNλ + Hλ(τ)) − F(JβNλ) Z 1 1 1 2 ⊗(2) = √ ∇F(JβNλ).H1(τ)+ (1−r) ∇ F(JβNλ+Hλ(τ)).H1(τ) dr. λ λ 0

For the rest, see Coutin and Decreusefond, “Stein’s Method for Brownian Approximations”.

92 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Bibliography I

Reference books Privault, Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales Nualart, The Malliavin Calculus and Related Topics Üstünel, “Analysis on Wiener Space and Applications” fBm Decreusefond, “Stochastic Calculus with Respect to Volterra Processes” Coutin and Decreusefond, “Volterra Stochastic Differential Equations with Singular Kernels”

93 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Bibliography II

Other approaches for fBm Feyel and de La Pradelle, “On Fractional Brownian Processes” Zähle, “On the link between fractional and stochastic calculus” Russo and Vallois, “Elements of stochastic calculus via regularization” Dirichlet forms and Malliavin calculus Bouleau, “Théorème de Donsker et formes de Dirichlet” Bouleau and Hirsch, Dirichlet forms and analysis on Wiener space Independent random variables Decreusefond and Halconruy, “Malliavin and Dirichlet structures for independent random variables”

94 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Bibliography III

Bernoulli random variables Privault, Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales Nourdin, Peccati, and Reinert, “Stein’s Method and Stochastic Analysis of Rademacher Functionals”

95 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Bouleau, Nicolas. “Théorème de Donsker et formes de Dirichlet”. In: Bulletin des Sciences Mathématiques 129.5 (2005), pp. 369–380. issn: 0007-4497. doi: 10.1016/j.bulsci.2004.09.005. Bouleau, Nicolas and Francis Hirsch. Dirichlet forms and analysis on Wiener space. Vol. 14. De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1991, pp. x+325. isbn: 3-11-012919-1. doi: 10.1515/9783110858389. Coutin, L. and L. Decreusefond. “Stein’s Method for Brownian Approximations”. In: Communications on Stochastic Analysis 7.3 (Sept. 2013). 00000, pp. 349–372. –.“Volterra Stochastic Differential Equations with Singular Kernels”. In: Stochastic Analysis and Mathematical Physics. Progress in Probability 50 (2001). Ed. by A.B. Cruzeiro and J.-C. Zambrini, pp. 39–50. Decreusefond, L. “Stochastic Calculus with Respect to Volterra Processes”. In: Annales de l’Institut Henri Poincaré (B) Probability and Statistics 41 (2005). 00000, pp. 123–149.

96 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Decreusefond, L. and Hélène Halconruy. “Malliavin and Dirichlet structures for independent random variables”. In: (July 25, 2017). arXiv: 1707.07915v1 [math.PR]. Feyel, D. and A. de La Pradelle. “On Fractional Brownian Processes”. In: Potential Anal. 10.3 (1999), pp. 273–288. issn: 0926-2601. Nourdin, Ivan, Giovanni Peccati, and Gesine Reinert. “Stein’s Method and Stochastic Analysis of Rademacher Functionals”. In: Electronic Journal of Probability 15 (2010), no. 55, 1703–1742. issn: 1083-6489. doi: 10.1214/EJP.v15-843. Nualart, D. The Malliavin Calculus and Related Topics. 00000. Springer–Verlag, 1995. Privault, Nicolas. Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Vol. 1982. Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2009. isbn: 978-3-642-02379-8.

97 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Russo, Francesco and Pierre Vallois. “Elements of stochastic calculus via regularization”. In: Séminaire de Probabilités XL. Vol. 1899. Lecture Notes in Math. Springer, Berlin, 2007, pp. 147–185. doi: 10.1007/978-3-540-71189-6_7. url: https://doi.org/10.1007/978-3-540-71189-6_7. Üstünel, A. S. “Analysis on Wiener Space and Applications”. In: arXiv:1003.1649 (Mar. 2010). arXiv: 1003.1649. Zähle, Martina. “On the link between fractional and stochastic calculus”. In: Stochastic dynamics (Bremen, 1997). Springer, New York, 1999, pp. 305–325. doi: 10.1007/0-387-22655-9_13. url: https://doi.org/10.1007/0-387-22655-9_13.

98 / 98 June 2018 Institut Mines-Télécom Malliavin calculus