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Malliavin Calculus 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 − jt − sj) 2 Independent and stationary increments Hölder (1/2 − ") 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 [ ] p sin(π(n+ 1 )t) ( )−λ p1 P nt P 2 Nλ pt 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 − ") Wη;p for 0 < η − 1/p < 1/2 ZZ jf (t) − f (s)jp kf kp = ds dt Wη;p 1+ηp [0;1]2 jt − sj 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 2 [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 ^ : 2 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 2 [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 − ") 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; !i ∗ )] = 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; !iW ∗;W : 10 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Identification ∗ ∗ emb ∗ emb W −−−!H 'H −−! W and H = I1;2 ∗ ht = emb "t must satisfy for ! 2 H ⊂ W and differentiable Z 1 ∗ _ !(t) = h"t; emb !iW ∗;W = hemb "t;!iH = ht(u)_!(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; !iW ∗;W 11 / 98 June 2018 Institut Mines-Télécom Malliavin calculus Fractional Brownian motion Definition For any H in (0; 1), fBH (t); t ≥ 0g 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 − jt − sj H H H H 2 where Γ(2 − 2H) cos(πH) V = : H πH(1 − 2H) stationary increments Hölder (H − ") 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 2 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−1/2j H−1/2 1 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 fRH (t;:); t 2 [0; 1]g 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) 2 IH+1/2;2 ⊂ W ∗ h" ; i ∗ = h " ; 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 ("t) = 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; !iW ∗;W Lemma δ(RH (t;:)) = BH (t) (δ(KH (1[0;t])); t 2 [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 2 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(! + "! ) − F(!) ? "!0 " The Itô map is not continuous on W , so not Fréchet differentiable A random variable F is defined up to a negligeable set.
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