Integration by parts and Quasi-invariance of Horizontal Wiener Measure on a foliated compact manifold

Transcription

1 Integration by parts and Quasi-invariance of Horizontal Wiener Measure on a foliated compact manifold Qi Feng (Based on joint works with Fabrice Baudoin and Maria Gordina) University of Connecticut Mathematical Finance Colloquium USC, 4/23/218

2 Outline of the Talk Motivation: Sub-Riemannian Structure Math Finance. Greek evaluation with Malliavin calculus. Large Deviation Principle. (variational representation) Totally geodesic Riemannian foliations. Quasi-invariance of horizontal Wiener measure. General idea. Main theorem. Examples. Clark-Ocone formula and Integration by parts formulas. Functional inequalities. Applications and future work.

3 Motivation of the talk.

4 Motivation I: Greek evaluation with Malliavin calculus For Black-Scholes model: dst x = rst x dt + σst x db t (GBM). Consider the European option with payoff function: φ = f (S T ). The option price is: V = E Q (e rt f (S T )). The Delta is: := V S = E(e rt f (S T )) V = S }{{} E(e rt f (S t ) ds T ) = ds }{{} e rt E(f (S T )δ( ds T (DS T ) 1 )) ds E I.B.P E(f (F )G) = E(f (F )H(F, G)), H(H, G) = δ(gh t (D γ F ) 1 ) DS T = T D t S T dt = σs T T, ds T ds = e rt S σt E(f (S T )B T ) Monte Carlo. = S T S, δ(1) = B T Elliptic case (BS): Fournié-Lasry-Lebuchoux-Lions-Touzi, 99, 1 Finance and Stochastics.

5 Motivation I: Greek evaluation with Malliavin calculus For Black-Scholes model: dst x = rst x dt + σst x db t (GBM). Consider the European option with payoff function: φ = f (S T ). The option price is: V = E Q (e rt f (S T )). The Delta is: := V S = E(e rt f (S T )) V = S }{{} E(e rt f (S t ) ds T ) = ds }{{} e rt E(f (S T )δ( ds T (DS T ) 1 )) ds E I.B.P E(f (F )G) = E(f (F )H(F, G)), H(H, G) = δ(gh t (D γ F ) 1 ) DS T = T D t S T dt = σs T T, ds T ds = e rt S σt E(f (S T )B T ) Monte Carlo. = S T S, δ(1) = B T Elliptic case (BS): Fournié-Lasry-Lebuchoux-Lions-Touzi, 99, 1 Finance and Stochastics. 2??? dxt x = b(xt x )dt + σ i (Xt x ) dbt, i Xt x R 3, [σ 1, σ 2 ] = z. i=1

6 Motivation II: More application with Malliavin calculus Integration by parts formula Clark-Ocone formula. T F = E[F ] + E[D t F F t ]dw (t) F = f (X t1,, X tn ), what if X t is the solution of the Purple SDE.? Application of Clark-Ocone formula. Karatzas-Ocone-Li, extension of Clark-Ocone formula. Karatzas-Ocone, Optimal portfolio. d what about? dxt x = b(xt x )dt + σ i (Xt x ) dbt, i Xt x R n. i=1 span{σ 1,, σ d, [σ i, σ j ], [σ i, [ [σ j, σ k ]]]} = R n. d < n.

7 Motivation III: large deviation principle (variational representation) Variational representation: Boué-Dupuis, 98. Ann. Prob. log Ee f (W ) = inf v E{1 2 1 v s 2 ds + f (W + v s ds)} Replace W with X we have LDP, explicit rate function obtained.

8 Motivation III: large deviation principle (variational representation) Variational representation: Boué-Dupuis, 98. Ann. Prob. log Ee f (W ) = inf v E{1 2 1 v s 2 ds + f (W + v s ds)} Replace W with X we have LDP, explicit rate function obtained. Idea of the proof Girsanov theorem Relative entropy function. what about the soluiton to Purple SDE? i.e. hypoelliptic. Ben Arous-Deuschel 94 CPAM.(Hypoelliptic). Baxendale-Strook, 88 PTRF. (manifold) Deuschel-Friz-Jacquier-Violante, 13, 14 CPAM. Stochastic Volatility, hypoelliptic.

9 sub-riemannian Geometry.

10 What is a foliation? Course description from IHP (7 ) for Steven Hurder A foliation is like an onion: you just peel space back layer by layer to see what s there. The word foliation is related to the French term feuilletage,as in the very tasty pastry mille feuilles.

11 Riemannian submersion Definition A smooth surjective map π : (M, g) (B, j) is called a Riemannian submersion if its derivative maps T x π : T x M T x B are orthogonal projections, i.e. for every x M, the map (T p(x)π ) T p(x) π : T p(x) B T p(x) B is the identity. A Riemannian submersion is said to have totally geodesic fibers if fore every b B, the set π 1 (b) is a totally geodesic submanifold of M.

12 Totally geodesic foliations. In general, for a manifold M n+m, we can have m-dimensional totally geodesic leaves and n-dimensional horizontal direction. The sub-riemannian structure is (g H, H) Hopf fibration U(1) S 2n+1 CP n. Riemannian submersion, π : M n+m B n. K-contact manifold, foliated by Reeb vector field. Sasakian manifold Generalized Hopf fibration.... Figure: Hopf fibration: Wikipedia

13 Brownian motion on totally geodesic foliations.

14 Construction of BM on a Riemannian manifold. Isometry u : (R n+m,, ) (T x M, g): e i u(e i ) Orthonormal frame bundle O(M) = x M O(M) x. Canonical projection π : O(M) M with πu = x. Horizontal lift π : H u O(M) T x M, π H i (x, u) = u(e i ). Figure: Rolling without slipping by B. K. Driver

15 Construction of horizontal Brownian motion on foliations. du t = n+m i=1 H i (U t ) db i t, U = (x, u) O(M), with πu = x π(u t ) = X t : a Brownian motion on Riemannian manifold M. An isometry u : (R n+m,, ) (T x M, g) will be called horizontal if u(r n {}) H x ; u({} R m ) V x. e 1,, e n, f 1,, f m canonical basis of R n+m u(e 1 ),, u(e n ), u(f 1 ),, u(f m ) basis for T M. A 1,, A n, V 1,, V m : lift of u(e i ), u(f i ) on T O(M). Horizontal frame bundle: O H (M). du t = n A i (U t ) dbt, i i=1 U O H (M), W t = π(u t ) is a horizontal Brownian motion on foliation M.

16 Construction of horizontal BM on Heisenberg group: I. B = (B 1 s,t, B 2 s,t, 1 2 ( t s t Bs,r 1 dbr 2 Bs,r 2 dbr 1 )) s db t = X db 1 t + Y db 2 t X = x 1 2 y z Y = y x z σ 1 = X, σ 2 = Y and [σ 1, σ 2 ] = σ 1 σ 2 σ 2 σ 2 = z. Purple SDE.

17 Construction of horizontal BM on Heisenberg group: II.

18 Horizontal BM on Totally geodesic foliations. (Hopf fibration) X t = e iθ (ω(t), 1), 1 + ω(t) 2 (Baudoin-Wang, PTRF17 ) Figure: Hopf fibration: Wikipedia

19 Quasi-invariance of Horizontal Wiener Measure.

20 General idea of quasi-invariance of Wiener measure Let µ be the Wiener measure on the path space W, and T : W W a transformation of W. The image measure µ T is defined by µ T (B) = µ(t 1 B) In general, the transformation T can be represented as a flow, and the flow is generated by a vector field Z on the path space W. du Z t (x) dt = Z(U Z t (x)), U Z (x) = x If (U Z t ) µ and µ are mutually absolutely continuous, we say µ is quasi-invariant under Z. Radon-Nikodym derivative: d(u Z t ) µ/dµ=?

21 Quasi-invariance of Wiener measure: progress R.H Cameron, W.T Martin, C-M Thm Ann. of Math I.V Girsanov Girsanov Thm Theory Probab. Appl, 196. L Gross Hilbert space Trans. AMS, 94 (196). Bruce K. Driver Compact Riemannian manifold for C 1 C-M functions. JFA, (1992), pp B.K Driver For pinned Brownian motion Trans. AMS Elton. P. Hsu Compact Riemannian manifold for all C-M paths. JFA (1995). Bruce K. Driver Heat kernel measures on loop groups, JFA (1997). E. P. Hsu Noncompact case JFA (22). E. P. Hsu, C. Ouyang, complete case. JFA (29). heat kernel measure and finite dimensional approximation F. Baudoin, M. Gordina and T. Melcher, heat kernel measures on infinite-dimensional Heisenberg groups. (Trans. AMS 13 ).

22 Cameron-Martin Theorem Theorem (CM Thm: we have U Z t (x) = x + tz.) Let µ denote the Wiener measure on the path space W = C ([, 1], R n ). Let h = (h 1,, h n ) be a path in L 2 [, 1] and define the translation T : R n R n by T (ω) = ω + ω (ω 1 + hds, h 1 ds,, ω n + Then µ T is absolutely continuous w.r.t. µ and h n ds) dµ T dµ (ω) = exp( n i=1 1 h i (s)dω i (s) 1 2 n 1 i=1 h 2 i (s)ds)

23 Cameron-Martin Theorem and horizontal Wiener measure. H = {h W (R n ) h is a.c. and h() =, T h (s) 2 ds < } (W t ) t 1 is the horizontal Brownian motion and the law µ W of W t on M will be referred to as the horizontal Wiener measure on M. Therefore, µ W is a probability measure on the space C x (M) of continuous paths w : [, 1] M, w() = x.

24 Quasi-invariance of Wiener measure on a Riemannian manifold. µ X : Wiener measure on W (M). // t : stochastic parallel translation along X, // t : T x M T Xt M. γ W(M) W(M) ζ t γ Not Diagram Proof I p h (γ) p h (ω) I ω W (R n ) W (R n ) ξ t ω For Presentation Only! dζ t γ dt ζ t γ = I ξ t ω I 1 = Z(ζ t γ) what is Z?

25 Quasi-invariance of Wiener measure on Riemannian manifolds. Theorem (B. Driver 1992, E.P. Hsu 1995 ) Let h H(T x M), and Z h be the µ X a.e. well defined vector field on W (M) given by Z h s (γ) =// s (γ)h s, for s [, 1] Then Z h admits a flow e tz h on W (M) and the Wiener measure µ X is quasi-invariant under the this flow. Figure: From: Bruce K. Driver, Curved Wiener Space Analysis

26 Quasi-invariance on compact foliated manifolds. Theorem (Baudoin,F. and Gordina, 217 ) Let h CM H (R n+m ). Consider the µ H X -a.e. defined process given by t vt h (ω) = h(t) + T IH (ω H ) s (A dωs H, Ah(s)). }{{} martingale part in vertical direction. and Z h be the µ H X a.e. well defined vector field on W (Mn+m ) given by Z h s (γ) = ˆΘ ε s(γ)v h s, for s [, 1] Then Z h admits a flow e tz h on W (M n+m ) and the horizontal Wiener measure µ H X is quasi-invariant under this flow.

27 Outline of the proof Recall that: ζ t γ = I ξ t ω I 1 : W H (M) W H (M), is generated by D v := Directional Derivative. ξv t ω = ω H + t p v (ξv λ (ω))dλ. p v (t):= I 1 Z h = v H (t) + 1 t 2 (Ric H) Us (Av H (s))ds 1 t ε J Vv(s)(Adωs H ) Us + ( t s Ω ) ε U τ (A dωτ H, Av(τ) + Vv(τ)) dωs H. For every fixed t R, the law P ζ t v of ζv t is equivalent to the horizontal Wiener measure P X with Radon-Nikodym derivative : dp ζ t v dp X (w) = dp ξt v dp H (I 1 w), w W H (M). [ dp ξ t T v (ω H ) = exp a dp s(ξ t t ω) dωs H 1 T ] a t H 2 s(ξ t ω) 2 ds. where a t = v H t + t c(ξλ )dλ + t b(ξλ )a λ dλ,

28 Quasi-invariance on foliated manifold: example No Diagram Proof At All. For Presentation Only! γ W(M n+m ) W(M n+m ) ζ t γ, ˆp h (γ) = e t ˆΘ ε s (γ)v h s I ˆp h (γ) σ W(B n ) W(B n ) η t σ, p h (σ) = e t// s (σ)hs I p h (σ) p h (ω) I I ω W (R n ) W (R n ) ξ t ω, p h (ω) = ω + th s s Heisenberg group : vs h (B) = (h 1 (s), h 2 (s), h 1 (τ)dbτ 2 h 2 (τ)dbτ 1 ). Hopf fibration : v h s = (h 1 (s), h 2 (s), s (( Θ ε τ ) 1 R) J Θ ε τ h(τ), Θ ε τ db τ H ).

29 Quasi-invariance of horizontal Wiener measure on Heisenberg group. p h (w) t ( =w t + t h(t), n i=1 h i (t)w i+n t h i+n (t)w i t + ) (h i (s) + 2ws)dh i i+n (s) (h i+n (s) + 2ws i+n )dh i (s) Let µ h H be the pushforward of µ H under p h. The density is explicitly given by dµ h H dµ H = exp ( 2n T i=1 h i(s)dw i s 1 2 T h (s) 2 R 2n ds ).

30 Clark-Ocone formula and Integration by Parts formulas.

31 Various Gradients Definition For F = f (w t1,..., w tn ) FC (C x (M))) we define Intrinsic gradient (Intrinsic derivative): D ε t F = n 1 [,ti ](t)θ ε t i d i f (W t1,, W tn ), t 1 i=1 Damped gradient (Damped Malliavin derivative): D ε t F = n 1 [,ti ](t)(τt ε ) 1 τt ε i d i f (X t1,, X tn ), t 1. i=1 Directional Derivative For an F-adapted, T x M-valued semimartingale (v(t)) t 1 with v() =, D v F = n i=1 d i f (W t1,, W tn ), Θ ε t i v(t i )

32 Clark-Ocone formula Clark-Ocone formula Baudoin-F.15, Baudoin-F.-Gordina17 Let ε >. Let F = f (W t1,, W tn ), f C (M). Then F = E x (F ) + 1 E x ( D ε s F F s ), Θ ε sdb s.

33 Clark-Ocone formula Clark-Ocone formula Baudoin-F.15, Baudoin-F.-Gordina17 Let ε >. Let F = f (W t1,, W tn ), f C (M). Then F = E x (F ) + 1 E x ( D ε s F F s ), Θ ε sdb s. Gradient Representation. Let F = f (W t1,, W tn ) FC (C x (M)). We have ( n ) de x (F ) = E x τt ε i d i f (W t1,, W tn ). i=1 Baudoin 14, sub-rie transverse symmetry dp t f (x) = E(τ ε t df (X t ))

34 Integration by parts formula for the damped gradient Theorem (Baudoin-F.15, Baudoin-F.-Gordina17) Suppose F FC (C x (M)) and γ CM H (M, Ω), then E x ( 1 ) ( D s ε F, Θ ε sγ (s) ds = E x F 1 ) γ (s), db s H. (2.1) Definition An F t -adapted absolutely continuous H x -valued process (γ(t)) t 1 such that γ() = and E x ( 1 γ (t) 2 H dt ) < will be called a horizontal Cameron-Martin process. The space of horizontal Cameron-Martin processes will be denoted by CM H (M, Ω). Recall the Cameron-Martin space. H = {h W (R n ) h is a.c. and h() =, T h (s) 2 ds < }

35 Integration by parts formula for directional derivative Theorem (Baudoin-F.-Gordina17) Suppose F FC (C x (M)) and v TW H (M, Ω), then E x (D v F ) = E x (F 1 v H(t) + 12 ( Θ εt) 1 Ric H Θεt v H (t), db t H ). Definition An F t -adapted T x M-valued continuous( semimartingale ) 1 (v(t)) t 1 such that v() = and E x v(t) 2 dt < will be called a tangent process if the process v(t) t ( Θ ε s) 1 T ( Θ ε s db s, Θ ε sv(s)) is a horizontal Cameron-Martin process. The space of tangent processes will be denoted by TW H (M, Ω).

36 Functional inequalities.

37 Functional inequalities Theorem (Baudoin-F,15,) For every cylindric function G FC (C x (M)) we have the following log-sobolev inequality. E x (G 2 ln G 2 ) E x (G 2 ) ln E x (G 2 ) 2e 3T (K+ κ ε ) E x ( T ) Ds ε G 2 εds.

38 Functional inequalities Theorem (Baudoin-F,15,) For every cylindric function G FC (C x (M)) we have the following log-sobolev inequality. E x (G 2 ln G 2 ) E x (G 2 ) ln E x (G 2 ) 2e 3T (K+ κ ε ) E x ( T Yang-Mills conditionδ H T ( ) = n i=1 X i T (X i, ) = Theorem (Baudoin-F,15,) Improved Log-Sobolev inequality Equivalent conditions: two-sided uniform Ricci curvature bound Log-Sobolev inequality Poincare inequality gradient estimates. ) Ds ε G 2 εds.

39 Concentration inequalities. we denote by d ε the distance associated with g ε Proposition ( Baudoin-F 15, under curvature bounds ) Let ε >. We have for every T > and r ( [ ] ) P x sup d ε (X t, x) E x sup d ε (X t, x) + r t T t T ( ) 1 = lim sup r r 2 ln P x sup d ε (X t, x) r t T Proposition (Baudoin-F 15, K= and bounds.) ( 1 lim sup r + r 2 ln P x lim inf r + 1 r 2 ln P x ( sup t T d(x t, x) r sup d(x t, x) r t T ) ) ( exp 1 2Te (K+ κ ε )T 1 2T, D 2nT, r 2 2Te (K+ κ ε )T )

40 Future work and applicaitons. Compact complete; Rough paths proof of Picard iteration; Path space on sub-riemannian manifold Loop space on sub-riemannian manifold; Totally geodesic foliations non totally geodesic foliations; Ricci flow on totally geodesic foliations, F. 17, 18. differential Harnack inequalities, monotonicity formulas. Stochastic Ricci flow and random matrices. Characterization of Ricci flow using Path space and Wasserstein space on sub-riemannian manifolds.

41 Future work and applicaitons. Compact complete; Rough paths proof of Picard iteration; Path space on sub-riemannian manifold Loop space on sub-riemannian manifold; Totally geodesic foliations non totally geodesic foliations; Ricci flow on totally geodesic foliations, F. 17, 18. differential Harnack inequalities, monotonicity formulas. Stochastic Ricci flow and random matrices. Characterization of Ricci flow using Path space and Wasserstein space on sub-riemannian manifolds. Quasi-invariance of horizontal Wiener measure. Girsanov Thm Is there any finance model that satisfies a Heisenberg group type stochastic differential equation? control problem. LDP, SV model, Malliavin calculus application.

42 Bibliography F. Baudoin and Q. Feng, Log-sobolev inequalities on the horizontal path space of a totally geodesic foliation, arxiv: (215). Revision for ECP. F. Baudoin, Q. Feng, and M. Gordina, Quasi-invariance of horizontal wiener measure on a compact foliated riemannian manifold,arxiv: (217). Revision for JFA.

43 Thanks for your attention!

44 Technical part: various connections. Bott connection π H ( R X Y ), X, Y Γ (H), π H ([X, Y ]), X Γ (V), Y Γ (H), X Y = π V ([X, Y ]), X Γ (H), Y Γ (V), π V ( R X Y ), X, Y Γ (V), Isomorphism g H (J Z (X ), Y ) x = g V (Z, T (X, Y )) x, Z Γ (V), X, Y Γ (H) Damped connection ε X Y = X Y T (X, Y ) + 1 ε J Y X, X, Y Γ (M). Adjoint connection of the damped connection ε X Y := ε X Y T ε (X, Y ) = X Y + 1 ε J X Y,

45 Technical part: Bochner-Weitzenböck identity ε = ( H T ε H) ( H T ε H) 1 ε J2 + 1 ε δ HT Ric H, Let f C (M), x M and ε >, then dlf (x) = ε df (x). (Baudoin-Kim-Wang. 16 CGT) Consequence dp t f = Qt ε df, P t = e tl, Qt ε = e t ε Clark-Ocone formula F = E x (F ) + T E x ( D ɛ s F F s ), Θ ε,sdb s H.

46 Technical part: stochastic parallel translation. Θ ε t : T X t M T x M is a solution to d[ Θ ε tα(x t )] = Θ ε t ε dx t α(x t ), τ ε t : T X t M T x M is a solution of d[τt ε α(x t )] (3.2) ( = τt ε dxt T ε dx t 1 ( 1 2 ε J2 1 ) ) ε δ HT + Ric H dt α(x t ), τ = Id, τ ε t = M ε tθ ε t. M ε t : Tx M Tx M, t, is the solution to ODE dm ε t = 1 ( 1 dt 2 Mε tθ ε t ε J2 1 ) ε δ HT + Ric H (Θ ε t) 1, (3.3) M ε = Id.

47 Technical part: example. Given a number ρ, suppose that G(ρ) is simply a connected three-dimensional Lie group whose Lie algebra g admits a basis {X, Y, Z} satisfying [X, Y ] = Z, [X, Z] = ρy, [Y, Z] = ρx. ρ =, ε = ˆΘ 1 1 B 2 ε t = 1, τt t = 1 B 1 t 1 1 e ρ 2 t t ρ ρ τt e 2 s db 2 s = e ρ 2 t t ρ e 2 s dbs 1 1