Dudley s representation theorem in infinite dimensions and weak characterizations of stochastic integrability

Transcription

1 Dudley s representation theorem in infinite dimensions and weak characterizations of stochastic integrability Mark Veraar Delft University of Technology London, March 31th, 214 Joint work with Martin Ondreját Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

2 Introduction (Ω, A, P) - a probability space F = (F t ) t - filtration W - a (cylindrical) Brownian motion with respect to F. Let I : L p F W (Ω; L 2 (R + )) L p (Ω, F W, P) be defined by Iφ = φ dw. (i) I is an isomorphism onto L p (Ω, F W, P) for p (1, ), (ii) I is a surjection when p < 1, (iii) I is an isomorphism onto the Hardy space H 1 when p = 1. Results in (i) are connected to the martingale representation theorem. In (ii) there is no uniqueness: there exists φ L F W (Ω; L 2 (, 1)): 1 φ dw = and 1/2 φdw = W (1/2) Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

3 Introduction (Ω, A, P) - a probability space F = (F t ) t - filtration W - a (cylindrical) Brownian motion with respect to F. Let I : L p F W (Ω; L 2 (R + )) L p (Ω, F W, P) be defined by Iφ = φ dw. (i) I is an isomorphism onto L p (Ω, F W, P) for p (1, ), (ii) I is a surjection when p < 1, (iii) I is an isomorphism onto the Hardy space H 1 when p = 1. Results in (i) are connected to the martingale representation theorem. In (ii) there is no uniqueness: there exists φ L F W (Ω; L 2 (, 1)): 1 φ dw = and 1/2 φdw = W (1/2) Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

4 Introduction (Ω, A, P) - a probability space F = (F t ) t - filtration W - a (cylindrical) Brownian motion with respect to F. Let I : L p F W (Ω; L 2 (R + )) L p (Ω, F W, P) be defined by Iφ = φ dw. (i) I is an isomorphism onto L p (Ω, F W, P) for p (1, ), (ii) I is a surjection when p < 1, (iii) I is an isomorphism onto the Hardy space H 1 when p = 1. Results in (i) are connected to the martingale representation theorem. In (ii) there is no uniqueness: there exists φ L F W (Ω; L 2 (, 1)): 1 φ dw = and 1/2 φdw = W (1/2) Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

5 History and main problem (i) is due to Itô 51, Kunita-Watanabe 67 and Garling 78. (ii) is due to Dudley 77 for p = and Garling 78 for p (, 1). (iii) is due to Dubins, Gilat, Jacka, Oblój, Revuz, Yor, etc. van Neerven-Weis-V. 7: E-valued extension of (i) under geometric condition UMD on the Banach space E. Main question in the first part of this talk: Does (ii) extend to the vector-valued setting? In other words: is it true that for every ξ L (Ω, F W ; E) there exists a φ L F W (Ω; L 2 (R + ; E)) such that φ dw = ξ? Even for Hilbert spaces E this was unknown. Applying Dudley s result in each coordinate does not work. Dudley s proof is based on order of the real numbers. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

6 History and main problem (i) is due to Itô 51, Kunita-Watanabe 67 and Garling 78. (ii) is due to Dudley 77 for p = and Garling 78 for p (, 1). (iii) is due to Dubins, Gilat, Jacka, Oblój, Revuz, Yor, etc. van Neerven-Weis-V. 7: E-valued extension of (i) under geometric condition UMD on the Banach space E. Main question in the first part of this talk: Does (ii) extend to the vector-valued setting? In other words: is it true that for every ξ L (Ω, F W ; E) there exists a φ L F W (Ω; L 2 (R + ; E)) such that φ dw = ξ? Even for Hilbert spaces E this was unknown. Applying Dudley s result in each coordinate does not work. Dudley s proof is based on order of the real numbers. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

7 History and main problem (i) is due to Itô 51, Kunita-Watanabe 67 and Garling 78. (ii) is due to Dudley 77 for p = and Garling 78 for p (, 1). (iii) is due to Dubins, Gilat, Jacka, Oblój, Revuz, Yor, etc. van Neerven-Weis-V. 7: E-valued extension of (i) under geometric condition UMD on the Banach space E. Main question in the first part of this talk: Does (ii) extend to the vector-valued setting? In other words: is it true that for every ξ L (Ω, F W ; E) there exists a φ L F W (Ω; L 2 (R + ; E)) such that φ dw = ξ? Even for Hilbert spaces E this was unknown. Applying Dudley s result in each coordinate does not work. Dudley s proof is based on order of the real numbers. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

8 History and main problem (i) is due to Itô 51, Kunita-Watanabe 67 and Garling 78. (ii) is due to Dudley 77 for p = and Garling 78 for p (, 1). (iii) is due to Dubins, Gilat, Jacka, Oblój, Revuz, Yor, etc. van Neerven-Weis-V. 7: E-valued extension of (i) under geometric condition UMD on the Banach space E. Main question in the first part of this talk: Does (ii) extend to the vector-valued setting? In other words: is it true that for every ξ L (Ω, F W ; E) there exists a φ L F W (Ω; L 2 (R + ; E)) such that φ dw = ξ? Even for Hilbert spaces E this was unknown. Applying Dudley s result in each coordinate does not work. Dudley s proof is based on order of the real numbers. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

9 History and main problem (i) is due to Itô 51, Kunita-Watanabe 67 and Garling 78. (ii) is due to Dudley 77 for p = and Garling 78 for p (, 1). (iii) is due to Dubins, Gilat, Jacka, Oblój, Revuz, Yor, etc. van Neerven-Weis-V. 7: E-valued extension of (i) under geometric condition UMD on the Banach space E. Main question in the first part of this talk: Does (ii) extend to the vector-valued setting? In other words: is it true that for every ξ L (Ω, F W ; E) there exists a φ L F W (Ω; L 2 (R + ; E)) such that φ dw = ξ? Even for Hilbert spaces E this was unknown. Applying Dudley s result in each coordinate does not work. Dudley s proof is based on order of the real numbers. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

10 Overview 1 Introduction Known representation theorems History and main problem 2 Dudley s result in infinite dimensions Main result Key Lemmas Universal representation theorem 3 Weak characterizations of stochastic integrability History and questions Known results Counterexample for p < 1 Positive result for p = 1 4 Doob s representation theorem 5 Summary and remarks Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

11 Dudley s result in infinite dimensions Theorem (Ondreját-V. 13) Let E be a Banach space and a < b. Let ξ : Ω E be strongly measurable. The following are equivalent: 1 there exists a stochastically integrable φ : [a, b] Ω E such that b a 2 ξ is σ(f t : t < b)-measurable. φ dw = ξ F W -measurability is not needed in this result. No measurability condition on ξ if b =. If additionally ξ L p (Ω; E) for p (, 1), then one can construct φ with additional p-integrability properties. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

12 Dudley s result in infinite dimensions One of the key Lemmas: Lemma Let a. Let η : Ω R be F a -measurable. Let τ = inf { r a : W (r) W (a) = η }. Then a τ < a.s. and 2/(πe) E min{ η / t, 1} P(τ a > t) E min{ η / t, 1}, t >. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

13 Universal representation theorem Theorem (Ondreját-V. 13) Let E be a separable Banach space. Assume F is P-countably generated. Then there exists a strongly predictable process φ : R + Ω E which is locally stochastically integrable and for every ξ L (Ω, F ; E) there exists an increasing sequence (n k ) k of natural numbers such that lim ζ(n k) = ξ k in L (Ω; E), where ζ(t) = t φ dw. Result is new even in the scalar setting. φ is only locally integrable, because lim t ζ(t) does not exist. There is also a version of the result on finite time intervals. Proof based on similar ideas as previous result. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

14 Universal representation theorem Theorem (Ondreját-V. 13) Let E be a separable Banach space. Assume F is P-countably generated. Then there exists a strongly predictable process φ : R + Ω E which is locally stochastically integrable and for every ξ L (Ω, F ; E) there exists an increasing sequence (n k ) k of natural numbers such that lim ζ(n k) = ξ k in L (Ω; E), where ζ(t) = t φ dw. Result is new even in the scalar setting. φ is only locally integrable, because lim t ζ(t) does not exist. There is also a version of the result on finite time intervals. Proof based on similar ideas as previous result. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

15 Weak characterizations Representation problems are closely connected with the problem of weak characterization of stochastic integrability. E be a Hilbert space, p [, ), φ a strongly F-progressively measurable E-valued process ξ L p (Ω; E) Assume φ, x L p (Ω; L 2 (, 1)) for all x E and 1 φ(s), x dw (s) = ξ, x, x E. (1) Does this imply that φ is stochastically integrable in E and, 1 φ(s) dw (s) = ξ? Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

16 Weak characterizations Representation problems are closely connected with the problem of weak characterization of stochastic integrability. E be a Hilbert space, p [, ), φ a strongly F-progressively measurable E-valued process ξ L p (Ω; E) Assume φ, x L p (Ω; L 2 (, 1)) for all x E and 1 φ(s), x dw (s) = ξ, x, x E. (1) Does this imply that φ is stochastically integrable in E and, 1 φ(s) dw (s) = ξ? Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

17 Weak characterizations Representation problems are closely connected with the problem of weak characterization of stochastic integrability. E be a Hilbert space, p [, ), φ a strongly F-progressively measurable E-valued process ξ L p (Ω; E) Assume φ, x L p (Ω; L 2 (, 1)) for all x E and 1 φ(s), x dw (s) = ξ, x, x E. (1) Does this imply that φ is stochastically integrable in E and, 1 φ(s) dw (s) = ξ? Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

18 Weak characterizations Answer is known in several cases: Finite dimensional case. For p > 1 this result is due to van Neerven-Weis-V. 7 In particular, the assumptions imply φ L p (Ω; L 2 (, 1; E)) and then also 1 φ(s) dw (s) = ξ (thus weak=strong). A version of the result holds for UMD spaces E. We will discuss remaining cases: p = 1 positive answer p < 1 negative answer Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

19 Weak characterizations Answer is known in several cases: Finite dimensional case. For p > 1 this result is due to van Neerven-Weis-V. 7 In particular, the assumptions imply φ L p (Ω; L 2 (, 1; E)) and then also 1 φ(s) dw (s) = ξ (thus weak=strong). A version of the result holds for UMD spaces E. We will discuss remaining cases: p = 1 positive answer p < 1 negative answer Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

20 Counterexample for p < 1 Theorem (Ondreját-V 13) Let E be an infinite dimensional Hilbert space. There exists a strongly progressive process φ : [, 1] Ω E with φ, x L p (Ω; L 2 (, 1)) for all x E and for all p [, 1), and 1 φ(t), x dw (t) = for all x E (2) but φ L 2 (,1;E) = almost surely. In particular, φ is not stochastically integrable. This provides a counterexample to the weak characterization of stochastic integrability. In this case ξ =. Construction based on technique of Dudley s representation theorem and 1 φ dw = with φ. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

21 Counterexample for p < 1 Theorem (Ondreját-V 13) Let E be an infinite dimensional Hilbert space. There exists a strongly progressive process φ : [, 1] Ω E with φ, x L p (Ω; L 2 (, 1)) for all x E and for all p [, 1), and 1 φ(t), x dw (t) = for all x E (2) but φ L 2 (,1;E) = almost surely. In particular, φ is not stochastically integrable. This provides a counterexample to the weak characterization of stochastic integrability. In this case ξ =. Construction based on technique of Dudley s representation theorem and 1 φ dw = with φ. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

22 Counterexample for p < 1 One of the key lemmas Lemma Let (ξ n ) n 1 be a sequence of independent [, )-valued random variables for which there is a constant C > such that P(ξ 1 > t) (t + 1) 1. For a sequence (c n ) n 1 in (, 1] the following are equivalent: 1 sup n 1 c n ξ n < a.s. 2 n 1 c n <. 3 for all p (1, ], (c n ξ n ) n 1 l p < a.s. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

23 Positive result for p = 1 Theorem (Ondreját-V. 13) Let E be a Hilbert space. Let φ : [, T ] Ω E be strongly progressively measurable. Assume φ, x L (Ω; L 2 (, T )) for all x E. Let ξ L 1 (Ω; E). If T φ, x dw = ξ, x, for all x E. then φ in L F (Ω; L2 (, T ; E)), and T φ dw = ξ. Moreover, for all r (, 1) one has φ L r (Ω; L 2 (, T ; E)). Version of the result holds for UMD spaces E. Proof based on Burkholder-Davis-Gundy for stochastic integrals. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

24 Doob s representation theorem Theorem (Ondreját-V. 13) Let E be a UMD Banach space. Let M : R + Ω E be a continuous local martingale. If there exists a progressively measurable φ : R + Ω E such that φ, x L (Ω; L 2 (R + )) for all x E, and for all x E, a.s. [ M, x ] t = t φ(s), x 2 ds, t R +, then φ L (Ω; γ(l 2 (R + ), E)), M is a local martingale with a.s. continuous paths and there exists a Brownian motion W on an extended probability space such that a.s. M t = t φ(s) dw (s), t R +. (3) Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14

25 Summary and remarks Dudley s representation theorem holds for any Banach space E Weak characterizations hold for processes φ which are weakly in L p (Ω; L 2 (, 1)) and random variables ξ L p (Ω; E) as long as p [1, ). For p < 1, there are counterexamples. Doob s representation theorem naturally extends to UMD spaces. What about weak characterizations for integrators different from Brownian motion? Thank you for your attention. Mark Veraar (TUDelft) Dudley s Representation theorem London, March 31th, / 14