Integral representations in models with long memory

Transcription

1 Integral representations in models with long memory Georgiy Shevchenko, Yuliya Mishura, Esko Valkeila, Lauri Viitasaari, Taras Shalaiko Taras Shevchenko National University of Kyiv 29 September 215, Ulm Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

2 Outline 1 Problem formulation 2 Representation for general integrators 3 Representation with respect to Gaussian processes 4 Representation with respect to fbm 5 Questions Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

3 Contents 1 Problem formulation 2 Representation for general integrators 3 Representation with respect to Gaussian processes 4 Representation with respect to fbm 5 Questions Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

4 Question Given some integrator X, which random variables ξ can be represented as with adapted φ? ξ = 1 φ s dx s Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

5 Question Given some integrator X, which random variables ξ can be represented as with adapted φ? ξ = 1 φ s dx s Motivation: financial models with continuous time. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

6 Question Given some integrator X, which random variables ξ can be represented as with adapted φ? ξ = 1 φ s dx s Motivation: financial models with continuous time. For X = W, the Wiener process: ξ = 1 φ s dw s with adapted φ L 2 ([, 1] Ω) iff ξ is W -measurable, centered, and square integrable; Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

7 Question Given some integrator X, which random variables ξ can be represented as with adapted φ? ξ = 1 φ s dx s Motivation: financial models with continuous time. For X = W, the Wiener process: ξ = 1 φ s dw s with adapted φ L 2 ([, 1] Ω) iff ξ is W -measurable, centered, and square integrable; ξ = 1 φ s dw s with adapted φ (ω) L 2 ([, 1]) a.s. iff ξ is W -measurable (Dudley (1977)). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

8 Fractional Brownian motion Definition The fractional Brownian motion (fbm) with Hurst index H (, 1) is a centered Gaussian process B H = {Bt H, t } with stationary increments and the covariance function [ ] E Bt H Bs H = 1 2 (t2h + s 2H t s 2H ). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

9 Fractional Brownian motion Definition The fractional Brownian motion (fbm) with Hurst index H (, 1) is a centered Gaussian process B H = {Bt H, t } with stationary increments and the covariance function [ ] E Bt H Bs H = 1 2 (t2h + s 2H t s 2H ). For H > 1/2 (the case we consider here) fbm has long memory. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

10 Fractional Brownian motion Definition The fractional Brownian motion (fbm) with Hurst index H (, 1) is a centered Gaussian process B H = {Bt H, t } with stationary increments and the covariance function [ ] E Bt H Bs H = 1 2 (t2h + s 2H t s 2H ). For H > 1/2 (the case we consider here) fbm has long memory. B H is almost surely Hölder continuous with any exponent γ < H. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

11 Contents 1 Problem formulation 2 Representation for general integrators 3 Representation with respect to Gaussian processes 4 Representation with respect to fbm 5 Questions Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

12 Integration For β (, 1) fractional derivatives ( D β a+ f ) ( 1 f (x) x (x) = Γ(1 β) (x a) β + β a ( ( D 1 β b g) (x) = e iπβ g(x) + (1 β) Γ(β) (b x) 1 β f (x) f (u) (x u) b x 1+β du g(x) g(u) du (u x) 2 β ) 1 (a,b) (x), ) 1 (a,b) (x). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

13 Integration For β (, 1) fractional derivatives ( D β a+ f ) ( 1 f (x) x (x) = Γ(1 β) (x a) β + β a ( ( D 1 β b g) (x) = e iπβ g(x) + (1 β) Γ(β) (b x) 1 β f (x) f (u) (x u) b Assume that D β a+ f L1 [a, b], D 1 β b g b L [a, b]; g b (x) = g(x) g(b). x 1+β du g(x) g(u) du (u x) 2 β ) 1 (a,b) (x), ) 1 (a,b) (x). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

14 Integration For β (, 1) fractional derivatives ( D β a+ f ) ( 1 f (x) x (x) = Γ(1 β) (x a) β + β a ( ( D 1 β b g) (x) = e iπβ g(x) + (1 β) Γ(β) (b x) 1 β f (x) f (u) (x u) b x 1+β du g(x) g(u) du (u x) 2 β ) 1 (a,b) (x), ) 1 (a,b) (x). Assume that D β a+ f L1 [a, b], D 1 β b g b L [a, b]; g b (x) = g(x) g(b). Then the fractional integral b a f (x)g.(x) is defined as b b f (x)dg(x) = e iπβ ( D β a+ f ) (x) ( D 1 β b g b ) (x)dx. a a Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

15 Let (Ω, F, P) be a complete probability space endowed with a P-complete filtration F = {F t, t [, 1]}, and X = {X t, t [, 1]} be F-adapted such that (H) Hölder continuity: X C θ [, 1] a.s. for some θ > 1/2. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

16 Let (Ω, F, P) be a complete probability space endowed with a P-complete filtration F = {F t, t [, 1]}, and X = {X t, t [, 1]} be F-adapted such that (H) Hölder continuity: X C θ [, 1] a.s. for some θ > 1/2. (S) Small ball estimate: there exist positive constants λ, µ, K 1, K 2 such that for all ε >, >, s [, 1 ] { } { P sup X t X s K 1 ε exp K 2 ε λ µ}. s t s+ Here C θ [, 1] denotes the class of Hölder continuous functions of order θ. The exponents are such that θ µ/λ. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

17 Let (Ω, F, P) be a complete probability space endowed with a P-complete filtration F = {F t, t [, 1]}, and X = {X t, t [, 1]} be F-adapted such that (H) Hölder continuity: X C θ [, 1] a.s. for some θ > 1/2. (S) Small ball estimate: there exist positive constants λ, µ, K 1, K 2 such that for all ε >, >, s [, 1 ] { } { P sup X t X s K 1 ε exp K 2 ε λ µ}. s t s+ Here C θ [, 1] denotes the class of Hölder continuous functions of order θ. The exponents are such that θ µ/λ. The assumptions are satisfied e.g. by fbm B H with H > 1/2 for θ < H, µ = 1, λ = 1/H. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

18 Lemma There exists an F-adapted process ϕ = {ϕ t, t [, 1]} such that ϕ C[, 1) and For any t < 1 the integral v t = t ϕ s dx s exists. lim t 1 v t = a.s. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

19 Lemma There exists an F-adapted process ϕ = {ϕ t, t [, 1]} such that ϕ C[, 1) and For any t < 1 the integral v t = t ϕ s dx s exists. lim t 1 v t = a.s. Theorem (Improper representation) Assume that F is left-continuous at 1. Then for any F 1 -measurable random variable ξ there exists an adapted process ψ C[, 1) such that For any t < 1 the integral v t = t ψ s dx s exists. lim t 1 v t = ξ a.s. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

20 Theorem (Proper representation) Let a random variable ξ be such that ξ = Z 1 for some adapted process Z C ρ [, 1] with ρ > µ/(λθ) 1 a.s. Then there exists an adapted process φ such that and almost surely. 1 ψ s dx s = ξ Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

21 Theorem (Proper representation) Let a random variable ξ be such that ξ = Z 1 for some adapted process Z C ρ [, 1] with ρ > µ/(λθ) 1 a.s. Then there exists an adapted process φ such that and almost surely. 1 ψ s dx s = ξ ξ = F (X s1,..., X sn ), where F : R n R is locally Hölder continuous with large enough exponent with respect to each variable. z t = F (X s1 t,..., X sn t). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

22 Theorem (Proper representation) Let a random variable ξ be such that ξ = Z 1 for some adapted process Z C ρ [, 1] with ρ > µ/(λθ) 1 a.s. Then there exists an adapted process φ such that and almost surely. 1 ψ s dx s = ξ ξ = F (X s1,..., X sn ), where F : R n R is locally Hölder continuous with large enough exponent with respect to each variable. z t = F (X s1 t,..., X sn t). ξ = G({X s, s [, 1]}), where G : C[, 1] R is locally Hölder continuous with respect to the supremum norm on C[, 1]. z t = G({X s t, s [, 1]}). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

23 Theorem (Proper representation) Let a random variable ξ be such that ξ = Z 1 for some adapted process Z C ρ [, 1] with ρ > µ/(λθ) 1 a.s. Then there exists an adapted process φ such that and almost surely. 1 ψ s dx s = ξ ξ = F (X s1,..., X sn ), where F : R n R is locally Hölder continuous with large enough exponent with respect to each variable. z t = F (X s1 t,..., X sn t). ξ = G({X s, s [, 1]}), where G : C[, 1] R is locally Hölder continuous with respect to the supremum norm on C[, 1]. z t = G({X s t, s [, 1]}). If F is left-continuous at 1, then ξ = 1 A, A F 1, satisfies the assumption any simple F 1 -measurable rv satisfies the assumption. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

24 Contents 1 Problem formulation 2 Representation for general integrators 3 Representation with respect to Gaussian processes 4 Representation with respect to fbm 5 Questions Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

25 Let X = {X t, t [, 1]} be a centered Gaussian process such that (A1) For some H 1 (, 1] and C 1 > and any s, t [, 1] E [ (X t X s ) 2 ] C 1 t s 2H 1. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

26 Let X = {X t, t [, 1]} be a centered Gaussian process such that (A1) For some H 1 (, 1] and C 1 > and any s, t [, 1] E [ (X t X s ) 2 ] C 1 t s 2H 1. (A2) For some H 2 (, 1] and C 2 > and any s, t [, 1] E [ (X t X s ) 2 ] C 2 t s 2H 2. Clearly, H 1 H 2. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

27 Let X = {X t, t [, 1]} be a centered Gaussian process such that (A1) For some H 1 (, 1] and C 1 > and any s, t [, 1] E [ (X t X s ) 2 ] C 1 t s 2H 1. (A2) For some H 2 (, 1] and C 2 > and any s, t [, 1] E [ (X t X s ) 2 ] C 2 t s 2H 2. Clearly, H 1 H 2. Also assume that the increments of X are positively correlated. (B + ) For any s 1, t 1, s 2, t 2 [, 1], s 1 t 1 s 2 t 2 E [ (X t1 X s1 )(X t2 X s2 ) ]. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

28 Let X = {X t, t [, 1]} be a centered Gaussian process such that (A1) For some H 1 (, 1] and C 1 > and any s, t [, 1] E [ (X t X s ) 2 ] C 1 t s 2H 1. (A2) For some H 2 (, 1] and C 2 > and any s, t [, 1] E [ (X t X s ) 2 ] C 2 t s 2H 2. Clearly, H 1 H 2. Also assume that the increments of X are positively correlated. (B + ) For any s 1, t 1, s 2, t 2 [, 1], s 1 t 1 s 2 t 2 E [ (X t1 X s1 )(X t2 X s2 ) ]. E.g. fbm B H with H > 1/2. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

29 Lemma (Small ball estimate) Under (A1), (A2), and (B + ), there exists C > such that for any >, t [, 1 ] and ε > small enough { } { } P sup t s t t + X t X s ε exp Cε 4 (2H 2+2)/H 1 2 2H 2. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

30 Lemma (Small ball estimate) Under (A1), (A2), and (B + ), there exists C > such that for any >, t [, 1 ] and ε > small enough { } { } P sup t s t t + X t X s ε exp Cε 4 (2H 2+2)/H 1 2 2H 2. Theorem (Proper representation) Assume that a centered Gaussian process X satisfies (A1), (A2), (B + ) with < 2H 1 1 < H 2 H 1, and let a random variable ξ be such that ξ = Z 1 for some adapted process Z C ρ [, 1] with ρ > (1+H 2)(H 1 H 2 ) H H 1. Then there exists an adapted process φ such that almost surely. 1 ψ s dx s = ξ Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

31 Contents 1 Problem formulation 2 Representation for general integrators 3 Representation with respect to Gaussian processes 4 Representation with respect to fbm 5 Questions Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

32 Reminder: fractional integral is defined as b a f (x)dg(x) = e iπβ b a ( D β a+ f ) (x) ( D 1 β b g b ) (x)dx; we defined it for D β a+ f L1 [a, b], D 1 β b g b L [a, b]. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

33 Reminder: fractional integral is defined as b a f (x)dg(x) = e iπβ b a ( D β a+ f ) (x) ( D 1 β b g b ) (x)dx; we defined it for D β a+ f L1 [a, b], D 1 β b g b L [a, b]. Idea: for β (1 H, 1) D 1 β BH 1 (x) C(ω) 1 x H+β 1 log(1 x) 1/2. 1 Define for some µ > 1/2 the weight ρ(x) = (1 x) H+β 1 log(1 x) µ. Let a function f : [, 1] R be such that D β + f L1 ([, 1], ρ); this will be our class of admissible integrands. Then the extended fractional integral 1 is well-defined. f (t)db H t = e iπβ 1 ( D β + f ) (x) ( D 1 β 1 BH 1 ) (x)dx Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

34 Assumption There exists an adapted process Z such that Z 1 = ξ and for some a > 1 Z 1 Z t C(ω) log(1 t) a, for all t [, 1). Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

35 Assumption There exists an adapted process Z such that Z 1 = ξ and for some a > 1 Z 1 Z t C(ω) log(1 t) a, for all t [, 1). Theorem (Proper representation) Let ξ satisfy the above assumption. Then there exists an adapted process ψ such that 1 ψ s db H s = ξ holds with the integral defined in the extended fractional sense. Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

36 Contents 1 Problem formulation 2 Representation for general integrators 3 Representation with respect to Gaussian processes 4 Representation with respect to fbm 5 Questions Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

37 Which random variables can be represented in the form ξ = with adapted ψ C[, 1]? 1 ψ s dx s Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

38 Which random variables can be represented in the form with adapted ψ C[, 1]? ξ = 1 ψ s dx s Which random variables can be represented in the form ξ = 1 with t ψ s dx s bounded from below? ψ s dx s Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

39 Which random variables can be represented in the form with adapted ψ C[, 1]? ξ = 1 ψ s dx s Which random variables can be represented in the form ξ = 1 ψ s dx s with t ψ s dx s bounded from below? Which adapted processes can be represented in the form ξ t = t ψ s dx s? Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18

40 Which random variables can be represented in the form with adapted ψ C[, 1]? ξ = 1 ψ s dx s Which random variables can be represented in the form ξ = 1 ψ s dx s with t ψ s dx s bounded from below? Which adapted processes can be represented in the form ξ t = t ψ s dx s? What about H < 1/2? Georgiy Shevchenko (Kyiv University) Integral representations with long memory 29 September / 18