Topics in fractional Brownian motion

Topics in fractional Brownian motion Esko Valkeila Spring School, Jena 25.3. 2011

We plan to discuss the following items during these lectures: Fractional Brownian motion and its properties. Topics in fractional Brownian motion 2/52

We plan to discuss the following items during these lectures: Fractional Brownian motion and its properties. Representation results for fractional Brownian Motions. Topics in fractional Brownian motion 2/52

We plan to discuss the following items during these lectures: Fractional Brownian motion and its properties. Representation results for fractional Brownian Motions. Characterization theorems for fbm. On stochastic integrals with respect to fbm. Topics in fractional Brownian motion 2/52

Fractional Brownian motion Basic set-up Here X is a stochastic process, defined on (Ω, F, P); we shall assume that X 0 = 0, X is square integrable and centered: EX t = 0. The sample paths of X are continuous. If we speak about the filtrations, we assume always that the filtration F = (F t ) t 0 is generated by the process X: F t = F X t = σ{x s : s t}. Topics in fractional Brownian motion 3/52

Fractional Brownian motion Definition A process X is a fractional Brownian motion with index H (0, 1), if in addition it is a Gaussian process with covariance E (X t X s ) = 1 2 ( t 2H + s 2H s t 2H) for s, t 0. (1) Topics in fractional Brownian motion 4/52

Fractional Brownian motion Properties We denote by B H fractional Brownian motion with index H. It follows from (1) that B H has stationary increments, the increments have positive correlation for H > 1 2, negative correlation for H < 1 2, and they are independent, if H = 1 2. B H is self-similar with index H: the distribution of B H at, 0 t 1 is the same as the distribution of a H B H t, 0 t 1. Topics in fractional Brownian motion 5/52

Fractional Brownian motion History Fractional Brownian motion was introduced by Kolmogorov in connection to his work related to turbulence. Kolmogorov s work was published in 1940. Kolmogorov gave a spectral representation of fbm using an orthogonally scattered Gaussian measure. In 1969 Molchan and Golosov proved a Girsanov theorem for Kolmogorov process. They also gave a representation theorem for Kolmogorov process in terms of standard Brownian motion. Topics in fractional Brownian motion 6/52

Fractional Brownian motion More properties By standard arguments Kolmogorov s continuity theorem gives that B H is Hölder-continuous on compacts: B H s B H t C t s β for s, t [0, T]. (2) The sequence b H k := BH k BH, k 1 is an erdogic sequence, k 1 and we have that 1 n n k=1 f(b H k ) L 1 (P) Ef(b H 1 ); (3) where f is a Borel measurable function R R. We assume also that f(b H 1 ) L1 (P). Note that in (3) b H is a random variable with 1 standard normal distribution. The sequence b H = (b H k ) k 1 is called fractional Gaussian noise. Topics in fractional Brownian motion 7/52

Fractional Brownian motion First results The property described in (3) and self-similarity allows us to obtain results of the following type: Consider the interval [0, T], and put t k = Tk n for k = 0,..., n. Then n k=1 and hence as n, B H t k B H t k 1 1 H Law = T n n b H k 1 H, k=1 n B H t k B H t k 1 1 P H TE b H 1 1 H. (4) k=1 Topics in fractional Brownian motion 8/52

Fractional Brownian motion More results If H = 1 2, then B 1 2 is the standard Brownian motion W, and the property (4) is the well-known property n (W tk W tk1 ) 2 P T. k=1 of the quadratic variation of the standard Brownian motion.. Let Π = { t = (ti ) n i=0 : 0 = t 0 < t 1 < < t n = T : n 1} and define the random variable, the 1/H- variation of fbm V 1/H, by T V 1/H T = sup t Π n B H t i B H t i 1 1 H. i=1 Then V 1/H T = + P-a.s. For the proof we refer to Pratelli (2011). Topics in fractional Brownian motion 9/52

FBm is not a semimartingale BDM-theorem We will show directly that fractional Brownian motion is not a semimartingale. Recall that K n is a simple predictable process, if K n s = k n k=1 α n k 1 (t k 1,t k ](s), α n k F t k 1 and 0 = t 0 < t 1 < < t kn = T Topics in fractional Brownian motion 10/52

FBm is not a semimartingale BDM-theorem, II According to the Bichteler Dellacherie Mokobodski- theorem the process X is a semimartingale if and only if for any sequence of predictable simple processes K n with the property we also have (K n X) T = sup K n P s 0 (5) s T k n k=1 α n k (X t k X tk 1 ) P 0 Topics in fractional Brownian motion 11/52

FBm is not a semimartingale Take now X = B H and put t k = kt n, k = 0, 1,..., n. In the definition of K n take now α n = 0 and for k = 2,..., n put 1 α n k = n2h 1 (B H t k 1 B H t k 2 ). The Hölder continuity of B H gives now that our special K n satisfies (5): sup K n P s 0. s T Topics in fractional Brownian motion 12/52

FBm is not a semimartingale End of the argument On the other hand the self-similarity of B H gives [b H k = BH k BH k 1 ] n (K n B H ) T = n 2H 1 (B H t k 1 B H t k 2 )(B H t k B H t k 1 ) Law = T 2H 1 n k=2 n b H k 1 bh k T 2H (2 2H 1 1). k=2 But 2 2H 1 1 0 for H (0, 1) \ 1 2, and the BDM-theorem gives that B H is not a semimartingale for H 1 2. Topics in fractional Brownian motion 13/52

Transformations Introduction The following is by Mandelbrot and van Ness (1968). Assume that w is a two-sided Brownian motion on the real line. Then the process ( ( ) ) c(h) (t s) H 1 2 + 1 ( s)h 2 + dw s t 0 is an H fractional Brownian motion; here x + = max(x, 0). Topics in fractional Brownian motion 14/52

Transformations Introduction, II Pipiras and Taqqu (2002) proved the following extension to the previous display: b K is a two-sided K fractional Brownian motion. Then ( ( c(h, K) (t s) H K + ) ) ( s)h K + db K s t 0 is an H fractional Brownian motion. We will give some recent results on the corresponding finite interval representations. Topics in fractional Brownian motion 15/52

Transformations Basic set-up Let Γ(α) be the gamma function: Γ(α) = The beta function is defined by B(α, β) = We have the connection 1 0 0 e v v α 1 dv, α > 0. (1 v) α 1 v β 1 dv, α, β > o. B(α, β) = Γ(α)Γ(β) Γ(α + β). (6) Topics in fractional Brownian motion 16/52

Transformations Basic set-up,ii Put A = R \ N 0. Using recursion Γ(α + 1) = Γ(α) one can extend Γ to all α A. Topics in fractional Brownian motion 17/52

Transformations Basic set-up,ii Put A = R \ N 0. Using recursion Γ(α + 1) = Γ(α) one can extend Γ to all α A. Molchan & Golosov For all H (0, 1) it holds that ( t ) (B H d t ) t [0, ) = z H (t, s)dw s Here, 0 < s < t <, and for H > 1 2, z H (t, s) = 0 [0, ) C(H) t 1 Γ(H 1 2 H 2 )s u H 1 2 (u s) H 3 2 du. s. (7) Topics in fractional Brownian motion 17/52

Fractional Brownian motion and standard Brownian motion Molchan & Golosov, 1969 For H 1 2, 0 < s < t < we have z H (t, s) = C(H) Γ(H + 1 2 ) ( t s C(H) 1 Γ(H 1 2 H 2 )s ) H 1 2 (t s) H 1 2 t s u H 3 2 (u s) H 1 2 du. Moreover, 2HΓ(H + 1 2 C(H) = )Γ( 3 2 H) Γ(2 2H) 1 2. Topics in fractional Brownian motion 18/52

Material for transformations For a, b, c and variable z define the Gauss hypergemetric function with (a) k (b) k z k 2F 1 (a, b, c, z) = (c) k k. k=0 Here (a) 0 = 1 and (a) k = a (a + 1) (a + k 1) is the Pochhammer symbol; we also assume that c A. [Decreusefond & Üstünel, 1999] For H (0, 1) and 0 < s < t we have that z H (t, s) = C(H) Γ(H + 1 2 )(t s)h 1 2 2F 1 ( 1 2 H, 1, H + 1 2, s t t ). Topics in fractional Brownian motion 19/52

More material for transformations Fractional calculus over [0, T] For a n- fold integral we have, using Fubini s theorem T T s s n 1 T s 1 f(u)duds 1... ds n 1 = 1 Γ(n) T s f(u)(u s) n 1 du. The right hand side of the previous display is well-defined for every α > 0. Topics in fractional Brownian motion 20/52

More material for transformations Fractional calculus over [0, T], II Let T > 0. The right-sided Riemann-Liouville fractional integral operator of order α over [0, T] is defined by (I α T f)(s) = 1 T Γ(α) s f(u)(u s)α 1 du, s (0, T), if α (0, 1); f(s), s (0, T), if α = 0. Topics in fractional Brownian motion 21/52

More material for transformations Fractional calculus over [0, T], III The right-sided Riemann-Liouville fractional derivative operator of order α over [0, T] is defined by d (D α T f)(s) = ds (I1 α T f)(s), s (0, T), if α (0, 1); d ds f(s), s (0, T), if α = 1; f(s), s (0, T), if α = 0. Topics in fractional Brownian motion 22/52

More material for transformations Fractional calculus over [0, T], III We put, α (0, 1]: I α T f = Dα T f. We have, for f L 1 ([0, T]), α, β 0: I α T Iβ T f = Iα+β T f. The derivative operator D α T is defined for such functions f, which satisfy f = I β T g for some g L 1 ([0, T]) and β α. Topics in fractional Brownian motion 23/52

More material for transformations Fractional calculus over [0, T], IV We have, f L 1 ([0, T]), 0 α β <, α 1: D α T Iβ T f = Iβ α f. (8) T We have D α T Iα T f = f, but the reverse does not hold, and we have I α T Dα T f f. We have then the following: Let H (0, 1), and T > 0. Then for all s, t (0, T], s t z H (t, s) = C(H)s 1 2 H (I H 1 2 T H 1 2 1[0,t) ) (s). Topics in fractional Brownian motion 24/52

Wiener integrals of [0, T] For H (0, 1) and T > 0, define power-weighted fractional integral operators for s (0, T) by and (K H f)(s) := (K H T f)(s) = C(H)s 1 2 H (I H 1 2 T H 1 2 f ) (s) ( ) (K H, f)(s) := (K H, T f)(s) = C(H) 1 s H 1 2 I 1 2 H H 1 2 T f (s). Topics in fractional Brownian motion 25/52

Useful results The operators K H and K H, are mutually inverse in the following sense: Let H > 1 2. Then K H, K H f = f, f L 2 ([0, T]), (9) and K H K H, 1 [0,t) = 1 [0,t), t [0, T]. (10) Moreover, let H < 1 2. Then K H K H, f = f, f L 2 ([0, T]), (11) and K H, K H 1 [0,t) = 1 [0,t), t [0, T]. (12) Topics in fractional Brownian motion 26/52

Wiener integrals Using the previous results we have ( T ) (B H t ) t [0,T] = (K H T 1 [0,t))(s)dW s 0 t [0,T] Let E T be the space of elementary functions f on [0, T]: n f(s) = a i 1 [0,ti )(s), s [=, T], i=1. (13) where a i R, t i [0, T], i = 1,..., n. The Wiener integral with respect to B H is defined by T n I H T (f) = f(s)dbs H = a i B H t i. 0 i=1 Topics in fractional Brownian motion 27/52

Wiener integrals Using (13) we have the equality T 0 f(s)db H s = T 0 (K H f)(s)dw s. (14) By considering (14) and the classical Wiener isometry, we define the space of time domain Wiener integrands by { Λ T (H) = f : [0, T] R : H 1 2 f( ) L 1 ([0, T]) and T 0 } (K H f)(s) 2 ds < Topics in fractional Brownian motion 28/52

Wiener integrals For H < 1 2, K H is a weighted fractional derivative operator, so its arguments must be suffiently smooth. We define Λ T (H) = { f : [0, T] R : φ f L 2 ([0, T]) such that f = K H, φ f }. The space E T is dense in Λ T (H) with respect to the scalar product (f, g) ΛT (H) = (K H f, K H g) L 2 ([0,T]), f, g Λ T (H). Topics in fractional Brownian motion 29/52

Transformations, finally For H > 1 2 one can show that Λ T(H) is not complete. But for H < 1 2 Λ T (H) is complete. The next shows that standard Brownian motion can be obtained from fractional Brownian motion: Let H (0, 1). Then ) ( t (W t ) t [0,T] = z H (t, s)dbh s where for T > 0 and s, t [0, T], it holds that z H (t, s) = ( K H, T 1 [0,t)) (s) = C(H) 1 Γ( 3 2 H)(t s) 1 2 H 2F 1 0, t [0,T] ( 1 2 H, 1 2 H, 3 2 H, s t ) 1 [0,t) (s) s Topics in fractional Brownian motion 30/52

Transformation between two fractional Brownian motions The following results have been established by Jost (2006): Let (B H t ) t [0,T] be an H fractional Brownian motion and let K (0, 1). Then there exists a unique (up to modification) K fractional Brownian motion (B K t ) t [0,T], such that we have T B H t = C(H)C(K) ( 1 s 1 H K I H K H+K 1) t (s)dbs K 0 t ( = C(K, H) (t s) H K 2F 1 0 1 K H, H K, 1 + H K, s t s ) d Topics in fractional Brownian motion 31/52

Simple connection The next corollary is a surprising result: Let (B H t ) t [0,T] be an H fractional Brownian motion. Then there exists a unique (up to modification) 1 H fractional Brownian motion (B 1 H t ) t [0,T] such that we have ( B H t = 2H Γ(2H)Γ(3 2H) ) 1 2 t 0 (t s) 2H 1 db 1 H s. Topics in fractional Brownian motion 32/52

Comments of transformations The above transformations are also ω by ω, or in other words pathwise integrals [either using Young integration theory or classical integration by parts theory]. There is a recent big survey of Picard (2011), which includes the above results, and much more. Topics in fractional Brownian motion 33/52

Applications The representation theorems have been applied to Girsanov theorems. Limit theorems. Moment inequalities for stopped fractional Brownian motions. To develop stochastic integrals with respect to fractional Brownian motion. For more details on these results we refer to Mishuras book (2008). Next we apply the representation theorem to obtain a Lévy type of characterization for fractional Brownian motion. Topics in fractional Brownian motion 34/52

Characterization theorems Characterization of Brownian motion Recall the Lévy characterization of Brownian motion: a square integrable process X is a standard Brownian motion if and only if X and X 2 t t are F X martingales. For fractional Brownian motion B H we have the following two results. Discussion Lévy theorem is a fundamental result in classical stochastic analysis: it is used in Girsanov theorems, filtering, martingale approach to diffusion theory, functional central limit theorems... Topics in fractional Brownian motion 35/52

Characterization of Brownian motion The proof of the Lévy theoremis one of the favorites in the first course of stochastic analysis. It goes as follows: If the process X is a standard Brownian motion, then the processes X and X 2 t t are martingales, because X has independent increments. Conversely, using Itô formula one can show that the conditional characteristic function of the increment is E(e iλ(x t X s ) F X s ] = e λ2 2 (t s). From this one obtains that the process X has independent Gaussian increments, and so X is a standard Brownian motion. Topics in fractional Brownian motion 36/52

Lévy theorem for fractional Brownian motion The next shows that it is possible to extend the similar characterization to fractional Brownian motion. [Mishura, E.V. 2011] A continuous process X is an H- fractional Brownian motion if and only if (a) The process X is Hölder-continuous for all β < H. (b) For every t > 0 and t k = kt n we have that n 2H 1 k (X tk X tk 1 ) 2 P t 2H. (c) The process M t = t 0 s 1 2 H (t s) 1 2 H dx s is a martingale. Topics in fractional Brownian motion 37/52

Another characterization Originally we wanted to replace the weighted quadratic variation property by the following property of fbm X: n X tk X tk 1 1 P H c H t; k=1 but this is difficult to work with. Recently Hu, Nualart and Song (2009) have replaced the weighted quadratic variation with this 1/H- variation condition, but in order to complete the proof they need additional assumption for H > 1 2 : the bracket [M, M] is absolutely continuous with respect to the Lebesgue measure. Topics in fractional Brownian motion 38/52

On the proof We work with the filtration F X, i.e. the filtration generated by the process X: F X t := σ(x s : s t). Assume that X is an H-fBm. Then it is Hölder up to H, and to check the weighted quadratic property use self similarity and ergodicity of the associated fractional Gaussian noise. If X is an H-fBm, then the process M defined in item (c) is a martingale. Topics in fractional Brownian motion 39/52

On the proof Assume now that the process X satisfies (a) (c). Idea: to show that M is a Gaussian martingale with bracket [M] t = c H t 2 2H. Then going back to X by inverse tranforms one can conclude that X is a fbm. Now we describe the proof of the reverse implication. Assume that the process X satisfies (a), (b) and (c) in the Theorem. Topics in fractional Brownian motion 40/52

First observation Let s t be such that n s t is an integer for big enough n and let R t be the the set of all s < t with this property. The set R t is dense on [0, t] and if s R t, then along a subsequence ñ ñ 2H 1 ñ k=ñ s t 1 (X tk X tk 1 ) 2 L 1 (P) t 2H 1 (t s). Topics in fractional Brownian motion 41/52

First observation We show that the following asymptotic expansion holds n 2H 1 = n 2H 1 n k=n s t +1 n k=n s t +1 (X tk X tk 1 ) 2 (15) t s ( h t k (u)) 2 d[m]u + o P (1). In (15) h t k is a deterministic sequence of functions and o P(1) means convergence to zero in probability. This is proved for H > 1 2 and H < 1 2 separately. Topics in fractional Brownian motion 42/52

On the proof Note that since an H- fractional Brownian motion B H also satisfies (a), (b) and (c) it also satisfies the asymptotic expansion n ( ) n 2H 1 B H t k B H 2 t k 1 = n 2H 1 c H n k=n s t +1 k=n s t +1 t s ( h t k (u)) 2 s 1 2H ds + o P (1) with the same sequence of functions h t k. Secondly, we will show that [M] Leb and the density ρ t (u) = d[m] u du satisfies 0 < c ρ t (u) C < with some constants c, C. Topics in fractional Brownian motion 43/52

On the proof Last steps After this, we can finish the proof: P lim n n 2H 1 n = P lim n n 2H 1 n = t 2H 1 (t s) t k=n s t +1 s t k=n s t +1 s ( h t k (s)) 2 ρ t (u)du ( h t k (u)) 2 d[m]u Topics in fractional Brownian motion 44/52

On the proof Last steps As explained earlier, at the same time we have t 2H 1 (t s) = = P lim n n 2H 1 n P lim n 2H 1 c H k=n s t +1 (B H t k B H t k 1 ) 2 = n k=n s t +1 t s ( h t k (s)) 2 u 1 2H du. Topics in fractional Brownian motion 45/52

On the proof Last steps, II Since the set R t is a dense set on the interval [0, t] we can conclude from the above that ρ t (u) = c H (2 2H)u 1 2H. This means that the martingale M is a Gaussian martingale with the bracket [M] u = c H u 2 2H and the process X is an H-fBm. For the proof we use the representation results. For H > 1 2 we use the following representation: t ( t ) X t = c H s H 1 2 (s u) H 3 2 ds dm u ; 0 u Topics in fractional Brownian motion 46/52

On the proof Last steps, III The previous implis that X tk X tk 1 = tk 2 0 tk 1 tk f t k (s)dm s + t k 2 f t k (s)dm s + g t k (s)dm s, t k 1 where f t k (u) = t k v H 1 2 (v u) H 3 2 dv and t k 1 g t k (u) = t k u vh 1 2 (v u) H 3 2 dv. Topics in fractional Brownian motion 47/52

On the proof Last steps, IV Consider the term t k 2 f t 0 k (s)dm s, for example. With the Itô formula one can write tk 2 ( f t k (s)dm s) 2 = 0 tk 2 tk 2 u (f t k (s))2 d[m] s + 2 f t k (u) f t k (s)dm sdm u. 0 0 0 Put n S n := n 2H 1 tk 2 k= s t n+2 0 (f t k (s))2 d[m] s. Topics in fractional Brownian motion 48/52

On the proof Last steps, V We show that t ct 2H 1 u 2H 1 d[m] u P lim S n Ct 4H 2 ([M] t [M] s ). (16) s In addition we show that the weighted sum of the double integrals is o P (1). The other terms with bracket [M] we estimate only from above, and this estimation gives similar upper bound as the right hand side of (16). Again the weighted sums of double integrals are o P (1) as well as the weighted sums of the crossproducts of martingale integrals. This proves the asymptotic expansion in the case of H > 1 2. Topics in fractional Brownian motion 49/52

On the proof The case of H < 1 2 Assume now that the Hurst index satisfies H < 1 2. Now we use the representation X t = t z(t, 0 s)dw s, with W t = t 0 sh 1/2 dm s, and the kernel is ( s ) 1/2 H z(t, s) = (t s) H 1/2 t t +(1/2 H)s 1/2 H u H 3/2 (u s) H 1/2 du. s Topics in fractional Brownian motion 50/52

On the proof The case of H < 1 2 The ideology of the proof is the same, but since the kernel is different, we must repeat the arguments for the case H > 1 2, with corresponding modifications. Topics in fractional Brownian motion 51/52

Summary of the first lecture fbm is not a semimartingale. Topics in fractional Brownian motion 52/52

Summary of the first lecture fbm is not a semimartingale. Transformation from K-fBm to H-fBm. In particular, from 1 H-fBm to H-fBm: ( B H t = 2H Γ(2H)Γ(3 2H) ) 1 2 t 0 (t s) 2H 1 db 1 H s. Topics in fractional Brownian motion 52/52

Summary of the first lecture fbm is not a semimartingale. Transformation from K-fBm to H-fBm. In particular, from 1 H-fBm to H-fBm: ( B H t = 2H Γ(2H)Γ(3 2H) Characterization of fbm. ) 1 2 t 0 (t s) 2H 1 db 1 H s. Topics in fractional Brownian motion 52/52