Grey Brownian motion and local times

Transcription

1 Grey Brownian motion and local times José Luís da Silva 1,2 (Joint work with: M. Erraoui 3 ) 2 CCM - Centro de Ciências Matemáticas, University of Madeira, Portugal 3 University Cadi Ayyad, Faculty of Sciences Semlalia, Marrakech, Morocco International Conference on Stochastic Analysis and Applications. Hammamet 14-19, Supported by: CCM - PEst-OE/MAT/UI0219/2011 José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

2 Outline Outline 1 Motivation 2 Grey Brownian motion 3 Representations of gbm 4 On the increments of gbm 5 Local Times 6 References José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

3 Motivation 1 Grey Brownian motion is a stochastic process which a) is self-similar, b) has stationary increments, c) is completely determined knowing its expectation and second moment d) is a non-gaussian process. 2 The marginal probability density function of the grey Brownian motion process is the fundamental solution of the stretched time-fractional diffusion equation: u(x, t) = u 0 (x) + 1 Γ(β) t 0 α β s α β 1( t α β s α ) β β 1 2 u(x, s) ds x2 which provides stochastic models for (slow/fast)-anomalous diffusions. Mura et al. [2008], Mura [2008], Mainardi et al. [2010], Schneider [1992]. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

4 Grey Brownian motion Grey noise space Goal 1 Show the existence of a square integrable grey Brownian motion local times (Berman s criterium). 2 Show that grey Brownian motion local times admits a weak-approximation by the number of crossings associated to a regularized convolution. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

5 Grey Brownian motion Grey noise space Definition (Grey noise measure) Define the By Minlos theorem there is probability measure µ α,β, 0 < α < 2, 0 < β 1 on (S (R), B), with characteristic functional e i w,ϕ dµ α,β (w) := E β ( 1 ) 2 ϕ 2 α, ϕ S(R), (1) S (R) where ϕ 2 α := C(α) R ( πα ϕ(x) 2 x 1 α dx, C(α) := Γ(α + 1) sin 2 and E β is the Mittag-Leffler (entire) function E β (z) = n=0 z n Γ(βn + 1), z C. The probability space (S (R), B, µ α,β ) is called grey noise space. ) José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

6 Grey Brownian motion Generalized stochastic process We now consider the generalized stochastic process X α,β defined canonically on the (S (R), B, µ α,β ), called grey noise for ϕ S(R) by Properties X α,β (ϕ) : S (R) R, w X α,β (ϕ)(w) := w, ϕ. 1 Characteristic function: E (e ) ( ) iθx α,β(ϕ) = E β θ2 2 ϕ 2 α. 2 Moments: E ( { 0, k = 2n + 1, Xα,β k (ϕ)) = (2n)! 2 n Γ(βn+1) ϕ 2n α, k = 2n. 3 For any f H α, we have X α,β (f) L 2 (µ α,β ) and X α,β (f) 2 L 2 (µ α,β ) = 1 Γ(β + 1) f 2 α. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

7 Grey Brownian motion Grey Brownian motion - gbm Grey Brownian motion - gbm Since 1 [0,t) H α with 1 [0,t) 2 α = t α we may define the generalized stochastic process X α,β ( 1 [0,t) ) as an element in L 2 (µ α,β ). Definition (Grey Brownian motion) The stochastic process ( Bα,β (t) ) t 0 = ( X α,β ( 1 [0,t) ) ) t 0 is called gbm - grey Brownian motion. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

8 Grey Brownian motion Grey Brownian motion - gbm Properties of gbm 1. B α,β (0) = 0 a.s., for any t 0, E ( B α,β (t) ) = 0 and 2. The covariance function is E ( B 2 α,β (t)) = 1 Γ(β + 1) tα. E ( B α,β (t)b α,β (s) ) = 1 1 ( t α + s α t s α). 2 Γ(β + 1) 3. Hölder continuity E( Bα,β (t) B α,β (s) p) = C β,p t s pα/2. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

9 Grey Brownian motion Grey Brownian motion - gbm 4. Characteristic function of the increments E ( e iθ(b α,β(t) B α,β (s)) ) = E β ( θ2 t s α 2 ). 5. Self-similarity property: B α,β (at) d = a α/2 B α,β (t), t 0, a > 0. We summarize all these properties in the following: Theorem (Mura and Mainardi [2009]) For any 0 < α < 2 and 0 < β 1 the process B α,β (t), t 0 is H-self-similar with stationary increments (H-sssi), with H = α/2. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

10 Grey Brownian motion Grey Brownian motion - gbm Special classes From what have seen, ( B α,β (t) ) forms a class of H-sssi stochastic t 0 processes indexed by 0 < α < 2 and 0 < β 1. This class includes: 1 Fractional Brownian motion (β = 1) B α,1. 2 Brownian motion (α = β = 1) B 1,1. 3 An H-sssi process (0 < α = β < 1) B α,α with H = α/2. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

11 Grey Brownian motion Grey Brownian motion - gbm Theorem (Finite dimensional distributions) Let B α,β be a gbm, then for any collection X = { B α,β (t 1 ),..., B α,β (t n ) } the characteristic function is given by E(e i(θ,x) ) = E β ( 1 ) 2 θ Σ α θ, θ R n, Σ α = (t α i + t α j t i t j α ) n i,j=1 and the joint probability density function is given by: θ R n f α,β (θ, Σ α ) = (2π) n 2 τ n 2 e θ Σ 1 α θ 2τ M β (τ) dτ, (2) det Σα where the M-Wright density function M β is such that (LM β )(s) = E β ( s). ( ) E.g., for β = 1 2, we have M 1 (τ) = 1 2 2π exp( τ 2 /2). 0 José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

12 Grey Brownian motion Grey Brownian motion - gbm Plot of M-Wright function for β [0, 1/2] Transition : M 0 ( x ) = exp( x ) M 1 ( x ) = 1 exp( x 2 /2) 2 2π José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

13 Grey Brownian motion Grey Brownian motion - gbm Plot of M-Wright function for β [1/2, 1] Transition : M 1 ( x ) = 1 exp( x 2 /2) M 1 ( x ) = δ(x ± 1) 2 2π José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

14 Grey Brownian motion Grey Brownian motion - gbm The previous result together with the Kolmogorov extension theorem allows us to define the gbm in an abstract probability space (Ω, F, P ). Theorem Let X(t), t 0, be a stochastic process on a probability space (Ω, F, P ), such that 1 X(t) has covariance matrix Σ α and finite-dimensional distributions f α,β as in (2), 2 E ( X 2 (t) ) = 2 Γ(β+1) tα, 0 < α < 2, 0 < β 1, 3 X(t) has stationary increments, then X(t), t 0 is grey Brownian motion. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

15 Representations of gbm Normal variance mixture Theorem (Normal variance mixture) Let B α,β (t), t 0 be the gbm, then B α,β (t) = d Y β X α (t), t 0, 0 < β 1, 0 < α < 2, (3) where X α (t), t 0 is a standard fbm with Hurst parameter H = α 2, Y β is an independent non-negative r.v. with pdf M β (τ), τ 0. Remark (Advantages) The representation (3) is very interesting since: 1 Many question related to gbm B α,β may be reduced to questions concerning the fbm X α which is easier since it is Gaussian. 2 The factorization is also suitable for path-simulations once we have a method to generate the r.v. Y β. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

16 Representations of gbm Normal variance mixture Subordination Grey Brownian motion admits also to other representations in terms of subordination but these representations are valid only for one dimensional distributions. 1 We have B α,β (t) 1-dim == B(S β (t α )), B is a standard BM and S(t), t 0 be a β-stable subordinator. 2 The grey Brownian motion B α,β is represented as B α,β (t) 1-dim == B H (A 1/α t α/β ), H = α 2, where the process A t has 1-dim. probability density function f At (x) = t β M β (xt β ), x, t 0. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

17 Representations of gbm Question Question Consider the fractional Poisson measure with characteristic function C β,α (ϕ) :=E β α e iϕ(x) 1 x α 1 dx R d and the process X β,α (t) :=, 1 [0,t), t 0. Obtain a representation for X β,α (t) d =? José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

18 On the increments of gbm 1 We now consider the increments of gbm B α,β : Z α,β,ε (t) := ε α/2 (B α,β (t + ε) B α,β (t)), ε > 0, 0 t 1 in order to study the convergence of λ{t [0, 1], Z α,β,ε (t) x}, ε 0. 2 This is equivalent to find the limit of the t-characteristic function 1 lim ε 0 0 e iuz α,β,ε(t) dt. 3 This question is related to the moment problem, i.e., study the limit lim Y α,β,ε,k := lim ε 0 ε Zα,β,ε k (t) dt, k N. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

19 On the increments of gbm We obtain the following result which will be useful for the approximation of the occupation measure. Proposition For any t [0, 1], almost surely we have t N has distribution N(0, 1). 0 Zα,β,ε k k/2 (s) ds tyβ E(N k ), ε 0, Theorem 1. For a.s. for all x R we have λ{t [0, 1], Z α,β,ε (t) x} P ( Y β N x), ε 0, where N is a standard normal distribution. 2. For a.s. for each interval I R + and all x R, we have λ{t I, Z α,β,ε (t) x} λ(i)p ( Y β N x ), ε 0. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

20 On the increments of gbm The above result is also valid in a more general context. Define the convolution approximation of B α,β by Bα,β ε = ψ ε B α,β, where ψ ε (t) = 1 ( ) t ε ψ, ε ψ is a bounded variation function with support in [ 1, 1] and ψ(t) dt = 1. Define R Z α,β,ε (t) := ε 1 α/2 d dt Bε α,β (t), t 0. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

21 On the increments of gbm Theorem 1. For almost sure for all x R we have λ { t [0, 1], Zα,β,ε (t) x } P (C ψ Yβ N x), ε 0, (4) where C ψ is given by C ψ = ( /2 u v α dψ(u) dψ(v)) For almost sure for each interval I R + and all x R, it follows from (4) that λ { t I, Zα,β,ε (t) x } λ(i)p (C ψ Yβ N x), ε 0. (5) José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

22 Local Times Definition (Occupation measure) For a measurable function f : I R, I a Borel set in [0, 1], we define the occupation measure µ f on I by µ f (B) := 1 B (f(s)) ds, B B(R). I Interpreting [0, 1] as a time set this is the amount of time spent by f in B during the time period I. Definition (Occupation density) We say that f has an occupation density on I if µ f is absolutely continuous with respect to the Lebesgue measure λ and denote it by L f (, I), in explicit, for any x R, L f (x, I) = dµ f dλ (x). José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

23 Local Times Definition (Berman s criterium: Existence of LT) A stochastic process X admits a local times if and only if (see or Berman [1969, Lemma 3.1] or Geman and Horowitz [1980, Thm 21.9] R E ( e iλ(x(t) X(s))) ds dt dλ <. (6) José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

24 Local Times Theorem 1 The gbm process B α,β admits a λ-square integrable local time L B α,β(, I) almost surely. 2 As a consequence of the existence of the local time L B α,β(, I), we obtain the occupation formula f(b α,β (s)) ds = f(x)l B α,β (x, I) dx, a.s. I I José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

25 Local Times Proof. For the gbm B α,β we have = = 1 1 R R E ( e iλ(b α,β(t) B α,β (s)) ) ds dt dλ ( ) E β λ2 t s α ds dt dλ ( ds dt E t s α/2 β r 2 ) dr. R } {{ } (B) } {{ } (A) (A) : (B) : 1 t s α/2 ds dt = 8 (2 α)(4 α). ( E β r 2 ) dr < R José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

26 Local Times Theorem For any continuous bounded real function f and any bounded interval I, almost surely, we have ε 1 α/2 π 2 C 1 ψ R f(x)c Bε α,β(x, I) dx Yβ f(x)l ε 0 R B α,β (x, I) dx. Here B ε α,β is the regularized gbm Bε α,β := ψ ε B α,β and C ψ is defined by C ψ = ( ) 1/2 u v α dψ(u) dψ(v) and C Bε α,β(x, I) is the number of crossing at level x of Bα,β ε interval I: } C Bε α,β(x, I) := # {t I : Bα,β ε (t) = x. in the José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

27 Local Times Proof. Step 1: For any continuous bounded function f and Banach-Kac formula we have ε 1 α/2 f(x)c Bε α,β(x, I) dx = ε 1 α/2 f(bα,β ε (t)) d dt Bε α,β (t) dt. R Now apply a standard trick (add and subtract) ( = f(b ε α,β (t)) f(b α,β (t)) ) ε 1 α/2 d I dt Bε α,β (t) dt + f(b α,β (t)) d ε1 α/2 dt Bε α,β (t) dt. I Step 2: Since B α,β and f are continuous it follows that, almost surely lim f(b α,β ε (t)) f(b α,β(t)) = 0 sup ε 0 t I I José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

28 Local Times and So we have lim ε 0 I sup ε>0 I d ε1 α/2 dt Bε α,β (t) dt <. ( f(b ε α,β (t)) f(b α,β (t)) ) ε 1 α/2 d dt Bε α,β (t) dt = 0. Step 3: For the other integral, f(b α,β (t)) d ε1 α/2 dt Bε α,β (t) dt lim ε 0 we use the previous result: t 0 I Z α,β,ε (s) ds ty 1/2 β E ( N ), ε 0. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

29 Local Times So that f(b α,β (t)) d ε1 α/2 dt Bε α,β (t) dt = lim ε 0 I 2 π C ψ Yβ f(b α,β (t)) dt. I The result of the theorem follows from the occupation formula: 2 = π C ψ Yβ f(x)l B α,β (x, I) dx. R We have ε 1 α/2 π 2 C 1 ψ R f(x)c Bε α,β(x, I) dx Yβ f(x)l ε 0 R B α,β (x, I) dx. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

30 Local Times Summary I The grey Brownian motion B α,β is a α 2 -self-similar, stationary increments process and the marginal pdf solves the stretched time-fractional diffusion equation. This class of processes includes Fractional Brownian motion, Brownian motion and other α 2 -sssi process as special cases. gbm admits different representations: Normal variance mixture B α,β (t) = Y β X α (t), t 0, where X α is a standard fbm, H = α 2 and Y β is an independent non-negative r.v. pdf M β. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

31 Local Times Summary II Multi-variate elliptic distribution X = R β A α S, where R β 0 is a radial r.v., A α is such that Σ α = A α A α and S is the uniform distribution on the sphere {x R n : x = 1}. Brownian motion subordinator B α,β (t) B(S β (t α )), t 0. Fractional Brownian motion subordinator B α,β (t) B H (A 1/α t α/β ), H = α 2, t 0. The gbm process B α,β admits a λ-square integrable local time L B α,β(, I) almost surely and as a consequence we obtain the occupation formula. The number of crossings C Bε α,β(x, I) of gbm weakly converges to the local times L B α,β(, I) almost surely. José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

32 References References I S. M. Berman. Local times and sample function properties of stationary Gaussian processes. Transactions of the American Mathematical Society, 137: , J. L. Da Silva and M. Erraoui. Grey Brownian motion local time: Existence and weak-approximation. Preprint, Univeristy of Madeira, URL D. Geman and J. Horowitz. Occupation densities. Ann. Probab., 8(1):1 67, ISSN F. Mainardi, A. Mura, and G. Pagnini. The M-Wright function in time-fractional diffusion processes: A tutorial survey. Int. J. Differ. Equ., pages Art. ID , 29, ISSN A. Mura. Non-Markovian Stochastic Processes and their Applications: From Anomalous Diffusions to Time Series Analysis. PhD thesis, Bologna, A. Mura and F. Mainardi. A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. Integral Transforms Spec. Funct., 20(3-4): , ISSN URL José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33

33 References References II A. Mura, M. S. Taqqu, and F. Mainardi. Non-Markovian diffusion equations and processes: analysis and simulations. Phys. A, 387(21): , ISSN URL W. R. Schneider. Grey noise. In Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), pages Cambridge Univ. Press, Cambridge, Thank you José Luís Silva (CCM - UMa) Grey Brownian motion local times Hammamet / 33