Covariance structure of continuous time random walk limit processes

Transcription

1 Covariance structure of continuous time random walk limit processes Alla Sikorskii Department of Statistics and Probability Michigan State University Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments A workshop on the occasion of the retirement of Francesco Mainardi Bilbao, Basque Country, Spain November 6-8, 2013

2 Abstract Processes obtained by changing the time in Lévy processes to positive non-decreasing stochastic processes are useful in many applications. The use of time-changed processes in modeling often requires the knowledge of their second order properties such as the correlation function. This paper provides the explicit expression for the correlation function for time-changed Lévy processes. These processes include limits of continuous time random walks (CTRW). Several examples useful in applications are discussed.

3 Acknowledgments Joint work with Nikolai N. Leonenko, School of Mathematics, Cardiff University Mark M. Meerschaert, Department of Statistics and Probability, Michigan State University MMM was partially supported by NSF grants DMS and DMS , and NIH grant R01-EB New book: M. M. Meerschaert and A. Sikorskii (2012) Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics 43, De Gruyter, Berlin, 2012, ISBN

4 Random walk Suppose that J n are independent and identically distributed (IID) random variables with mean µ = E[J n ] = 0 and finite variance σ 2 = Var(J n ) = E[(J n µ) 2 ]. The random walk S(n) = J J n represents the location of a randomly selected particle at time n. Then n 1/2 S [nt] X (t), a Brownian motion with mean zero and variance σ 2 t, which gives a stochastic model for particle motion in the long time limit.

5 CTRW Now suppose that the IID particle jumps are separated by IID waiting times W n, with P(W n > t) t α /Γ(1 α) for 0 < α < 1 as t. Then the particle arrives at location S n at time T n = W W n. The number of jumps by time t > 0 is given by the renewal process N t = max{n 0 : T n t}. The extended central limit theorem (Theorem 4.5, Meerschaert and Sikorskii (2012)) yields n 1/α T [nt] L(t), a standard α-stable subordinator with E[e sl(t) ] = e tsα for all s, t > 0. Since {N t n} = {T n t}, a continuous mapping argument yields n α N nt Y (t) = inf{u > 0 : L(u) > t}, an inverse α-stable subordinator.

6 CTRW limits The CTRW S(N t ) gives the particle location at time t > 0, with long time limit n α/2 S(N nt ) X (Y (t)), a time changed Brownian motion (Meerschaert and Scheffler (2004)). A very general class of CTRW models was considered in Meerschaert and Scheffler (2008), with triangular arrays of jumps and waiting times {S (c) (cu), T (c) (cu))} u 0 converging to a Lévy process {(A(u), L(u))} u 0 as c in Skorokhod s J 1 topology on D([0, ), R d R + ). If A and L have no common points of discontinuity a.s., then the CTRW {X (c) (t) = S (c) (N (c) t ) with N (c) t = max{n 0 : T (c) (n) t} converges to A(Y (t)) in Skorokhod s M 1 topology on D([0, ), R d ).

7 Applications The CTRW is used as a model of anomalous diffusion in physics, finance, hydrology, and other fields (Mainardi, Gorenflo and Vivoli (2007), Mainardi, Mura, Pagnini and Gorenflo (2008), Metzler and Klafter (2000), Scalas (2006), Benson, Wheatcraft and Meerschaert (2000)). Recent work in finance produced random activity time models,where the calendar time is replaced with a stochastic process that represents the activity time. (Howison and Lamper (2001), Heyde and Leonenko (2005)). Time change allows to obtain distributions that are heavier-tailed than Gaussian. In many applications (Chechkin, Gorenflo and Sokolov (2002), Sokolov, Chechkin and Klafter (2004), Janczura and Wy lomańska (2009)), it is useful to obtain second order properties of the time changed process, including the correlation function.

8 Existing result For the case where X (t) is Brownian motion and L(t) is a standard stable subordinator with index 0 < α < 1, a formula for the correlation function has been obtained by Janczura and Wy lomańska (2009), using the result of Magdziarz who showed that X (Y (t)) is a martingale with respect to a suitably defined filtration. Using martingale property, Janczura and Wy lomańska computed that for 0 < s < t: ( s ) α/2 corr(x (Y (t)), X (Y (s))) =. t We use a different method of proof for a more general setting and show that in the case when mean of the outer process is not zero, the asymptotic behavior of the correlation changes.

9 Theorem Suppose that X (t), t 0 is a homogeneous Lévy process with X (0) = 0 and finite variance, and Y (t) is a non-decreasing process independent of X, with finite mean U(t) = EY (t) and finite variance. Then the mean of the process Z = X (Y (t)) is the variance is E[Z(t)] = U(t)E[X (1)], Var[Z(t)] = [EX (1)] 2 Var[Y (t)] + U(t) Var[X (1)], and the covariance function Cov[Z(t), Z(s)] = Var[X (1)]U(min(t, s))+[ex (1)] 2 Cov[Y (t), Y (s)]. Remark. The main technical issue in using the above formulas is computing function U.

10 Examples Consider a subordinator L with Laplace transform where the Laplace exponent φ(s) = µs + E[e sl(t) ] = e tφ(s), s 0. (0, ) The time change is the inverse process Y (1 e sx )ν(dx), s 0. Y (t) = inf {u 0 : L(u) > t}, t 0. The process Y (t), t 0, is nondecreasing, and its sample paths are almost surely continuous if L(t) is strictly increasing.

11 Renewal function Meerschaert and Scheffler (2006) and Veillette and Taqqu (2010) show that the renewal function U(t) = E[Y (t)] has Laplace transform Ũ(s) = 0 U(t)e st dt = 1 sφ(s), where φ is Laplace exponent of L. Thus, U characterizes the inverse process Y, since φ characterizes L. For example, the second moment is EY 2 (t) = t the covariance function of Y is given by Cov[Y (t 1 ), Y (t 2 )] = min(t 1,t 2 ) 0 0 2U(t τ)du(τ) (U(t 1 τ)+u(t 2 τ))du(τ) U(t 1 )U(t 2 ). For many inverse subordinators, Ũ can be written explicitly, but its inversion may be difficult.

12 Example: inverse stable subordinator Suppose L is standard α-stable subordinator with index 0 < α < 1, so that the Laplace exponent is φ(s) = s α for all s > 0. Then from Bingham (1971), the inverse stable subordinator has a Mittag-Leffler distribution: In this case [ E e sy (t)] = n=0 ( st α ) n Γ(αn + 1) = E α( st α ). Ũ(s) = 1 s α+1 and U(t) = E[Y (t)] = t α Γ(1 + α). When the outer process X (t) is a homogeneous Poisson process, the time changed process X (Y (t)) is fractional Poisson process (FPP) (Mainardi, Gorenflo and Scalas (2004), Mainardi, Gorenflo, Vivoli (2007), Laskin (2003), Repin and Saichev (2000), Uchaikin, Cahoy, Sibatov (2008), Beghin and Orsingher (2009), Meerschaert, Nane, Vellaisamy (2011)).

13 The covariance function of the time-changed process The covariance function of Z(t) = X (Y (t)) is Cov[Z(t), Z(s)] = sα Var[X (1)] Γ(1 + α) [EX [ ] (1)]2 + Γ(1 + α) 2 αs 2α B(α, α + 1) + F (α; s, t) where B(a, b; x) := x 0 u a 1 (1 u) b 1 du is the incomplete beta function, B(a, b; 1) = B(a, b), and F (α; s, t) := αt 2α B(α, α + 1; s/t) (ts) α = α (s/t)α+1 α + 1 Note that F (α; s, t) 0 as t, hence Cov[Z(t), Z(s)] sα Var[X (1)] Γ(1 + α) + O((s/t) α+2 ), + s2α [EX (1)] 2 Γ(1 + 2α) as t.

14 The correlation function asymptotics The asymptotic behavior of the correlation depends on whether the outer process has zero mean. If E[X (1)] 0 as for the FPP, then for any s > 0 fixed we have where C(α, s) = corr[z(t), Z(s)] t α C(α, s) as t, ( ) 1 Γ(2α) 1 1 [ αγ(α) 2 α Var[X (1)] Γ(1 + α)[ex (1)] 2 + αs α Γ(1 + 2α) On the other hand, if E[X (1)] = 0, then the correlation function is ( s ) α/2 corr[z(t), Z(s)] =. t Thus the correlation function of Z(t) falls off like a power law t α when E[X (1)] 0, and even more slowly, like the power law t α/2 when E[X (1)] = 0. In either case, the non-stationary time-changed process Z(t) exhibits long range dependence. ].

15 Remark If E[X (1)] = 0, the time-changed process Z(t) = X (Y (t)) also has approximately uncorrelated increments: Since asymptotics of Cov[Z(t), Z(s)] does not depend on t, we have Var[Z(s)] = Cov[Z(s), Z(s)] Cov[Z(s), Z(t)] and hence, since the covariance is additive, Cov[Z(s), Z(t) Z(s)] 0 for 0 < s < t and t large. Approximately uncorrelated increments together with long range dependence is a hallmark of financial data, and hence this process can be useful to model such data. Since the outer process X (t) can be any Lévy process, the distribution of the time-changed process Z(t) = X (Y (t)) can take many forms.

16 Inverse stable mixture Now consider a mixture of standard α-stable subordinators with Laplace exponent φ(s) = 1 0 q(w)s w dw = 0 (1 e sx )l q (x)dx, where q(w) is a probability density on (0, 1), and the density l q (x) of the Lévy measure is given by l q (x) = 1 0 wx w 1 Γ(1 w) q(w)dw. Such mixtures are used to model ultraslow diffusion, see Sokolov, Chechkin and Klafter (2004), Kovács and Meerschaert (2006). They also arise in the theory of time-fractional diffusion models of accelerating subdiffusion, see Mainardi, Mura, Pagnini and Gorenflo (2008), Chechkin, Gorenflo and Sokolov (2002).

17 Mixture continued The α-stable subordinator corresponds to the choice q(w) = δ(w α), where δ( ) is the delta function. The model q(w) = C 1 δ(w α 1 ) + C 2 δ(w α 2 ), C 1 + C 2 = 1, with α 1 < α 2 was considered in Chechkin, Gorenflo and Sokolov (2002). The subordinator L in this case is the sum of two independent stable subordinators with φ(s) = C 1 s α 1 + C 2 s α 2, so that 1 Ũ(s) = s(c 1 s α 1 + C2 s α. 2 ) Using the properties of two-parameter Mittag-Leffler function (Podlubny (1994)), we can explicitly invert this Laplace transform to get U(t) = E[Y (t)] = tα 1 C 1 E α1 α 2,α 1 +1( C 2 t α 1 α 2 /C 1 ).

18 The time changed process Then the time-changed process Z(t) = X (Y (t)) has mean E[Z(t)] = tα 1 E[X (1)] C 1 E α1 α 2,α 1 +1( C 2 t α 1 α 2 /C 1 ). When E[X (1)] = 0, the time changed process has zero mean and the variance is Var[Z(t)] = Var[X (1)] C 1 t α 1 E α1 α 2,α 1 +1( C 2 t α 1 α 2 /C 1 ). In the case when the outer process is Brownian motion, this expression for the mean square displacement of the time-changed process was obtained in Chechkin, Gorenflo and Sokolov (2002) using a different method.

19 Asymptotic behavior of the correlation When E[X (1)] = 0, for a fixed s and t corr[z(t), Z(s)] C 3 (s, α 1, α 2 )t α2/2, where ( C2 Γ(α 2 + 1)s α 1 E α1 α C 3 (s, α 1, α 2 ) = 2,α 1 +1( C 2 s α 1 α 2 ) 1/2 /C 1 ). When t is fixed and s 0 corr[z(t), Z(s)] C 4 (t, α 1, α 2 )s α1/2, where C 4 (t, α 1, α 2 ) = ( Γ(α 1 + 1)t α 1 E α1 α 2,α 1 +1( C 2 t α 1 α 2 /C 1 ) ) 1/2. C 1

20 Case of non-zero mean of the outer process For the case when E[X (1)] 0, we can also explicitly compute the variance of the time-changed process: Var[Z(t)] = Var[X (1)] tα 1 (E[X (1)]) 2 t2α 1 C 2 1 C 1 E α1 α 2,α 1 +1( C 2 t α 1 α 2 /C 1 )+ (2E (1) α 1 α 2,α 1 +α 2 +1 ( C 2t α 1 α 2 /C 1 ) E α1 α 2,α 1 +1( C 2 t α 1 α 2 /C 1 ) 2 ), where E (1) α,β (y) = d dy E α,β(y) is the derivative (k = 0, 1, 2,... ) of two-parameter Mittag-Leffler function.

21 Concluding remarks When Y is the hitting time of a subordinator L, then the Laplace transform Ũ of the renewal function U(t) = EY (t) is expressed via the Laplace exponent of L, and U completely determines the process Y (and its moments and correlations). Inversion of Ũ to get U explicitly is possible for some applications. Alternatively, Karamata s Tauberian Theorem can be used to determine the asymptotics of moments and correlations for the process Z(t) = X (Y (t)).

22 Some references F. Mainardi, R. Gorenflo and E. Scalas (2004) A fractional generalization of the Poisson processes.vietnam Journ. Math., 32, F. Mainardi, R. Gorenflo, A. Vivoli (2007) Beyond the Poisson renewal process: A tutorial survey. J. Comput. Appl. Math., 205, F. Mainardi, A. Mura, G. Pagnini, R. Gorenflo (2008) Time-fractional diffusion of distributed order. J. Vib Control, 14: L. Beghin and E. Orsingher (2009) Fractional Poisson processes and related random motions. Electron. J. Probab., 14, A. V. Chechkin, R. Gorenflo and I. M. Sokolov (2002) Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Physical Review E, 66, R. Metzler and J. Klafter (2000) The random walk s guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 339(1), M. M. Meerschaert, E. Nane and P. Vellaisamy (2011) The fractional Poisson process and the inverse stable subordinator, Electronic Journal of Probability, 16, Paper no. 59, pp M. M. Meerschaert and A. Sikorskii (2012) Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics 43, De Gruyter, Berlin, 2012, ISBN