Fractional Brownian motion and the Markov Property

Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This representation leads naturally to: - An efficient algorithm to approximate the process. - An infinite dimensional ergodic theorem which applies to functionals of the type $integral_0^t phi(V_h(s)) ds $ where $V_h(s)=integral_0^t h(t-u) dB_u$ and $B$ is a standard Brownian motion.


Introduction
Fractional Brownian motion of Hurst parameter H ∈ (0, 1) is a real centered Gaussian process (W H t ; t ≥ 0 ) of covariance One easily sees that it starts from zero, W H 0 = 0 almost surely, it has stationary increments (W H t+s − W H s ; t ≥ 0 ) Fractional Brownian motion enjoys the following scaling (self-similarity) property The increments are independent if and only if H = 1 2 (Brownian motion).If H < 1  2 , they are negatively correlated, and if H > 1 2 they are positively correlated.Furthermore it is a long memory process, since, for H = 1  2 , the covariance between far apart increments decrease to zero as a power law: Date: March 30, 2022.1991 Mathematics Subject Classification.60FXX,60J25,60G15,65U05,26A33,60A10 .
Key words and phrases.Gaussian processes, Markov Processes, Numerical Approximation, Ergodic Theorem.
Fractional Brownian motion has been used as a model of signal or noise in various domains: geophysical data [10,3], communication processes [12,11], see also the references therein.Therefore it is important to provide good numerical approximating schemes.
Problem I: Approximation of fractional Brownian motion.We shall not linger on the first (naive) idea that may cross our mind.We simulate an n-dimensional Gaussian random vector with the same disribution as (W H s 1 , . . ., W H sn ) and we perform a linear or spline interpolation.This amounts to computing the square root of the n × n covariance matrix and is computationnaly inneficient.A second solution is to use the representation introduced by Mandelbrot and Van Ness [13]: where B is a standard Brownian motion on the real line.Without any loss in generality, we can say that the problem is to approximate the finite, but long, memory part: where (B t ; t ≥ 0 ) is a standard Brownian motion.The naïve way to do this is to fix a small step |∆ n | = 1 n for the partitions we use Hence, if t = r/n, then V (t) is approximated by where For the next step, t + ∆t = t + 1 n = (r + 1)/n, we have Therefore, all we have to do is to compute the h(i/n), generate the ξ i , and keep them all in memory for the next step.
Unfortunately, if we want to keep the approximation error low (that is n big), this amounts to a lot of data to keep in memory.
The natural idea is to impose a threshold, that is to compute h(i/n) and generate ξ only for 1 ≤ i ≤ M n .This may work well for smooth functions h, but not for where 1 2 < α < 1, since h explodes near zero, and thus our partition is not dense enough near t !Unfortunately this example is important since it is in essence fractional Brownian motion of index H = α − 1 2 .In [7],Carmona, Coutin and Montseny derived an approximation scheme from a representation formula inspired by the diffusive input-output model of fractional integration developped by Audounet, Montseny adn Mbodje [1,2].More precisely Fubini's stochastic theorem implies that if h has the "spectral" representation where µ is a positive measure on (0, +∞), finite on compact sets, and such that (1 ∧ x − 1 2 ) µ(dx) < +∞, then where Observe that for fixed x ≥ 0, (Y x t ; t ≥ 0 ) is an Ornstein-Uhlenbeck process of parameter x, that is a Gaussian semi-martingale Markov process solution of the stochastic differential equation x ≥ 0) .The idea behind the approximation scheme is to first do a spatial approximation where π = {η i } i is a chosen partition of K converging to identity, and c i are coefficients depending only on h.This approximation is uniformly good with respect to time, that is sup t≤T |V h (t) − V π h (t)| goes to zero when π grows.The second step is the time dicretization, which can be performed in the following way : This second step is a classical one : see e.g. the book of Bouleau and Lepingle [5], chapter V, section B, pages 271-278, where it is stated that to get a precision of ǫ, one has to consider at most a constant C times N = log( 1 ǫ ) time points.Nevertheless, when one needs to do the whole family of approximations (4), one needs to control precisely the dependence of the constant C = C(η i , T ) with respect to the space and time variables.The classical method, combined with Gronwall's lemma, gives an exponential upper bound for the L p -norm of the approximation error C(η, T ) ≤ c 1 (T, p)η exp(c 2 (T, p)η p ).Therefore, we have to use another time discretization scheme for which we have polynomial constants (see e.g.[6] or [8]).The first step amounts to approximate a linear functional of an infinite Markov process by a linear functional of a finite dimensional Markov process.The secret of the success of this method is that for a large class of functions h, this dimension can be kept low : for a precison of ǫ, the cardinal of the partition π(ǫ) is roughly We do not say that this approximation method is the best, but we want to stress the fact that the Markov property of the underlying process Y allows an algorithm that is very economic in memory allocation.You only have to store N (ǫ) values at each time step, then simulate one Gaussian random variable, and then update these values for the next step (according to rule (4)).
The example of fractional Brownian motion.We can take advantage not only of the Gaussian nature of the processes Y x , but also of their semimartingale nature (via Itô's formula).We have been able to give exact L 2 and almost sure convergence rates for fractional Brownian motion of index H = α − 1 2 ∈ (0, 1 2 ).Indeed, to obtain the precision we may choose the compact and a geometric partition Furthermore, if ǫ n decreases fast enough to 0, then, we have the uniform almost sure convergence Theoretical results (see [7] or [8]) show then that to obtain a precision of ǫ = 10 −3 , we need to consider a few hundred points.Practical results show that forty points are enough.Furthermore, we have been able to check that (almost surely) the numerical approximations stick exponentially fast to the actual random process (and this happens uniformly on compact sets), when ǫ n goes faster to 0 than indicated in (5).
Problem II : an ergodic theorem.It is well known (Birkhoff's ergodic theorem) that for a stationary process U, we have the convergence, almost sure and in L 1 , as soon as φ is an integrable Borel function: Of course, we first have to check the ergodicity, that is that the invariant σ-field is almost surely trivial, which is not so easy to do in general (see e.g.Rozanov [14], page ???).But what we actually want to do is to answer the following question.Given (V (t) ; t ≥ 0 ) a real Gaussian process, a priori non stationary, such that V r = (V (t + r) ; t ≥ 0 ) converges in distribution to the stationary process U. We want to show that, as soon as φ is an integrable Borel function: E [|φ(U(0))|] < +∞, we have the convergence, almost sure and in L 1 , To be more precise, given h ∈ L 2 (R + ) and (B t , t ∈ R) a standard Brownian motion, we let We shall show that (7) holds if h is the Laplace transform of a positive measure µ, finite on compact sets and such that

The Markov and ergodic properties of the underlying infinite dimensional process
Let B = (B t ; t ≥ 0 ) be a standard Brownian motion.Given x ≥ 0, Y x = (Y x t ; t ≥ 0 ) is the Ornstein-Uhlenbeck process driven by B of parameter x (starting from 0): It is also the solution of the stochastic differential equation x + y (x, y > 0) .
The natural candidate for a limiting process is Y ∞ the centered Gaussian process of covariance Proposition 1.The process Y = (Y t ; t ≥ 0 ) is an infinite dimensional Markov process.
Proof.The strong Markov property is quite easy to prove.Indeed, let T be an almost surely finite stopping time (with respect to (F t ; t ≥ 0 ) the completed, hence right-continuous, filtration generated by the Brownian motion B).Then, B = ( Bt = B t+T − B T ; t ≥ 0 ) is a standard Brownian motion independent of (B u , u ≤ T ), and (Y x t+T , ; t ≥ 0 ) is the solution of the stochastic differential equation Hence, for any bounded measurable function ψ, and (Y x,y t ; t ≥ 0 ) is the Ornstein-Uhlenbeck process driven by B, of parameter x, starting from y.
Proposition 2. Suppose that µ is a sigma-finite measure on (0, +∞) such that, for a p > 1, Then, 1.For every y ∈ L 1 (µ), under P y , we have the convergence in distribution 2. The only bounded invariant functions are constant. Proof.
(1) We shall first prove the convergence of characteristic functions, since this will characterize the distribution of a possible limit (see [4]), and then prove the tightness of the family of distributions of (Y t , ; t ≥ 0 ).The characteristic function is defined by Observe that χ, Y t is, under P y , a Gaussian random variable of mean χ(x)y(x)e −xt µ(dx) −→ t→∞ 0 by dominated convergence and of variance This last integral is finite, Indeed, since ab ≤ 1 2 (a 1 +b 2 ),we can majorize it by Therefore, χ, Y t converges in distribution to a centered Gaussian variable with variance χ(x)χ(y) µ(dx)µ(dy) x + y , and this is easily seen to be the variance of χ, Y ∞ .
We shall now prove the tightness of the family of laws of the processes (Y t , t ≥ 1).We need to introduce a little topology: we consider that for every t > 0, Y t is a random variable taking its values in L 1 (µ) endowed with the topology σ(L 1 , L ∞ ); this is the coarsest topology for which the functions are continuous.Indeed, Let us notice that, by construction, under P y the law of Y t is the law , under P 0 of Y t + (x → e −xt y(x)).Since this last function converges weakly to the null function, we can suppose y = 0, without any loss in generality.We have the sequence of inequalities, for t ≥ 1, This last quantity goes uniformly in t ≥ 1 to zero as M → ∞ , since the integral is finite.
(2) (i) First of all, we have to prove that if f is a bounded function depending on a finite number of coordinate, then Indeed, we have, for a bounded measurable function g : R n → R, and But we know that if X = (X 1 , . . ., X n ) is a non degenerate centered Gaussian vector, then we have the absolute continuity relationship with Λ a continuous bounded function.Applying this to ( χ i , Y t , 1 ≤ i ≤ n), for t > 0, yields Now if y p → y weakly in L 1 (µ), then, for each i, χ i , y p (.)e −t.→ χ i , y(.)e −t. and P t f (y p ) → P t f (y).Therefore, if t > 0, then P t f is a continuous bounded function.Since, under P y , Y t d −→ t→∞ Y ∞ , the law of Y ∞ is an invariant measure for the semi-group (P t ; t ≥ 0 ), and we have (ii) Now, Let f : L 1 (µ) → R be a bounded invariant function; that is, Observe that the sigma field F = σ(χ, χ ∈ L ∞ (µ)), is generated by the algebra of sets depending on a finite number of coordinate that is the smallest algebra containings the sets {y : χ, y ∈ B} , χ ∈ L ∞ (µ), B borel set of R. Therefore, the measure P Y∞ , the law of Y ∞ , being finite, there exists bounded functions (f n , g n , n ∈ N) depending only on a finite number of coordinate such that We take the limit as t → +∞ in Lemma 3. Let µ be a sigma-finite measure, and φ be a strictly positive φ > 0, integrable function.Then, for every M ≥ 0, p, q > 1 such taht 1 = 1/p + 1/q, K = f : |f | p φ p/q dµ ≤ M is a relatively compact set of L 1 (µ) endowed with the weak topology σ(L 1 , L ∞ ).
Proof.Following Theorem IV.8.9 of Dunford-Schwarz [9] we have to prove that 1.K is a bounded subset of L 1 .2. If E n is a sequence of measurable sets that decreases to ∅, then sup f ∈K En f dµ −→ n→∞ 0.
By Hölder's inequality, for f in K .
Taking E 0 = Ω the whole space, we see that K is bounded in L 1 .The dominated convergence theorem gives that 1 En φ dµ −→ n→∞ 0, so the convergence is uniform for f in K.
Proposition 4. Suppose that µ is a sigma-finite measure on (0, +∞) such that for a p > 1, Then, for any measurable ψ : where the convergence is almost sure and in L 1 .In particular, h where the convergence takes place almost surely and in L 1 .
Proof.The second part of the proposition is an obvious consequence of the first part: we only need to take ψ(y) = φ( y(x) dµ(x)) and to use Fubini-Tonelli's theorem To prove the first part, we consider the invariant sigma-field I, and f (y) = P y ((t → Y t ) / ∈ F ). Birkhoff's ergodic theorem implies that E [f (Y ∞ )] = 0 (and that the convergence also takes place in L 1 ).The invariance of f is an easy consequence of the Markov property of (Y t ; t ≥ 0 ) since for all t ≥ 0: F = θ −1 t (F ) where θ t denotes the shift operator.Proposition 2 implies that f is a constant, and thus, for all y ∈ L 1 (µ), f (y) = 0.All we need to show now is that the invariant sigma-field I is almost surely trivial.Let Z be a bounded I measurable random variable.We have, for all t, Z = Z • θ t and therefore the function f (y) = E y [Z] is constant since Taking expectations under P y yields Proposition 2 implies that f is constant, say f (y) = c for every y ∈ L 1 (µ).Injecting this in relation (11) yields, that under P y E [Z | F t ] = c (t ≥ 0) .
Taking limits as t → +∞ gives finally that Z = c a.s.