Communications on Stochastic Analysis

Volume 7 | Number 4 Article 3

12-1-2013 Local time of a multifractional Aissa Sghir

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Recommended Citation Sghir, Aissa (2013) "Local time of a multifractional Gaussian process," Communications on Stochastic Analysis: Vol. 7 : No. 4 , Article 3. DOI: 10.31390/cosa.7.4.03 Available at: https://digitalcommons.lsu.edu/cosa/vol7/iss4/3 Communications on Stochastic Analysis Serials Publications Vol. 7, No. 4 (2013) 523-533 www.serialspublications.com

LOCAL TIME OF A MULTIFRACTIONAL GAUSSIAN PROCESS

AISSA SGHIR*

Abstract. In this paper, a new multifractional Gaussian process is defined by a integral representation. We prove an approximation in law of this process and we prove that the covariance function of this process permits us to obtain a generalization of the sub-fractional [5] by a decomposition in law into the sum of our process and the standard multifractional Brownian motion [1]. We prove also the existence and the joint continuity of the local time of our process.

1. Introduction Let XH := {XH ; t ≥ 0} be the Gaussian process defined by: t Z +∞ − − H+1 H − θt 2 Xt = (1 e )θ dWθ, 0 where H ∈ (0, 2) and W := {Wt ; t ≥ 0} is a standard Brownian motion. This process was introduced by Lei and Nualart [10] in order to obtain a decomposi- tion in law of the bifractional Brownian motion [9] and was used later by Bardina and Bascompte [2] and Ruiz de Chavez and Tudor [11] in order to obtain a de- H { H ≥ } composition in law of the sub-fractional Brownian motion S := St ; t 0 with parameter H ∈ (0, 2). This process was introduced by Bojdecki et al. [5]. It is a continuous centered Gaussian process, starting from zero, with covariance function: 1 E(SH SH ) = tH + sH − [(t + s)H + |t − s|H ]. t s 2 In this paper, firstly, we introduce a new multifractional Gaussian process which generalize the process XH by substituting to the parameter H a function H(.) such that H(t) ∈ (0, 2) and we prove an approximation in law of this process. Secondly, we prove that the covariance function of this process permits us to obtain a generalization of the sub-fractional Brownian motion by a decomposition in law into the sum of our process and the standard multifractional Brownian motion [1]. We prove also under some assumptions on H(.), the existence, the joint continuity and the H¨olderregularity of the local time of our process. Our idea is inspired from the work of Boufoussi et al. [6] in case of multifractional Brownian motion. We use the concept of local nondeterminism for Gaussian process introduced by

Received 2012-8-10; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 60G18; Secondary 60J55. Key words and phrases. Multifractional Gaussian process, sub-fractional Brownian motion, decomposition in law, local time, local nondeterminism. * This research is supported by the Doe Foundation. 523 524 AISSA SGHIR

Berman [4] and the analytic method used by Berman [3] for the calculation of the moments of local time.

2. The Multifractional Gaussian Process Definition 2.1. Let H : [0, +∞[→ (0, 2) be a function satisfying: there exists finite and positive constants β, C2 and C3 such that β β C2|t − s| ≤ |H(t) − H(s)| ≤ C3|t − s| , for all s, t ≥ 0. (2.1) The right hand side of (2.1) means that H(.) is β-H¨oldercontinuous.

≥ H(t) { H(t) ≥ } Definition 2.2. For t 0, we denote by X := Xt ; t 0 the multifrac- tional Gaussian process with the multifractional function H(t) defined by: Z +∞ H(t) − − H(t)+1 − θt 2 Xt = (1 e )θ dWθ, 0 where the integration is taken in the mean square sense. An approximation in law of our process can be obtained by the same argument used by Bardina and Bascompte [2] in case of the process XH .

Proposition 2.3. Let {Ns ; s ≥ 0 } be a standard Poisson process and let θ ∈ (0, π) ∪ (π, 2π). Then the processes,  Z  +∞   H(.) 2 −st − H(t)+1 X = (1 − e )s 2 cos θN 2s ds, t ∈ [0,T ] ε 2 ε 0 ε and  Z  +∞   H(.) 2 −st − H(t)+1 X˜ = (1 − e )s 2 sin θN 2s ds, t ∈ [0,T ] ε 2 ε 0 ε converge in law, in the sense of the finite dimensional distributions, toward two independent processes with the same law that XH(.). The following result concerns the regularity property of XH(.). Lemma 2.4. Let T > 0 fixed. Assume that H(.) is H¨oldercontinuous with ex- ponent β ∈ (0, 1). There exists a finite and positive constant C, such that for all t, s ∈ [0,T ], we have E| H(t) − H(s)|2 ≤ | − |2β Xt Xs C t s . To prove Lemma 2.4, we need the following result. Lemma 2.5. Let T > 0 fixed and [µ, ν] ⊂ (0, 2). There exists two positive and 1 2 0 ∈ finite constants Cµ,ν and Cµ,ν such that for all λ, λ [µ, ν], 1 | − 0|2 ≤ E λ − λ0 2 ≤ 2 | − 0|2 Cµ,ν λ λ sup [Xt Xt ] Cµ,ν λ λ . (2.2) t∈[0,T ] Proof. We have Z +∞ 0 0 − − λ+1 − λ +1 E λ − λ 2 − θt 2 2 − 2 2 [Xt Xt ] = (1 e ) (θ θ ) dθ. 0 MULTIFRACTIONAL GAUSSIAN PROCESS 525

Using the theorem on finite increments, ([7], Corollary 2.6.1), for the function − x+1 0 0 x → θ 2 ; x ∈ (λ, λ ), there exists ξ ∈ (λ, λ ) such that

0 − λ+1 − λ +1 1 − ξ+1 0 θ 2 − θ 2 = | ln(θ)|θ 2 |λ − λ |, 2 therefore Z +∞ E λ − λ0 2 1| − 0|2 − −θt 2 2 −(ξ+1) [Xt Xt ] = λ λ (1 e ) ln (θ)θ dθ 4 0 n Z 1 1 ≤ |λ − λ0|2 sup (1 − e−θt)2 ln2(θ)θ−(ξ+1)dθ 4 t∈[0,T ] 0 Z +∞ o + (1 − e−θt)2 ln2(θ)θ−(ξ+1)dθ 1 n Z 1 1 ≤ |λ − λ0|2 sup (1 − e−θt)2 ln2(θ)θ−(ν+1)dθ 4 t∈[0,T ] 0 Z +∞ o + (1 − e−θt)2 ln2(θ)θ−(µ+1)dθ . 1 Consequently E λ − λ0 2 ≤ 2 | − 0|2 sup [Xt Xt ] Cµ,ν λ λ . t∈[0,T ] Changing the sup by the inf, we complete the proof of Lemma 2.5.  Remark 2.6. The theorem on finite increments for the function x → ex, implies that ey − ex < ey(y − x), ∀y > x, therefore, there exists a finite and positive constant C1 such that for any H ∈ (0, 1) and all s, t ∈ [a, b] ⊂ (0, +∞), we have E H − H 2 ≤ | − |2 [Xt Xs ] C1 t s , (2.3) where Z +∞ −2aθ 1−H C1 = e θ dθ. 0 Now we are ready to prove Lemma 2.4. Proof of Lemma 2.4. Using the elementary inequality (a + b)2 ≤ 2(a2 + b2), we obtain h i E| H(t) − H(s)|2 ≤ E| H(t) − H(t)|2 E| H(t) − H(s)|2 Xt Xs 2 Xt Xs + Xs Xs . The first term on the right hand side of the previous expression is the variance of the increments of the process XH of parameter H(t). Therefore (2.3) implies that E| H(t) − H(t)|2 ≤ | − |2 Xt Xs C t s . Moreover, by virtue of (2.2), we have E| H(t) − H(s)|2 ≤ | − |2 Xs Xs C H(t) H(s) ≤ C|t − s|2β. 526 AISSA SGHIR

Consequently E| H(t) − H(s)|2 ≤ | − |2β | − |2(1−β) Xt Xs C t s [1 + t s ] ≤ C|t − s|2β. The proof of Lemma 2.4 is done.  In the next result, we give the formula of the covariance function our process. The proof follows the lines of that given by Bardina and Bascompte [2] in case of the process XH . Proposition 2.7. The process XH(.) is Gaussian, centered, and its covariance function is ( 0 Γ(1−H ) H0 H0 − H0 ∈ H(t) H(s) H0 [t + s (t + s) ], if H(.) (0, 1), E(X X ) = 0 0 0 0 t s Γ(2−H ) H − H − H ∈ H0(H0−1) [(t + s) t s ], if H(.) (1, 2). 0 H(t)+H(s) where H = 2 . As application, we obtain a generalization of the sub-fractional Brownian mo- tion SH by a decomposition in law into the sum of our process and the standard multifractional Brownian motion [1]. The covariance function of this last process was given by Ayyache et al. [1] as follows: E H(t) H(s) H0 H0 − | − |H0 (Bt Bs ) = D(H(t),H(s))[t + s t s ], where p Γ(x + 1)Γ(y + 1) sin(πx/2) sin(πy/2) D(x, y) = . 2Γ((x + y)/2 + 1) sin(π(x + y)/4) Theorem 2.8. The centered multifractional Gaussian SH(.) process with a mul- tifrcational function H(t) ∈ (0, 1) defined by the decomposition in law: SH(t) = BH(t) + C(H(.))XH(t), q t t t H(t) H(.) H(.) where C(H(t)) = 2Γ(1−H(t)) and B and X are independent, is a gener- alization of the sub-fractional Brownian motion SH with parameter H ∈ (0, 1).

3. Local Time and Local Nondeterminism We begin this section by the definition of local time. For a complete survey on local time, we refer to Geman and Horowitz [8] and the references therein. Let X := {Xt ; t ≥ 0} be a real-valued separable random process with Borel sample functions. For any Borel set B ⊂ R+, the occupation measure of X on B is defined as µB(A) = λ{s ∈ B ; Xs ∈ A}, ∀A ∈ B(R), + where λ is the one-dimensional Lebesgue measure on R . If µB is absolutely continuous with respect to Lebesgue measure on R, we say that X has a local time on B and define its local time, L(B,.), to be the Radon-Nikodym derivative of µB. Here, x is the so-called space variable and B is the time variable. By standard monotone class arguments, one can deduce that the local time have a MULTIFRACTIONAL GAUSSIAN PROCESS 527 measurable modification that satisfies the occupation density formula: for every Borel set B ⊂ R+ and every measurable function f : R → R+, Z Z

f(Xt)dt = f(x)L(B, x)dx. B R Sometimes, we write L(t, x) instead of L([0, t], x). Here is the outline of the analytic method used by Berman [3] for the calculation of the moments of local time. For fixed sample function at fixed t, the Fourier transform on x of L(t, x) is the function Z F (u) = eiuxL(t, x)dx. R Using the density of occupation formula, we get Z t F (u) = eiuXs ds. 0 Therefore, we may represent the local time as the inverse Fourier transform of this function, i.e., Z Z  +∞ t 1 − L(t, x) = eiu(Xs x)ds du. (3.1) 2π −∞ 0 We end this section by the definition of the concept of local nondeterminism, (LND for short). Let J be an open interval on the t axis. Assume that {Xt ; t ≥ 0} is a zero mean Gaussian process without singularities in any interval of the length δ, for some δ > 0, and without fixed zeros, i.e., there exists δ > 0, such that  E 2 ∈ P (Xt) > 0, for t J. ( ): 2 E[Xt − Xs] > 0, whenever 0 < |t − s| < δ, To introduce the concept of LND, Berman [4] defined the relative conditioning error { − } V ar Xtp Xtp−1 /Xt1 , ...Xtp−1 Vp = { − } , V ar Xtp Xtp−1 where for p ≥ 2, t1 < ... < tp are arbitrary ordered points in J. We say that the process X is LND on J if for every p ≥ 2,

lim inf Vp > 0. (3.2) c→0+ 0

4. Existence and Joint Continuity of Local Time The purpose of this section is to present sufficient conditions for the existence of the local time of XH(.). Furthermore, using the LND approach, we show that the local time of XH(.) have a jointly continuous version. Theorem 4.1. Assume H(.) satisfying (2.1) with β ∈ (0, 1). On each time- interval [a, b] ⊂ [0, ∞),XH(.) admits a local time which satisfies Z L2([a, b], x)dx < ∞ a.s. R For the proof of Theorem 4.1, we need the following lemma. This result on the regularity of the increments of XH(.) will be the key for the existence and the joint continuity of local times. Lemma 4.2. Assume H(.) satisfying (2.1) with β ∈ (0, 1). There exists two positive and finite constants δ > 0 and C such that E H(t) − H(s) 2 ≥ | − |2β [Xt Xs ] C t s , (4.1) for all s, t ≥ 0 such that |t − s| < δ. 2 ≥ 1 2 − 2 Proof. By virtue of the elementary inequality (a + b) 2 a b , we get 1 E|XH(t) − XH(s)|2 ≥ E|XH(t) − XH(s)|2 − E|XH(t) − XH(t)|2. t s 2 s s t s Therefore (2.1), (2.2) and (2.3) implies that " # 1 C C2 E|XH(t) − XH(s)|2 ≥ |t − s|2β µ,ν − C |t − s|2(1−β) . t s 2 1

Since β < 1, we can choose δ > 0 small enough such that for all t, s ≥ 0 with | − | t s < δ, we have " # 1 C C2 µ,ν − C |t − s|2(1−β) > 0. 2 1 Indeed, it suffices to choose

!1/2(1−β) 1 C C2 δ < µ,ν ∧ 1 . 2C1 Finally, E H(t) − H(s) 2 ≥ | − |2 [Xt Xs ] C t s , | − | with t s < δ and " # 1 C C2 C = µ,ν − C δ2(1−β) . 2 1 The proof of Lemma 4.2 is done.  MULTIFRACTIONAL GAUSSIAN PROCESS 529

Proof of Theorem 4.1. It is well known by Berman [3] that, for a jointly measur- able zero-mean Gaussian process X := {X(t); t ∈ [0,T ]} with bounded variance, the variance condition Z Z T T −1/2 E[X(t) − X(t)]2 dsdt < ∞ 0 0 is sufficient for the local time L(t, u) of X to exist on [0,T ] a.s. and to be square integrable as a function of u. For any [a, b] ⊂ [0, ∞), and for I = [a0, b0] ⊂ [a, b] such that |b0 − a0| < δ, according to (4.1), we have, Z Z Z Z  −1/2 E H(t) − H(s) 2 | − |−β [Xt Xs ] dsdt < C t s dsdt. I I I I The last integral is finite, then XH(.) possesses, on any interval I ⊂ [a, b] with length |I| < δ, a local time which is square integrable as function of u. Finally, since [a, b] is a finite interval, we can obtain the local time on [a, b] by a patch-up ∪n | − | procedure, i.e. we partitionP [a, b] into i=1[ai−1, ai], such that ai ai−1 < δ, and n  define L([a, b], x) = i=1 L([ai−1, ai], x), where a0 = a and an = b. Proposition 4.3. Assume H(.) is derivable and H¨oldercontinuous with β ∈ (0, 1). Then for every ε > 0, and any T > ε, XH(.) is locally LND on [ε, T ]. H(.) E H(t) 2 Proof. Notice that X is a zero mean Gaussian process, and (Xt ) > 0 for all t ∈ [ε, T ]. Moreover, thanks to (4.1) also the second point in (P) is satisfied on [ε, T ]. It remains to show that XH(.) satisfies (3.2). We have H(θ) ≤ ⊂ ≤ ≥ σ(Xθ , θ s) σ(Wθ , θ s), for all s 0. Then, for any t > s,     H(t) − H(s) H(θ) ≤ ≥ H(t) − H(s) ≤ V ar Xt Xs /Xθ , θ s V ar Xt Xs /Wθ , θ s . R s −θt − H(t)+1 The measurability of (1 − e )θ 2 dW with respect to σ(W , θ ≤ s) and R 0 θ θ +∞ − − H(t)+1 − θt 2 ≤ the fact that s (1 e )θ dWθ is independent of σ(Wθ , θ s), (by the independence of the increments of the Brownian motion), implies that   H(t) − H(s) H(θ) ≤ V ar Xt Xs /Xθ , θ s Z +∞ h i2 −θt − H(t)+1 −θs − H(s)+1 ≥ (1 − e )θ 2 − (1 − e )θ 2 dθ. s Making use of the theorem on finite increments for the function: x → fθ(x) = −θx − H(x)+1 (1 − e )θ 2 for x ∈ (s, t), there exists ξ ∈ (s, t) such that Z    +∞ V ar XH(t) − XH(s)/XH(u) , u ≤ s ≥ |t − s|2 inf (f 0 (x))2dθ . t s u ∈ θ x [ε,T ] T

Therefore the relative prediction error Vp given by (3.2) is at least equal to R  +∞ 0 2 infx∈[ε,T ] (f (x)) dθ  T θ  R ∞ , + 0 2 supx∈[ε,T ] 0 (fθ(x)) dθ 530 AISSA SGHIR which is bounded away from 0, as t tends to s, and the proof is complete. 

Remark 4.4. When applying the LND to the estimation of the moments of local times of XH(.), the condition t > ε can be circumvented by slightly adjusting the arguments. Indeed, we can consider X = XH(.) +Y , where Y is a standard normal random variable independent of XH(.). Since the occupation density of X has the H(.) E 2 E H(t) 2 E 2 same local properties as that of X and (Xt) = (Xt ) + (Y ) > 0 for all t ∈ [0,T ], we can use the LND on the whole interval [0,T ] for XH(.). We end this section by the following main result. Theorem 4.5. Assume H(.) is derivable and satisfying (2.1) with β ∈ (0, 1) and let δ the constant appearing in Lemma 4.2. For any integer p ≥ 2 there exists a finite positive constant Cp such that, for any t ≥ 0, any h ∈ (0, δ), all x, y ∈ R, 1−β and any 0 < ξ < 2β , hp(1−β) E[L(t + h, x) − L(t, x)]p ≤ C , (4.2) p Γ(1 + p(1 − β)) E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p hp(1−β(1+ξ)) ≤ C |y − x|pξ . (4.3) p Γ(1 + p(1 − β(1 + ξ))) Proof. We will prove only (4.3), the proof of (4.2) is similar. It follows from (3.1) that for any x, y ∈ R, t, t + h ≥ 0 and for any integer p ≥ 2, E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p

Z Z p   p p Y P H(s ) Y Y 1 − − i p u X j iyuj − ixuj × E j=1 j sj = p [e e ] e duj dsj. (2π) p Rp [t,t+h] j=1 j=1 j=1 Using the elementary inequality |1 − eiθ| ≤ 21−ξ|θ|ξ for all 0 < ξ < 1 and any θ ∈ R, we obtain E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p ≤ (2ξπ)−pp!|x − y|pξ Z Z Yp Xp Yp Yp | |ξE H(tj ) uj [exp(i ujXtj )] duj tj, (4.4) Rp t

uj = vj − vj+1, j = 1, ..., p − 1; up = vp. Then the linear combination in the exponent in (4.4) is transformed according to Xp Xp u XH(tj ) = v (XH(tj ) − XH(tj )), j tj j tj tj −1 j=1 j=1 MULTIFRACTIONAL GAUSSIAN PROCESS 531

H(.) where t0 = 0. Since X is a Gaussian process, the characteristic function in (4.4) has the form    Xp  1  H(tj ) H(tj )  exp − V ar vj(X − X − ) . (4.5) 2 tj tj 1 j=1 Since |x − y|ξ ≤ |x|ξ + |y|ξ for all 0 < ξ < 1, it follows that Yp pY−1 ξ ξ ξ ξ |uj| ≤ (|vj| + |vj+1| )|vp| . (4.6) j=1 j=1 p−1 Moreover,Q the last product is at most equalP to a finite sum of 2 terms of the p p | |ξεj form j=1 xj , where j = 0, 1 or 2 and j=1 j = p.  2 Let us write for simply σ2 = E XH(tj ) − XH(tj ) . Combining the result of j tj tj −1 Proposition 3.1, (4.5) and (4.6), we get that the integral in (4.4) is dominated by the sum over all possible choices of ( , ...,  ) ∈ {0, 1, 2}m of the following terms 1 m  Z Z Yp Xp Yp Cp | |ξj − 2 2 vj exp vj σj dtjdvj, Rp 2 t

Let us denote   Z Yp Xp Yp Cp | |ξj − 2 J(p, ξ) = xj exp xj dxj. Rp 2 j=1 j=1 j=1 Consequently E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p Z Yp − − ≤ | − |pξ 1 ξj J(p, ξ)Cp y x σj dt1....dtp. (4.7) t

Since, (tj − tj−1) < 1, for all j ∈ {2, ..., p}, we have

−β(1+ξj ) −β(1+2ξ) (tj − tj−1) ≤ (tj − tj−1) , ∀j ∈ {0, 1, 2}. 532 AISSA SGHIR

1 − 1 Since by Hypothesis 0 < ξ < 2β 2 , the integral in (4.8) is finite. Moreover, by an elementary calculation, for all p ≥ 1, h > 0 and bj < 1, Z Q Yp P p p Γ(1 − b ) −b p− b j=1 j (s − s − ) j ds ...ds = h j=1 j P , j j 1 1 p Γ(1 + h − p b ) t

Remark 4.6. (1) The process XH has infinitely differentiable trajectories and it is well-known that in this case the local time does not exist because the occupation measure is singular. (2) We believe that the same arguments used in this paper can be used for the bifractional Brownian motion and the Gaussian process introduced in Sghir [12].

Acknowledgment. The author would like to thank the anonymous referee for her/his careful reading of the manuscript and useful comments.

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AISSA SGHIR: Laboratoire de Modelisation´ Stochastique et Deterministe´ et URAC 04, Faculte´ des Sciences, Oujda, 60000, MAROC E-mail address: [email protected]