$L^p$ uniform random walk-type approximation for fractional Brownian motion with Hurst exponent $0

In this note, we prove an $L^p$ uniform approximation of the fractional Brownian motion with Hurst exponent $0<H<\frac{1}{2}$ by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice $\epsilon_k$ for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is $O(\epsilon_k^{p(1-2\lambda)+ 2(\delta-1)})$ whenever $\max\{0,1-\frac{pH}{2}\}<\delta<1$, $\lambda \in \big(\frac{1-H}{2}, \frac{1}{2} + \frac{\delta-1}{p}\big)$.


Introduction
The fractional Brownian motion B H (henceforth abbreviated by FBM) with Hurst exponent H ∈ (0, 1) is the zero mean Gaussian process with covariance function E[B H (t)B H (s)] = 1 2 [s 2H + t 2H − |t − s| 2H ]. It turns out that FBM is the only continuous Gaussian process which is self-similar with stationary increments. There are many applications of FBM in sciences, including Physics, Biology, Hydrology, network research, Finance; see Biagini et al. [2] and other references therein. In Probability theory, FBM is the canonical model of Gaussian process which exhibits non-trivial increment correlations (for H = 1 2 ) and it is still amenable to rigorous modelling by means of Malliavin calculus and rough path techniques. The goal of this note is to present L p uniform approximations for the FBM with Hurst exponent 0 < H < 1 2 by means of a family of continuous time random walks. The study of approximations of FBM (in the sense of weak convergence) dates back from 1970s with the pioneering works of Davydov [5] and Taqqu [22]. Since then, many authors have been proposed alternative weak approximation methods based on correlated random walks, random wavelet series, Poisson processes, etc. In this direction, we refer the reader to Bardina et al [1], Delgado and Jolis [6], Enriquez [9], Klüppelberg and Kühn [13] and Li and Dai [17] and other references therein. Almost sure uniform approximations for FBM have been studied by many authors in different contexts via transport processes, series representations, etc. In this direction, we refer the reader to Garzon et al [10], Hong et al [11], Dzhaparidze and Van Zanten [8], Chen and Dong [4], Igloi [12] and other references therein. Other approximations in L p (Ω × [0, T ]) were proposed by Decreusefond and Ustunel [7] and L p -estimates (not uniform in time) by Mishura [20].
In this work, we present an L p -approximation for a given FBM with Hurst exponent H ∈ (0, 1 2 ) in the supremum norm over a given time interval [0, T ]. Motivated by the stochastic analysis of non-Markovian phenomena in the rough regime, we restrict our analysis to the most delicate case 0 < H < 1 2 . An important step in our analysis is a pathwise representation B H = Λ H (B) of the FBM with respect to Brownian motion B, where Λ H is a suitable bounded linear operator from the space of λ-Hölder continuous functions (with 1 2 − H < λ < 1 2 ) to the space of continuous functions equipped with the sup norm. The representation (see Theorem 2.1) is a simple consequence of the classical Volterra-type representation and suitable regularization procedures on singular kernels. Our approximation is a functional of Λ H applied to a skeletal continuous-time random walk A k (previously suggested by F. Knight [14]) imbedded on a given Brownian motion B which satisfies Date: August 6, 2020.
for a given sequence {ǫ k ; k ≥ 1} such that ǫ k ↓ 0 as k → +∞. For a given sequence {ǫ k ; k ≥ 1} realizing (1.1), our approximation scheme admits a rate of convergence of order O(ǫ . From the perspective of numerical analysis, one advantage of our approximation scheme is the possibility to simulate FBM by only simulating the first time Brownian motion hits ±1 (see e.g [3,19]) and a Bernoulli random variable which must be composed with suitable singular deterministic integrals. From a theoretical perspective, such type of approximation plays a key role in the stochastic analysis of processes adapted to FBM via the methodology presented in Leão, Ohashi and Simas [15] in the context of functional stochastic calculus. In particular, the main result of this short note (Theorem 3.1) is one of the key arguments to tackle non-Markovian optimal stochastic control problems driven by FBM in the rough regime 0 < H < 1 2 as showed in Theorems 6.2 and 6.3 in Leão, Ohashi and Souza [16].
We stress that a random walk-type (almost sure) approximation based on Mandelbrot-van Ness representation was studied by Szabados [21]. He gave an approximation of FBM for H ∈ ( 1 4 , 1) with convergence rate O(n − min(H− 1 4 , 1 4 ) 2log2 log n) at the nth approximation step. Moreover, his convergence is established with respect to some FBM. In contrast, we study the problem with respect to a given FBM with H ∈ (0, 1 2 ) and we are interested in L p estimates in the supremum norm. Finally, we remark that the scheme introduced in this article is expected to work for approximations of the FBM in the p-variation topology for p > 1 H > 2. We leave this investigation to a future project. The remainder of this note is organized as follows. Section 2 presents a pathwise representation of FBM with H ∈ (0, 1 2 ) which is an important step in our approximation. Section 3 presents the proof of the main result of this note, namely Theorem 3.1.

2.
A pathwise representation of FBM with H ∈ (0, 1 2 ) Throughout this note, (Ω, F, P) denotes a filtered probability space equipped with a one-dimensional standard Brownian motion B where F := (F t ) t≥0 is the usual P-augmentation of the filtration generated by B under a fixed probability measure P. For a real-valued function f : where 0 < T < +∞ is a fixed terminal time.
In the sequel, we derive a pathwise FBM representation for 0 < H < 1 2 which will play a key role in constructing our approximation scheme for FBM. It is a well-known fact that the FBM can be represented w.r.t a Brownian motion B as follows In the sequel, if f : [0, T ] → R, we denote for every f ∈ C λ 0 . Proof. In the sequel, C is a constant which may differ from line to line and we fix 1 2 This concludes the proof.
In order to deal with singularities, for a given ǫ > 0, we set for 0 < s < t. Moreover, Moreover, and Proof. In the sequel, C is a constant which may differ from line to line. Let us fix t ∈ (0, T ]. We observe that for each s ∈ (0, t). Moreover, for 0 < ǫ < 1 and 0 < λ < 1 2 , we have and E B p λ < ∞ for every p > 1, we observe the right-hand side of (2.7) belongs to We claim there exists p > 1 such that By using Hölder's inequality, ∀ǫ ∈ (0, 1), for a constant C which depends on λ, H, p, α and β. This shows (2.9). By using (2.4) and (2.9) into (2.8), we conclude (2.6). Finally, there exists a constant C (which only depends on H) such that for every ǫ > 0, where the right-hand side of (2.10) belongs to L 2 ([0, t]). This concludes the proof.
We are now able to prove a pathwise representation for the FBM with 0 < H < 1 2 with respect to a given standard Brownian motion. Proof. We start by recalling that any FBM B H with exponent 0 < H < 1 2 can be represented by B H (t) = t 0 K H (t, s)dB(s) for some Brownian motion B. Next, the proof is an almost immediate consequence of Lemmas 2.1, 2.2 and a routine use of integration by parts formula. For sake of completeness, we give the details. We fix t ∈ (0, T ] and a Brownian motion B. For 0 < ǫ < 1, |K ǫ H (t, t − ǫ)| + |K ǫ H (t, 0)| < ∞ and integration by parts yields In other words, We observe 3. An L p uniform approximation for FBM in terms of a continuous-time random walk In this section, we present the rate of convergence of our approximation scheme w.r.t a given FBM B H with exponent 0 < H < 1 2 . For this purpose, we make use of Theorem 2.1 as follows. For a Brownian motion B realizing B H = Λ H B via Theorem 2.1, we construct a class of pure jump processes driven by suitable waiting times which describe its local behavior: we set T k 0 := 0 and is an arbitrary sequence such that ǫ k ↓ 0 as k → +∞. The strong Markov property yields the family (T k n ) n≥0 is a sequence of stopping times where the increments {∆T k n := T k n − T k n−1 ; n ≥ 1} is an i.i.d sequence with the same distribution as T k 1 . By the Brownian scaling property, ∆T k 1 = ǫ 2 k τ (in law) where τ = inf{t > 0; |B(t)| = 1} is an absolutely continuous variable with mean equals one and with all finite moments (see Section 5.3.2 in [19]). Then, we define the continuous-time random walk A k The size of the jumps {σ k n ; n ≥ 1} are given by for every k ≥ 1. In the sequel, we sett k := max{T k n ; T k n ≤ t}, and we define (3.5)t + k := min{T k n ;t k < T k n } ∧ T andt − k := max{T k n ; T k n <t k } ∨ 0, where we set max ∅ = −∞. By construction,t k ≤ t <t + k a.s for each t ≥ 0. Let us define Clearly, B k H is a pure jump process of the form

Let us define
Lemma 3.1. If 1 2 − H < λ < 1 2 and 0 < ε < H, then there exists a constant C which only depends on H such that Proof. In the sequel, C is a constant which may differ from line to line and we fix 1 2 − H < λ < 1 2 , 0 < ε < H. First of all, we observe To keep notation simple, we denote At first, we observe A k − B +,λ ≤ B λ a.s and A k − B T k 1 −,λ ≤ B λ a.s for every k ≥ 1. Furthermore, we have We observe s) a.s. The following estimates hold true a.s where ∆t k :=t k −t − k . Summing up (3.8), (3.9) and (3.10), we arrive at the following estimate almost surely for every k ≥ 1. Let us now estimate the second term in the right-hand side of (3.7). At first, we notice Summing up (3.6), (3.7), (3.15) and (3.11), we conclude the proof.
The following result is an immediate consequence of Lemmas 2 and 3 in [3].