Approximation Of Fractional Brownian Motion Research Articles

Shadow Price Approximation for the Fractional Black Scholes Model

In this work, we used Tran Hung Thao’s approximation of fractional Brownian motion to approximate the shadow price of the fractional Black Scholes model. In the case to maximize expectation of the utility function in a portfolio optimization problem under transaction cost, the shadow price is approximated by a Markovian process and semimartingale.

Stationary Wong–Zakai Approximation of Fractional Brownian Motion and Stochastic Differential Equations with Noise Perturbations

In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H∈(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions.

A generative model for fBm with deep ReLU neural networks

We provide a large probability bound on the uniform approximation of fractional Brownian motion with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a N-dimensional Gaussian vector, with bounds on the network design (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of O(N−H) is possible with log⁡(N) hidden layers and O(Nlog⁡N) parameters. Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction and in the general fractional Brownian motion case (H≠1/2), on the Lemarié-Meyer wavelet representation. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.

Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

The characterization of intermittency in turbulence has its roots in the refined similarity hypotheses of Kolmogorov, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in an attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, we investigate the Gaussian multiplicative chaos formalism, which requires the construction of a log-correlated stochastic process X_{t}. The fractional Gaussian noise of Hurst parameter H=0 is of great interest because it leads to a log correlation for the logarithm of the process. Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a stochastic model: X_{t}=∫_{0}^{∞}Y_{t}^{x}k(x)dx, where Y_{t}^{x} is an Ornstein-Uhlenbeck process with speed of mean reversion x and k is a kernel. A regularization of k(x) is required to ensure stationarity, finite variance, and logarithmic autocorrelation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models. To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature X_{t}^{N}=∑_{i=1}^{N}ω_{i}Y_{t}^{x_{i}}, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.

A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1]of Hurst indexH∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM forH∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Approximation of Fractional Brownian Motion by Martingales

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.

Convergence rate of multiple fractional Stratonovich type integral for Hurst parameter less than 1/2

In this paper, we have investigated the problem of the convergence rate of the multiple integral { ∫ 0 T . ⋯ ∫ 0 T f ( t 1 , ⋯ , t n ) d B t 1 H , π } , where f ɛ C n+1([0, T] n) is a given function, π is a partition of the interval ([0, T]) and { B t i H , π } is a family of interpolation approximation of fractional Brownian motion B H t with Hurst parameter H<1/2. The limit process is the multiple Stratonovich integral of the function f. In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is Δ 2 H in the mean square sense, where Δ is the norm of the partition generating the approximations.

Semimartingale approximation of fractional Brownian motion and its applications

The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black–Scholes model by a model driven by semimartingales, and a European option pricing formula is found.

A strong uniform approximation of fractional Brownian motion by means of transport processes

We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter H , and we derive a rate of convergence, which becomes better when H approaches 1 / 2 . The construction is based on the Mandelbrot–van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation.

Approximation of fractional Brownian motion by Wiener integrals

Approximation of fractional Brownian motion by Wiener integrals

Strong approximation of fractional Brownian motion by moving averages of simple random walks

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to Mandelbrot and van Ness (SIAM Rev. 10 (1968) 422) as a self-similar Gaussian process W ( H) ( t) with stationary increments. Here self-similarity means that (a −HW (H)(at) : t⩾0) = d (W (H)(t) : t⩾0) , where H∈(0,1) is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by Révész (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990) and then by Szabados (Studia Sci. Math. Hung. 31 (1996) 249–297). This approach is quite natural and elementary, and as such, can be extended to more general situations. Based on this, here we use moving averages of a suitable nested sequence of simple random walks that almost surely uniformly converge to fractional Brownian motion on compacts when H∈( 1 4 ,1) . The rate of convergence proved in this case is O(N − min(H−1/4,1/4) log N) , where N is the number of steps used for the approximation. If the more accurate (but also more intricate) Komlós et al. (1975,1976) approximation is used instead to embed random walks into ordinary Brownian motion, then the same type of moving averages almost surely uniformly converge to fractional Brownian motion on compacts for any H∈(0,1). Moreover, the convergence rate is conjectured to be the best possible O(N −H log N) , though only O(N − min(H,1/2) log N) is proved here.