Smooth approximations for fractional and multifractional fields

A Ferguson-Klass-LePage series representation of multistable multifractional processes and related processes

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. We present here a construction of multifractional multistable processes, based on the Ferguson-Klass-LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse Ornstein-Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also compute the finite dimensional distributions of those processes. Fi…

Spectral analysis for some multifractional Gaussian processes

We study the small ball asymptotics problem in $L_2$ for two generalizations of the fractional Brownian motion with variable Hurst parameter. To this end, we perform careful analysis of the singular values asymptotics for associated integral operators.

基于 Haar 的多分形过程的构造、性质与模拟

On Construction, Properties and Simulation of Haar-Based Multifractional Processes

本文提出基于 Haar 小波级数表示的高斯 Haar 多分形过程（GHBMP），扩展了可处理的时变 Hurst 函数范围，并通过理论分析与模拟验证了模型在不连续 Hurst 函数情形下的适用性。

Girsanov Theorem for Multifractional Brownian Processes

In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a regularity function. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the last 20 years. We develop a multifractional derivative operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. W…

Stochastic Calculus with respect to multifractional Brownian motion

Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is a Gaussian extension of fBm that allows to control the pointwise regularity of the paths of the process and to decouple it from its long range dependence properties. This generalization is obtained by replacing the constant Hurst parameter H of fBm by a function h(t). Multifractional Brownian motion has proved useful in many applications, including the ones…

Fractional Brownian fields, duality, and martingales

In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited ``duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and…

多分数随机过程的模拟与分析（R包Rmfrac）

Simulation and Analysis of Multifractional Stochastic Processes with R Package Rmfrac

本文介绍了Rmfrac包，用于模拟和分析多分数过程，提供Hurst函数估计、局部分形维数计算及可视化功能。

Stochastic Calculus with respect to Gaussian Processes: Part I

Stochastic integration \textit{wrt} Gaussian processes has raised strong interest in recent years, motivated in particular by its applications in Internet traffic modeling, biomedicine and finance. The aim of this work is to define and develop a White Noise Theory-based anticipative stochastic calculus with respect to all Gaussian processes that have an integral representation over a real (maybe infinite) interval. Very rich, this class of Gaussian processes contains, among many others, Volterra processes (and thus fractional Brownian motion) as well as processes the regularity of which varies…

Fine regularity of Lévy processes and linear (multi)fractional stable motion

In this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional s…

Tempered Fractional Brownian Motion Revisited Via Fractional Ornstein-Uhlenbeck Processes

Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied.

Weak convergence to derivatives of fractional Brownian motion

It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that, for any non-negative integer $M$, derivatives of order $m=0,1,\dots,M$ of the normalized fractional process with respect to the fractional parameter $d$, jointly converge weakly to the corresponding derivatives of fractional Brownian motion. As an illustration we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.

Utility maximization in Wiener-transformable markets

We consider a utility maximization problem in a broad class of markets. Apart from traditional semimartingale markets, our class of markets includes processes with long memory, fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called "constant" and "variable depth" memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. We introduce the notio…

Replication of Wiener-transformable stochastic processes with application to financial markets with memory

We investigate Wiener-transformable markets, where the driving process is given by an adapted transformation of a Wiener process. This includes processes with long memory, like fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called `constant' and `variable depth' memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. Motivated by integral …

Stochastic 2-microlocal analysis

A lot is known about the Hölder regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows in particular to predict the evolution of regularity under the action of (pseudo-) differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow, under fairly genera…

Generalized fractional smoothness and $L_p$-variation of BSDEs with non-Lipschitz terminal condition

We relate the $L_p$-variation, $2\le p < \infty$, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.

A note on Malliavin fractional smoothness for Lévy processes and approximation

Assume a Lévy process $X$ on the time interval $[0,1]$ that is an $L_2$-martingale and let $Y$ be either its stochastic exponential or $X$ itself. We consider Riemann-approximations of certain stochastic integrals driven by $Y$ and relate the $L_2$-approximation rates to the Malliavin fractional smoothness of the integral to be approximated. The Malliavin fractional smoothness is described by Besov spaces generated with the real interpolation method.

A smooth component of the fractional Brownian motion and optimal portfolio selection

We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the sense that it is differentiable in mean square. We consider fractional Brownian motion and stochastic integrals generated by the Riemann sums. As an example of applications, this results is used to find an optimal pre-programmed strategy in the mean-variance setting for a Bachelier type market model driven by a fractional Brownian motion.

From almost sure local regularity to almost sure Hausdorff dimension for Gaussian fields

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,ρ)$ are bounded from above using the local Hölder exponent at $t_0$. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdor…

Existence and smoothness of the density of the solution to fractional stochastic integral Volterra equations

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu-Nualart and Nualart-Saussereau where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space.

混合分数布朗运动的漂移有多平滑？

How smooth is the drift of the mixed fractional Brownian motion?

本文研究混合分数布朗运动的漂移在Doob-Meyer分解中的连续性，证明其导数在特定Hölder指数下连续。

On fractional smoothness and $L_p$-approximation on the Gaussian space

We consider Gaussian Besov spaces obtained by real interpolation and Riemann-Liouville operators of fractional integration on the Gaussian space and relate the fractional smoothness of a functional to the regularity of its heat extension. The results are applied to study an approximation problem in $L_p$ for $2\le p<\infty$ for stochastic integrals with respect to the $d$-dimensional (geometric) Brownian motion.

Stationary increments harmonizable stable fields: upper estimates on path behaviour

Studying sample path behaviour of stochastic fields/processes is a classical research topic in probability theory and related areas such as fractal geometry. To this end, many methods have been developed since a long time in Gaussian frames. They often rely on some underlying "nice" Hilbertian structure, and can also require finiteness of moments of high order. Therefore, they can hardly be transposed to frames of heavy-tailed stable probability distributions. However, in the case of some linear non-anticipative moving average stable fields/processes, such as the linear fractional stable sheet…

On the fractional stochastic integration for random non-smooth integrands

The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations; differentiability of Hölder property of any kind or degree is not required …

Smoothness of Local Times and Self-Intersection Local Times of Gaussian Random Fields

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Malliavin differentiability and regularity of densities in semi-linear stochastic delay equations driven by weighted fractional Brownian motion

In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.

Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density

In this paper we study a class of stochastic partial differential equations in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, driven by a Gaussian noise white in time and correlated in space. The differential operator is a fractional derivative operator. We show the existence, uniqueness and Hölder's regularity of the solution. Then by means of Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure.

具有随机初始条件的分数阶Riesz-Bessel方程核平滑解的多尺度渐近分析

Multiscale Asymptotic Analysis of Kernel-Smoothed Solutions to Fractional Riesz-Bessel Equations with Random Initial Conditions

本文研究具有经典或循环长程依赖随机初始条件的分数阶Riesz-Bessel方程，通过核平滑方法获得非退化极限场，并证明其弱收敛到时空非平稳的高斯随机场。

$L^{p}$-solutions of the Navier-Stokes equation with fractional Brownian noise

We study the Navier-Stokes equations on a smooth bounded domain $D\subset \mathbb R^d$ ($d=2$ or 3), under the effect of an additive fractional Brownian noise. We show local existence and uniqueness of a mild $L^p$-solution for $p>d$.

分数边值问题与弹性粘性布朗运动，II：光滑域上的非局部动态边界条件

Fractional Boundary Value Problems and elastic sticky Brownian motions, II: Non-local dynamic boundary conditions on smooth domains

本文研究了分数边值问题中的非局部动态边界条件，通过时间变换描述分数粘性条件和相关边界行为，揭示了布朗运动在边界上无限均时停留的陷阱效应。

一类带分数噪声的抛物型SPDE光滑稳定流形的存在性

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

本文针对由非线性乘性分数噪声驱动的抛物型SPDE，通过插值理论构造合适函数空间并利用离散化Lyapunov-Perron型算子的唯一不动点，证明了其局部稳定流形的存在性与光滑性，从而推进了Lu-Schmalfuß猜想的研究。

Fractional smoothness of functionals of diffusion processes under a change of measure

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [σσ^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t)_{t\in [0,T]}$ be the associated $\R^d$-valued diffusion process on some appropriate $(Ω,\cF,\Q)$. For $p\in [2,\infty)$ and a measure $d¶=λ_T d\Q$, where $λ_T$ satisfies the Muckenhoupt condition $A_α$ for $α\in (1,p)$, we relate the behavior of $\|g(X_T)-…