Weighted sub-fractional Brownian motion was introduced as a centered Gaussian process with covariance
$$ Q_{a,b}(s,t)=\frac{1}{1-b}\int_0^{s\wedge t}u^a[(s-u)^b+(t-u)^b-(s+t-2u)^b]\,du. $$
This kernel was proved to be positive definite for $a>-1$ and $b\in[0,2]\setminus{1}$, and also for $a>-1$, $-1<b<0$, and $a+b+1\ge0$. In this paper, we replace the power weight $u^a$ by a measurable non-negative function $f:\mathbb R_+\to\mathbb R_+$ satisfying
$$ \int_0^t f(u)(t-u)^b\,du<\infty,\qquad t>0, $$
and consider the kernel
$$ R_{f,b}(s,t)=\frac{1}{1-b}\int_0^{s\wedge t}f(u)[(s-u)^b+(t-u)^b-(s+t-2u)^b]\,du. $$
We prove that, for $b>-1$, $b\ne1$, the kernel $R_{f,b}$ is positive definite for every such function $f$ if and only if $b\in[0,1)\cup(1,2]$. For every $b\in(-1,0)\cup(2,\infty)$, we construct a non-negative function $f$ satisfying the integrability condition for which $R_{f,b}$ is not positive definite. The case $b=1$ is obtained as a logarithmic limit. For the associated Gaussian process, we study Hölder regularity, total and quadratic variation, non-stationarity, and long-range dependence. For $b\in(0,1)\cup(1,2]$, we also define differential equations driven by this process and establish pathwise convergence of the corresponding Euler approximation, together with strong $L^p$-rates under the additional regularity assumptions stated below. The logarithmic boundary $b=1$ is included in the covariance theory but is not part of the pathwise-equation and Euler results.
Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion
Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose …
In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of $0$. We show the existence, uniqueness and positivity of the solutions. The estimations of moments, including the negative power moments, are given. Based on these estimations, strong convergence of the positivity preserving drift-implicit Euler-type scheme is proved, and optimal convergence rate is obtained. By using Lamperti transformation, we show th…
We introduce a semi-implicit Euler-Maruyama approximation which preservers the non-colliding property for some class of non-colliding particle systems such as Dyson Brownian motions, Dyson-Ornstein-Uhlenbeck processes and Brownian particles systems with nearest neighbour repulsion, and study its rates of convergence in both $L^p$-norm and path-wise sense.
In this article we propose a new explicit Euler-type approximation method for stochastic differential equations (SDEs). In this method, Brownian increments in the recursion of the Euler method are replaced by suitable bounded functions of the Brownian increments. We prove strong convergence rate one-half for a large class of SDEs with polynomial coefficient functions whose local monotonicity constant grows at most like the logarithm of a Lyapunov-type function.
In this paper, we establish the theory of chaos propagation and propose an Euler-Maruyama scheme for McKean-Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst exponent $H \in (0,1)$. Meanwhile, upper bounds for errors in the Euler method is obtained. A numerical example is demonstrated to verify the theoretical results.
A number of recent new proofs of Euler's celebrated identity are presented, all of which are probabilistic in nature and depend on the conformal invariance of planar Brownian motion.
We propose discrete random-field models that are based on random partitions of $\mathbb{N}^2$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.
The purpose of this paper is to establish the convergence in distribution of the normalized error in the Euler approximation scheme for stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form $(t-s)^α$, where $α\in (-\frac 12, \frac 12)$.
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q -- L $ρ$ drift coefficient when the condition d $ρ$ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 -- (d $ρ$ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove …
In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.
The Numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion
School of Sciences, Great Bay University（大湾大学理学院）；School of Mathematics, University of Science and Technology of China（中国科学技术大学数学系）；School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Department of Mathematics and Department of Physics, Great Bay University（大湾大学数学系和物理系）；School of Mathematics, Southeast University（东南大学数学学院）
In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work \cite{Yoshida2013}, which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.
The strong convergence rate of the Euler scheme for SDEs driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter $H\in(\frac13,\frac12)$ and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the $2$D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order $2H$ for the case that the drift coefficient has bounded derivatives up…
This paper provides yet another look at the mixed fractional Brownian motion (fBm), this time, from the spectral perspective. We derive an approximation for the eigenvalues of its covariance operator, asymptotically accurate up to the second order. This in turn allows to compute the exact $L_2$-small ball probabilities, previously known only at logarithmic precision. The obtained expressions show an interesting stratification of scales, which occurs at certain values of the Hurst parameter of the fractional component. Some of them have been previously encountered in other problems involving su…
Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion
CIMFAV - Ingemat, Facultad de Ingeniería, Universidad de Valparaíso（瓦尔帕莱索大学工程学院）；Aix-Marseille Univ, CNRS, AMSE（艾克斯-马赛大学）；Centro de Matemática, Facultad de Ciencias, Universidad de la República（乌拉圭共和国大学理学院数学中心）
