Search

Let B H be a fractional Brownian motion with Hurst index 1 2 ≤ H < 1 . In this paper, we consider the linear self-attracting diffusion X t H = B t H - θ ∫ 0 t ∫ 0 s X s H - X u H d u d s + ν t with X 0 H = 0 , where θ > 0 and ν ∈ R are two unknown parameters. The process is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We introduce some laws of large numbers associated with the self-attracting diffusion. In particular, we show that the convergences 1 θ T 3 - 2 H ∫ 0 T Y t H d B t H , 1 T 3 - 2 H ∫ 0 T Y t H 2 d t ⟶ H θ - 2 H 3 - 2 H Γ (2 H) hold almost surely and in L 2 (Ω) , as T → ∞ , where Y t H = ∫ 0 t (X t H - X s H) d s and the integral ∫ 0 T Y t H d B t H is the Young integral. As some applications, we study asymptotic behaviors of the least squares estimators of θ and ν under the continuous observation. [ABSTRACT FROM AUTHOR]

Let BH = {BtH, t ⩾ 0} be a fractional Brownian motion with Hurst index H ∈ (0, 1). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to BH as follows: ∫ 0 t u s ⋄ d − B s H = lim ε ↓ 0 1 ε ∫ 0 t u s ⋄ (B s + ε H − B s H) d s ∫ 0 t u s ⋄ d ° B s H = lim ε ↓ 0 1 2 ε ∫ 0 t u s ⋄ (B s + ε H − B (s − ε) ∨ 0 H) d s in probability, where ⋄ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of BH for all H ∈ (0, 1). [ABSTRACT FROM AUTHOR]

Let u (t , x) be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X (t , x) , t ≥ 0 , x ∈ R with initial condition u (0 , x) ≡ 0 , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u (t , x) . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u (t , x) is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 for every real number x. However, the many properties of the process x ↦ u (⋅ , x) are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u (⋅ , x) , t ↦ u (t , ⋅) . We show that x ↦ u (⋅ , x) admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with "Itô's integral", but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we consider the least squares estimators of the Ornstein-Uhlenbeck process with a constant drift d X t = (θ 1 − θ 2 X t) d t + d Z t with X0 = x0, where θ1, θ2 are two unknown parameters with θ2 > 0 and Z is a strictly symmetric α-stable motion on ℝ with the index α ∈ (1, 2). We construct the least squares estimators of θ1 and θ2 based on the discrete observation, and discuss the strong consistency and asymptotic distributions of the two estimators. Finally, we give some numerical calculus and simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the large time behavior of the solution to the fractional heat equation ∂ u ∂ t (t , x) = - (- Δ) β / 2 u (t , x) + u (t , x) ∂ d + 1 W ∂ t ∂ x 1 ⋯ ∂ x d , t > 0 , x ∈ R d , where β ∈ (0 , 2) and the noise W(t, x) is a fractional Brownian sheet with indexes H 0 , H 1 , ... , H d ∈ (1 2 , 1) . By using large deviation techniques and variational method, we find a constant M 1 such that for any integer p ≥ 1 and α 0 β + α < β , lim t → ∞ t - 2 β - β α 0 - α β - α log E u (t , x) p = p 2 β - α β - α α H 2 β β - α M 1 , where α 0 = 2 - 2 H 0 , α = ∑ j = 1 d (2 - 2 H j) and α H = ∏ i = 0 d H i (2 H i - 1) . [ABSTRACT FROM AUTHOR]

In this paper, we discuss a class of neutral retarded stochastic functional differential equations driven by a fractional Brownian motion on Hilbert spaces. We develop a C 0 -semigroup theory of the driving deterministic neutral system and formulate the neutral time delay equation under consideration as an infinite-dimensional stochastic system without time lag and neutral item. Consequently, a criterion is presented to identify a strictly stationary solution for the systems considered. In particular, the ergodicity of the strictly stationary solution is studied. Subsequently, the ergodicity behavior of non-stationary solution for the systems considered is also investigated. We present an example which can be explicitly determined to illustrate our theory in the work. [ABSTRACT FROM AUTHOR]

In this paper, we consider the large deviation principle for a class of space-time fractional stochastic heat equation ∂ t β u ε ( t , x ) = − ν ( − Δ ) α 2 u ε ( t , x ) + I t 1 − β f ( u ε ( t , x ) ) + ε I t 1 − β [ W ˙ H ( t , x ) ] , $$\begin{array}{} \displaystyle \partial^\beta_tu^\varepsilon(t,x)=-\nu(-\Delta)^{\frac\alpha 2}u^\varepsilon(t,x)+I_t^{1-\beta}f(u^\varepsilon(t,x))+ \sqrt{\varepsilon}I^{1-\beta}_t[\dot{W}^H(t,x)], \end{array}$$ where ẆH is a fractional white noise, ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂ t β $\begin{array}{} \displaystyle \partial^\beta_t \end{array}$ is the Caputo fractional integration operator, and − ( − Δ ) α 2 $\begin{array}{} \displaystyle -(-\Delta)^{\frac\alpha 2} \end{array}$ is the fractional power of Laplacian. Our proof is based on the weak convergence approach. [ABSTRACT FROM AUTHOR]

We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic heat equation of the form where $\frac{\partial^{2}B}{\partial t\,\partial x}$ is a fractional Brownian sheet with Hurst indices $H_{1},H_{2}\in(\frac{1}{2},1)$ and $\Delta _{\alpha}=-(-\Delta)^{\alpha/2}$ is a fractional Laplacian operator with $1<\alpha\leq2$ . In particular, when $H_{2}=\frac{1}{2}$ we show that the temporal process $\{u(t,\cdot),0\leq t\leq T\}$ admits a nontrivial p-variation with $p=\frac{2\alpha}{2\alpha H_{1}-1}$ and study its local nondeterminism and existence of the local time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic heat equation of the form where W is a fractional Brownian sheet, $\Delta+\Delta_{\alpha}$ is a pseudo differential operator on ${\mathbb {R}}$ which gives rise to a Lévy process consisting of the sum of a Brownian motion and an independent symmetric α-stable process, and $f:[0,T]\times\mathbb{R}\times\mathbb {R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness, Hölder regularity and density estimate of the solution. [ABSTRACT FROM AUTHOR]

Let ${W}=\lambda B+\nu B^{H}$ be a mixed-fractional Brownian motion with Hurst index $0< H<\frac{1}{2}$ and $\lambda,\nu\neq0$ . In this paper we study the quadratic covariation $[f({W}),{W}]^{(H)}$ defined by in probability, where f is a Borel function and $\eta_{s}=\lambda^{2}s+\nu^{2}s^{2H}$ . For some suitable function f we show that the quadratic covariation exists in $L^{2}(\Omega)$ and the Itô formula holds for all absolutely continuous function F with $F'=f$ , where the integral is the Skorohod integral with respect to W. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an approximation theorem for the Rosenblatt sheet, using martingale differences. The proof involves the tightness and identification of finite dimensional distributions. [ABSTRACT FROM AUTHOR]

Let $$B^H$$ be a fractional Brownian motion with Hurst index $$\frac{1}{2}\alpha $$ with compact support, where $${\mathscr {D}}^\alpha _{+}$$ denotes the right-handed Marchaud fractional derivative of order $$\alpha $$ . [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic heat equation with multiplicative bi-fractional Brownian sheet. Using the technique of Feynman-Kac formula and Malliavin calculus, we give an explicit formula of the weak solution and study the regularity. [ABSTRACT FROM AUTHOR]

In this article, we will prove the existence, uniqueness and Hölder regularity of the solution to the fractional stochastic partial differential equation of the form where $$\mathfrak {D}(x,D)$$ denotes the Markovian generator of stable-like Feller process, $$f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$$ is a measurable function, and $$\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)$$ is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287-2309, ). [ABSTRACT FROM AUTHOR]

We study the existence, uniqueness and Hölder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation laws. The equation is driven by double-parameter fractional noises. In addition, the existence and moment estimate are also obtained for the density of the law of such a solution. [ABSTRACT FROM AUTHOR]

Let $$B^{H_1 ,K_1 }$$ and $$B^{H_2 ,K_2 }$$ be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence where K is a standard Gaussian kernel function and the bandwidth parameter α satisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion $$B^{H_1 ,K_1 }$$. We also give the stable convergence of the sequence S by using the techniques of the Malliavin calculus. [ABSTRACT FROM AUTHOR]

Let SH be a sub-fractional Brownian motion with index 0 < H < 1/2. In this paper we study the existence of the generalized quadratic covariation [ f( SH), SH] (W) defined by $[f(S^H ),S^H ]_t^{(W)} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \tfrac{1} {{\varepsilon ^{2H} }}\int_0^t {\{ f(S_{s + \varepsilon }^H ) - f(S_s^H )\} (S_{s + \varepsilon }^H - S_s^H )ds^{2H} } , $ provided the limit exists in probability, where x ↦ f( x) is a measurable function. We construct a Banach space ℋ of measurable functions such that the generalized quadratic covariation exists in L2 provided f ∈ ℋ. Moreover, the generalized Bouleau-Yor identity takes the form [Figure not available: see fulltext.] for all f ∈ ℋ, where ℒ H( x, t) is the weighted local time of SH. This allows us to write the generalized Itô’s formula for absolutely continuous functions with derivative belonging to ℋ. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct a family of continuous stochastic processes that converges in law to the multiple Wiener–Itô integrals with respect to the subfractional Brownian motion with for the integrand in a rather general class of functions. We mainly use Donsker and Stroock approximations and the techniques of the multiple Wiener–Itô integral with respect to the Wiener process. [Copyright &y& Elsevier]

Abstract: In this paper, we mainly prove a central limit theorem for weighted local time of -modulus of fractional Brownian motion with Hurst parameter . Similar to Hu and Nualart (2009), based on techniques of stochastic analysis, the main ingredients of the proof are analogous to the asymptotic version of Knight’s theorem and the fractional Clark–Ocone formula for the -modulus of the weighted local time increments. [Copyright &y& Elsevier]

Let $${B^{{H_i},{K_i}}} = \{ B_t^{{H_i},{K_i}},t \ge 0\} ,{\rm{ }}i = 1,2$$ be two independent, d-dimensional bifractional Brownian motions with respective indices H ∈ (0, 1) and K ∈ (0, 1]. Assume d ⩾ 2. One of the main motivations of this paper is to investigate smoothness of the collision local time , where δ denotes the Dirac delta function. By an elementary method we show that l is smooth in the sense of Meyer-Watanabe if and only if min{ H K, H K} < 1/( d + 2). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence results of mild solutions for a class of stochastic integro-differential equations with nonlocal conditions and stochastic impulsive integro-differential equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for the existence of mild solutions are derived by means of Leray–Schauder nonlinear alternative. An example is provided to illustrate the theory. [Copyright &y& Elsevier]

Abstract: The present note is devoted to prove, by means of Malliavin calculus, the convergence in of some properly renormalized weighted quadratic variation of sub-fractional Brownian motion with parameter . [Copyright &y& Elsevier]

Abstract: This paper studies the functionals where is a one-dimension sub-fractional Brownian motion with index . It shows that there exists a constant such that -variation of the process () is equal to 0 if , where , , are the local time and weighted local time of , respectively. This extends the classical results for Brownian motion. [Copyright &y& Elsevier]

Let $B^{H_i ,K_i } = \{ B_t^{H_i ,K_i } ,t \geqslant 0\} $, i = 1, 2 be two independent bifractional Brownian motions with respective indices H ∈ (0, 1) and K ∈ (0, 1]. One of the main motivations of this paper is to investigate the smoothness of the collision local time, introduced by Jiang and Wang in 2009, $\ell _T = \int_0^T {\delta (B_s^{H_1 ,K_1 } - B_s^{H_2 ,K_2 } )ds} $ T > 0, where δ denotes the Dirac delta function. By an elementary method, we show that ℓ is smooth in the sense of the Meyer-Watanabe if and only if min{ H K, H K} < 1/3. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2< H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L 2 if 1/2< H<3/4. [ABSTRACT FROM AUTHOR]

Let X be a Cox–Ingersoll–Ross (CIR) process given by$\{\backslash rm\; d\}X\_t=(a+bX\_t)\backslash ,\{\backslash rm\; d\}t+c\backslash sqrt\{\backslash vert\; X\_t\backslash vert\}\backslash ,\{\backslash rm\; d\}B\_t$with X0=0, where a, c>0, $b\backslash in\{\backslash open\; R\}$ and B a standard Brownian motion starting at zero. We obtain some inequalities between the integral functional $J\_\{\backslash varphi\}(t)=\backslash int\_0^t\; \backslash varphi(X\_s)\; \backslash ,\; \{\backslash rm\; d\}s$, t≥0 and the maximal process sup0≤s≤tXs, t≥0, where x↦ϕ(x) a nonnegative continuous function with some suitable conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the pth mean almost periodic solution to a neutral stochastic evolution equation with infinite delay and Poisson jumps. We give a sufficient condition for the existence and uniqueness of pth mean almost periodic solution and the condition depends on the continuity of coefficients and the power of a fractional operator. We give an example to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

Let SH={StH,t≥0} be a sub-fractional Brownian motion with Hurst index 0. In this paper, we give a local law of the iterated logarithm of the form lim sups↓0|St+sH−StH|sH2log+log(1/s)=1, almost surely, for all t>0, where log+x=max{1,logx} for x≥0. As an application, we introduce the ΦH-variation of SH driven by ΦH(x):=[x/2log+log+(1/x)]1/H(x>0) with ΦH(0)=0. [ABSTRACT FROM AUTHOR]

Let U, V and W be three Hilbert spaces and let BH be a W-valued fractional Brownian motion with Hurst index H∈(12,1). In this paper, we consider the approximate controllability of the Sobolev-type fractional stochastic differential equation {Dtαc[Lx(t)]=Ax(t)+f(t,xt)+Bu(t)+σ(t)ddtBH(t),t∈(0,T],x(t)=ϕ(t),t∈(−∞,0], where Dαc is the Caputo fractional derivative of order α∈(1−H,1), the time history xt:(−∞,0]→xt(θ)=x(t+θ) with t>0 belonging to the phase space Bh, the control function u(⋅) is given in L2([0,T],V), B is a bounded linear operator from V into U. Under some suitable conditions, we show that the system is approximately controllable on [0,T] and we give an example to illustrate the theory. [ABSTRACT FROM AUTHOR]