排序方式: 共有28条查询结果,搜索用时 15 毫秒
1.
In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+\frac{\partial f}{\partial x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x},$$ where $1<\beta<\alpha< 2$, $W(t,x)$ is a fractional Brownian sheet, $\Delta_\theta:=-(-\Delta)^{\theta/2}$ denotes the fractional Lapalacian operator and $f:[0,T]\times \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness and H\"older regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle. 相似文献
2.
Litan Yan 《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(1-2):47-56
Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) σ ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , σ both are continuous on R and σ ( x )>0 if x p 0. Denote X 𝜏 * =sup 0 h t h 𝜏 | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n σ 2 ( x ) h | w ( x )| h K 2 | x | n σ 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J 𝜏 ) p h Á X 𝜏 * Á p h C p , n log 1 n +1 (1+ J 𝜏 ) p (0< p < n +1) hold for all stopping times 𝜏 . 相似文献
3.
In this paper, we consider the power variation of subfractional Brownian motion. As an application, we introduce a class of estimators for the index of a subfractional Brownian motion and show that they are strongly consistent. 相似文献
4.
Litan Yan 《Mathematische Nachrichten》2003,259(1):84-98
Let X = (Xt, ?t) be a continuous local martingale with quadratic variation 〈X〉 and X0 = 0. Define iterated stochastic integrals In(X) = (In(t, X), ?t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I0(t, X) = 1 and I1(t, X) = Xt. Let (??xt(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = supx∈? ??xt(X) and X* = supt≥0 |Xt|. In this paper, we shall establish various ratio inequalities for In(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants cn,p and Cn,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant Cn,p,γ depending only on n, p and γ, where Un is one of 〈In(X)〉1/2∞ and I*n(X), and V one of the three random variables X*, 〈X〉1/2∞ and ??*∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
5.
In this paper, we study the fractional stochastic heat equation driven by fractional Brownian motions of the form
$$
du(t,x)=\left(-(-\Delta)^{\alpha/2}u(t,x)+f(t,x)\right)dt +\sum\limits^{\infty}_{k=1} g^k(t,x)\delta\beta^k_t
$$
with $u(0,x)=u_0$, $t\in[0,T]$ and $x\in\mathbb{R}^d$, where $\beta^k=\{\beta^k_t,t\in[0,T]\},k\geq1$ is a sequence of i.i.d. fractional Brownian motions with the same Hurst index $H>1/2$ and the integral with respect to fractional Brownian motion is Skorohod integral. By adopting the framework given by Krylov, we prove the existence and uniqueness of $L_p$-solution to such equation. 相似文献
6.
Methodology and Computing in Applied Probability - In this paper, we consider the least squares estimators of the Ornstein-Uhlenbeck process with a constant drift... 相似文献
7.
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 i ∈ (0, 1) and K i ∈ (0, 1]. Assume d ? 2. One of the main motivations of this paper is to investigate smoothness of the collision local time $${l_T} = \int_0^T {\delta (B_s^{{H_1},{K_1}} - B_s^{{H_2},{K_2}}} )ds,{\rm{ }}T > 0,$$ , where δ denotes the Dirac delta function. By an elementary method we show that l T is smooth in the sense of Meyer-Watanabe if and only if min{H 1 K 1,H 2 K 2} < 1/(d + 2). 相似文献
8.
Let Open image in new window be the weighted local time of fractional Brownian motion B H with Hurst index 1/2?H?1. In this paper, we use Young integration to study the integral of determinate functions Open image in new window As an application, we investigate the weighted quadratic covariation \([f\big(B^H\big),B^H]^{(W)}\) defined by where the limit is uniform in probability and t k ?=?kt/n. We show that it exists and provided f is of bounded p-variation with \(1\leq p<\frac{2H}{1-H}\). Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.
相似文献
$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $
9.
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. 相似文献
10.
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, 2009).
相似文献
$$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$