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Search Results: 1 - 10 of 53054 matches for " David Nualart "
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Kolmogorov and Probability Theory
Nualart, David
Arbor : Ciencia, Pensamiento y Cultura , 2004,
Abstract: No disponible
Some Processes Associated with Fractional Bessel Processes
Yaozhong Hu,David Nualart
Mathematics , 2004,
Abstract: Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process. It\^{o}'s formula for the fractional Brownian motion leads to the equation $ R_{t}=\sum_{i=1}^{d}\int_{0}^{t}\frac{B_{s}^{i}}{R_{s}}% dB_{s}^{i}+H(d-1)\int_{0}^{t}\frac{s^{2H-1}}{R_{s}}ds . $ In the Brownian motion case ($H=1/2$), $X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{% R_{s}}dB_{s}^{i}$ is a Brownian motion. In this paper it is shown that $X_{t}$ is \underbar{not} a fractional Brownian motion if $H\not=1/2$. We will study some other properties of this stochastic process as well.
Rough Path Analysis Via Fractional Calculus
Yaozhong Hu,David Nualart
Mathematics , 2006,
Abstract: Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously differentiable function such that $f'$ is $\lambda$-H\"oldr continuous for some $\lambda>\frac1\beta-2$. Under some further smooth conditions on $f$ the integral is a continuous functional of $x$, $y$, and the tensor product $x\otimes y$ with respect to the H\"{o}lder norms. We derive some estimates for these integrals and we solve differential equations driven by the function $y$. We discuss some applications to stochastic integrals and stochastic differential equations.
Notes on the two-dimensional fractional Brownian motion
Fabrice Baudoin,David Nualart
Mathematics , 2006, DOI: 10.1214/009117905000000288
Abstract: We study the two-dimensional fractional Brownian motion with Hurst parameter $H>{1/2}$. In particular, we show, using stochastic calculus, that this process admits a skew-product decomposition and deduce from this representation some asymptotic properties of the motion.
Central limit theorems for multiple Skorohod integrals
Ivan Nourdin,David Nualart
Mathematics , 2007,
Abstract: In this paper, we prove a central limit theorem for a sequence of iterated Shorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed.
Regularity of the density for the stochastic heat equation
Carl Mueller,David Nualart
Mathematics , 2007,
Abstract: We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.
Parameter estimation for fractional Ornstein-Uhlenbeck processes
Yaozhong Hu,David Nualart
Mathematics , 2009,
Abstract: We study a least squares estimator $\hat {\theta}_T$ for the Ornstein-Uhlenbeck process, $dX_t=\theta X_t dt+\sigma dB^H_t$, driven by fractional Brownian motion $B^H$ with Hurst parameter $H\ge \frac12$. We prove the strong consistence of $\hat {\theta}_T$ (the almost surely convergence of $\hat {\theta}_T$ to the true parameter ${% \theta}$). We also obtain the rate of this convergence when $1/2\le H<3/4$, applying a central limit theorem for multiple Wiener integrals. This least squares estimator can be used to study other more simulation friendly estimators such as the estimator $\tilde \theta_T$ defined by (4.1).
Central limit theorem for the modulus of continuity of the Brownian local time in $L^3(\mathbb{R})$
Yaozhong Hu,David Nualart
Mathematics , 2009,
Abstract: The purpose of this note is to prove a central limit theorem for the $L^3$-modulus of continuity of the Brownian local time using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulus of the Brownian local time.
CLT for an iterated integral with respect to fBm with H > 1/2
Daniel Harnett,David Nualart
Mathematics , 2012, DOI: 10.1080/17442508.2013.774403
Abstract: We construct an iterated stochastic integral with fractional Brownian motion with H > 1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment theorem of Nualart and Peccati, we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart.
Central limit theorem for functionals of two independent fractional Brownian motions
David Nualart,Fangjun Xu
Mathematics , 2012,
Abstract: We prove a central limit theorem for functionals of two independent $d$-dimensional fractional Brownian motions with the same Hurst index $H$ in $(\frac{2}{d+1},\frac{2}{d})$ using the method of moments.
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