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Jingjun Guo

Bio: Jingjun Guo is an academic researcher from Lanzhou University. The author has contributed to research in topics: Fractional Brownian motion & Order (ring theory). The author has an hindex of 2, co-authored 2 publications receiving 23 citations.

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TL;DR: In this article, the existence of high-order derivatives (k-th order) of intersection local time was shown to be true for two independent d-dimensional fractional Brownian motions.
Abstract: In this article, we obtain sharp conditions for the existence of the high-order derivatives (k-th order) of intersection local time $$ \widehat{\alpha }^{(k)}(0)$$ of two independent d-dimensional fractional Brownian motions $$B^{H_1}_t$$ and $$\widetilde{B}^{H_2}_s$$ of Hurst parameters $$H_1$$ and $$H_2$$ , respectively. We also study their exponential integrability.

15 citations

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TL;DR: In this article, the existence of high order derivatives of intersection local time derivatives of two independent d-dimensional fractional Brownian motions with Hurst parameters was shown to be true.
Abstract: In this article, we obtain sharp conditions for the existence of the high order derivatives ($k$-th order) of intersection local time $ \widehat{\alpha}^{(k)}(0)$ of two independent d-dimensional fractional Brownian motions $B^{H_1}_t$ and $\widetilde{B}^{H_2}_s$ with Hurst parameters $H_1$ and $H_2$, respectively. We also study their exponential integrability.

12 citations


Cited by
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TL;DR: In this article, the derivatives of local time for a ( 2, d ) -Gaussian field Z = { Z ( t, s ) = X t H 1 − X ˜ s H 2, s, t ≥ 0 }, where X H 1 and X H 2 are two independent processes from a class of d-dimensional centered Gaussian processes satisfying certain local nondeterminism property.

11 citations

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TL;DR: In this paper, Jeganathan et al. derived conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order limit theorem.
Abstract: We consider empirical processes associated with high-frequency observations of a fractional Brownian motion (fBm) $X$ with Hurst parameter $H\in (0,1)$, and derive conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order (possibly uniform) limit theorem. We devote specific emphasis to the `zero energy' case, corresponding to a kernel whose integral on the real line equals zero. Our asymptotic results are associated with explicit rates of convergence, and are expressed either in terms of the local time of $X$ or of its \blue{derivatives}: in particular, the full force of our finding applies to the `rough range' $0< H < 1/3$, on which the previous literature has been mostly silent. The {\color{black}use of the derivatives} of local times for studying the fluctuations of high-frequency observations of a fBm is new, and is the main technological breakthrough of the present paper. Our results are based on the use of Malliavin calculus and Fourier analysis, and extend and complete several findings in the literature, e.g. by Jeganathan (2004, 2006, 2008) and Podolskij and Rosenbaum (2018).

10 citations

Journal ArticleDOI
TL;DR: In this article, under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained under the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a pairing technique are employed.

8 citations

Journal ArticleDOI
TL;DR: In this article, the existence and Holder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion were studied.
Abstract: We consider the existence and Holder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where $$k=(k_1,k_2,\ldots , k_d)$$ . Moreover, we show a limit theorem for the critical case with $$H=\frac{2}{3}$$ and $$d=1$$ , which was conjectured by Jung and Markowsky [7].

7 citations

Posted Content
TL;DR: In this paper, the authors construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations and employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the roughness and non-Markovianity of the driving process.
Abstract: In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds to the Nash-Moser principle combined with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

6 citations