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Register a new userA Hoeffdings inequality for uniformly ergodic diffusion process
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In this note, we present a version of Hoeffdings inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffdings inequality for discrete-time uniformly ergodic Markov chain, the proof relies on techniques ranging from martingale theory to classical Hoeffdings lemma as well as the notion of deviation kernel of diffusion process. We present two examples to illustrate our results. In the first example we consider large deviation probability on the occupation time of the Jacobi diffusion, a popular process used in modelling of exchange rates in mathematical finance, while in the second example we look at the exponential functional of a finite interval analogue of the Ornstein-Uhlenbeck process introduced by Kessler and S{o}rensen (1999).
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We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $pi$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{e} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernsteins type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.
In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Levy process with finite Levy measure, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.
In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $Delta$n = sup i=0,...,n--1 (t n i+1 -- t n i) $rightarrow$ 0. We introduce an estimator of $theta$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $Delta$n $rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$Delta$ 3 n $rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $theta$ is feasible under the condition that n$Delta$ k n $rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{e}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.
We consider a controlled second order differential equation which is partially observed with an additional fractional noise. we study the asymptotic (for large observation time) design problem of the input and give an efficient estimator of the unknown signal drift parameter. When the input depends on the unknow parameter, we will try the one-step estimation procedure using the Newton-Raphson method.
A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to $1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.
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