In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity usin
g Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.
In this paper we address an open question formulated in [17]. That is, we extend the It{^o}-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and F the solution of the Fokker-Planck equation
associated to X, to random mappings f. To this end we provide new results on a class of adpated and non-adapted Fokker-Planck SPDEs and BSPDEs.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded cd
processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs.
In this paper we prove that every random variable of the form $F(M_T)$ with $F:real^d toreal$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $mathcal{F}$ admits an exact integral representati
on with respect to $M$, that is, without any orthogonal component. This representation holds true regardless any regularity assumption on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$ and show they can be solved without an orthogonal component. To this end, we extend first existence results for such BSDEs under a general filtration and then obtain regularity properties such as differentiability for the solution process.
We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{alpha, beta}$ with Hurst parameter $(alpha, beta) in (0,1)^2$. When $0<alpha leq 1-frac{1}{2q}$ or $0<beta leq 1-frac{1}{2q}$
a central limit theorem holds for the renormalized Hermite variations of order $qgeq 2$, while for $1-frac{1}{2q}<alpha, beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.
We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an appli
cation, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$.
Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes using their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we deriv
e an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.
In recent years, infinite-dimensional methods have been introduced for the Gaussian channels estimation. The aim of this paper is to study the application of similar methods to Poisson channels. In particular we compute the Bayesian estimator of a Po
isson channel using the likelihood ratio and the discrete Malliavin gradient. This algorithm is suitable for numerical implementation via the Monte-Carlo scheme. As an application we provide an new proof of the formula obtained recently by Guo, Shamai and Verduu relating some derivatives of the input-output mutual information of a time-continuous Poisson channel and the conditional mean estimator of the input. These results are then extended to mixed Gaussian-Poisson channels.
We combine Steins method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by P
eccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.
We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts fo
rmula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424].
