7 research outputs found
Malliavin and dirichlet structures for independent random variables
On any denumerable product of probability spaces, we construct a Malliavin
gradient and then a divergence and a number operator. This yields a Dirichlet
structure which can be shown to approach the usual structures for Poisson and
Brownian processes. We obtain versions of almost all the classical functional
inequalities in discrete settings which show that the Efron-Stein inequality
can be interpreted as a Poincar{\'e} inequality or that Hoeffding decomposition
of U-statistics can be interpreted as a chaos decomposition. We obtain a
version of the Lyapounov central limit theorem for independent random variables
without resorting to ad-hoc couplings, thus increasing the scope of the Stein
method
The insider problem in the trinomial model: a discrete-time jump process approach
In an incomplete market underpinned by the trinomial model, we consider two
investors : an ordinary agent whose decisions are driven by public information
and an insider who possesses from the beginning a surplus of information
encoded through a random variable for which he or she knows the outcome.
Through the definition of an auxiliary model based on a marked binomial
process, we handle the trinomial model as a volatility one, and use the
stochastic analysis and Malliavin calculus toolboxes available in that context.
In particular, we connect the information drift, the drift to eliminate in
order to preserve the martingale property within an initial enlargement of
filtration in terms of the Malliavin derivative. We solve explicitly the agent
and the insider expected logarithmic utility maximisation problems and provide
a hedging formula for replicable claims. We identify the insider expected
additional utility with the Shannon entropy of the extra information, and
examine then the existence of arbitrage opportunities for the insider.Comment: 29 page
Kernel Selection in Nonparametric Regression
In the regression model , where has a
density , this paper deals with an oracle inequality for an estimator of
, involving a kernel in the sense of Lerasle et al. (2016), selected via
the PCO method. In addition to the bandwidth selection for kernel-based
estimators already studied in Lacour, Massart and Rivoirard (2017) and Comte
and Marie (2020), the dimension selection for anisotropic projection estimators
of and is covered.Comment: 23 page
On a Projection Least Squares Estimator for Jump Diffusion Processes
This paper deals with a projection least squares estimator of the drift
function of a jump diffusion process computed from multiple independent
copies of observed on . Risk bounds are established on this
estimator and on an associated adaptive estimator. Finally, some numerical
experiments are provided.Comment: 19 pages, 3 figure
Insiderâs problem in the trinomial model: a discrete jump process point of view
In an incomplete market underpinned by the trinomial model, we consider two investors: an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a surplus of information encoded through a random variable for which he or she knows the outcome. Through the definition of an auxiliary model based on a marked binomial process, we handle the trinomial model as a volatility one, and use the stochastic analysis and Malliavin calculus toolboxes available in that context. In particular, we connect the information drift, i.e. the drift to eliminate in order to preserve the martingale property within an initial enlargement of filtration in terms of Malliavinâs derivative. We solve explicitly the agent and the insider expected logarithmic utility maximization problems and provide a Ocone-Karatzas type formula for replicable claims. We identify insiderâs expected additional utility with the Shannon entropy of the extra information, and examine then the existence of arbitrage opportunities for the insider
Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a chaotic expansion for square-integrable (marked binomial) functionals, prior to the elaboration of a Markov-Malliavin structure within this framework. We take advantage of the new formalism to deal with two main applications. First, we revisit the Chen-Stein method for the (compound) Poisson approximation which we perform in the paradigm of the built Markov-Malliavin structure, before studying in the second one the problem of portfolio optimisation in the trinomial model
Robust density estimation with the L1-loss. Applications to the estimation of a density on the line satisfying a shape constraint
We solve the problem of estimating the distribution of presumed i.i.d.\ observations for the total variation loss. Our approach is based on density models and is versatile enough to cope with many different ones, including some density models for which the Maximum Likelihood Estimator (MLE for short) does not exist. We mainly illustrate the properties of our estimator on models of densities on the line that satisfy a shape constraint. We show that it possesses some similar optimality properties, with regard to some global rates of convergence, as the MLE does when it exists. It also enjoys some adaptation properties with respect to some specific target densities in the model for which our estimator is proven to converge at parametric rate. More important is the fact that our estimator is robust, not only with respect to model misspecification, but also to contamination, the presence of outliers among the dataset and the equidistribution assumption. This means that the estimator performs almost as well as if the data were i.i.d.\ with density in a situation where these data are only independent and most of their marginals are close enough in total variation to a distribution with density . {We also show that our estimator converges to the average density of the data, when this density belongs to the model, even when none of the marginal densities belongs to it}. Our main result on the risk of the estimator takes the form of an exponential deviation inequality which is non-asymptotic and involves explicit numerical constants. We deduce from it several global rates of convergence, including some bounds for the minimax L1-risks over the sets of concave and log-concave densities. These bounds derive from some specific results on the approximation of densities which are monotone, convex, concave and log-concave. Such results may be of independent interest