A perturbation analysis of some Markov chains models with time-varying parameters

We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of V-geometrically ergodic Markov kernels indexed by a real parameter u, we give conditions under which the invariant probability distribution is differentiable with respect to u, in the sense of signed measures. Our results also complete the existing literature for the perturbation analysis of Markov chains, in particular when exponential moments are not finite. Our conditions are checked on several original examples of locally stationary processes such as integer-valued autoregressive processes, categorical time series or threshold autoregressive processes

Kolmogorov-Chentsov theorem and differentiability of random fields on manifolds

A version of the Kolmogorov-Chentsov theorem on sample differentiability and H\"older continuity of random fields on domains of cone type is proved, and the result is generalized to manifolds.Comment: 8 pages. Potential Analysis, February 201

(Non)Differentiability and Asymptotics for Potential Densities of Subordinators

For subordinators with positive drift we extend recent results on the structure of the potential measures and the renewal densities. Applying Fourier analysis a new representation of the potential densities is derived from which we deduce asymptotic results and show how the atoms of the Levy measure translate into points of (non)smoothness.Comment: 27 pages, appeared in Electronic Journal of Probability 201

On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets

We study the two-times differentiability of the value functions of the primal and dual optimization problems that appear in the setting of expected utility maximization in incomplete markets. We also study the differentiability of the solutions to these problems with respect to their initial values. We show that the key conditions for the results to hold true are that the relative risk aversion coefficient of the utility function is uniformly bounded away from zero and infinity, and that the prices of traded securities are sigma-bounded under the num\'{e}raire given by the optimal wealth process.Comment: Published at http://dx.doi.org/10.1214/105051606000000259 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical expansions and locally uniform Fréchet differentiability

Estimators which have locally uniform expansions are shown in this paper to be asymptotically equivalent to M-estimators. The M-functionals corresponding to these M-estimators are seen to be locally uniformly Fréchet differentiable. Other conditions for M-functionals to be locally uniformly Fréchet differentiable are given. An example of a commonly used estimator which is robust against outliers is given to illustrate that the locally uniform expansion need not be valid