A maximum a posteriori estimator for trajectories of diffusion processes

Let x _t be a diffusion process observed via a noisy sensor, whose output is y_t We consider the problem of evaluating the maximum a posteriori trajectory {x_s0≤ s ≤ t Based on results of Stratonovich [1] and Ikeda-Watanabe [2], we show that this estimator is given by the solution of an appropriate variational problem which is a slight modification of the "minimum energy" estimator. We compare our results to the non-linear filtering theory and show that for problems which possess a finite dimensional solution, our approach yields also explicit filters. For linear diffusions observed via linear sensors, these filters are identical to the Kalman-filter     

Extension of the ito calculus via the malliavin calculus

The Ito formula is extended to the tempered distributions "evaluated" on the trajectories of a nondegenerate Ito process in the sense of P. Malliavin. To do this the Ito integral is extended to vector-valued adapted distributions on Wiener space. Also a Galerkin type approximation using the Skorohod integral or the divergence operator is given for the diffusion processes. At the final section we give a sufficient condition for the existence of a smooth density for the filtering of nonlinear diffusions with the help of the techniques of the Malliavin calculus and the theory of nuclear spaces.     

Asymptotic expansion of Bayes estimators for small diffusions

Summary Using the Malliavin calculus we derived asymptotic expansion of the distributions of the Bayes estimators for small diffusions. The second order efficiency of the Bayes estimator is proved.     

Reconstruction of Gray-Scale Images

We present an algorithm to reconstruct gray scale images corrupted by noise. We use a Bayesian approach. The unknown original image is assumed to be a realization of a Markov random field on a finite two dimensional region Z². This image is degraded by some noise, which is assumed to act independently in each site of and to have the same distribution on all sites. For the estimator we use the mode of the posterior distribution: the so called maximum a posteriori (MAP) estimator. The algorithm, that can be used for both gray-scale and multicolor images, uses the binary decomposition of the intensity of each color and recovers each level of this decomposition using the identification of the problem of finding the two color MAP estimator with the min-cut max-flow problem in a binary graph, discovered by Greig et al. (1989). Experimental results and a detailed example are given in the text. We also provide a web page where additional information and examples can be found.     

Integral curves from noisy diffusion MRI data with closed-form uncertainty estimates

We present a method for estimating the trajectories of axon fibers through diffusion tensor MRI (DTI) data that provides theoretically rigorous estimates of trajectory uncertainty. We develop a three-step estimation procedure based on a kernel estimator for a tensor field based on the raw DTI measurements, followed by a plug-in estimator for the leading eigenvectors of the tensors, and a plug-in estimator for integral curves through the resulting vector field. The integral curve estimator is asymptotically normal; the covariance of the limiting Gaussian process allows us to construct confidence ellipsoids for fixed points along the curve. Complete trajectories of fibers are assembled by stopping integral curve tracing at locations with multiple viable leading eigenvector directions and tracing a new curve along each direction. Unlike probabilistic tractography approaches to this problem, we provide a rigorous, theoretically sound model of measurement uncertainty as it propagates from the raw MRI data, to the tensor field, to the vector field, to the integral curves. In addition, trajectory uncertainty is estimated in closed form while probabilistic tractography relies on sampling the space of tensors, vectors, or curves. We show that our estimator provides more realistic trajectory uncertainty estimates than a more simplified prior approach for closed-form trajectory uncertainty estimation due to Koltchinskii et al. (Ann Stat 35:1576–1607, 2007) and a popular probabilistic tractography method due to Behrens et al. (Magn Reson Med 50:1077–1088, 2003) using theory, simulation, and real DTI scans.     

Self-stabilizing Processes in Multi-wells Landscape in ℝ d -Invariant Probabilities

The aim of this work is to analyze the stationary measures for a particular class of non-Markovian diffusions, the self-stabilizing processes. All the trajectories of such a process attract each other. This permits to exhibit a non-uniqueness of the stationary measures in the one-dimensional case, see Herrmann and Tugaut (Stoch. Process. Their Appl. 120(7):1215–1246, 2010). In this paper, the extension to general multi-wells lansdcape in general dimension is provided. Moreover, the approach for investigating this problem is different and needs fewer assumptions. The small-noise limit behavior of the invariant probabilities is also given.     

Bidimensional random effect estimation in mixed stochastic differential model

In this work, a mixed stochastic differential model is studied with two random effects in the drift. We assume that N trajectories are continuously observed throughout a large time interval [0, T]. Two directions are investigated. First we estimate the random effects from one trajectory and give a bound of the \(L^2\)-risk of the estimators. Secondly, we build a nonparametric estimator of the common bivariate density of the random effects. The mean integrated squared error is studied. The performances of the density estimator are illustrated on simulations.     

The detection and estimation of the change point in a disccrete-time stochastic system

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution     

On nonlinear coupled diffusions in competition systems

A class of reaction-diffusion systems modeling plant growth with spatial competition in saturated media is presented. We show, in this context, that standard diffusion can not lead to pattern formation (Diffusion Driven Instability of Turing). Degenerated nonlinear coupled diffusions inducing free boundaries and exclusive spatial diffusions are proposed. Local and global existence results are proved for smooth approximations of the degenerated nonlinear diffusions systems which give rise to long-time pattern formations. Numerical simulations of a competition model with degenerate/non degenerate nonlinear coupled diffusions are performed and we carry out the effect of the these diffusions on pattern formation and on the change of basins of attraction.     

Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to handle the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.     

Iterated Stochastic Processes: Simulation and Relationship with High Order Partial Differential Equations

In this paper, we consider the composition of two independent processes: one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler scheme to simulate the trajectories of the corresponding iterated processes on a fixed time interval. This algorithm is natural and can be implemented easily. We show that it converges almost surely, uniformly in time, with a rate of convergence of order 1/4 and propose an estimation of the error. We then extend the well known Feynman-Kac formula which gives a probabilistic representation of partial differential equations (PDEs), to its higher order version using iterated processes. In particular we consider general position processes which are not necessarily Markovian or are indexed by the real line but real valued. We also weaken some assumptions from previous works. We show that intertwining diffusions are related to transformations of high order PDEs. Combining our numerical scheme with the Feynman-Kac formula, we simulate functionals of the trajectories and solutions to fourth order PDEs that are naturally associated to a general class of iterated processes.     

Edgeworth expansion for the pre-averaging estimator

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.     

On Solvability of Integro-Differential Equations

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.

     

Stability of nonlinear regime-switching jump diffusion

Motivated by networked systems, stochastic control, optimization, and a wide variety of applications, this work is devoted to systems of switching jump diffusions. Treating such nonlinear systems, we focus on stability issues. One of the distinct features considered here is that the switching process depends on the jump diffusions. First asymptotic stability in the large is obtained. Then the study on exponential p$p$-stability is carried out. Connection between almost surely exponential stability and exponential p$p$-stability is exploited. Also presented are smooth-dependence on the initial data. Using the smooth-dependence, necessary conditions for exponential p$p$-stability are derived. Then criteria for asymptotic stability in distribution are provided. A couple of examples are given to illustrate our results.     

Strong Averaging Along Foliated Lévy Diffusions with Heavy Tails on Compact Leaves

This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed Lévy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order \(p\geqslant 2\), perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of Lévy rotations on the unit circle subject to perturbations by a planar Lévy-Ornstein-Uhlenbeck process is carried out in detail.     

Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions

The problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated. We propose an estimator which is sharp adaptive on scales of Sobolev smoothness classes. The analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values—such as the geometry of the diffusion coefficient—of the prototypical drift estimation problem for a large class of multidimensional diffusion processes. We further sketch generalizations of our results to arbitrary diffusions satisfying suitable Bernstein-type inequalities.     

Data driven time scale in Gaussian quasi-likelihood inference

We study parametric estimation of ergodic diffusions observed at high frequency. Different from the previous studies, we suppose that sampling stepsize is unknown, thereby making the conventional Gaussian quasi-likelihood not directly applicable. In this situation, we construct estimators of both model parameters and sampling stepsize in a fully explicit way, and prove that they are jointly asymptotically normally distributed. High order uniform integrability of the obtained estimator is also derived. Further, we propose the Schwarz (BIC) type statistics for model selection and show its model-selection consistency. We conducted some numerical experiments and found that the observed finite-sample performance well supports our theoretical findings.

     

Diffusions in random environment and ballistic behavior

In this article we investigate the ballistic behavior of diffusions in random environment. We introduce conditions in the spirit of (T) and (T^′) of the discrete setting, cf. [A.-S. Sznitman, On a class of transient random walks in random environment, Ann. Probab. 29 (2) (2001) 723–764; A.-S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment, Probab. Theory Related Fields 122 (4) (2002) 509–544], that imply, when d2, a law of large numbers with non-vanishing limiting velocity (which we refer to as ‘ballistic behavior’) and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides examples of diffusions in random environment with ballistic behavior, beyond what was previously known.     

Stability of regime-switching diffusions

This work is devoted to stability of regime-switching diffusion processes. After presenting the formulation of regime-switching diffusions, the notion of stability is recalled, and necessary conditions for p$p$-stability are obtained. Then main results on stability and instability for systems arising in approximation are presented. Easily verifiable conditions are established. An example is examined as a demonstration. A remark on linear systems is also provided.     

Some Notes on Large Deviations of Markov Processes

In this paper we shall characterize the large deviation principles (abbreviated to LDP) of Donsker-Varadhan of a Markov process both for the weak convergence topology and for the τ-topology, by means of a hyper-exponential recurrence property. A Lyapunov criterion for this type of recurrence property is presented. These results are applied to countable Markov chains, unidimensional diffusions, elliptic or hypoelliptic diffusions on Rienmannian manifolds. Several counter-examples are equally presented. Received July 20, 1998, Accepted March 25, 1999