Past Stochastic Analysis Seminar

My goal is to estimate unknown parameters in the vector field of a rough differential equation, when the expected signature for the driving force is known and we estimate the expected signature of the response by Monte Carlo averages.

I will introduce the "expected signature matching estimator" which extends the moment matching estimator and I will prove its consistency and asymptomatic normality, under the assumption that the vector field is polynomial.  Finally, I will describe the polynomial system one needs to solve in order to compute this estimatior.

In this talk we will review some recent results on the long-time/large-scale, weak-friction asymptotics for the one dimensional Langevin equation with a periodic potential. First we show that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We also show that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. Furthermore we prove that the same result is valid for a whole one parameter family of space/time rescalings. We also present a new numerical method for calculating the diffusion coefficient and we use it to study the multidimensional problem and the problem of Brownian motion in a tilted periodic potential.

If a distribution, say F, has all moments finite, then either F is unique (M-determinate) in the sense that F is the only distribution with these moments, or F is non-unique (M-indeterminate).  In the latter case we suggest a method for constructing a Stieltjes class consisting of infinitely many distributions different from F and all having the same moments as F.  We present some shocking examples involving distributions such as N, LogN, Exp and explain what and why.  We analyse conditions which are sufficient for F to be M-determinate or M-indeterminate.  Then we deal with recent problems from the following areas:

 

(A)  Non-linear (Box-Cox) transformations of random data.

(B) Distributional properties of functionals of stochastic processes.

(C) Random sums of random variables.

 

If time permits, some open questions will be outlined.  The talk will be addressed to colleagues, including doctoral and master students, working or having interests in the area of probability/stochastic processes/statistics and their applications. 

A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced by C. Chevalier and F Debbasch, both in a heuristic and analytic way.  Roughly speaking, they are characterised by the existence at each (proper) time (of the moving particle) of a (local) rest frame where the random part of the acceleration of the particle (computed using the time of the rest frame) is brownian in any spacelike direction of the frame.

I will explain how the tools of stochastic calculus enable us to give a concise and elegant description of these random paths on any Lorentzian manifiold.  A mathematically clear definition of the the one-particle distribution function of the dynamics will emerge from this definition, and whose main property will be explained.  This will enable me to obtain a general H-theorem and to shed some light on links between probablistic notions and the large scale structure of the manifold.

All necessary tools from stochastic calculus and geometry will be explained.

Numerical computer codes implementing physics based models are the backbone of today's mechanical/aerospace engineering analysis and design methods. Such computational codes can be extremely expensive consisting of several millions of degrees of freedom. However, large models even with very detailed physics are often not enough to produce credible numerical results because of several types of uncertainties which exist in the whole process of physics based computational predictions. Such uncertainties include, but not limited to (a) parametric uncertainty (b) model inadequacy; (c) uncertain model calibration error coming from experiments and (d) computational uncertainty. These uncertainties must be assessed and systematically managed for credible computational predictions. This lecture will discuss a random matrix approach for addressing these issues in the context of complex structural dynamic systems. An asymptotic method based on eigenvalues and eigenvectors of Wishart random matrices will be discussed. Computational predictions will be validated against laboratory based experimental results.

The parabolic Anderson model is the Cauchy problem for the heat equation with random potential.  It offers a case study for the possible effects that a random, or irregular environment can have on a diffusion process.  In this talk I review results obtained for an extreme case of heavy-tailed potentials, among the effects we discuss our intermittency, strong localisation and ageing.

This talk is based on a joint work with Zhan Shi: We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou (2005). Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn (1988). Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k≥1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and coloured in space.

Given a foliation of a compact manifold, leaves of which are equipped with a Riemannian metric, one can consider the associated "leafwise"

Brownian motion, and study its asymptotic properties (such as asymptotic distribution, behaviour of holonomy maps, etc.).

Lucy Garnet studied such measures, introducing the notion of a harmonic measure -- stationary measure of this process; the name "harmonic" comes from the fact that a measure is stationary if and only if with respect to it integral of every leafwise Laplacian of a smooth function equals zero (so, the measure is "harmonic" in the sense of distributions).

It turns out that for a transversally conformal foliation, unless it possesses a transversally invariant measure (which is a rather rare case), the associated random dynamics can be described rather precisely. Namely, for every minimal set in the foliation there exists a unique harmonic measure supported on it -- and this gives all the possible ergodic harmonic measures (in particular, there is a finite number of them, and they are always supported on the minimal sets).

Also, the holonomy maps turn out to be (with probability one) exponentially contracting -- so, the Lyapunov exponent of the dynamics is negative. Finally, for any initial point almost every path tends to one of the minimal sets and is asymptotically distributed with respect to the corresponding harmonic measure -- and the functions defining the probabilities of tending to different sets form a base in the space of continuous leafwise harmonic functions.

An interesting effect that is a corollary of this consideration is that for transversally conformal foliations the number of the ergodic harmonic measures does not depend on the choice of Riemannian metric on the leaves. This fails for non-transversally conformal foliations:

there is an example, recently constructed in a joint with S.Petite (following B.Deroin's technique).

Stochastic geometry gradually becomes a necessary theoretical tool to model and analyse modern telecommunication systems, very much the same way the queuing theory revolutionised studying the circuit switched telephony in the last century. The reason for this is that the spatial structure of most contemporary networks plays crucial role in their functioning and thus it has to be properly accounted for when doing their performance evaluation, optimisation or deciding the best evolution scenarios.  The talk will present some stochastic geometry models and tools currently used in studying modern telecommunications.  We outline specifics of wired, wireless fixed and ad-hoc systems and show how the stochastic geometry modelling helps in their analysis  and optimisation.

Random polymers are used to model various physical ( Ising inter- faces, wetting, etc.) and biological ( DNA denaturation, etc.) phenomena They are modeled as a one dimensional random walk (Xn), with excursion length distribution

P(E1 = n) = (n)=nc, c > 1, and (n) a slowly varying function. The polymer gets a random reward, whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (Vn). Depending on the relation be- tween the mean value of the disorder Vn and the temperature, the polymer might prefer to stick on the interface (pinning) or undergo a long excursion away from it (depinning).

In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the 'Pinning' model and also its relation to other directed polymer models

We study the class of Azema-Yor processes which are of the form F(M_t)-f(M_t)(X_t-M_t), where F'=f, X_t is a semimartingale with no positive jumps and M_t is its running maximum. We show that these processes arise as unique strong solutions to the Bachelier SDE which we also show is equivalent to the DrawDown SDE. The proofs are greatly simplified thanks to (algebraic) group property of the set of AY processes indexed by functions. We then restrict our attention to the case when X is a martingale. It turns out that the AY martingales are the only local martingales of the form H(X_t,M_t) for a Borel function H. Furthermore, they can also be characterised by their optimal

properties: all uniformly integrable martingales whose maximum dominates a given target are dominated by an AY martingale in the concave ordering of terminal values. We mention how these results find direct applications in portfolio optimisation/insurance theory.

Joint work with Laurent Cararro and Nicole El Karoui

In this talk, we give the asymptotics estimates for the heat kernel and its gradient estimates on H-type groups. Moreover, we get gradient estimates for the heat semi-group.

We discuss some connections between various notions of rationality in the face of uncertainty and the theory of convex risk measures, both in a static and a dynamic setting.