Past Stochastic Analysis Seminar

We treat the first n terms of general orthogonal series evolving with n as the partial sum process, and proved that under Menshov-Rademacher condition, the partial sum process can be enhanced into a geometric 2-rough process. For Fourier series, the condition can be improved, with an equivalent condition on limit function identified.

We consider the forest fire process on Z: on each site, seeds and matches fall at random, according to some independent Poisson processes. When a seed falls on a vacant site, a tree immediately grows. When a match falls on an occupied site, a fire destroys immediately the corresponding occupied connected component. We are interested in the asymptotics of rare fires. We prove that, under space/time re-scaling, the process converges (as matches become rarer and rarer) to a limit forest fire process.
Next, we consider the more general case where seeds and matches fall according to some independent stationary renewal processes (not necessarily Poisson). According to the tail distribution of the law of the delay between two seeds (on a given site), there are 4 possible scaling limits.
We finally introduce some related coagulation-fragmentation equations, of which the stationary distribution can be more or less explicitely computed and of which we study the scaling limit.

We will consider a branching Brownian motion where particles have a drift $-\rho$, binary branch at rate $\beta$ and are killed if they hit the origin. This process is supercritical  if $\beta>\rho^2/2$ and we will discuss the survival probability in the regime as criticality is approached. (Joint work with Elie Aidekon)

We describe a construction of the Brownian measure on Jordan curves with respect to the Weil-Petersson metric. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in the complex plane requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S1). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales.

To identify the Martin boundary for a transient Markov chain with Green's function G(x,y), one has to identify all possible limits Lim G(x,y_n)/G(0,y_n) with y_n "tending to infinity". For homogeneous random walks, these limits are usually obtained from the exact asymptotics of Green's function G(x,y_n). For non-homogeneous random walks, the exact asymptotics af Green's function is an extremely difficult problem. We discuss several examples where Martin boundary can beidentified by using large deviation technique. The minimal Martin boundary is in general not homeomorphic to the "radial"  compactification obtained by Ney and Spitzer for homogeneous random walks in Z^d : convergence of a sequence of points y_n toa point on the Martin boundary does not imply convergence of the sequence y_n/|y_n| on the unit sphere. Such a phenomenon is a consequence of non-linear optimal large deviation trajectories.

 In the recent series of papers Kleban, Simmons, and Ziff gave a non-rigorous computation  (base on Conformal Field Theory) of probabilities of several connectivity events for critical percolation. In particular they showed that the probability that there is a percolation cluster connecting two points on the boundary and a point inside the domain can be factorized in therms of pairwise connection probabilities. We are going to use SLE techniques to rigorously compute probabilities of several connectivity events and prove the factorization formula.

Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they have also significant geometrical applications. Particular cases of planar maps are p-angulations, where each face (meaning each component of the complement of edges) has exactly p adjacent edges. Random planar maps have been used in theoretical physics, where they serve as models of random geometry.Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces.More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all p-angulations with n vertices. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. Both in the case p=3 and when p>3 is even, we prove that the resulting random metric spaces converge as n tends to infinity to a universal object called the Brownian map. This convergence holds in the sense of the Gromov-Hausdorff distance between compact metric spaces. In the particular case of triangulations (p=3), this solves an open problem stated by Oded Schramm in his 2006 ICM paper. As a key tool, we use bijections between planar maps and various classes of labeled trees

We consider nonlinear stochastic wave equations in dimension d\le 3.

Using Malliavin Calculus, we give upper bounds for the small eigenvalues of the inverse of two point densities.These provide a rate of degeneracy when points go close to each other.  Then, we analyze the consequences of this result on lower estimates for hitting probabilities. 

We define a notion of discrete Ricci curvature for a metric measure space by looking at whether "small balls are closer than their centers are". In a Riemannian manifolds this gives back usual Ricci curvature up to scaling. This definition is very easy to apply in a series of examples such as graphs (eg the discrete cube has positive curvature). We are able to generalize several Riemannian theorems in positive curvature, such as concentration of measure and the log-Sobolev inequality. This definition also allows to prove new theorems both in the Riemannian and discrete case: for example improved bounds on spectral gap of the Laplace-Beltrami operator, and fast convergence results for some Markov Chain Monte Carlo methods

We establish a large deviations principle for the block sizes of a uniformly random non-crossing partition. As an application we obtain a variational formula for the maximum of the support of a compactly supported probability measure in terms of its free cumulants, provided these are all non-negative. This is useful in free probability theory, where sometimes the R-transform is known but cannot be inverted explicitly to yield the density.

The vacant set is the set of vertices not visited by a random walk on a graph G before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit'

when the graph grows), and the relation of the problem to the random interlacement percolation.  I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.

Duality methods can be very powerful tools for the analysis of stochastic

processes. However, there seems to be no general theory available

yet. In this talk, I will discuss and aim to clarify various notions

of duality, give some recent rather striking examples (applied to

stochastic PDEs, interacting particle systems and combinatorial stochastic

processes)

and try to give some systematic insight into the type of questions

that can in principle be tackled. Finally, I will try to provide you

with some intuition for this fascinating technique.

The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and how this led to the solution of a problem dating back to the 50's: whether the the central limit theorem is driven by an analogue of the second law of thermodynamics.

This problem is classically addressed by the so-called Skorohod Embedding problem. We instead develop a stochastic control approach. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples beyond the known classical ones. In particular, we solve completely the case of finitely many given marginals.

In 1986 Kardar, Parisi, and Zhang proposed a stochastic PDE for the motion of driven interfaces,
in particular for growth processes with local updating rules. The solution to the 1D KPZ equation
can be approximated through the weakly asymmetric simple exclusion process. Based on work of 
Tracy and Widom on the PASEP, we obtain an exact formula for the one-point generating function of the KPZ
equation in case of sharp wedge initial data. Our result is valid for all times, but of particular interest is
the long time behavior, related to random matrices, and the finite time corrections. This is joint work with 
Tomohiro Sasamoto.

: Backward error analysis is a technique that has been extremely successful in understanding the behaviour of numerical methods for ordinary differential equations.  It is possible to fit an ODE (the so called modified equation) to a numerical method to very high accuracy. Backward error analysis has been of particular importance in the numerical study of Hamiltonian problems, since it allows to approximate symplectic numerical methods by a perturbed Hamiltonian system, giving an approximate statistical mechanics for symplectic methods. 

Such a systematic theory in the case of numerical methods for stochastic differential equations (SDEs) is currently lacking. In this talk we will describe a general framework for deriving modified equations for SDEs with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities will also be discussed, as well as the use of modified equations  as a tool for constructing higher order methods for stiff stochastic differential equations.

This is joint work with A. Abdulle (EPFL). D. Cohen (Basel), G. Vilmart (EPFL).

Recently, Liang, Lyons and Qian developed a new methodology for the study of backward stochastic differential equations (BSDEs) on general filtered probability spaces. Their approach is based on the analysis of a particular class of functional differential equations, where the driver of the equation does not depend only on the present, but also on the terminal value of the solution.

The purpose of this work is to study fully coupled systems of forward functional differential equations, which are related to a broad class of fully coupled forward-backward stochastic dynamics with respect to general filtrations. In particular, these systems of functional differential equations have a more homogeneous structure with respect to the underlying forward-backward problems, allowing to partly avoid the conflicting nature between the forward and backward components.

Another advantage of the approach is that its generality allows to consider many other types of forward-backward equations not treated in the classical literature: this is shown with the help of several examples, which have interesting applications to mathematical finance and are related to parabolic integro-partial differential equations. In the second part of the talk, we introduce a numerical scheme for the approximation of decoupled systems, based on a time discretization combined with a local iteration approach.

I will introduce the notion of a nonlinear Levy process, discuss basic well-posednes, SDE links and the connection with interacting particles. The talk is aimed to be an introduction to the topic of my recent CUP monograph 'Nonllinear Markov processes and kinetic equations'.

The paper analyses structural models for the evaluation of risky debt following H.E. LELAND [2], with an approach of optimal stopping problem (for instance cf. N. EL KAROUI [1]) and within a more general context: a dividend is paid to equity holders, moreover a different tax schedule is introduced, depending on the firm current value. Actually, an endogenous default boundary is introduced and a nonlinear convex tax schedule allowing for a possible switching in tax benefits. The aim is to find optimal capital structure such that the failure is delayed, meaning how to decrease the failure level VB, anyway preserving D debtholders and E equity holders’interests: for the firm VB is needed as low as possible, for the equity holder, an optimal equity is requested, finally an optimal coupon C is asked  for the total value.

Keywords: corporate debt, optimal capital structure, default,

Numerical Approximations of Non-linear Stochastic Systems. Abstract:  The explicit solution of stochastic differential equations (SDEs can be found only in a few cases. Therefore, there is a need fo accurate numerical approximations that could, for example, enabl  Monte Carlo Simulations. Convergence and stability of these methods are well understood for SDEs with Lipschit  continuous coefficients. Our research focuses on those situations wher  the coefficients of the underlying SDEs are non-Lipschitzian  It was demonstrated in the literature,  that in this case using the classical methods we may fail t  obtain numerically computed paths that are accurate for small step-sizes, or to obtain qualitative information about the behaviour of numerical methods over long time intervals. Our work addresses both of these issues, giving a customized analysis of the most widely used numerical methods.

We consider parabolic scalar conservation laws perturbed by a (conservative) noise. Large deviations are investigated in the singular limit of jointly vanishing viscosity and noise. The model is supposed to feature the same behavior of "asymmetric" particles systems (e.g. TASEP) under Euler scaling.

A first large deviations principle is obtained in a space of Young measures. A "second order" large deviations principle is then discussed, including connections with the Jensen and Varadhan functional. As time allows, more recent "long correlation" models will be treated.