Past Stochastic Analysis Seminar

We consider one-dimensional Brownian motion conditioned (in a suitable

sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.5860... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). I will also describe other conditionings of Brownian motion in which this principle of entropic repulsion manifests itself.

Joint work with Itai Benjamini.

Particle current is the net number of particles that pass an observer who moves with a deterministic velocity V. Its fluctuations in time-stationary interacting particle systems are nontrivial and draw serious attention. It has been known for a while that in most models diffusive scaling and the corresponding Central Limit Theorem hold for this quantity. However, such normal fluctuations disappear for a particular value of V, called the characteristic speed.

For this velocity value, the correct scaling of particle current fluctuations was shown to be t1/3 and the limit distribution was also identified by K. Johansson in 2000 and later by P. L. Ferrari and H. Spohn in 2006. These results use heavy combinatorial and analytic tools, and their application is limited to a few particular models, one of which is the totally asymmetric simple exclusion process (TASEP). I will explain a purely probabilistic, more robust approach that provides the t2/3-scaling of current variance, but not the limit distribution, in (non-totally) asymmetric simple exclusion (ASEP) and some other particle systems. I will also point out a key feature of the models which allows the proof of such universal behaviour.

Joint work with Júlia Komjáthy and Timo Seppälläinen)

We construct a kinetic SDE in the state variables (position,velocity), where the spatial dependency in the drift term of the velocity equation is a conditional expectation with respect to the position. Those systems are introduced in fluid mechanic by S. B. Pope and are used in the simulation of complex turbulent flows. Such simulation approach is known as Probability Density Function (PDF) method .

We construct a PDF method applied to a dynamical downscaling problem to generate fine scale wind : we consider a bounded domain D. A weather prediction model solves the wind field at the boundary of D (coarse resolution). In D, we adapt a Lagrangian model to the atmospheric flow description and we construct a particles algorithm to solve it (fine resolution).

In the second part of the talk, we give a (partial) construction of a Lagrangian SDE confined in a given domain and such that the corresponding Eulerian velocity at the boundary is given. This problem is related to stochastic impact problem and existence of trace at the boundary for the McKean-Vlasov equations with specular boundary condition

The Cameron-Martin theorem is a fundamental result in stochastic analysis. We will show that the Wiener measure on a geometrically and stochastically complete Riemannian manifold is quasi-invariant. This is a complete a complete generalization of the classical Cameron-Martin theorem for Euclidean space to Riemannian manifolds. We do not impose any curvature growth conditions.

The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).

The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) tries to maximise this quantity.

For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).

In the first part of the talk, we fit the Hua-Pickrell measure (which is a two parameters deformation of the Haar measure) on the unitary group and the Ewens measure on the symmetric group in a same framework. We shall see that in the unitary case, the eigenvalues follow a determinantal point process with explicit hypergeometric kernels. We also study asymptotics of these kernels. The techniques used rely upon splitting of the Haar measure and sampling techniques. In the second part of the talk, we provide a matrix model for the circular Jacobi ensemble, which is the sampling used for the Hua-Pickrell measure but this time on Dyson's circular ensembles. In this case, we use the theory of orthogonal polynomials on the unit circle. In particular we prove that when the parameter of the sampling grows with n, both the spectral measure and the empirical spectral measure converge weakly in probability to a non-trivial measure supported only by one piece of the unit circle.

We consider a critical, measure-valued branching diffusion ξ in Rd, where the branching is continuous and the spatial motion is given by the heat flow. For d ≥ 2 and fixed t > 0, ξt is known to be an a.s. singular random measure of Hasudorff dimension 2. We explain how it can be approximated by Lebesgue measure on ε-neighbourhoods of the support. Next we show how ξt can be approximated in total variation near n points, and how the associated Palm distributions arise in the limit from elementary conditioning. Finally we hope to explan the duality between moment and Palm measures, and to show how the latter can be described in terms of discrete “Palm trees.”

Euler equations for incompressible fludis describe the evolution of the divergence-free velocity of a non-viscous fluid (when viscosity is present, we have the well-known Navier-Stokes equations). V. Arnold discovered that they correspond to geodesic equations in the space of volume-preserving diffeomorphisms but several exemples show that it is not always possible to solve the corresponding variational problems inducing minimal energy displacements. A solvable relaxed version, in a non-deterministic setting (measures on the path space, with possible splitting of the particles), has been introduced by Y. Brenier who intensively studied the problem. Together with M. Bernot and A. Figalli we founded new solutions and characterization results. In the talk I'll present the most interesting features of the problem and of its solutions.

Mike Giles recently came up with a very general technique that improves the fundamental complexity of Monte Carlo simulation in the context where stochastic differential equations are simulated numerically. I will discuss some work with Mike Giles and Xuerong Mao that extends the theoretical support for this approach to the case of financial options without globally Lipschitz payoff functions. I will also suggest other application areas where this multi-level approach might prove valuable, including stochastic computation in cell biology.

Probabilistic methods for the solution of Backward Stochastic Differential Equations (BSDE) provide us with a new approach to the problem of approximating the solution of a semi-linear PDE. Utilizing on the Markovian nature of these BSDE’s we show how one may consider the problem of numerical solutions to BSDEs within the area of weak approximations of diffusions. To emphasize this point, we suggest an algorithm based on the Cubature method on Wiener space of Lyons - Victoir. Instead of using standard discretization techniques of BSDE’s, we choose to work with the actual flow. This allows to take advantage of estimates on the derivatives of the solution of the associated semi-linear PDE and hence, we recover satisfactory convergence estimates.

An important and challenging problem in mathematical finance is how to choose a pricing measure in an incomplete market, i.e. how to find a probability measure under which expected payoffs are calculated and fair option prices are derived under some notion of optimality.

The notion of q-optimality is linked to the unique equivalent martingale measure (EMM) with minimal q-moment (if q > 1) or minimal relative entropy (if q=1). Hobson's (2004) approach to identifying the q-optimal measure (through a so-called fundamental equation) suggests a relaxation of an essential condition appearing in Delbaen & Schachermayer (1996). This condition states that for the case q=2, the Radon-Nikodym process, whose last element is the density of the candidate measure, is a uniformly integrable martingale with respect to any EMM with a bounded second moment. Hobson (2004) alleges that it suffices to show that the above is true only with respect to the candidate measure itself and extrapolates for the case q>1. Cerny & Kallsen (2008) however presented a counterexample (for q=2) which demonstrates that the above relaxation does not hold in general.

The speaker will present the general form of the q-optimal measure following the approach of Delbaen & Schachermayer (1994) and prove its existence under mild conditions. Moreover, in the light of the counterexample in Cerny & Kallsen (2008) concerning Hobson's (2004) approach, necessary and sufficient conditions will be presented in order to determine when a candidate measure is the q-optimal measure.

We present the ideas of Malliavin calculus in the context of rough differential equations (RDEs) driven by Gaussian signals. We then prove an analogue of Hörmander's theorem for this set-up, finishing with the conclusion that, for positive times, a solution to an RDE driven by Gaussian noise will have a density with respect to Lebesgue measure under Hörmander's conditions on the vector fields.

Let u be a vector field on a bounded domain in R^3. The absolute boundary condition states that both the normal part of u and the tangential part of curl(u) vanish on the boundary. After motivating the use of this condition in the context of the Navier Stokes equation, we prove local (in time) existence with this boundary behaviour. This work is together with Dr. Z. Qian and Prof. G. Q. Chen, Northwestern University.

We will analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. These results will be applied to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and graphs with prescribed degree distribution. We will sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. If time allows, we will discuss related open problems. This is a joint work with Marc Lelarge (INRIA & Ecole Normale Supérieure).

I will present two new results in the context of the title. Both are joint work with B. Veto.

1. In earlier work a limit theorem with $t^{2/3}$ scaling was established for a class of self repelling random walks on $\mathbb Z$ with long memory, where the self-interaction was defined in terms of the local time spent on unoriented edges. For combinatorial reasons this proof was not extendable to the natural case when the self-repellence is defined in trems of local time on sites. Now we prove a similar result for a *continuous time* random walk on $\mathbb Z$, with self-repellence defined in terms of local time on sites.

2. Defining the self-repelling mechanism in terms of the local time on *oriented edges* results in totally different asymptotic behaviour than the unoriented cases. We prove limit theorems for this random walk with long memory.

I will take as my starting point a problem which is classical in

population genetics: we wish to understand the distribution of numbers

of individuals in a population who carry different alleles of a

certain gene. We imagine a sample of size n from a population in

which individuals are subject to neutral mutation at a certain

constant rate. Every mutation gives rise to a completely new type.

The genealogy of the sample is modelled by a coalescent process and we

imagine the mutations as a Poisson process of marks along the

coalescent tree. The allelic partition is obtained by tracing back to

the most recent mutation for each individual and grouping together

individuals whose most recent mutations are the same. The number of

blocks of each of the different possible sizes in this partition is

called the allele frequency spectrum. Recently, there has been much

interest in this problem when the underlying coalescent process is a

so-called Lambda-coalescent (even when this is not a biologically

``reasonable'' model) because the allelic partition is a nice example

of an exchangeable random partition. In this talk, I will describe

the asymptotics (as n tends to infinity) of the allele frequency

spectrum when the coalescent process is a particular Lambda-coalescent

which was introduced by Bolthausen and Sznitman. It turns out that

the frequency spectrum scales in a rather unusual way, and that we

need somewhat unusual tools in order to tackle it.

This is joint work with Anne-Laure Basdevant (Toulouse III).

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a linearly edge-reinforced random walk on these graphs

is recurrent. Furthermore, we derive bounds for the probability that

the edge-reinforced random walk hits the boundary of a large box

before returning to its starting point.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.

We describe individual based continuous models of random evolutions and discuss some effects of competitions in these models. The range of applications includes models of spatial ecology, genetic mutation-selection models and particular socio-economic systems. The main aim of our presentation is to establish links between local characteristics of considered models and their macroscopic behaviour

In this talk we revisit the setting of Bouchard, Touzi, and Zeghal (2004).

For an incomplete market and a non-smooth utility function U defined on the whole real line we study the problem:

sup E [U(XTx,θ – B)]

θΘ(S)

Here B is a bounded contingent claim and Xx,θ represents the wealth process with initial capital x generated by portfolio θ. We study the case when the portfolios are constrained in a closed convex cone.

For the case without constraints and with a smooth utility function the solution method is to approximate the utility function and look at the same problem on a bounded negative domain. However, when one attempts to solve this bounded domain problem for a non-smooth utility function, the standard methods of proof cannot be applied. To circumvent this difficulty the idea of quadratic inf-convolution was introduced in Bouchard, Touzi, and Zeghal (2004). This method is mathematically appealing but leads to lengthy and technical proofs.

We will show that despite the presence of constraints, the dependence on quadratic inf-convolution can be removed. We will also show the existence of a constrained replicating portfolio for the optimal terminal wealth when the filtration is generated by a Brownian motion. This provides a natural generalisation of the results of Karatzas and Shreve (1998) to the whole real line.

The convergence of Markov processes to stationary distributions is a basic topic of introductory courses in stochastic processes, and the theory has been thoroughly developed. What happens when we add killing to the process? The process as such will not converge in distribution, but the survivors may; that is, the distribution of the process, conditioned on survival up to time t, converges to a "quasistationary distribution" as t goes to infinity.

This talk presents recent work with Steve Evans, proving an analogue of the transience-recurrence dichotomy for killed one-dimensional diffusions. Under fairly general conditions, a killed one-dimensional diffusion conditioned to have survived up to time t either escapes to infinity almost surely (meaning that the probability of finding it in any bounded set goes to 0) or it converges to the quasistationary distribution, whose density is given by the top eigenfunction of the adjoint generator.

These theorems arose in solving part of a longstanding problem in biological theories of ageing, and then turned out to play a key role in a very different problem in population biology, the effect of unequal damage inheritance on population growth rates.