Past Stochastic Analysis Seminar

Abstract: Burgers equation is a quasilinear partial differential equation (PDE), proposed in 1930's to model the evolution of turbulent fluid motion, which can be linearized to the heat equation via the celebrated Cole-Hopf transformation. In the first part of the talk, we study in detail general versions of stochastic Burgers equation with random coefficients, in both forward and backward sense. Concerning the former, the Cole-Hopf transformation still applies and we reduce a forward stochastic Burgers equation to a forward stochastic heat equation that can be treated in a “pathwise" manner. In case of deterministic coefficients, we obtain a probabilistic representation of the Cole-Hopf transformation by associating the backward Burgers equation with a system of forward-backward stochastic differential equations (FBSDEs). Returning to random coefficients, we exploit this representation in order to establish a stochastic version of the Cole-Hopf transformation. This generalized transformation allows us to find solutions to a backward stochastic Burgers equation through a backward stochastic heat equation, subject to additional constraints that reflect the presence of randomness in the coefficients. In both settings, forward and backward, stochastic Feynman-Kac formulae are derived for the solutions of the respective stochastic Burgers equations, as well. Finally, an application that illustrates the obtained results is presented to a pricing/hedging problem arising from mathematical finance.

In the second part of the talk, we study a class of stochastic saddlepoint systems, represented by fully coupled FBSDEs with infinite horizon, that gives rise to a continuous time rational expectations / consol rate model with random coefficients. Under standard Lipschitz and monotonicity conditions, and by means of the contraction mapping principle, we establish existence, uniqueness and dependence on a parameter of adapted solutions. Making further the connection with quasilinear backward stochastic PDEs (BSPDEs), we are led to the notion of stochastic viscosity solutions. A stochastic maximum principle for the optimal control problem of a large investor is also provided as an application to this framework.

This is joint work with N. Frangos, X.- I. Kartala and A. N. Yannacopoulos*

The standard Taylor series approach to the higher-order approximation of vector SDEs requires simulation of iterated stochastic integrals, which is difficult. The talk will describe an approach using methods from optimal transport theory which avoid this difficulty in the case of non-degenerate diffusions, for which one can attain arbitrarily high order pathwise approximation in the Vaserstein 2-metric, using easily generated random variables.

If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations, that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs in this setting and outline some novel applications of this approach.

The star-triangle transformation is used to obtain an equivalence extending over a set bond percolation models on isoradial graphs. Amongst the consequences are box-crossing (RSW) inequalities and the universality of alternating arms exponents (assuming they exist) for such models, under some conditions. In particular this implies criticality for these models.

(joint with Geoffrey Grimmett)

I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure.

In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation

Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one.

Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly.

We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D.

We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations.

Abstract: A central question in the theory of random walks on groups is how symmetries of the underlying space gives rise to structure and rigidity of the random walks. For example, for nilpotent groups, it is known that random walks have diffusive behavior, namely that the rate of escape, defined as the expected distance of the walk from the identity satisfies E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn| ~= n. (~= meaning upto (multiplicative) constants )

In this work, for every 3/4 <= \beta< 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^\beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.

Previous examples were only of exponents of the form 1-1/2^k or 1 , and were based on lamplighter (wreath product) constructions.(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups , can be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. (Paper available at http://arxiv.org/abs/1203.6226)

No previous knowledge of random walks,automaton groups or wreath products is assumed.

Abstract: Hairer recently had the remarkable insight that Lyons' theory of rough paths can be used to make sense of nonlinear SPDEs that were previously ill-defined due to spatial irregularities. Since rough path theory deals with the integration of functions defined on the real line, the SPDEs studied by Hairer have a one-dimensional spatial index variable. I will show how to combine paraproducts, a notion from functional analysis, with ideas from the theory of controlled rough paths, in order to develop a formulation of rough path theory that works in any index dimension. As an application, I will present existence and uniqueness results for an SPDE with multidimensional spatial index set, for which previously it was not known how to describe solutions. No prior knowledge of rough paths or paraproducts is required for understanding the talk. This is joint work with Massimiliano Gubinelli and Peter Imkeller.

(Joint work with P.E. Chaudru de Raynal and F. Delarue)

Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity.

An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity.

In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm.

We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests.

References

[1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010.

[2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

Abstract: We study measures on half planar maps that satisfy a natural domain Markov property. I will discuss their classification and some of their geometric properties. Joint work with Gourab Ray.

Abstract: We consider the inverse problem of recovering u from a noisy, indirect observation We adopt a Bayesian approach, in which the aim is to determine the posterior distribution _y on the unknown u, given some prior information about u in the form of a prior distribution _0,together with the observation y. We are interested in the question of posterior consistency, which is the characterization of the behaviour of _y as more data become available. We work in a separable Hilbert space X, assuming a Gaussian prior _0 = N(0; _ 2C0). The theory is developed using two concrete problems: i) a family of linear inverse problems in which we want to _nd u from y where y = A

We consider the fractional Laplacian perturbed by the gradient operator b(x)\nabla for various classes of vector fields b. We construct end estimate the corresponding semigroup.

I will speak about the of weak (and the existence and uniqueness of strong solutions) to the stochastic
Landau-Lifshitz equations for multi (one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire. The talk is based on a joint works with B. Goldys and T. Jegaraj.

I will report on joint work in progress with David Aldous, concerning a curious random metric space on the plane which can be constructed with the help of an improper Poisson line process.

We study the asymptotic behavior of deterministic dynamics of many interacting particles with random initial data in the limit where the number of particles tends to infinity. A famous example is hard sphere flow, we restrict our attention to the simpler case where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density $f_0(u,v)$ depending on $d$-dimensional position $u$ and velocity $v$. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are $L^\infty$ in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.

Abstract: By using the Skorohod equation we derive an
iteration procedure which allows us to solve a class of reflected backward
stochastic differential equations with non-linear resistance induced by the
reflected local time. In particular, we present a new method to study the
reflected BSDE proposed first by El Karoui et al. (1997).

Abstract: In the talk we first introduce the level set equation for the evolution by mean curvature flow, explaining the main difference between the standard Euclidean case and the horizontal evolution.

Then we will introduce a stochastic representation formula for the viscosity solution of the level set equation related to the value function of suitable associated stochastic controlled ODEs which are motivated by a concept of intrinsic Brownian motion in Carnot-Caratheodory spaces.

Abstract: We consider an implicit finite difference
scheme on uniform grids in time and space for the Cauchy problem for a second
order parabolic stochastic partial differential equation where the parabolicity
condition is allowed to degenerate. Such equations arise in the nonlinear
filtering theory of partially observable diffusion processes. We show that the
convergence of the spatial approximation can be accelerated to an arbitrarily
high order, under suitable regularity assumptions, by applying an extrapolation
technique.

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount
of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly,but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which
means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ``dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ``spine'' decomposition for the fragmentation, and Markov renewal theory.

This is joint work with Bénédicte Haas (Paris-Dauphine).

The Schramm-Loewner evolution (SLE(\kappa)) is a family of random fractal curves that arise in a natural way as scaling limits of interfaces in critical models in statistical physics. The SLE curves are constructed by solving the Loewner differential equation driven by a scaled Brownian motion. We will give an overview of some of the almost sure properties of SLE curves, concentrating in particular on properties that can be derived by studying the the geometry of growing curve locally at the tip. We will discuss a multifractual spectrum of harmonic measure at the tip, regularity in the capacity parameterization, and continuity of the curves as the \kappa-parameter is varied while the driving Brownian motion sample is kept fixed.

This is based on joint work with Greg Lawler, and with Steffen Rohde and Carto Wong.

There are two main statistical mechanical models of ferromagnetism: the simpler and better-understood Ising model, and the more realistic and more challenging classical Heisenberg model, where the spins are in the 2-sphere instead of in {-1,+1}. In dimensions one and two, the classical Heisenberg model with nearest-neighbor interactions has no phase transition, but in three dimensions it has been intractable.

To shed some light on the qualitative behavior of the 3D Heisenberg model, we use the versatile tools of mean-field theory and Stein's method in recent work with Elizabeth Meckes, studying the Heisenberg model on a complete graph with the number of vertices going to infinity. Our results include detailed descriptions of the magnetization, the empirical spin distribution, the free energy, and a second-order phase transition.

abstract:In this talk, we describe how to compute the critical point for various lattice models of planar statistical physics. We will first introduce the percolation, Ising, Potts and random-cluster models on the square lattice. Then, we will discuss how critical points of these different models are related. In a final part, we will compute the critical point of these models. This last part harnesses two main ingredients that we will describe in details: duality and sharp threshold theorems. No background is necessary.

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma

conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. We show that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure. The mixing measure is interpreted as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer.

This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010).

(Joint work with Pierre Tarrès.)

We will give a quick overview of the semigroup perspective on splitting schemes for S(P)DEs which present a robust, "easy to implement" numerical method for calculating the expected value of a certain payoff of a stochastic process driven by a S(P)DE. Having a high numerical order of convergence enables us to replace the Monte Carlo integration technique by alternative, faster techniques. The numerical order of splitting schemes for S(P)DEs is bounded by 2. The technique of combining several splittings using linear combinations which kills some additional terms in the error expansion and thus raises the order of the numerical method is called the extrapolation. In the presentation we will focus on a special extrapolation of the Lie-Trotter splitting: the symmetrically weighted sequential splitting, and its subsequent extrapolations. Using the semigroup technique their convergence will be investigated. At the end several applications to the S(P)DEs will be given.

Based on ideas from rough path analysis and operator splitting, the Kusuoka-Lyons-Victoir scheme provides a family of higher order methods for the weak approximation of stochastic differential equations. Out of this family, the Ninomiya-Victoir method is especially simple to implement and to adjust to various different models. We give some examples of models used in financial engineering and comment on the performance of the Ninomiya-Victoir scheme and some modifications when applied to these models.

Pathwise Holder convergence with optimal rates is proved for the implicit Euler scheme associated with semilinear stochastic evolution equations with multiplicative noise. The results are applied to a class of second order parabolic SPDEs driven by space-time white noise. This is joint work with Sonja Cox.

"The model of Vertex Reinforced Random Walk (VRRW) on Z goes back to Pemantle & Volkov, '99, who proved a result of localization on 5 sites with positive probability. They also conjectured that this was the a.s. behavior of the walk. In 2004, Tarrès managed to prove this conjecture. Then in 2006, inspired by Davis'paper '90 on the edge reinforced version of the model, Volkov studied VRRW with weight on Z. 

He proved that in the strongly reinforced case, i.e. when the weight sequence is reciprocally summable, the walk localizes a.s. on 2 sites, as expected. He also proved that localization is a.s. not possible for weights growing sublinearly, but like a power of n. However, the question of localization remained open for other weights, like n*log n or n/log n, for instance. In the talk I will first review these results and formulate more precisely the open questions. Then I will present some recent results giving partial answers. This is based on joint (partly still on-going) work with Anne-Laure Basdevant and Arvind Singh."

We find the order of growth of the typical number of components of zero sets of smooth random functions of several real variables. This might be thought as a statistical version of the (first half of) 16th Hilbert problem. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution.

Joint work with Fedor Nazarov.

It is well known that electrical resistance arguments provide (usually) the best method for determining whether a graph is transient or recurrent. In this talk I will discuss a similar characterization of 'sub-diffusive behaviour' -- this occurs in spaces with many obstacles or traps.

The characterization is in terms of the energy of functions in annuli.

We consider the approximation of the marginal distribution of solutions of stochastic partial differential equations by splitting schemes. We introduce a functional analytic framework based on weighted spaces where the Feller condition generalises. This allows us to apply the theory of strongly continuous semigroups. The possibility of achieving higher orders of convergence through cubature approximations is discussed.

Applications of these results to problems from mathematical finance (the Heath-Jarrow-Morton equation of interest rate theory) and fluid dynamics (the stochastic Navier-Stokes equations) are considered. Numerical experiments using Quasi-Monte Carlo simulation confirm the practicality of our algorithms.

Parts of this work are joint with J. Teichmann and D. Veluscek.

I will discuss properties of stochastic differential equations and numerical algorithms for sampling Gibbs (i.e smooth) measures. Methods such as Langevin dynamics are reliable and well-studied performers for molecular sampling.   I will show that, when the objective of simulation is sampling of the configurational distribution, it is possible to obtain a superconvergence result (an unexpected increase in order of accuracy) for the invariant distribution.   I will also describe an application of thermostats to the Hamiltonian vortex method in which the energetic interactions with a bath of weak vortices are treated as thermal fluctuations

This is joint work with P. Caputo and D. Chafai. In this talk, we
will consider various probability distributions on the set of stochastic
 matrices with n states and on the set of Laplacian/Kirchhoff
matrices on n states. They will arise naturally from the conductance model on
n states with i.i.d conductances. With the help of random matrix
theory, we will study the spectrum of these processes.

To study turbulence,B. Mandelbrot introduced a random fractal which is called

now Mandelbrot percolation or fractal percolation. The construction is as follows:

given an integer M _ 2 and a probability 0

This talk is devoted to Talagrand's transport-entropy inequality and its deep connections to the concentration of measure phenomenon, large deviation theory and logarithmic Sobolev inequalities. After an introductive part on the field, I will present recent results obtained with P-M Samson and C. Roberto establishing the equivalence of Talagrand's inequality to a restricted version of the Log-Sobolev inequality. If time enables, I will also present some works in progress about transport inequalities in a discrete setting.

Abstract: First we provide a survey on the long-time behaviour of stochastic delay equations with bounded memory, addressing existence and uniqueness of invariant measures, Lyapunov spectra, and exponential growth rates.

Then, we study the very simple one-dimensional equation $dX(t)=X(t-1)dW(t)$ in more detail and establish the existence of a deterministic exponential growth rate of a suitable norm of the solution via a Furstenberg-Hasminskii-type formula.

Parts of the talk are based on joint work with Martin Hairer and Jonathan Mattingly. 

 Abstract:  In this talk we will introduce a new particle approximation scheme to solve the stochastic filtering problem. This new scheme makes use of the Kusuoka-Lyons-Victoir (KLV) method to approximate the dynamics of the signal. In order to control the computational cost, a partial sampling procedure based on the tree based branching algorithm (TBBA) is performed. The novelty of the method lies in the fact that the weights used in the TBBA are computed combining the cubature weights and the filtering weights. In this way, we can avoid the sample degeneracy problem inherent to particle filters. We will also present some simulations showing the performance of the method.

It is a general belief that the heat kernels on fractals should exhibit highly oscillatory behaviors as opposed to the classical case of Riemannian manifolds.

For example, on a class of finitely ramified fractals, called (affine) nested fractals, a canonical ``Brownian motion" has been constructed and its transition density (heat kernel) $p_{t}(x,y)$ satisfies $c_{1} \leq t^{d_{s}/2} p_{t}(x,x) \leq c_{2}$ for $t \leq 1$ for any point $x$ of the fractal; here $d_{s}$ is the so-called spectral dimension. Then it is natural to ask whether the limit of this quantity as $t$ goes to 0 exists or not, and it has been conjectured NOT to exist by many people.

In this talk, I will present partial affirmative answers to this conjecture. First, for a general (affine) nested fractal, the non-existence of the limit is shown to be true for a ``generic" (in particular, almost every) point. Secondly, the same is shown to be valid for ANY point of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).

Abstract : The question adressed in this talk is the following one : how are the extreme eigenvalues of a matrix X moved by a small rank perturbation P of X ?
We shall consider this question in its generic apporach, i.e. when the matrices X and P are chosen at random independently and in isotropic ways.
We shall give a general answer, uncovering a remarkable phase transition phenomenon: the limit of the extreme eigenvalues of the perturbed matrix differs from the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. We also examine the consequences of this eigenvalue phase transition on the associated eigenvectors and generalize our results to examine the case of multiplicative perturbations or of additive perturbations for the singular values of rectangular matrices.