Probability Seminar, Division of Applied Mathematics, Brown University

Schedule for Fall 2009:

• Speaker: Konstantinos Spiliopoulos, Division of Applied Mathematics, Brown University, Tuesday 15 Sep 2009
  Title: Reaction Diffusion Equations with Nonlinear Boundary Conditions in Narrow Domains and Wave Front Propagation
  
  Absract: We will consider the second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions. Using probabilistic methods we show that the solution of such a problem converges as $\epsilon \downarrow 0$ to the solution of a standard reaction-diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front. In addition, an important and interesting problem, which is related to the previous one, is the Wiener process with instantaneous reflection in a narrow tube which, in contrast to before, is assumed to be non-smooth asymptotically.



• Speaker: Martin Hairer, Courant Institute New York University, Tuesday 6 October 2009
  Title: Ergodicity of Markov processes: A marriage of topology and measure theory
  
  Abstract: One very widely used criterion in the theory of Markov chains states that if a Markov operator has the strong Feller property and is topologically irreducible, then it can have at most one invariant measure. While this criterion is very useful in finite-dimensional situations, it fails for many infinite-dimensional problems. In this talk, we will present two different generalisations of the strong Feller property that can be applied to a much larger class of problems. These include semilinear parabolic stochastic PDEs, stochastic delay equations, and diffusions driven by coloured noise.



• Speaker: Markus Fischer, Brown University, Tuesday 13 October 2009
  Title: Large deviation properties of weakly interacting Ito processes
  
  Abstract: Consider the following model for a system of N weakly interacting particles: N stochastic differential equations (SDEs) with coefficients all of the same functional form describe the state evolution of the particles; particles interact through the empirical measure of their states at any given time. In the diffusion case, it is known that the sequence of empirical measures converges, as N tends to infinity, to the weak solution of an associated McKean-Vlasov equation and that it satisfies a large deviation principle. We will present the derivation of a Laplace principle equivalent to the large deviation principle, using the Dupuis-Ellis weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type; to illustrate this, we will present the case of SDEs with delay. This is joint work with Amarjit Budhiraja and Paul Dupuis



• Speaker: Rafail Khasminskii, Department of Mathematics, Wayne State University, Friday 23 October 2009, Time: 2-3pm, Room 110, 182 George St. NOTE:Special Day and Time
  Title: METHOD OF LYAPUNOV FUNCTIONS FOR ANALYSIS OF EXTINCTION AND EXPLOSION IN MARKOV CHAINS
  
  Abstract: We derive some sufficient conditions for testing the extinction a.s.,non-extinction with positive probability, explosion and non-explosion of time-nonhomogeneous Markov chains with a countable state space. The method of Lyapunov functions is used for this purpose. Several theorems concerned with such sufficient conditions are proven for a general class of time-nonhomogeneous Markov chains. Then they are applied to some problems in the time-nonhomogeneous birth-death processes and branching Markov processes. It is joint work with P.L.Chow..



• Speaker: Richard Sowers, Department of Mathematics, University of Illinois at Urbana-Champaign, Tuesday 27 October 2009
  Title: Large Deviations and Collateralized Debt Obligations, or: How I learned to stop worrying and love rare financial events.
  
  Abstract: We discuss collateralized debt obligations and tranched pools of credit assets through the lens of large deviations theory. We find that this provides a convenient set of tools for understanding some of the complexities involved in structured financial products.



• Speaker: Jan Vecer, Department of Statistics Columbia University , Tuesday 10 November 2009
  Title: Change of Numeraire with Perspective Mapping
  
  Abstract: Typical contingent claims such as options are written on two or more underlying assets. Each of the underlying assets can be chosen as a numeraire for the purposes of pricing and hedging as long as the price of such asset is positive. This leads to at least two alternative formulations of the pricing problem, depending on the number of available reference assets with a positive price that enter a given contract. We show that the prices when expressed under different numeraires are connected by a functional relationship known as perspective mapping. This technique of computing prices under different reference assets is more general than simply computing the prices as expected discounted payoffs under the martingale measure associated with a given numeraire since it works also in situations when the reference asset does not have a corresponding martingale measure. For instance, an asset that represents the maximum price in the payoff of lookback options does not have a martingale measure, but the price of the contract with respect to the maximum can still be expressed using perspective mapping. This method applies for a general evolution of the price process. We give examples of the relationship of the pricing measures in the binomial model, the diffusion model, and the L\'evy jump model. We give two formulations of the pricing problem for European and American options, and three formulations of the problem for exotic options such as quantos, lookbacks, or Asians. In diffusion models, we obtain partial differential equations that correspond to the pricing problem.



• Speaker: Mark Kelbert, Physical Sciences, Mathematics Department, Swansea University, England , Tuesday 17 November 2009
  Title: Probabilistic representations for solutions of higher-order elliptic equations
  
  Abstract: Consider the so-called Lauricella problem with an integer $m\geq 1$ $$ \begin{array}{lcll} (\frac{1}{2}\Delta+V)^m u &=& 0,&\quad x\in D,\smallskip\\ (\frac{1}{2}\Delta+V)^{k} u&=&(-1)^{k}g_{k},&\quad x\in\partial D. \end{array} \eqno (1) $$ where $k=0,\dots, m-1$, $(\frac{1}{2}\Delta+V)^{0}:=id$, $D\subset {\bf R}^d$ is a bounded domain with a regular boundary $\partial D$, $V\in C(\overline{D})$ and $g_k\in C(\partial {D})$. Let $\lambda_1=\lambda_1(D,V)$ denote the principle Dirichlet eigenvalue of $-\frac{1}{2}\Delta-V$ in $D$. Under condition $\lambda_1>0$ we prove existence, uniqueness and different bounds for solution of problem (1). The main tool is a probabilistic representation of the solution in terms of iterated stochastic integral.



• Speaker: Giovanna Nappo, Dipartimento di Matematica, Universita degli Studi LA SAPIENZA, Itali , Tuesday 24 November 2009
  Title: CONVERGENCE IN NONLINEAR FILTERING FOR STOCHASTIC DELAY SYSTEMS
  
  Abstract: We study an approximation scheme for a nonlinear filtering problem when the (unobservable) state process {X(t)} is the solution of a stochastic delay diffusion equation, and the observation process {Y(t)} is a noisy function of the segment process, i.e. of X(s) for s in [t-r,t], where r is a constant. The rate of convergence of this scheme is computed when the approximating state is the piecewise linear Euler-Maruyama scheme, and the observation process is a noisy function of (the piecewise constant segment of) the approximating state. The proof is based on a general technique, which can be used also for other classes of partially observable systems, and on upper bounds (extending previously known ones) for the moments of the modulus of continuity of an Ito process. Based on joint works with A. Calzolari, P. Florchinger, and M. Fischer.



• Speaker: Harold Kushner, Division of Applied Mathematics, Brown University , Tuesday 1 December 2009
  Title: Large deviations for two-time-scale systems
  
  Abstract: We consider the problem of large deviations for two-time-scale reflected diffusion processes, possibly with delays in the dynamics, via the weak convergence approach. The main difficulty is in characterizing the limit equation and we try to resolve this problem..

Schedule for Spring 2010:

• Speaker: Francois Delarue, Laboratoire J.A. Dieudonne, Universite de Nice Sophia-Antipolis, France , Thursday 14 January 2010 Time: 10-11 NOTE:Special Day and Time
  
  Title: Density Estimates for a Random Noise Propagating through a Chain of Differential Equations
  
  Absract: We here provide two sided bounds for the density of the solution of a system of n differential equations, the first one being forced by a non-degenerate random noise and the n-1 other ones being degenerate. The system formed by the n equations satisfies a suitable Hormander condition: the second equation feels the noise plugged into the first equation, the third equation feels the noise transmitted from the first to the second equation and so on..., so that the noise propagates one way through the system. When the coefficients of the system are Lipschitz continuous, we show that the densityof the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem



• Speaker: J. Song University of Kansas (Lawrence), Tuesday 26 January 2010 Time: 10-11 NOTE:Special Time
  
  Title: Fractional martingales and characterization of the fractional Brownian motion
  
  Absract: We introduce the notion of fractional martingale as the fractional derivative of order \alpha of a continuous local martingale, where \alpha \in (-1/2,1/2), and we show that it has a nonzero finite variation of order 2/(1+2\alpha) , under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Levy???????????i???o?????????i???????????i???s characterization theorem for the fractional Brownian motion.



• Speaker: Kostas Kardaras, Department of Mathematics and Statistics, Boston University , Tuesday 2 February 2010
  
  Title: Pricing and hedging barrier options in diffusion models via 3-Dimensional Bessel bridges
  
  Absract: Due to the discontinuous payoff of barrier options, finite difference methods typically lead to large error for the price function and spatial derivatives near expiry date and the barrier. Furthermore, usual Monte-Carlo estimators for their price and sensitivities typically have significant variance. In this work, we consider alternative representations for barrier option prices in terms of the 3-dimensional Bessel bridge, and show how this leads to better estimators, especially for short maturities where we are able to increase the estimator efficiency dramatically. We also discuss the related problem of efficient estimation of the density of first-passage times for diffusions. Even though the density estimation problem is essentially non-parametric, our method achieves (the typical Monte-Carlo) square-root order of convergence.



• Speaker: Philip Protter, School of Operations Research and Information Engineering, Cornell University , Thursday 11 February 2010 NOTE:Cancelled due to weather
  
  Title: The Mathematical Modeling of Financial Bubbles
  
  Absract: Despite Ben Bernanke's recent remark to the effect that it is hard to tell if one is in a bubble or not in real time, over the last 10 years there has been much progress in the mathematical modeling, and at least theoretical detection in real time, of financial bubbles. We review the recent progress in the mathematical modeling of financial bubbles.



• Speaker: Michael Tretyakov, Department of Mathematics, University of Leicester, England , Tuesday 16 February 2010
  
  Title: Geometric Integrators for Diffusions
  
  Absract: In many applications (and especially in molecular dynamics) it is important to effectively compute means of a given function with respect to invariant laws of diffusions, i.e. ergodic limits. To evaluate such ergodic limits, one faces a number of computational challenges since systems of interest are usually very large and have to be integrated over very long time intervals. In such situations geometric integrators play an important role as they allow us to simulate dynamical systems on long time intervals with high accuracy. We illustrate construction and use of geometric integrators in the case of Langevin thermostats for rigid body dynamics. Computational errors arising in evaluating ergodic limits are also discussed. The talk is based on joint works with G.N. Milstein, Yu.M. Repin, R.L. Davidchack, and R. Handel.



• Speaker: Carl Mueller, Department of Mathematics, University of Rochester , Tuesday 2 March 2010
  
  Title: Nonuniqueness for some stochastic PDE
  
  Absract: The superprocess is one of the most widely studied models in probability. It arises as a limit of population processes which depend on space as well as time. One long-standing question involves the uniqueness of the stochastic PDE which describes the superprocess. Due to randomness, standard results about uniqueness of PDE do not apply. We will describe joint work with Barlow, Mytnik, and Perkins, in which we prove nonuniqueness for the equation describing the superprocess. Our results generalize to several related equations.



• Speaker: Weiqing Ren, Courant Institute of Mathematical Sciences New York University , Tuesday 6 April 2010
  
  Title: The string method for computing transition pathways in complex systems
  
  Absract: Many problems in material sciences, physics, chemistry and biology can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. The system is confined in metastable states for long times before making transitions from one metastable state to another. The disparity of time scales makes the study of transition pathways and transition rates a very challenging task. The string method developed by E, Ren and Vanden-Eijnden is an efficient way of identifying transition mechanisms and transition rates between metastable states in systems with complex energy landscapes. In this talk, I will discuss the string method and its applications to micromagnetics and the isomerization of alanine dipeptide.



• Speaker: Sergey Lototsky, University of Southern California , Tuesday 20 April 2010
  
  Title: Parameter Estimation in Stochastic Equations That are Second-Order in Time
  
  Absract: Consider the following problems:
  1. Given an undamped harmonic oscillator driven by additive Gaussian white noise, estimate oscillator's frequency from the observations of the oscillations;
  2. Given an undamped wave equation driven by additive space-time white noise, estimate the propagation speed from the observations of the solution.
  It turns out that the the first (one-dimensional) problem is not necessarily easier to study than the second (infinite-dimensional) problem. The objective of the talk is to study the asymptotic properties of the maximum likelihood estimator in both problems and to discuss various generalizations.



• Speaker: Hidehiro Kaise (Graduate School of Information Science, Nagoya University) , Thursday 22 April 2010, 11-12, NOTE:Special Day-Time
  
  Title: Partially observed H-infinity control with maximum running cost
  
  Absract: H-infinity control is a robust control theory where problems can be formulated as zero-sum dynamic games between the controller and the antagonistic deterministic disturbance. It is known that partially observed H-infinity control with an integral running cost can be reduced to a perfectly observed differential game with the information state, which is an infinite-dimensional sufficient statistics for the state of the system. In this talk, we consider an H-infinity control problem with a maximum running cost under partial observations. We first reduce the problem to a perfectly observed infinite-dimensional differential game with a maximum running cost of the information state. Then, by using dynamic programming on the information state and introducing a viscosity notion in an infinite-dimensional space, we show that the value function of the differential game is a viscosity solution of the dynamic programming partial differential equation of a quasivariational type in the infinite-dimensional space.



• Speaker: Isaac M. Sonin, Dept. of Mathematics and Statistics,Univ. of North Carolina at Charlotte , Thursday 22 April 2010, 4:15-5:15pm, ROOM: 110, NOTE:Special Day-Time
  
  Title: Gittins Index and Related Optimization Problems
  
  Absract: The celebrated Gittins index, its generalizations and related techniques play an im- portant role in applied probability models, resource allocation problems, optimal portfolio management problems as well as other problems of nancial mathematics. It is well known that 1) a connection exists between the Ratio (cycle) maximization problem, the Kathehakis-Veinot (KV) Restart Problem and the Whittle family of Retire- ment Problems, and 2) that their key characteristics, the classical Gittins index, the KV index, and the Whittle index are equal. These indices were generalized by the author (Statistics and Probability Letters, 2008) in such a way that it is possible to use the so called State Elimination algorithm, developed earlier to solve the problem of Optimal Stopping of Markov Chains to calculate this common index : The main goal of this talk is to demonstrate that the equality of these indices is a special case of a similar equality for three simple abstract optimization problems. By an abstract optimization problem we mean a problem with maximization over an abstract set of indices U without any specics about this set.

Schedule for Fall 2010:

• Speaker: Patrick Dondl, University of Bonn Tuesday 7 Sep 2010
  Title: Pinning and depinning of interfaces in random media
  
  Absract: We consider the evolution of an interface, modeled by a parabolic equation, in a random environment. The randomness is given by a distribution of smooth obstacles of random strength. To provide a barrier for the moving interface, we construct a positive, steady state supersolution. This construction depends on the existence, after rescaling, of a Lipschitz hypersurface separating the domain into a top and a bottom part, consisting of boxes that contain at least one obstacle of sufficient strength. We prove this percolation result.
  Furthermore, we examine the existence of a solution propagating with positive velocity in a random field with non-bounded random obstacle strength.
  This work shows the emergence of a rate independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment.
  Joint work with N. Dirr (Bath University) and M. Scheutzow (TU Berlin).



• Speaker: Reuven Rubinstein, Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Haifa, Israel Tuesday 14 Sep 2010
  Title: Stochastic Enumeration Method for Self-Avoiding Walks
  
  Absract: We present a new method for counting self-avoiding walks (SAW's), called the stochastic enumeration (SE) method. SE presents a natural generalization of the classic sequential importance sampling method for SAW's, called one-step-look-ahead (OSLA). We discuss the convergency properties of SE and present numerical studies demonstrating its superiority as compared to OSLA and the classic splitting method.



• Speaker: Konstantinos Spiliopoulos, Division of Applied Mathematics, Brown University, Tuesday 28 Sep 2010
  Title: Large Deviations and Fast Simulation for Multiscale Diffusions and Rough Energy Landscapes
  
  Absract: We discuss the large deviations principle and the problem of designing asymptotically optimal importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. There are three possible regimes depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter. We use weak convergence methods which provide us with convenient representations for the action functional for all three regimes, and then we use these representations to study their similarities and differences. Furthermore, we derive a control that nearly achieves the large deviations lower bound at the prelimit level. This control is useful for designing efficient importance sampling schemes. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit, and apart from the smallness of the noise, an additional reason for this is the presence of the fast oscillating coefficients. These results have applications in chemical physics and biology. Examples will be provided.
  Joint work with P. Dupuis and H. Wang.



• Speaker: Xiang Zhou, Division of Applied Mathematics, Brown University, Tuesday 12 Oct 2010
  Title: Noise-induced transition pathway in non-gradient systems
  
  Absract: The dynamical systems perturbed by weak noise have profound transition behaviors at long time scale. Such rare events has exponentially small probability but critically contribute to lots of interesting physical phenomena, such as phase transition, subcritical instability, etc. This talk mainly focuses on the transition scenarios in non-gradient systems where no energy landscape exists. Our approach is based on the framework of the large deviation principle (Freidlin-Wentzell theory). The applications include the stochastic Lorentz model, Kuramoto-Sivashinsky PDE, etc. If time permits, the attack of classic problem of turbulent transition, a subcritical bifurcation, will be discussed as one of the most important applications.
  These are joint works with Weinan E, Xiaoliang Wan, Weiqing Ren.



• Speaker: Sunil Chhita, Department of Mathematics, Brown University, Tuesday 19 Oct 2010
  Title: Particle Systems arising from an Anti-Ferromagnetic Ising Model
  
  Absract: We present a low temperature anisotropic anti-ferromagnetic 2D Ising model through the guise of a certain dimer model. This model also has a bijection with a one-dimensional particle system equipped with creation and annihilation. In the thermodynamic limit, we give the phase diagrams, which determines two significant values (the independent and critical value). At independence, we show that the scaling window measure for the particle system is a Continuum Noisy Voter Model. We also give results for the behavior in the scaling window at criticality, such as the stationary distribution of the particles.



• Speaker: Harold Kushner, Division of Applied Mathematics, Brown University , Tuesday 2 November 2010
  Title: Controlled Stochastic Delay Equations: Modeling and Approximations
  
  Absract: The modeling of general controlled stochastic delay equations requires some non classical approaches to basic issues, such as the structure of the class of admissible controls, extensions of the notion of relaxed controls, limits of sequences of controlled delayed Poisson measures, and methods of approximation of controls, as well as numerical procedures. The problems and methods of solutions will be outlined.



• Speaker: Jeremy Quastel, Department of Mathematics, University of Toronto, Canada , Tuesday 9 November 2010
  Title: Random growth and directed polymers
  
  Absract: About 30 years ago it was observed by physicists that a large class of one dimensional random systems (the KPZ universality class) have highly anomalous, but universal, fluctuation behaviour. We will describe some of the models, and survey recent progress in proving some of the conjectured scalings and limiting distributions for the Kardar-Parisi-Zhang equation itself, and the associated continuum random polymer.



• Speaker: Hyejin Kim, Department of Mathematics, University of Michigan , Tuesday 23 November 2010
  Title: Weak convergence of one dimensional Markov processes governed by the W. Feller generalized second order differential operator in closed intervals.
  
  Absract: In case of one spatial variable, we consider the question of the weakest possible topology, which provides convergence of the solutions. Convergence of solutions of initial-boundary value problems is equivalent to weak convergence of corresponding diffusion processes. In general, the infinitesimal generator of limiting processes need not to be a classical second order elliptic differential operator. Continuous Markov processes corresponding to the generalized second order differential operators introduced by W. Feller can appear as limiting processes. We provide necessary and sufficient conditions that guarantee the convergence of solutions of initial-boundary value problems for parabolic equations.



• Speaker: Chia Ying Lee, Division of Applied Mathematics, Brown University , Tuesday 30 November 2010
  Title: Randomization of Forcing in Large Systems of PDEs for Improvement of Energy Estimates.
  
  Absract: We consider a class of stochastic PDEs (SPDEs) driven by purely spatial white noise, for which the numerical computation of the energy is desired. Our paper compares the efficiency of two different bases of expansion of white noise, one of a local scale and the other of a ???large scale,?????i??¨¨ for approximating the energy of the SPDE, and we will show that the latter basis dramatically improves the approximation of the energy. Such problems with a local scale basis arise in applications such as electromagnetic wave propagation with incoherent sources, but current approaches to computing the energy have found a roadblock in the sheer size of the problem. Thus, knowledge of the improved efficiency of a large scale basis becomes useful in vastly reducing computational cost while attaining highly accurate approximations of the energy.
  This is joint work with Professors Boris Rozovsky and Hao Min Zhou.



• Speaker: Svetlana Anulova, Russian Academy of Sciences, Institute of Problems in Control , Tuesday 7 December 2010 NOTE:Cancelled
  Title: A Multidimensional Comparison Theorem for Solutions of the Skorokhod Problem in a Wedge with Applications to Control of a Group of Independent Identical Agents.
  
  Absract: A problem of controlling a group of independent identical agents is considered. Such problems arise in mathematical economics (MacKean --- Shepp), in robotics (swarms of mini-robots). The behavior of an agent is described by a Wiener process with a controlled drift term. The principal setup. A multidimensional BM with a controlled drift in an orthant (the sum of drift coordinates is bounded from above). Criterion: maximize the boundary hitting time.
  The general method of the Bellman equation (Krylov) cannot be applied to this problem. The problem is reduced to a comparison theorem for reflected processes in a wedge (a powerful generalization of the classic theorem, cf. Ikeda + Watanabe book). The solution is found by the comparison method for solutions of stochastic differential equations using the technique of the Skorokhod reflection operators. The optimal policy admits a realization as a decentralized group control algorithm.

Schedule for Spring 2011:

• Speaker: Frederick E. Daum, Raytheon, MA, Tuesday 1 February 2011 Time: 11-12
  
  Title: Exact particle flow for nonlinear filters
  
  Absract: We have solved the well known and important problem of "particle degeneracy" using a new theory, called particle flow. Our filter is four orders of magnitude faster than standard particle filters for any given number of particles, and we required many orders of magnitude fewer particles to achieve the same filter accuracy. Our filter beats the EKF accuracy by several orders of magnitude for difficult nonlinear problems. Our theory uses exact particle flow to compute Bayes?? rule, rather than a pointwise multiply. We do not use resampling or proposal densities or importance sampling or any MCMC method. But rather, we design the particle flow with the solution of a linear first order highly underdetermined PDE, like the Gauss divergence law in electromagnetics. We study over a dozen methods to solve this PDE, including: (1) irrotational flow (i.e., the gradient of the solution of Poisson's equation); (2) incompressible flow; (3) optimal control; (4) Gauss' and Hertz' variational method; (5) direct integration in terms of d-1 arbitrary functions; (6) complete integrals (i.e., in terms of d arbitrary constants); (7) Fourier transform of the divergence form of our PDE; (8) Gromov's h-principle; (9) generalized inverse of linear differential operator; (10) separation of variables; (11) generalized method of characteristics; (12) generalized Jacobi's method, and hybrids of the above.



• Speaker: Jingchen Liu, Department of Statistics, Columbia University , Tuesday 15 February 2011 Time: 11-12
  
  Title: Some Asymptotic Results and Computational Methods of Gaussian Random Fields
  
  Absract: Gaussian processes and multivariate Gaussian random vectors constitute a cornerstone of probability models in many disciplines. In this talk, we focus on the extreme behavior of Gaussian processes and their applications. In particular, we consider the tails of random variables that can be written as convex functionals of a Gaussian process including suprema and integrals of convex functions. We present asymptotic approximations of the tail probabilities and the density functions of such random variables and develop efficient Monte Carlo algorithms to compute them. The emphasis of this talk is the usage of change-of-measure-based techniques in the analysis of extreme behavior of random fields.



• Speaker: Ramon van Handel, Operations Research & Financial Engineering, Princeton University , Thursday 24 February 2011 Time: 11-12 NOTE:Special Day
  
  Title: The ergodic theory of nonlinear filters.
  
  Absract: The goal of nonlinear filtering is to estimate the state of a Markov process given noisy and incomplete observations. In many applications, one is interested in understanding the performance of the nonlinear filter and related numerical algorithms over a long time horizon, which requires an understanding of the ergodic theory of nonlinear filters. Such a theory was first developed in a classic paper of H. Kunita (1971). Unfortunately, the key part of the proof in this paper contains a serious measure theoretic error, which lies at the heart of the ergodicity problem.
  In this talk, I will discuss recent progress in understanding the stability theory of nonlinear filters. I will introduce the central measure theoretic identity and outline its proof under very general assumptions by means of the theory of Markov chains in random environments. I will also discuss two surprising counterexamples where the filter fails to be ergodic. The upshot is that the ergodicity of classical nonlinear filtering problems is now largely resolved, but nonlinear filtering problems with infinite dimensional signals (such as those appearing in weather prediction or data assimilation) remain a mystery.



• Speaker: Toshio Mikami, Hiroshima University, Japan , Thursday 24 March 2011 Time: 11-12 NOTE:Special Day
  
  Title: Particle transport by marginal problem of Stochastic control.
  
  Absract: The mass or particle transport in quantum mechanics by the variational problem was first considered by Schrodinger. This idea was developed in Nelson's stochastic quantization by the theory of stochastic control by many authors. We have been developing this idea to include the optimal mass transportation theory. In this talk we introduce our result obtained in the last several years: zero noise limit of h-path processes, duality theorem, marginal problem of stochastic processes, etc.



• Speaker: Szymon Peszat, Institute of Mathematics, Polish Academy of Sciences, Cracow, Poland , Tuesday 12 April 2011 Time: 11-12
  
  Title: L\'evy--Ornstein--Uhlenbeck transition semigroup as a second quantized operator.
  
  Absract: Let $\mu$ be an invariant measure for the transition semigroup $(P_t)$ of the Markov family defined by the Ornstein--Uhlenbeck type equation $$ \d X= AX\d t + \d L $$ on a Hilbert space $E$, driven by a L\'evy process $L$. It is shown that for any $t\ge 0$, $P_t$ considered on $L^2(\mu)$ is a second quantized operator on a Poisson Fock space of $\e ^{At}$. From this representation it follows that the transition semigroup corresponding to the equation on $E=\mathbb{R}$, driven by an $\alpha$-stable noise $L$, $\alpha\in (0,2)$, is neither compact nor symmetric.



• Speaker: Paul Dupuis, Division of Applied Mathematics, Brown University , Tuesday 19 April 2011 Time: 11-12
  
  Title: Accelerating Monte Carlo--What does the Donsker-Varadhan theory have to say?
  
  Absract: The theory of large deviations has played an important role in the development of Monte Carlo methods for estimating quantities defined in terms of a specific rare event, such as ruin probabilities or buffer overflow probabilities. In this setting the associated large deviation theory is that of small random perturbations of deterministic systems, also referred to as the Freidlin??Wentsel theory. However, rare events also play an important role when estimating functionals of an invariant distribution, where straightforward simulation will converge very slowly when parts of the state space do not communicate well. Problems of this sort are common in statistical inference, engineering and the physical sciences, and much effort has gone into the development of methods to accelerate the convergence of Monte Carlo. We will argue that the Donsker??Varadhan theory, which describes the large deviation properties of the empirical measure of a Markov chain, gives insight into the analysis of existing schemes and the design of new methods for this class of problems.



• Speaker: Sergey Lototsky, University of Southern California , Tuesday 19 April 2011: NOTE TIME:4-5 in 37 MANNING
  
  Title: Wick product in the stochastic Burgers equation: a curse or a cure?
  
  Absract: It has been known for a while that a nonlinear equation driven by singular noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form, whence the cure. The analysis is based on the study of the coefficients of the chaos expansion of the solution at different stochastic scales. This is joint work with Sivaditya Kaligotla.



• Speaker: Arnaud Debussche, D?§|partement de Math?§|matiques de l'ENS Cachan Bretagne , Tuesday 26 April 2011 Time: 11-12
  
  Title: The 3D stochastic Navier-Stokes equations: existence of markov semigroup and density of solutions.
  
  Absract: We first give a brief overview of a series of results on the construction of Markov semigroup for the 3D Navier-Stokes equation. Since uniqueness is an open problem, the construction is not obvious. We show that under appropriate conditions on the non degeneracy of the noise, it is possible to do this in a constructive way and obtain a transition semigroup which is (and even strong) Feller. This is the result of joint works with G. Da Prato and C. Odasso. Exponential convergence to equilibrium follows from an argument due to C. Odasso.
  Then we consider physically more realistic noises which do not satisfy the above non degeneracy condition and prove that the projections of the distribution onto finite dimensional spaces of any (weak) solution have a density with respect to the Lebesgue measure. Since Malliavin calculus does not seem to be applicable, we use and extend a recent idea due to N. Fournier and J. Printems. This is a joint work with M. Romito.

Schedule for Fall 2011:

• Speaker: Anastasia Papavasileiou, Department of Statistics, Warwick University, England , Tuesday September 6, 2011 Time: 11-12
  
  Title: Statistical modelling of stochastic systems through rough paths
  
  Absract: Stochastic processes modelled by differential equations driven by rough paths (RDEs) are very general. They include diffusions but also non-Markovian processes such as the ones modelled by stochastic delay equations. First, I will discuss the basic ideas of the theory of rough paths, as developed by T. Lyons. Then, I will present the methodology for performing statistical inference for such models.



• Speaker: Ruth Williams, Department of Mathematics, University of California, San Diego , Tuesday 20 September 2011 Time: 11-12
  
  Title: Control of Stochastic Processing Networks
  
  Absract: Stochastic processing networks (SPNs) are a significant generalization of conventional queueing networks that allow for flexible scheduling through dynamic sequencing and alternate routing. SPNs arise naturally in a variety of applications in operations management and their control and analysis present challenging mathematical problems. One approach to these problems, via approximate diffusion control problems, has been outlined by J. M. Harrison. Various aspects of this approach have been developed mathematically, including a reduction in dimension of the diffusion control problem. However, other aspects have been less explored, especially, solution of the diffusion control problem, derivation of policies by interpretating such solutions, and limit theorems that establish optimality of such policies in a suitable asymptotic sense.
  In this talk, for a concrete class of networks called parallel server systems which arise in service network and computer science applications, we explore previously undeveloped aspects of Harrison's scheme and illustrate the use of the approach in obtaining simple control policies that are nearly optimal. Identification of a graphical structure for the network, an invariance principle and properties of local times of reflecting Brownian motion, will feature in our analysis. The talk will conclude with a summary of the current status and description of open problems associated with the further development of control of stochastic processing networks.
  This talk will draw on aspects of joint work with M. Bramson, M. Reiman, S. Bell, W. Kang and V. Pesic..



• Speaker: Konstantinos Spiliopoulos, Division of Applied Mathematics, Brown Univeristy, Tuesday 4 October 2011 Time: 11-12
  
  Title: Default clustering in large portfolios and most likely path to failure
  
  Absract: The past several years have made clear the need to better understand the behavior of risk in large interconnected financial systems. Interconnections often make a system robust, but they can act as conduits for risk. Even things which may seemingly be unrelated may become related as risk restrictions may, for example, force a sale of one type of well-performing asset to compensate for the poor behavior of another asset. The financial crises of the past several years is replete with such interactions.
  In this talk, I will discuss a way to study how different types of risk compete and interact in a large-scale system. An empirically motivated system of interacting point processes modeling the dynamics of correlated default events in the financial market is introduced. In the model, a name defaults at a stochastic intensity that is influenced by an idiosyncratic risk process, a systematic risk process common to all names, and past defaults. A law of large numbers for the loss from default is proven and then used for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. The density of the limiting measure is shown to solve a non-linear SPDE, and the moments of the limiting measure are shown to satisfy an infinite system of SDEs. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Large deviation arguments are then used to identify the way that atypically large (i.e., ``rare'') default clusters are most likely to occur. The results give insights into how different sources of default correlation interact to generate typical and atypical portfolio losses.



• Speaker: Di Liu (Richard), Depatment of Mathematics, Michigan State Univeristy. , Tuesday 18 October 2011 Time: 11-12
  
  Title: Numerical methods for stochastic bio-chemical reacting networks with multiple time scales
  
  Absract: Multiscale and stochastic approaches play a crucial role in faithfully capturing the dynamical features and making insightful predictions of cellular reacting systems involving gene expression. Despite their accuracy, the standard stochastic simulation algorithms are necessarily inefficient for most of the realistic problems with a multiscale nature characterized by multiple time scales induced by widely disparate reactions rates. In this talk, I will discuss some recent progress on using asymptotic techniques for probability theory to simplify the complex networks and help to design efficient numerical schemes.



• Speaker: Jianfeng Lu, Courant Institute of Mathematical Sciences,New York University , Tuesday 1 Movember 2011 Time: 11-12, NOTE: This seminar will take place at 37 Manning
  
  Title: Diffusion complexes and geometry of energy landscape
  
  Absract: We will discuss some new perspectives from geometry on understanding activated processes and complex networks. For diffusion processes, we introduce the diffusion complexes. They provide geometric information about transition paths and transition interfaces for overdamped dynamics on smooth potential surface. On the discrete level, we extend the geometric perspective to Markov chains to explore complex networks, from social networks to transition networks in protein folding.



• Speaker: David Gamarnik, MIT Sloan School of Management , Tuesday 8 Movember 2011 Time: 11-12
  
  Title: Correlation decay property and inference in Markov Random Fields.
  
  Absract: Loosely speaking, a stochastic system exhibits the correlation decay property if correlations between components of the system decay as a function of the distance between the components. The notion appears primarily in the context of Gibbs measures in statistical physics and Markov random field inference, but also appears to have interesting algorithmic applications.
  We illustrate how the correlation decay property can be used for designing a polynomial time deterministic approximation algorithm, which we call cavity expansion algorithm, for the problem of computing the partition function of a Markov random field. Prior algorithms for this problem rely primarily on the Monte Carlo simulation technique and thus suffer from the sampling error. One can view the cavity expansion algorithm as a corrected Belief Propagation algorithm applied to graphs with loops. We will illustrate our method on a stylized Markov random field model: counting partial matchings in a graph - a well known #P hard problem. Then we will demonstrate the practicality of our approach for the problem on inference on some lattice models. Specifically, we compute the entropy of monomer-dimer coverings of a lattice, improving earlier estimations by several orders of magnitude.



• Speaker: Natesh Pillai, Department of Statistics, Harvard University , Tuesday 15 November 2011 Time: 11-12
  
  Title: Optimal Scaling of MCMC algorithms.
  
  Absract: MCMC (Markov Chain Monte Carlo) algorithms are an extremely powerful set of tools for sampling from complex probability distributions. Understanding and quantifying their behavior in high dimensions thus constitute an essential part of modern statistical inference. In this regard, most of the research efforts so far were focussed on obtaining estimates for the mixing times of the corresponding Markov chain.
  In this talk we offer a new perspective for studying the efficiency of commonly used algorithms. We will discuss optimal scaling of MCMC algorithms in high dimensions where the key idea is to study the properties of the proposal distribution as a function of the dimension. This point of view gives us new insights on the behaviour of the algorithm, such as precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space.
  In the first part of the talk, we will describe the main ideas and discuss recent results on high dimensional target measures arising in the context of statistical inference for mathematical models representing physical phenomena. In the second part of the talk, we discuss the Hybrid Monte Carlo Algorithm (HMC) and answer a few open questions about its efficiency in high dimensions. We will also briefly discuss applications to parallel tempering, Gibbs samplers and conclude with concrete problems for future work.



• Speaker: Minh-Binh TRAN, Departement de Mathematiques, Institut Galilee, Universite Paris 13 , Tuesday 22 November 2011 Time: 11-12
  
  Title: Numerical schemes for a system of stochastic differential equations.
  
  Absract: The theory of forward–backward stochastic differential equations is a very active field of research since the first work of Pardoux and Peng and Antonelli came out in the early 1990s. These equations appear in a large number of application fields in stochastic control and financial mathematics. Such systems strongly couple a forward stochastic differential equation with a backward one. From here, there are two directions to solve this system. The first trend is to solve the system by using the decoupling technique combining with some probability methods to avoid treating the PDEs directly. The second trend is to solve directly the PDEs derived from the system. We present a new approach based on the second trend to solve this system parallely.



• Speaker: Camelia Pop, Department of Mathematics, Rutgers University , Tuesday 29 November 2011 Time: 11-12
  
  Title: Mimicking theorem for generalized Heston-like processes
  
  Absract: We consider the initial value problem on H^d, where H^d = R^{d-1} x (0,8),
  -Lu = -u_t + x_d a_{ij}(t,x) u_{x_ix_j} + b_i(t,x) u_{x_i} + c(t,x) u = f , in (0,8) x H^d, u (0, )=g, in H^d.
  The coefficients are assumed locally Holder continuous with respect to suitable metrics, the matrix (a_{ij}) is uniformly elliptic, and the coefficients may grow linearly in |x| away from the degenerate boundary, {x_d=0}. We prove existence, uniqueness, and regularity of solutions in weighted Holder spaces which incorporate both the degeneracy at {x_d=0} and the linear growth of the coefficients. We use these results to build generalized Heston processes which match the 1-dimensional marginal distributions of a certain class of Ito processes. The mimicking process is the unique weak solution to a stochastic differential equation and it possesses the strong Markov property.
  This is joint work with P. Feehan.

Schedule for Spring2012:

• Speaker: Terry Lyons, Mathematical Institute, University of Oxford, England , Thursday February 9, 2012 Time: 3:15-4:15 NOTE: DIFFERENT DAY- COLLOQUIUM TALK
  
  Title: The expected signature of a stochastic process. Some new PDE's
  
  Absract: How can one describe a probability measure of paths? And how should one approximate to this measure so as to capture the effect of this randomly evolving system. Markovian measures were efficiently described by Stroock and Varadhan through the Martingale problem. But there are many measures on paths that are not Markovian and a new tool, the expected signature provides a systematic ways of describing such measures in terms of their effects. We explain how to calculate this expected signature I the case of the measure on paths corresponding to a Brownian motion started at a point x in the open set and run until it leaves the same set. A completely new (at least to the speaker) PDE is needed to characterise this expected signature. Joint work with Ni Hao.



• Speaker: Rafail Khasminskii, Department of Mathematics, Wayne State University, Tuesday February 14, 2012 Time: 11-12
  
  Title: STABILITY OF REGIME-SWITCHING STOCHASTIC DIFFERENTIAL EQUATIONS
  
  Absract:



• Speaker: Wei Zhou, School of Mathematics, University of Minnesota, Tuesday February 21, 2012 Time: 3-4
  
  Title: A probabilistic approach to regularity of fully nonlinear degenerate elliptic equations
  
  Absract: We discuss the concept and motivation of quasi-derivatives and give an example constructed by random time change, Girsanov's theorem and Levy's theorem. Then we use this probabilistic method to investigate the regularity of the probabilistic solution of the Dirichlet problem for degenerate elliptic equations, from linear cases to fully nonlinear cases. In each Dirichlet problem we consider, the probabilistic solution is the unique solution in our setting.



• Speaker: Dr. Mikhail Neklyudov, Mathematisches Institut, Universität Tübingen, Tuesday February 28, 2012 Time: 3-4, CANCELLED
  
  Title: The role of noise in finite ensembles of nanomagnetic particles
  
  Absract: The dynamics of finitely many nanomagnetic particles is described by the stochastic Landau-Lifshitz-Gilbert equation. We show that the system relaxes exponentially fast to the unique invariant measure which is described by a Boltzmann distribution. Furthermore, we provide Arrhenius type law for the rate of the convergence to the distribution. Then, we discuss two implicit discretizations to approximate transition functions both, at finite and infinite times: the first scheme is shown to inherit the geometric `unit-length' property of single spins, as well as the Lyapunov structure, and is shown to be geometrically ergodic; moreover, iterates converge strongly with rate for finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at optimal rate for all finite times. We use a general result of Shardlow and Stuart to then conclude convergence to the invariant measure of the limiting problem for both discretizations. Computational examples will be reported to illustrate the theory. This is a joint work with A. Prohl.



• Speaker: Markos Katsoulakis, Department of Mathematics and Statistics , UMass, Amherst, Tuesday March 6, 2012 Time: 3-4
  
  Title: Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms.
  
  Absract: In this talk we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting as means of decomposing efficiently and systematically the computational load and communication between multiple processors.The spatial decomposition of the Markov operator of the KMC algorithm into a hierarchy of operators, corresponds to processors in the parallel architecture. Based on this operator decomposition, we formulate Fractional Step Approximation schemes by employing the Trotter Theorem and its random variants; these schemes, (a) are run independently on each processor through a serial KMC simulation on each fractional step time-window, giving rise to partially asynchronous algorithms, and (b) are characterized by the communication schedule between processors. Furthermore, the proposed mathematical framework allows us to rigorously justify the numerical and statistical consistency of the proposed algorithms by showing goal-orianted error estimates for quantities of interest. The approach also provides a systematic evaluation of the balance between error and processor communication in the parallel algorithms.



• Speaker: Alexei Novikov, Department of Mathematics, Penn State, Tuesday March 20, 2012 Time: 3-4
  
  Title: Exit times of diffusions with incompressible drift
  
  Absract: Consider a Brownian particle in a prescribed time-intependent incompressible flow in a bounded domain. We investigate how the strength of the flow and its geometric properties affect the expected exit time of the particle. The two main questions we analyze in this talk are as follows. 1. Incompressible flows are known to enhance mixing in many contexts, but do they also always decrease the exit time? We prove that the answer is no, unless the domain is a disk. 2. Suppose the flow is cellular with amplitude A, and the domain is of size L. What could be said about the exit time when both L and A are large? We prove that there are two characteristic regimes: a) if L << A^4, then the exit time from the entire domain is compatible with the exit time from a single flow cell, and it can be determined from the Freidlin–Wentzell theory; b) if L>> A^4, then the problem `homogenizes' and the exit time is determined by the effective diffusivity of cellular flows.



• Speaker: Wendell Fleming, Division of Applied Mathematics, Brown University, Tuesday April 3, 2012 Time: 3-4
  
  Title: Max-plus stochastic processes and control
  
  Absract: The Maslov idempotent calculus provides a framework for a variety of asymptotic problems, including large deviations for Markov diffusions described by stochastic differential equations. The asymptotic limit is described through a deterministic optimization problem. This limit retains a "stochastic" interpretation, in which expectations are linear with respect to "max-plus" addition and scalar multiplication. The first part of the lecture will discuss max-plus stochastic differential equations, with associated Hamilton-Jacobi PDEs and variational inequalities. The second part of the lecture is concerned with controlled max-plus stochastic differential equations and associated two-controller, zero sum differential games. As an example, the solution to the max-plus version of the classical Mertion optimal consumption problem is given.



• Speaker: Kevin Leder, Industrial and Systems Engineering, University of Minnesota, Tuesday April 10, 2012 Time: 3-4
  
  Title: Evolutionary Dynamics of Tumor Growth and Recurrence
  
  Absract: In the first part of my talk I will discuss a supercritical branching process model that we introduced to study to study intra-tumor diversity (i.e. the variation amongst the cells within a single tumor). In the asymptotic (t- > infinity) regime, we study growth properties of the model and in addition ecological measures of diversity. In the second part of my talk I will discuss the use of branching processes to study the timing of drug resistance mediated recurrence in treated tumor populations. In the large initial population regime I will discuss approximations to the time at which the tumor population recurs and the time at which the resistant population first dominates the total population.



• Speaker: Atilla Yilmaz, Department of Mathematics, Bogazici University (Istanbul, Turkey) , Tuesday April 24, 2012 Time: 3-4
  
  Title: Harmonic functions, h-transform and large deviations for RWRE
  
  Absract: In the context of random walk in a random environment (RWRE), large deviation principles (LDPs) are classified according to two criteria: (i) which random variable they are concerned with, and (ii) whether it is under the quenched measure (i.e., when the environment $\omega$ is frozen) or the averaged measure (i.e., when $\omega$ is integrated out). The most commonly studied random variable is the mean velocity of the walk. Such LDPs are called level-1. Level-2 refers to the empirical measures for the environment (from the point of view of the particle) and k-future steps of the walk (for some fixed $k\geq 0$). Finally, if all future steps are recorded, then the corresponding LDPs are known as level-3 (or process level). The so-called contraction principle lets us derive (and give variational formulas for the rate functions of) lower level LDPs from higher level ones. In some cases, we know that these variational formulas have unique minimizers which correspond to Markov processes whose kernels are obtained from the original RWRE kernel by a generalization of the h-transform technique of J.L. Doob, involving a tilting by harmonic functions. In this talk, I will give a survey of the cases where this connection between large deviations and h-transform has been established. The results for dimensions $d=1$, $d=2,3$ and $d\geq 4$ are rather different in flavor.



• Speaker: Russell Lyons, Department of Mathematics, Indiana University , Thursday May 17, 2012 Time: 4-5
  
  Title: Random orderings and unique ergodicity of automorphism groups
  
  Absract: Is there a natural way to put a random total ordering on the vertices of a finite graph? Natural here means that all finite graphs get an isomorphism-invariant random ordering and induced subgraphs get the random ordering that is inherited from the larger graph. Thus, the uniformly random ordering is natural; are there any others? What if we restrict to certain kinds of graphs? What about finite hypergraphs or finite metric spaces? We discuss these questions and sketch how their answers give unique ergodicity of corresponding automorphism groups; for example, in the case of graphs, the group is the automorphism group of the infinite random graph. This is joint work with Omer Angel and Alexander



• Speaker: William A. Massey Department of Operations Research and Financial Engineering Princeton University , Friday May 18, 2012 Time: 11-12 (?)
  
  Title: Gaussian Skewness Approximation for Dynamic Rate Multi-Server Queues with Abandonment
  
  Absract: The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally as staffing issues for these systems in response to predictable increasing demand. Mathematically, this type of asymptotic scaling gives us the fluid and diffusion limits. These limits suggest a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are computed from a two-dimensional dynamical system for the limiting fluid process and variance of the diffusion process. Recent work has shown that a modified version of these differential equations can be used to obtain better Gaussian estimates of the original queueing system. In this paper, we introduce a new three-dimensional dynamical system that improves on all these approaches. Using Hermite polynomials, we construct a distribution from a quadratic function of a Gaussian random variable to estimate the mean, variance, and third cumulative moment of the dynamic queueing process. This is joint work with Jamol Pender of Princeton University.