"Approximation for the Quasipotential for the 2-D Stochastic Navier-Stokes Equations and Applications to the Exit Problem"
Time: 11:00 AM
Location: Hill 705
Abstract: We are dealing with the Navier-Stokes equation in a bounded regular domain $D$ of $mathbb{R}^2$, perturbed by an additive Gaussian noise $partial w^{Q_delta}/partial t$, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $deltadownarrow 0$, so that the noise converges to the white noise in space and time. For every $delta>0$ we introduce the large deviation action functional $S^delta_{0,T}$ and the corresponding quasi-potential $V_delta$ and, by using arguments from relaxation and $Gamma$-convergence, we show that $V_delta$ converges to $V=V_0$, in spite of the fact that the Navier-Stokes equation has no meaning, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional $V$ is explicitly computed.
Finally, we apply these results to estimate the asymptotics of the expected exit time of the solution of the stochastic Navier-Stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system.
Monday, April 22nd
Yu Wang, Columbia University
"Small Perturbation Solutions of the Complex Monge-Ampére Equation"
Time: 11:00 AM
Location: Hill 705
Abstract: In this talk, we discuss the regularity of the solutions u to the complex Monge-Ampére equation that are uniformly close to quadratic polynomials. Our main tool is Savin’s small perturbation theorem. By study these solutions, we produce two regularity results for the complex Monge-Ampére equation.
The first result is a Liouville-type theorem. It considers uniqueness of the global solutions on $mathbb{C}^{n}$ with certain growth at infinity. The second result is a $C^{2,alpha}$-estimate of $W^{2,p}$-solutions. It is a perturbation result of Blocki-Dinew’s estimate of $W^{2,p}$-solutions.
Monday, April 15th
Kei Kobayashi, Tufts University
"Time-changed Stochastic Processes and Associated Fractional Order PDEs "
Time: 11:00 AM
Location: Hill 705
Abstract: It is known that the transition probabilities of a classical Brownian motion satisfy the associated forward Kolmogorov equation, which is a diffusion equation involving a first-order time derivative. In many applications, however, Kolmogorov type equations with fractional order time derivatives are employed to model "sub-diffusions," in which particles spread more slowly than the classical Brownian motion predicts. Generally speaking, stochastic processes describing such sub-diffusions are obtained by applying a time-change to classical processes such as Brownian motion and Levy processes, where the simplest time-change to be considered is the generalized inverse, or equivalently, the first hitting time process, of a stable subordinator.
In this talk, I will first introduce properties of the time-change, followed by discussions on derivations of some important classes of fractional order PDEs. A lot of interesting papers on time-changed processes have been published, some of which will be presented along with related open problems during the talk.
Monday, April 8th
Christopher Evans, University of Missouri
"A Wong-Zakai Approximation Scheme for Reflected Stochastic Differential Equations"
Time: 11:00 AM
Location: Hill 705
Abstract: In a series of famous papers E. Wong and M. Zakai showed that the solution to a Stratonovich SDE is the limit of the solutions to a corresponding ODE driven by the piecewise-linear interpolation of the driving Brownian motion. In particular, this implies that solutions to Stratonovich SDE "behave as we would expect from ODE theory". Working with my PhD adviser, Daniel Stroock, we have shown that a similar approximation result holds, in the sense of weak convergence of distributions, for reflected Stratonovich SDE.
Monday, April 1st
Olympia Hadjiliadis, Dept. of Mathematics, Graduate Center of CUNY
" Drawdowns, Last Passage time Distributions and Applications to Online Trading and Quickest Detection"
Time: 11:00 AM
Location: Hill 705
Abstract: In this work we derive analytical formulas for the joint distribution of the drawdown, the last visit time of the maximum of a process preceding the drawdown and the maximum of the process under general diffusion dynamics.
The initial motivation of this work arises in the financial risk management of drawdowns. Drawdowns measure the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level. Thus, drawdowns capture the time of distress and have as such been used as path dependent measures of risk. However, in order to encapsulate the time of distress one needs to measure the duration of time between the drawdown and the last time at which the maximum was achieved. We call this time the speed of market crash and study its distribution under general diffusion dynamics in detail. We further examine the sensitivity of the speed of market crash to the drift parameter of a drifted Brownian motion model. Our results suggest that the speed of market crash can serve as sensitive online estimator of changes in the drift which can be used in algorithmic trading. We finally discuss the connection the drawdown and its speed to the Cumulative sum statistic and its speed of reaction and in particular to the problem of optimal quickest detection and identification of a drift.
Monday, March 25th
Ruoting Gong, Rutgers University
"Small-time Asymptotics and Expansions of Option Prices under Lévy-based Models"
Time: 11:00 AM
Location: Hill 705
Abstract: This talk is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Lévy-based models. During the last decade, Lévy processes and other stochastic processes with jumps have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes. In the first part of the talk, I shall explain some basic concepts and preliminary results of Lévy processes, which are needed in the later results. I shall also provide some motivations for using Lévy processes in financial modeling. In the second part, I shall present some recent results on the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several exponential Lévy models.
Monday, March 11th
Fei Xing, University of Tennessee
" Large time asymptotics of Ornstein-Uhlenbeck process in Poisson potential Date"
Time: 11:00 AM
Location: Hill 705
Abstract: We consider the following random motion in random media (RMRM) model: An Ornstein-Uhlenbeck(O-U) process travels in d-dimensional space, where the space is occupied by homogeneous Poisson point processes as random media. The O-U process accumulates energy from the Poisson random media during the journey. Exponential moments of the accumulated energy turn out to have interesting connections in various areas, such as solutions of certain parabolic PDEs, trapping probabilities of random polymers, etc. In this talk, I would like talk about the long time asymptotics of these exponential moments, in two regimes: the quenched case and the annealed case.
Monday, February 25th
Christian Houdre, School of Mathematics, Georgia Institute of Technology
"Asymptotics for the length in some longest common and/or increasing subsequence problems"
Time: 11:00 AM
Location: Hill 705
Abstract: I will first provide a panorama of various recent and not so recent results (due to various authors) on the asymptotics, in mean, variance and limiting law, for the length of some subsequence problems. Then, I will describe a recent result in the following framework: Let $X_1, X_2,dots, X_n,dots$ and $Y_1, Y_2,dots, Y_ndots$ be two independent sequences of iid random variables taking their values in a common ordered alphabet. Let LCI$_n$ be the length of the longest common and increasing subsequence of $X_1,dots, X_n$ and $Y_1,dots, Y_n$. As $n$ grows without bound, and when properly centered and normalized, LCI$_n$ is shown to converge, in distribution, towards a Brownian functional that we identify.
Monday, February 4th
Paul Feehan, Rutgers University
"A Perron method for existence of solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps"
Time: 2:00 PM
Location: Hill 525
Abstract: We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using new a version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (1993) in mathematical finance. Existence of a solution to the Dirichlet problem on a half-ball, where the operator becomes degenerate on the flat boundary and a Dirichlet condition is only imposed on the spherical boundary, provides the key additional ingredient required for our Perron method. Surprisingly, proving existence of a solution to this Dirichlet problem with "mixed" boundary conditions on a half-ball is a harder problem than one might expect. Due to the difficulty in developing a global Schauder estimate and due to compatibility conditions arising where the "degenerate" and "non-degenerate boundaries" touch, one cannot directly apply the continuity or approximate solution methods. However, in dimension two, there is a holomorphic map from the half-disk to the infinite strip in the complex plane and one can extend this definition to higher dimensions to give a diffeomorphism from the half-ball to the infinite "slab". The solution to the Dirichlet problem on the half-ball can thus be converted to a Dirichlet problem on the slab, albeit for an operator which now has exponentially growing coefficients. The required Schauder regularity theory and existence of a solution to the Dirichlet problem on the slab can nevertheless be obtained using previous work of the author and Camelia Pop in arXiv:1210.6727. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, suitable concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates developed by the author in arXiv:1204.6613.
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