Probability Seminar, Fall 2014

 

Time: Tuesdays, 12:30pm - 1:30pm

Room: MSB 213

 

Dec 2, 2014: Nathanaël Berestycki, Statistical Laboratory, University of Cambridge

 

Title: Liouville Brownian motion

 

I will introduce and discuss a canonical notion of Brownian motion in random planar geometry, called Liouville Brownian motion. I will explain the construction and discuss some of its basic properties, for instance related to its heat kernel and to the time spent in the thick points of the Gaussian Free Field. Time permitting I will also discuss a derivation of the KPZ formula based on the Liouville heat kernel (joint work with C. Garban. R. Rhodes and V. Vargas).

 


 

Nov 25, 2014: Paul Jung, Department of Mathematics, University of Alabama at Birmingham

 

Title: Random walks at random times

 

Kesten and Spitzer (1979) introduced random walks in random scenery (RWRS) which are collective reward processes where a random walker collects a random reward (or scenery) at each site it visits. If the walker visits a site N times, it collects the same reward N times thus leading to correlations in the collective reward process. Cohen and Samorodnitsky (2006) studied a certain renormalization of RWRS and proposed self-similar, symmetric alpha-stable processes, which generalize fractional Brownian motion, as their scaling limits. The limiting processes have self-similarity exponents H>1/alpha.


We consider a modification of RWRS in which a sign associated to the reward (scenery) alternates upon successive visits of the random walk. The resulting process is what we call a random walk at random time, and it generalizes the so-called iterated random walk. We will discuss renormalizations of this discrete process, and in particular, show that the alternating scenery can lead to limiting processes which have self-similarity exponents H<1/alpha.

 


 

Nov 18, 2014: Tyler Gomez, Department of Mathematics, University of Central Florida

 

Title: Modeling random measures via iterated Dirichlet processes

 

This presentation identifies the task of simulating a random probability measure with exact data on a finite sample space. We do this by sequential imputation of a Dirichlet process, meaning we weigh the validity of our simulations by the likelihood of their generating subsequent data. There are many advantages to such a method, including the ability to simulate of a Dirichlet-distributed probability measure by generating independent gamma-distributed random variables and the marginal distributions of a Dirichlet probability vector being independent and beta-distributed. Moreover, we will discuss the implications of applying certain weights to our priors and the occurrence of “unlikely events” in our data. Finally, we shall demonstrate that in the limit, our weighted simulations yield the desired conditional expectations given the exact data.

 


 

Nov 11, 2014: No seminar (Veterans Day)

 


 

Nov 4, 2014: Jiongmin Yong, Department of Mathematics, University of Central Florida

 

Title: Stochastic control, mathematical finance, and BSDEs II

 

Stochastic optimal control problem is investigated by variational method. First, the problem is observed from an abstract angle. Then adjoint equation is introduced, Pontragin type maximum principle will be presented in which forward-backward stochastic differential equations (FBSDEs) are used.

 


 

Oct 21, 2014: Jiongmin Yong, Department of Mathematics, University of Central Florida

 

Title: Stochastic control, mathematical finance, and BSDEs I

 

We introduce some standard stochastic control models, for which, Black-Scholes-Merton model is a special case. Controllability problem and optimal control problem will be formulated. It turns out that European contingent claim pricing problem is a controllability problem and is equivalent to the solvability of a backward stochastic differential equation. The well-known Black-Scholes option pricing formula is a consequence of the nonlinear Feynman-Kac formula.

 


 

Oct 7, 2014: Jingrui Sun, University of Science and Technology of China, Hefei, Anhui, China

 

Title: Linear quadratic stochastic differential games: open-loop and closed-loop saddle points, Part II

 

This is a continuation of the previous talk.

 


 

Sep 30, 2014: Jingrui Sun, University of Science and Technology of China, Hefei, Anhui, China

 

Title: Linear quadratic stochastic differential games: open-loop and closed-loop saddle points, Part I

 

In this talk and its sequel, we consider a linear quadratic stochastic two-person zero-sum differential game. The existence of an open-loop saddle point is characterized by the existence of an adapted solution to a linear forward-backward stochastic differential equation with constraints, together with a convexity-concavity condition, and the existence of a closed-loop saddle point is characterized by the existence of a regular solution to a Riccati differential equation. It turns out that there is a significant difference between open-loop and closed-loop saddle points.

 


 

Sep 23, 2014: Jie Zhong, Department of Mathematics, University of Central Florida

 

Title: Stochastic evolution equations with constant coefficients

 

While solvability of a single stochastic hyperbolic or parabolic equation is well known, the problem remains mostly open for stochastic evolution systems. The paper investigates well-posedness and stability in Sobolev spaces on R^d of the initial value problem for systems of stochastic evolution equations with constant coefficients and multiplicative time-only Gaussian white noise. A general criterion for well-posedness is derived in terms of sums of certain Kronecker products of the system matrices, and a stochastic analogue of the Petrowski parabolicity condition is proposed.

 


 

Sep 9, 2014: Jie Zhong, Department of Mathematics, University of Central Florida

 

Title: Second order evolution equations and Wiener chaos approach

 

In this paper, we investigate the well-posedness of stochastic second order equations with unbounded damping operators. We use the Skorohod integral with respect to Gaussian noise as the driving random source, which allow us to cover time-only, space-only or space-time noises.