In this talk we discuss a continuous time optimization problem which combines two conflicting objectives of maximizing
the portfolio wealth up to a level and minimizing the conditional value-at-risk of the portfolio wealth loss. The standard utility maximization theory cannot be applied to solve the problem as the associated utility function is not differentiable nor strictly concave and does not satisfy Inada's condition. We use the dual control method to show that there is a classical solution to the Hamilton-Jacobi-Bellman equation and that the optimal value function is smooth if the optimal control satisfies an exponential moment condition. We find the closed-form optimal feedback control and optimal value function for a wealth maximization problem.

An option trading overview covers option basics, option valuations, option risk management and basic option trading principle. It provides some practical information to those audience who interesting in financial engineering and financial risk management. Some basic option concepts are needed.

I present an analog of the 2-Wasserstein metric in a non-commuatative probability setting arising in quantum mechanics that has been developed by Jan Maas and myself. It will be shown that the Femionic Ornstein-Uhlenbeck equation is the gradient flow for the relative entropy with respect to the equilibrium state, and analogy with a well known result of Otto, Kinderleher and jordan, and we prove a sharp non-commutative Talagrand inequality as well. We shall also discuss a range of open problems.

Kinetic theory describes the evolution of dilute gas of particles interacting through binary collisions. The equations of kinetic theory are non-linear evolution equations (various forms of the Boltzmann equation for a probability density. In 1956, Mark Kac proposed a strategy for investigating such equations via a direct analysis of a stochastic model, different from, and simpler than, the underlying physical collision model, but still leading to the same equations.
There has been much recent progress on Kac's program. We shall discuss recent work on determining the rate of equilibriation
in this model. Until now, this had only had been made only Maxwellian molecules, or, in a singular result of Villani, for super-hard collisions Recent work of myself, Carlen and Loss extends this progress to the physically significant hard-sphere case, as will be explained in this lecture. I will also discuss recent results of Mischler and Mouhot on propagation of chaos for this model that makes use of the new rate information.

Prof. Hall is among the world's most prolific and highly-cited authors in both probability and statistics. Mathscinet lists him with more than 500 publications as of January 2008. He has made very substantial and important contributions to nonparametric statistics, in particular for curve estimation and resampling: the bootstrap method, smoothing, density estimation, and bandwidth selection. He has worked on numerous applications across fields of economics, engineering, physical science and biological science. Prof. Hall has also made groundbreaking contributions to surface roughness measurement using fractals. In probability theory he has made many contributions to limit theory, spatial processes and stochastic geometry. His paper "Theoretical comparison of bootstrap confidence intervals" (Annals of Statistics, 1988) has been reprinted in the Breakthroughs in Statistics collection.

In this talk I will present some recent results on the stability of Dirichlet heat kernel estimates for non-local operators under two kinds of perturbations: gradient perturbation and non-local Feynman-Kac perturbations. This talk is based on joint work with Zhen-Qing Chen and Panki Kim.

In this series of lectures, I will first introduce two classes of first order Hamilton-Jacobi-Bellman (HJB) equations in the space of probability measures, and then develop a well posed-ness theory for both the Cauchy and resolvent type problems.
In the first lecture, I will derive two Hamiltonians using probabilistic large deviation theory applied to mean-field models from statistical mechanics, provide their connections with optimal controlled PDEs in space of measures, and explain some a priori estimates at microscopic level using probability theory.
In the second lecture, I will introduce necessary background about optimal mass transportation theory and use the language of such theory to prove uniqueness (comparison principle) result for HJB equations involving one of the Hamiltonians (the nicer one).
In the third lecture, I will improve the comparison principle result in the last lecture by regularization method and obtain continuity result of the solution under a reasonably weak topology (the Wasserstein-2 metric).
In the fourth lecture, I will introduce new ingredients to show the uniqueness (comparison principle) of HJB equations for the second Hamiltonian which has a strong singular term. I will also explain the relation of such equation to the Onsager-Joyce-Montgomery theory of 2-D turbulent vortex flows.

The multiple testing procedure plays an important role in detecting the presence of spatial signals for large-scale imaging data. Typically, the spatial signals are sparse but clustered. This paper provides empirical evidence that for a range of commonly used control levels, the conventional FDR procedure can lack the ability to detect statistical significance, even if the p-values under the true null hypotheses are independent and uniformly distributed; more generally, ignoring the neighboring information of spatially structured data will tend to diminish the detection effectiveness of the FDR procedure. This paper first introduces a scalar quantity to characterize the extent to which the “lack of identification phenomenon” (LIP) of the FDR procedure occurs.Second, we propose a new multiple comparison procedure, called FDR_L, to accommodate the spatial information of neighboring p-values, via a local aggregation of p-values. Theoretical properties of the FDR_L procedure are investigated under weak dependence of p-values. It is shown that the FDR_L procedure alleviates the LIP of the FDR procedure, thus substantially facilitating the selection of more stringent control levels. Simulation evaluations indicate that the FDR_L procedure improves the detection sensitivity of the FDR procedure with little loss in detection specificity. The computational simplicity and detection effectiveness of the FDR_L procedure are illustrated through a real dataset.

The prevalent cohort sampling design is frequently adopted to study the natural history of a disease. Important examples include epidemiological studies of disease etiology (Simon, 1980), cancer screening trials (Zelen and Feinleib, 1969), and HIV prevalent cohort studies (Lagakos et al, 1988). In addition to right censoring due to loss to follow-up, survival data collected via prevalent sampling are subject to left truncation, as those who fail before the recruitment time are not observable.In this talk, I will present a pseudo-profile likelihood method for survival data arising from length-biased sampling, where the survival times are right censored and left truncated by uniformly distributed random truncation times. The proposed method does not involve estimation of the censoring distribution; hence it is consistent even when the censoring time is correlated with covariates. Simulation studies show that the proposed estimator yields significant improvement in efficiency over the popular maximum partial likelihood estimator. To illustrate the proposed method, statistical analyses using data from the Canadian Study of Health and Aging (CSHA)and the CDC AIDS Blood Transfusion Study will be reported. (This is joint work with Drs. Jing Qin and Dean Follmann.)

Variational atmospheric data assimilation is a variational minimization problem subject to a set of nonlinear partial differential equations and is very important for improving numerical weather prediction at various time scales from tornados to hurricanes. It is very large and complex optimization problem. In this presentation, we review a few important features of a Space and Time Multiscale Analysis System (STMAS) developed at NOAA Earth System Research Laboratory for improving variational data assimilation and forecasts. We also review several outstanding challenges in general data assimilation in variational or Kalman filter. Some applications and numerical results will be shown