We study the classic problem of choosing a prior distribution for a location parameter β = (β1, . . . , βp) as p grows large. First, we study the standard “global-local shrinkage” approach, based on scale mixtures of normals. Two theorems are presented which characterize certain desirable properties of shrinkage priors for sparse problems. Next, we review some recent results showing how Levy processes can be used to generate infinite-dimensional versions of standard normal scale-mixture priors, along with new priors that have yet to be seriously studied in the literature. This approach provides an intuitive framework both for generating new regularization penalties and shrinkage rules, and for performing asymptotic analysis on existing models.

http://www.researchgate.net/profile/David_Heckerman/publication/228923701_Shrink_globally_act_locally_Sparse_Bayesian_regularization_and_prediction/links/5547741b0cf2e2031b36b898.pdf

Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction

http://directory.umm.ac.id/Data%20Elmu/jurnal/S/Stochastic%20Processes%20And%20Their%20Applications/Vol90.Issue1.2000/959.pdf

Chaotic and predictable representations for Levy processes

In this paper we show the existence and uniqueness of a solution for backward stochastic differential equations driven by a Levy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for Levy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equation provide an analogue of the famous Black-Scholes partial differential equation and thus can be used for the purpose of option pricing in a Levy market.

/pdf/backward-stochastic-differential-equations-and-feynman-kac-exyva1alpj.pdf

Backward stochastic differential equations and Feynman-Kac formula for Levy processes, with applications in finance

The risk-neutral process is modeled by a four parameter self-similar process of independent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with 610 free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices.

Self-decomposability and option pricing

We use L´ evy processes to generate joint prior distributions, and therefore penalty functions, for a location parameter b = (b1;:::;bp) as p grows large. This generalizes the class of local-global shrinkage rules based on scale mixtures of normals, illuminates new connections among disparate methods, and leads to new results for computing posterior means and modes under a wide class of priors. We extend this framework to large-scale regularized regression problems where p > n, and provide comparisons with other methodologies.

/pdf/local-shrinkage-rules-levy-processes-and-regularized-3ioezytog3.pdf

Local shrinkage rules, Lévy processes and regularized regression

Canonical form, self-decomposability, and the Esscher transforms of Meixner processes are discussed. Mixed Meixner processes are defined and characterized as Markov processes or semimartingales. Ornstein-Uhlenbeck and selfsimilar processes of Meixner type are also described.

B. Grigelionis

Papers

Processes of Meixner type