Filtration shrinkage, strict local martingales and the F\"{o}llmer measure

When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated F\"{o}llmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.Comment: Published in at http://dx.doi.org/10.1214/13-AAP961 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Matrix-valued Bessel processes

This paper introduces a matrix analog of the Bessel processes, taking values in the closed set $E$ of real square matrices with nonnegative determinant. They are related to the well-known Wishart processes in a simple way: the latter are obtained from the former via the map $x\mapsto x^\top x$. The main focus is on existence and uniqueness via the theory of Dirichlet forms. This leads us to develop new results of potential theoretic nature concerning the space of real square matrices. Specifically, the function $w(x)=|\det x|^\alpha$ is a weight function in the Muckenhoupt $A_p$ class for $-11$). The set of matrices of co-rank at least two has zero capacity with respect to the measure $m(dx)=|\det x|^\alpha dx$ if $\alpha>-1$, and if $\alpha\ge 1$ this even holds for the set of all singular matrices. As a consequence we obtain density results for Sobolev spaces over (the interior of) $E$ with Neumann boundary conditions. The highly non-convex, non-Lipschitz structure of the state space is dealt with using a combination of geometric and algebraic methods.Comment: This is the final version appearing in EJ

Spectral Characterization of functional MRI data on voxel-resolution cortical graphs

The human cortical layer exhibits a convoluted morphology that is unique to each individual. Conventional volumetric fMRI processing schemes take for granted the rich information provided by the underlying anatomy. We present a method to study fMRI data on subject-specific cerebral hemisphere cortex (CHC) graphs, which encode the cortical morphology at the resolution of voxels in 3-D. We study graph spectral energy metrics associated to fMRI data of 100 subjects from the Human Connectome Project database, across seven tasks. Experimental results signify the strength of CHC graphs' Laplacian eigenvector bases in capturing subtle spatial patterns specific to different functional loads as well as experimental conditions within each task.Comment: Fixed two typos in the equations; (1) definition of L in section 2.1, paragraph 1. (2) signal de-meaning and normalization in section 2.4, paragraph

Polynomial Diffusions and Applications in Finance

This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Uniqueness of polynomial diffusions is established via moment determinacy in combination with pathwise uniqueness. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Examples include the unit ball, the product of the unit cube and nonnegative orthant, and the unit simplex.Comment: This article is forthcoming in Finance and Stochastic

M5-branes and Matrix Theory

We consider supermembranes ending on M5-branes, with the aim of deriving the appropriate matrix theories describing different situations. Special attention is given to the case of non-vanishing (selfdual) C-field. We identify the relevant deformation of the six-dimensional super-Yang-Mills theory whose dimensional reduction is the matrix theory for membranes in the presence of M5-branes. Possible applications and limitations of the models are discussed.Comment: 10 pp., plain tex. Contribution to the proceedings of the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'03, Dubna, Russia, July 24-29, 2003) and of the 9th Adriatic Meeting "Particle Physics and the Universe" (Dubrovnik, Croatia, September 4-14, 2003

Polynomial diffusions on compact quadric sets

Polynomial processes are defined by the property that conditional expectations of polynomial functions of the process are again polynomials of the same or lower degree. Many fundamental stochastic processes, including affine processes, are polynomial, and their tractable structure makes them important in applications. In this paper we study polynomial diffusions whose state space is a compact quadric set. Necessary and sufficient conditions for existence, uniqueness, and boundary attainment are given. The existence of a convenient parameterization of the generator is shown to be closely related to the classical problem of expressing nonnegative polynomials---specifically, biquadratic forms vanishing on the diagonal---as a sum of squares. We prove that in dimension $d\le 4$ every such biquadratic form is a sum of squares, while for $d\ge6$ there are counterexamples. The case $d=5$ remains open. An equivalent probabilistic description of the sum of squares property is provided, and we show how it can be used to obtain results on pathwise uniqueness and existence of smooth densities.Comment: Forthcoming in Stochastic Processes and their Application