Laplace transform identities for diffusions, with applications to rebates and barrier options

Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model

A visual criterion for identifying Ito diffusions as martingales or strict local martingales

It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

The Void Galaxy Survey: Optical Properties and H I Morphology and Kinematics

We have carefully selected a sample of 60 galaxies that reside in the deepest underdensities of geometrically identified voids within the SDSS. HI imaging of 55 galaxies with the WSRT reveals morphological and kinematic signatures of ongoing interactions and gas accretion. We probe a total volume of 485 Mpc^3 within the voids, with an angular resolution of 8 kpc at an average distance of 85 Mpc. We reach column density sensitivities of 5 x 10^19 cm^-2, corresponding to an HI mass limit of 3 x 10^8 M_sun. We detect HI in 41 galaxies, with total masses ranging from 1.7 x 10^8 to 5.5 x 10^9 M_sun. The upper limits on the 14 non-detections are not inconsistent with their luminosities, given their expected HI mass to light ratios. We find that the void galaxies are generally gas rich, low luminosity, blue disk galaxies, with optical and HI properties that are not unusual for their luminosity and morphology. The sample spans a range of absolute magnitudes (-16.1 > M_r > -20.4) and colors (0.06 < g-r < 0.87), and includes disk and irregular galaxies. We also identify three as early type galaxies, all of which are not detected in HI. All galaxies have stellar masses less than 3 x 10^10 M_sun, and many have kinematic and morphological signs of ongoing gas accretion, suggesting that the void galaxy population is still in the process of assembling. The small scale clustering in the void, within 600 kpc and 200 km/s, is similar to that in higher density regions, and we identify 18 HI rich neighboring galaxies in the voids. Most are within 100 kpc and 100 km/s of the targeted galaxy, and we find no significant population of HI rich low luminosity galaxies filling the voids, contrary to what is predicted by simulations.Comment: 34 pages, 33 figures (including Atlas in Appendix), accepted for publication in A

Only the Lonely: H I Imaging of Void Galaxies

Void galaxies, residing within the deepest underdensities of the Cosmic Web, present an ideal population for the study of galaxy formation and evolution in an environment undisturbed by the complex processes modifying galaxies in clusters and groups, as well as provide an observational test for theories of cosmological structure formation. We have completed a pilot survey for the HI imaging aspects of a new Void Galaxy Survey (VGS), imaging 15 void galaxies in HI in local (d < 100 Mpc) voids. HI masses range from 3.5 x 10^8 to 3.8 x 10^9 M_sun, with one nondetection with an upper limit of 2.1 x 10^8 M_sun. Our galaxies were selected using a structural and geometric technique to produce a sample that is purely environmentally selected and uniformly represents the void galaxy population. In addition, we use a powerful new backend of the Westerbork Synthesis Radio Telescope that allows us to probe a large volume around each targeted galaxy, simultaneously providing an environmentally constrained sample of fore- and background control sample of galaxies while still resolving individual galaxy kinematics and detecting faint companions in HI. This small sample makes up a surprisingly interesting collection of perturbed and interacting galaxies, all with small stellar disks. Four galaxies have significantly perturbed HI disks, five have previously unidentified companions at distances ranging from 50 to 200 kpc, two are in interacting systems, and one was found to have a polar HI disk. Our initial findings suggest void galaxies are a gas-rich, dynamic population which present evidence of ongoing gas accretion, major and minor interactions, and filamentary alignment despite the surrounding underdense environment.Comment: 53 pages, 18 figures, accepted for publication in AJ. High resolution available at http://www.astro.columbia.edu/~keejo/kreckel2010.pd

Arbitrage and deflators in illiquid markets

This paper presents a stochastic model for discrete-time trading in financial markets where trading costs are given by convex cost functions and portfolios are constrained by convex sets. The model does not assume the existence of a cash account/numeraire. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. In the presence of nonlinearities, the classical notion of arbitrage turns out to have two equally meaningful generalizations, a marginal and a scalable one. We study their relations to state price deflators by analyzing two auxiliary market models describing the local and global behavior of the cost functions and constraints

The Void Galaxy Survey

The Void Galaxy Survey (VGS) is a multi-wavelength program to study $\sim$60 void galaxies. Each has been selected from the deepest interior regions of identified voids in the SDSS redshift survey on the basis of a unique geometric technique, with no a prior selection of intrinsic properties of the void galaxies. The project intends to study in detail the gas content, star formation history and stellar content, as well as kinematics and dynamics of void galaxies and their companions in a broad sample of void environments. It involves the HI imaging of the gas distribution in each of the VGS galaxies. Amongst its most tantalizing findings is the possible evidence for cold gas accretion in some of the most interesting objects, amongst which are a polar ring galaxy and a filamentary configuration of void galaxies. Here we shortly describe the scope of the VGS and the results of the full analysis of the pilot sample of 15 void galaxies.Comment: 9 pages, 6 figures. This is an extended version of a paper to appear in "Environment and the Formation of Galaxies: 30 years later", Proceedings of Symposium 2 of JENAM 2010, eds. I. Ferreras, A. Pasquali, ASSP, Springer. Version with highres figures at http://www.astro.rug.nl/~weygaert/vgs_jenam_weygaert.col.pd

Cavity evolution in relativistic self-gravitating fluids

We consider the evolution of cavities within spherically symmetric relativistic fluids, under the assumption that proper radial distance between neighboring fluid elements remains constant during their evolution (purely areal evolution condition). The general formalism is deployed and solutions are presented. Some of them satisfy Darmois conditions whereas others present shells and must satisfy Israel conditions, on either one or both boundary surfaces. Prospective applications of these results to some astrophysical scenarios is suggested.Comment: 10 pages Revtex. To appear in Class. Quantum Grav

Alan Greenspan, the Quants and Stochastic Optimal Control

Alan Greenspan's paper (March 2010) presents his retrospective view of the crisis. His theme has several parts. First, the housing price bubble, its subsequent collapse and the financial crisis were not predicted either by the market, the FED, the IMF or the regulators in the years leading to the current crisis. Second, financial intermediation tried to function on too thin layer of capital - high leverage - owing to a misreading of the degree of risk embodied in ever more complex financial products and markets. Third, the breakdown was unpredictable and inevitable, given the 'excessive' leverage - or low capital - of the financial intermediaries. The proposed legislation for the 'reform' of the financial system requires that the FED "identify, measure, manage and mitigate risks to the financial stability of the United States". The focus is upon capital requirements or debt ratios. The 'Quants' ignored systemic risk and just focused upon risk transfer in very liquid markets. The FED, IMF, Treasury and the 'Quants'/market lacked the appropriate tools of analysis to answer the following questions: what is an optimal leverage or capital requirement that balances the expected growth against risk? What are theoretically founded early warning signals of a crisis? The author explains why the application of stochastic optimal control (SOC)/dynamic risk management is an effective approach to determine the optimal degree of leverage, the optimum and excessive risk and the probability of a debt crisis. The theoretically derived early warning signal of a crisis is the excess debt ratio, equal to the difference between the actual and optimal ratio. The excess debt starting from 2004-05 indicated that a crisis was most likely. This SOC analysis should be used by those charged with surveillance of financial markets