Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise

We consider the family of stochastic partial differential equations indexed by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) = \eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*} (t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\sigma$ and $b$ are smooth functions and $\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let p^\eps_{t,x} denote the density of the law of u^\eps(t,x) at a fixed point (t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0} \eps^2\log p^\eps_{t,x}(y) for a fixed $y\in\R$. The results apply to a class of stochastic wave equations with $d\in\{1,2,3\}$ and to a class of stochastic heat equations with $d\ge1$.Comment: 39 pages. Will be published in the book " Stochastic Analysis and Applications 2014. A volume in honour of Terry Lyons". Springer Verla

Fundamental Theorem of Asset Pricing under fixed and proportional transaction costs

We show that the lack of arbitrage in a model with both fixed and proportional transaction costs is equivalent to the existence of a family of absolutely continuous single-step probability measures, together with an adapted process with values between the bid-ask spreads that satisfies the martingale property with respect to each of the measures. This extends Harrison and Pliska's classical Fundamental Theorem of Asset Pricing to the case of combined fixed and proportional transaction costs

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.Comment: See journal reference for teh final published versio

Cosmology with the Laser Interferometer Space Antenna

The Laser Interferometer Space Antenna (LISA) has two scientific objectives of cosmological focus: to probe the expansion rate of the universe, and to understand stochastic gravitational-wave backgrounds and their implications for early universe and particle physics, from the MeV to the Planck scale. However, the range of potential cosmological applications of gravitational wave observations extends well beyond these two objectives. This publication presents a summary of the state of the art in LISA cosmology, theory and methods, and identifies new opportunities to use gravitational wave observations by LISA to probe the universe

SOME REMARKS ON ARBITRAGE AND PREFERENCES IN SECURITIES MARKET MODELS

We introduce the notion of a Market Free Lunch that depends on the preferences of all agents participating in the market. In semimartingale models of securities markets, we characterize No Arbitrage (NA) and No Free Lunch with Vanishing Risk (NFLVR) in terms of the Market Free Lunch and show that the difference between NA and NFLVR consists in the selection of the class of monotone, resp. monotone and continuous, utility functions that determines the absence of the Market Free Lunch. We also provide a direct proof of the equivalence between the absence of a Market Free Lunch, with respect to monotone concave preferences, and the existence of an equivalent (local/sigma) martingale measure

Evidence for a milliparsec separation supermassive binary black hole with quasar microlensing

peer reviewedWe report periodic oscillations in the 15-year long optical light curve of the gravitationally lensed quasar QJ0158-4325. The signal is enhanced during a high magnification microlensing event undergone by the fainter lensed image of the quasar, between 2003 and 2010. We measure a period of $P_{o}=172.6\pm0.9$ days. We explore four scenarios to explain the origin of the periodicity: 1- the high magnification microlensing event is due to a binary star in the lensing galaxy, 2- QJ0158-4325 contains a massive binary black hole system in its final dynamical stage before merging, 3- the quasar accretion disk contains a bright inhomogeneity in Keplerian motion around the black hole, and 4- the accretion disk is in precession. Among these four scenarios, only a supermassive binary black hole can account for both the short observed period and the amplitude of the signal, through the oscillation of the accretion disk towards and away from high-magnification regions of a microlensing caustic. The short measured period implies that the semi-long axis of the orbit is $\sim10^{-3}$pc, and the coalescence timescale is $t_{coal}\sim1000$ years, assuming that the decay of the orbit is solely powered by the emission of gravitational waves. The probability of observing a system so close to coalescence suggests either a much larger population of supermassive black hole binaries than predicted, or, more likely, that some other mechanism significantly increases the coalescence timescale. Three tests of the binary black hole hypothesis include: i) the recurrence of oscillations in photometric monitoring during any future microlensing events in either image, ii) spectroscopic detection of Doppler shifts (up to 0.01$c$), and iii) the detection of gravitational waves through Pulsar Timing Array experiments, such as the SKA, which will have the sensitivity to detect the $\sim$100 nano-hertz emission