Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging

We propose a general method to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one of type 0 and one of type 1, in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestors' only through the mother's characteristic. The dependency structure may be described by a transition probability $P(x,dy dz)$ which gives the probability that the pair of daughters' characteristics is around $(y,z)$, given that the mother's characteristic is $x$. Note that $y$, the characteristic of the daughter of type 0, and $z$, that of the daughter of type 1, may be conditionally dependent given $x$, and their respective conditional distributions may differ. We then speak of bifurcating Markov chains. We derive laws of large numbers and central limit theorems for such stochastic processes. We then apply these results to detect cellular aging in Escherichia Coli, using the data of Stewart et al. and a bifurcating autoregressive model.Comment: Published in at http://dx.doi.org/10.1214/105051607000000195 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Euler Scheme and Tempered Distributuions

Given a smooth R^d-valued diffusion, we study how fast the Euler scheme with time step 1/n converges in law. To be precise, we look for which class of test functions f the approximate expectation E[f(X^{n,x}_1)] converges with speed 1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class contains all tempered distributions, and all measurable functions with exponential growth. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.Comment: 26 page

Short communication: inversion of convex ordering: local volatility does not maximise the price of VIX futures

It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX future is strictly more expensive than in its associated local volatility model. More generally, in our model, strictly convex payoffs on a squared VIX are strictly cheaper than in the associated local volatility model. This corresponds to an inversion of convex ordering between local and stochastic variances, when moving from instantaneous variances to squared VIX, as convex payoffs on instantaneous variances are always cheaper in the local volatility model. We thus prove that this inversion of convex ordering, which is observed in the S&P 500 market for short VIX maturities, can be produced by a continuous stochastic volatility model. We also prove that the model can be extended so that, as suggested by market data, the convex ordering is preserved for long maturities

Experimental study of a low-order wavefront sensor for the high-contrast coronagraphic imager EXCEDE

The mission EXCEDE (EXoplanetary Circumstellar Environments and Disk Explorer), selected by NASA for technology development, is designed to study the formation, evolution and architectures of exoplanetary systems and characterize circumstellar environments into stellar habitable zones. It is composed of a 0.7 m telescope equipped with a Phase-Induced Amplitude Apodization Coronagraph (PIAA-C) and a 2000-element MEMS deformable mirror, capable of raw contrasts of 1e-6 at 1.2 lambda/D and 1e-7 above 2 lambda/D. One of the key challenges to achieve those contrasts is to remove low-order aberrations, using a Low-Order WaveFront Sensor (LOWFS). An experiment simulating the starlight suppression system is currently developed at NASA Ames Research Center, and includes a LOWFS controlling tip/tilt modes in real time at 500 Hz. The LOWFS allowed us to reduce the tip/tilt disturbances to 1e-3 lambda/D rms, enhancing the previous contrast by a decade, to 8e-7 between 1.2 and 2 lambda/D. A Linear Quadratic Gaussian (LQG) controller is currently implemented to improve even more that result by reducing residual vibrations. This testbed shows that a good knowledge of the low-order disturbances is a key asset for high contrast imaging, whether for real-time control or for post processing.Comment: 12 pages, 20 figures, proceeding of the SPIE conference Optics+Photonics, San Diego 201

High Contrast Imaging and Wavefront Control with a PIAA Coronagraph: Laboratory System Validation

The Phase-Induced Amplitude Apodization (PIAA) coronagraph is a high performance coronagraph concept able to work at small angular separation with little loss in throughput. We present results obtained with a laboratory PIAA system including active wavefront control. The system has a 94.3% throughput (excluding coating losses) and operates in air with monochromatic light. Our testbed achieved a 2.27e-7 raw contrast between 1.65 lambda/D (inner working angle of the coronagraph configuration tested) and 4.4 lambda/D (outer working angle). Through careful calibration, we were able to separate this residual light into a dynamic coherent component (turbulence, vibrations) at 4.5e-8 contrast and a static incoherent component (ghosts and/or polarization missmatch) at 1.6e-7 contrast. Pointing errors are controlled at the 1e-3 lambda/D level using a dedicated low order wavefront sensor. While not sufficient for direct imaging of Earth-like planets from space, the 2.27e-7 raw contrast achieved already exceeds requirements for a ground-based Extreme Adaptive Optics system aimed at direct detection of more massive exoplanets. We show that over a 4hr long period, averaged wavefront errors have been controlled to the 3.5e-9 contrast level. This result is particularly encouraging for ground based Extreme-AO systems relying on long term stability and absence of static wavefront errors to recover planets much fainter than the fast boiling speckle halo.Comment: 18 pages, 12 figures. Accepted for publication in PASP. The pointing control scheme for this system is described in a separate paper (Coronagraphic Low-Order Wave-Front Sensor: Principle and Application to a Phase-Induced Amplitude Coronagraph, The Astrophysical Journal, Volume 693, Issue 1, pp. 75-84 (2009)

Diffraction-limited polarimetric imaging of protoplanetary disks and mass-loss shells with VAMPIRES

Both the birth and death of a stellar system are areas of key scientific importance. Whether it's understanding the process of planetary formation in a star's early years, or uncovering the cause of the enormous mass-loss that takes place during a star's dying moments, a key to scientific understanding lies in the inner few AU of the circumstellar environment. Corresponding to scales of 10s of milli-arcseconds, these observations pose a huge technical challenge due to the high angular-resolutions and contrasts required. A major stumbling block is the problem of the Earth's own atmospheric turbulence. The other difficulty is that precise calibration is required to combat the extremely high contrast ratios and high resolutions faced. By taking advantage of the fact that starlight scattered by dust in the circumstellar region is polarized, differential polarimetry can help achieve this calibration. Spectral features can also be utilized

On-sky demonstration of low-order wavefront sensing and control with focal plane phase mask coronagraphs

The ability to characterize exoplanets by spectroscopy of their atmospheres requires direct imaging techniques to isolate planet signal from the bright stellar glare. One of the limitations with the direct detection of exoplanets, either with ground- or space-based coronagraphs, is pointing errors and other low-order wavefront aberrations. The coronagraphic detection sensitivity at the diffraction limit therefore depends on how well low-order aberrations upstream of the focal plane mask are corrected. To prevent starlight leakage at the inner working angle of a phase mask coronagraph, we have introduced a Lyot-based low-order wavefront sensor (LLOWFS), which senses aberrations using the rejected starlight diffracted at the Lyot plane. In this paper, we present the implementation, testing and results of LLOWFS on the Subaru Coronagraphic Extreme Adaptive Optics system (SCExAO) at the Subaru Telescope. We have controlled thirty-five Zernike modes of a H-band vector vortex coronagraph in the laboratory and ten Zernike modes on sky with an integrator control law. We demonstrated a closed-loop pointing residual of 0.02 mas in the laboratory and 0.15 mas on sky for data sampled using the minimal 2-second exposure time of the science camera. We have also integrated the LLOWFS in the visible high-order control loop of SCExAO, which in closed-loop operation has validated the correction of the non-common path pointing errors between the infrared science channel and the visible wavefront sensing channel with pointing residual of 0.23 mas on sky.Comment: 12 pages, 15 figures, Accepted and scheduled for publication in September 2015 issue of the PAS