http://web.stanford.edu/group/brunet/Cell1999

Akt Promotes Cell Survival by Phosphorylating and Inhibiting a Forkhead Transcription Factor

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Mitogen-activated protein kinases (MAPKs) are important signal transducing enzymes, unique to eukaryotes, that are involved in many facets of cellular regulation. Initial research concentrated on defining the components and organization of MAPK signalling cascades, but recent studies have begun to shed light on the physiological functions of these cascades in the control of gene expression, cell proliferation and programmed cell death.

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Mammalian MAP kinase signalling cascades

Cells undergo a variety of biological responses when placed in hypoxic conditions, including activation of signalling pathways that regulate proliferation, angiogenesis and death. Cancer cells have adapted these pathways, allowing tumours to survive and even grow under hypoxic conditions, and tumour hypoxia is associated with poor prognosis and resistance to radiation therapy. Many elements of the hypoxia-response pathway are therefore good candidates for therapeutic targeting.

Hypoxia — a key regulatory factor in tumour growth

Specificity of receptor tyrosine kinase signaling: Transient versus sustained extracellular signal-regulated kinase activation

Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm $|\cdot|$). Let $ r> 0 $ and let $(\alpha_n)_{n \geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq1}$ defined for every $n \geq1$ by $\rho(\alpha_n) = \max{|a|, a \in\alpha_n}$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup{|x|, x \in\operatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.

Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers

The mitogen activated protein kinase signal transduction pathway is inhibited by a natural product, the nordidemnin

With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. Under this assumption, we first show that uniqueness of the invariant distribution (weak confluence) of the duplicated system is essentially always true in the one-dimensional case. In the multidimensional case, we begin by exhibiting explicit counter-examples. Then, we provide a series of weak confluence criterions (of integral type) and also of a.s. pathwise confluence, depending on the drift and diffusion coefficients through a non-infinitesimal Lyapunov exponent. As examples, we apply our criterions to some non-trivially confluent settings such as classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component. We finally establish that the weak confluence property is connected with an optimal transport problem. As a main application, we apply our results to the optimization of the Richardson-Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.

Invariant distribution of duplicated diffusions and application to Richardson-Romberg extrapolation

The Quantization Tree algorithm has proven to be quite an efficient tool for the evaluation of financial derivatives with non-vanilla exercise rights as American-, Bermudan-or Swing options. Nevertheless, it relies heavily on a fast computation of the transition probabilities in the underlying Quantization Tree. Since this estimation is typically done by Monte-Carlo simulations, it is appealing to take advantage of the massive parallel computing capabilities of modern GPGPU-devices. We present in this article a parallel implementation of the transition probability estimation for a Gaussian 2-factor model in CUDA. Since we have to deal in this case with a huge amount of data and quite long MC-paths, it turned out that the naive pathwise parallel implementation is not optimal. We therefore present a time-layer wise parallelization which can better exploit the parallel computing power of GPGPU-devices by using faster memory structures.

Parallel implementation of a Quantization algorithm for pricing American style options on GPGPU

We propose a constructive proof for the sharp rate of optimal quadratic functional quantization and we tackle the asymptotics of the critical dimension for quadratic functional quantization of Gaussian stochastic processes as the quantization level goes to infinity, i.e. the smallest dimensional truncation of an optimal quantization of the process which is "fully" quantized. We first establish a lower bound for this critical dimension based on the regular variation index of the eigenvalues of the Karhunen-Loeve expansion of the process. This lower bound is consistent with the commonly shared sharp rate conjecture (and supported by extensive numerical experiments). Moreover, we show that, conversely, optimized quadratic functional quantizations based on this critical dimension rate are always asymptotically optimal (strong admissibility result).

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Gilles Pagès

Papers

Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers

The mitogen activated protein kinase signal transduction pathway is inhibited by a natural product, the nordidemnin

Invariant distribution of duplicated diffusions and application to Richardson-Romberg extrapolation

Parallel implementation of a Quantization algorithm for pricing American style options on GPGPU

Constructive quadratic functional quantization and critical dimension