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

We investigate a weighted Multilevel Richardson-Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions adapted from the one introduced in~[Lemaire-Pages, 2013] for regular Monte Carlo simulation. In a first result, we prove under weak confluence assumptions on the diffusion, that for any integer $R\ge2$, the procedure allows us to attain a rate $n^{\frac{R}{2R+1}}$ whereas the original algorithm convergence is at a weak rate $n^{1/3}$. Furthermore, this is achieved without any explosion of the asymptotic variance. In a second part, under stronger confluence assumptions and with the help of some second order expansions of the asymptotic error, we go deeper in the study by optimizing the choice of the parameters involved by the method. In particular, for a given $\varepsilon\textgreater{}0$, we exhibit some semi-explicit parameters for which the number of iterations of the Euler scheme required to attain a Mean-Squared Error lower than $\varepsilon^2$ is about $\varepsilon^{-2}\log(\varepsilon^{-1})$. Finally, we numerically this Multilevel Langevin estimator on several examples including the simple one-dimensional Ornstein-Uhlenbeck process but also on a high dimensional diffusion motivated by a statistical problem. These examples confirm the theoretical efficiency of the method.

Weighted Multilevel Langevin Simulation of Invariant Measures

Recursive computation of invariant distributions of Feller processes

Alpha-thrombin receptor and MAP kinases in the control of cell growth.

We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic equation $dY_t = - \sigma \sigma^\top \nabla V(Y_t) dt + a(t)\sigma(Y_t)dW_t + a(t)^2\Upsilon(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $\sigma : \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $\Upsilon$ is a correction term. This setting can be applied to optimization problems arising in Machine Learning. The case where $\sigma$ is a constant matrix has been extensively studied however little attention has been paid to the general case. We prove the convergence for the $L^1$-Wasserstein distance of $Y_t$ and of the associated Euler-scheme $\bar{Y}_t$ to some measure $\nu^\star$ which is supported by $\text{argmin}(V)$ and give rates of convergence to the instantaneous Gibbs measure $\nu_{a(t)}$ of density $\propto \exp(-2V(x)/a(t)^2)$. To do so, we first consider the case where $a$ is a piecewise constant function. We find again the classical schedule $a(t) = A\log^{-1/2}(t)$. We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.

Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise

We introduce a new approach to quantize the Euler scheme of an R d-valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of fast quantization in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously using a Newton-Raphson algorithm. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities (for the quantized process and for its components) using closed formulas. From the analytical point of view, we show that the induced quantization errors at the k-th discretization step t k is a cumulative of the marginal quantization error up to time t k. Numerical experiments are performed for the pricing of a Basket call option and a European call option in a Heston model to show the performances of the method.

Gilles Pagès

Papers

Weighted Multilevel Langevin Simulation of Invariant Measures

Recursive computation of invariant distributions of Feller processes

Alpha-thrombin receptor and MAP kinases in the control of cell growth.

Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise

Markovian and product quantization of an R^d -valued Euler scheme of a diffusion process with applications to finance