Highlights d Cities possess a consistent ''core'' set of non-human microbes d Urban microbiomes echo important features of cities and city-life d Antimicrobial resistance genes are widespread in cities d Cities contain many novel bacterial and viral species

We propose an unconstrained stochastic approximation method for finding the optimal change of measure (in an a priori parametric family) to reduce the variance of a Monte Carlo simulation. We consider different parametric families based on the Girsanov theorem and the Esscher transform (exponential-tilting). In [Monte Carlo Methods Appl. 10 (2004) 1-24], it described a projected Robbins-Monro procedure to select the parameter minimizing the variance in a multidimensional Gaussian framework. In our approach, the parameter (scalar or process) is selected by a classical Robbins-Monro procedure without projection or truncation. To obtain this unconstrained algorithm, we extensively use the regularity of the density of the law without assuming smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions as well as for diffusion processes.We illustrate the efficiency of our algorithm on several pricing problems: a Basket payoff under a multidimensional NIG distribution and a barrier options in different markets.

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We propose and analyze a Multilevel Richardson-Romberg (ML2R) estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg method introduced in [Pag07] and the variance control resulting from Multilevel Monte Carlo (MLMC) paradigm (see [Gil08,Hei01]). Thus, in standard frameworks like discretization schemes of diffusion processes, the root mean squared error (RMSE) ε > 0 can be achieved with our ML2R estimator with a global complexity of ε −2 log(1/ε) instead of ε −2 (log(1/ε)) 2 with the standard MLMC method, at least when the weak error

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme --with constant or decreasing step-- may explode and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems

In human cells, estrogenic signals induce cyclical association and dissociation of specific proteins with the DNA in order to activate transcription of estrogen-responsive genes. These oscillations can be modeled by assuming a large number of sequential reactions represented by linear kinetics with random kinetic rates. Application of the model to experimental data predicts robust binding sequences in which proteins associate with the DNA at several different phases of the oscillation. Our methods circumvent the need to derive detailed kinetic graphs, and are applicable to other oscillatory biological processes involving a large number of sequential steps.

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 RichardsonRomberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.

In this paper, we are interested in the exact simulation of a class of Piecewise Deterministic Markov Processes (PDMP). We show how to perform efficient thinning algorithms depending on the jump rate bound. For different types of jump rate bounds, we compare theoretically the mean number of generated (total) jump times and we compare numerically the simulation times. We use thinning algorithms on Hodgkin-Huxley models with Markovian ion channels dynamic to illustrate the results.