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