Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing

We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo algorithm, where $X$ is the solution of a stochastic differential equation and $f$ is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps $\delta$ and $\delta^{\beta},0<\beta<1$. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal $\beta$ (which turns out to be $\beta=1/2$), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.Comment: Published at http://dx.doi.org/10.1214/105051605000000511 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random generation of finitely generated subgroups of a free group

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity \O(n) in the RAM model and \O(n^2\log^2n) in the bitcost model

Policy packages for modal shift and CO2 reduction in Lille, France

This paper proposes different policy scenarios to cut CO2 emissions caused by the urban mobility of passengers. More precisely, we compare the effects of the ‘direct tool’ of carbon tax, to a combination of ‘indirect tools’ – not originally aimed at reducing CO2 (i.e. congestion charging, parking charges and a reduction in public transport travel time) in terms of CO2 impacts through a change in the modal split. In our model, modal choices depend on individual characteristics, trip features (including the effects of policy tools), and land use at origin and destination zones. Personal “CO2 emissions budgets” resulting from the trips observed in the metropolitan area of Lille (France) in 2006 are calculated and compared to the situation related to the different policy scenarios. We find that an increase of 50% in parking charges combined with a cordon toll of €1.20 and a 10% travel time decrease in public transport services (made after recycling toll-revenues) is the winning scenario. The combined effects of all the policy scenarios are superior to their separate effects

A resource-based Korean morphological annotation system Hyun-gue Huh Éric Laporte

We describe a resource-based method of morphological annotation of written Korean text. Korean is an agglutinative language. The output of our system is a graph of morphemes annotated with accurate linguistic information. The language resources used by the system can be easily updated, which allows users to control the evolution of the performances of the system. We show that morphological annotation of Korean text can be performed directly with a lexicon of words and without morphological rules.