Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions

In this paper we obtain Gaussian-type lower bounds for the density of solutions to stochastic differential equations (SDEs) driven by a fractional Brownian motion with Hurst parameter $H$. In the one-dimensional case with additive noise, our study encompasses all parameters $H\in(0,1)$, while the multidimensional case is restricted to the case $H>1/2$. We rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools.Comment: Published at http://dx.doi.org/10.1214/14-AOP977 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic behaviour of the density in a parabolic SPDE

Consider the density of the solution $X(t,x)$ of a stochastic heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$. In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable local integration by parts formula is developped.Malliavin Calculus, parabolic SPDE, large deviations, Taylor expansion of a density, exponential estimates of the tail probabilities, stochastic integration by parts formula

Multidimensional Quasi-Monte Carlo Malliavin Greeks

We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide flexibility for variance reduction. In contrast, the Malliavin approach exhibits a higher flexibility by providing a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fourni\'e et al and reduce both the computational cost and the variance. Moreover, we show that the parameters employed for variance reduction can be obtained \textit{on the flight} in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and Exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market.Comment: 22 pages, 6 figure

Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We assume a Black-Scholes market with time-dependent volatilities and show how to compute the deltas by the aid of the Malliavin Calculus, extending the procedure employed by Montero and Kohatsu-Higa (2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. We present a new and fast Cholesky algorithm for block matrices that makes the Linear Transformation even more convenient. Moreover, we propose a new-path generation technique based on a Kronecker Product Approximation. This construction returns the same accuracy of the Linear Transformation used for the computation of the deltas and the prices in the case of correlated asset returns while requiring a lower computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004).Comment: 16 page

Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise

We consider the family of stochastic partial differential equations indexed by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) = \eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*} (t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\sigma$ and $b$ are smooth functions and $\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let p^\eps_{t,x} denote the density of the law of u^\eps(t,x) at a fixed point (t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0} \eps^2\log p^\eps_{t,x}(y) for a fixed $y\in\R$. The results apply to a class of stochastic wave equations with $d\in\{1,2,3\}$ and to a class of stochastic heat equations with $d\ge1$.Comment: 39 pages. Will be published in the book " Stochastic Analysis and Applications 2014. A volume in honour of Terry Lyons". Springer Verla

On the wellposedness of some McKean models with moderated or singular diffusion coefficient

We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Meleard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Holder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamics and a nondegenerate diffusion term in the position dynamics

Toward optimal implementation of cancer prevention and control programs in public health: A study protocol on mis-implementation

Abstract Background Much of the cancer burden in the USA is preventable, through application of existing knowledge. State-level funders and public health practitioners are in ideal positions to affect programs and policies related to cancer control. Mis-implementation refers to ending effective programs and policies prematurely or continuing ineffective ones. Greater attention to mis-implementation should lead to use of effective interventions and more efficient expenditure of resources, which in the long term, will lead to more positive cancer outcomes. Methods This is a three-phase study that takes a comprehensive approach, leading to the elucidation of tactics for addressing mis-implementation. Phase 1: We assess the extent to which mis-implementation is occurring among state cancer control programs in public health. This initial phase will involve a survey of 800 practitioners representing all states. The programs represented will span the full continuum of cancer control, from primary prevention to survivorship. Phase 2: Using data from phase 1 to identify organizations in which mis-implementation is particularly high or low, the team will conduct eight comparative case studies to get a richer understanding of mis-implementation and to understand contextual differences. These case studies will highlight lessons learned about mis-implementation and identify hypothesized drivers. Phase 3: Agent-based modeling will be used to identify dynamic interactions between individual capacity, organizational capacity, use of evidence, funding, and external factors driving mis-implementation. The team will then translate and disseminate findings from phases 1 to 3 to practitioners and practice-related stakeholders to support the reduction of mis-implementation. Discussion This study is innovative and significant because it will (1) be the first to refine and further develop reliable and valid measures of mis-implementation of public health programs; (2) bring together a strong, transdisciplinary team with significant expertise in practice-based research; (3) use agent-based modeling to address cancer control implementation; and (4) use a participatory, evidence-based, stakeholder-driven approach that will identify key leverage points for addressing mis-implementation among state public health programs. This research is expected to provide replicable computational simulation models that can identify leverage points and public health system dynamics to reduce mis-implementation in cancer control and may be of interest to other health areas

Peripheral, Central and Behavioral Responses to the Cuticular Pheromone Bouquet in Drosophila melanogaster Males

Pheromonal communication is crucial with regard to mate choice in many animals including insects. Drosophila melanogaster flies produce a pheromonal bouquet with many cuticular hydrocarbons some of which diverge between the sexes and differently affect male courtship behavior. Cuticular pheromones have a relatively high weight and are thought to be — mostly but not only — detected by gustatory contact. However, the response of the peripheral and central gustatory systems to these substances remains poorly explored. We measured the effect induced by pheromonal cuticular mixtures on (i) the electrophysiological response of peripheral gustatory receptor neurons, (ii) the calcium variation in brain centers receiving these gustatory inputs and (iii) the behavioral reaction induced in control males and in mutant desat1 males, which show abnormal pheromone production and perception. While male and female pheromones induced inhibitory-like effects on taste receptor neurons, the contact of male pheromones on male fore-tarsi elicits a long-lasting response of higher intensity in the dedicated gustatory brain center. We found that the behavior of control males was more strongly inhibited by male pheromones than by female pheromones, but this difference disappeared in anosmic males. Mutant desat1 males showed an increased sensitivity of their peripheral gustatory neurons to contact pheromones and a behavioral incapacity to discriminate sex pheromones. Together our data indicate that cuticular hydrocarbons induce long-lasting inhibitory effects on the relevant taste pathway which may interact with the olfactory pathway to modulate pheromonal perception