Malliavin calculus for backward stochastic differential equations and application to numerical solutions

In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the $L^p$-H\"{o}lder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained $L^p$-H\"{o}lder continuity results. The main tool is the Malliavin calculus.Comment: Published in at http://dx.doi.org/10.1214/11-AAP762 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Management of a Eutrophied Coastal Ecosystem: Balancing Agricultural and Municipal Abatement Measures

Agriculture and municipal wastewater are the principal sources of eutrophying nutrients in many water ecosystems. We develop a model which considers the characteristics of agricultural and municipal nutrient abatement. The model explicitly accounts for the investment needed to set up wastewater treatment facilities, and makes it possible to determine the optimal timing of investment as well as the optimal agricultural and municipal abatement levels. We apply the model to the Finnish coastal waters of the Gulf of Finland. Our results indicate that substantial savings in abatement costs and the damage associated with eutrophication could be obtained by constructing the facilities needed to process all the wastewaters entering the coastal ecosystem. The optimal timing of investment is shown to hinge on both the economic and ecological characteristics of the ecosystem.Resource /Energy Economics and Policy,

Temporal Topic Analysis with Endogenous and Exogenous Processes

We consider the problem of modeling temporal textual data taking endogenous and exogenous processes into account. Such text documents arise in real world applications, including job advertisements and economic news articles, which are influenced by the fluctuations of the general economy. We propose a hierarchical Bayesian topic model which imposes a "group-correlated" hierarchical structure on the evolution of topics over time incorporating both processes, and show that this model can be estimated from Markov chain Monte Carlo sampling methods. We further demonstrate that this model captures the intrinsic relationships between the topic distribution and the time-dependent factors, and compare its performance with latent Dirichlet allocation (LDA) and two other related models. The model is applied to two collections of documents to illustrate its empirical performance: online job advertisements from DirectEmployers Association and journalists' postings on BusinessInsider.com

Bayesian functional linear regression with sparse step functions

The functional linear regression model is a common tool to determine the relationship between a scalar outcome and a functional predictor seen as a function of time. This paper focuses on the Bayesian estimation of the support of the coefficient function. To this aim we propose a parsimonious and adaptive decomposition of the coefficient function as a step function, and a model including a prior distribution that we name Bayesian functional Linear regression with Sparse Step functions (Bliss). The aim of the method is to recover areas of time which influences the most the outcome. A Bayes estimator of the support is built with a specific loss function, as well as two Bayes estimators of the coefficient function, a first one which is smooth and a second one which is a step function. The performance of the proposed methodology is analysed on various synthetic datasets and is illustrated on a black P\'erigord truffle dataset to study the influence of rainfall on the production

Ensemble updating of binary state vectors by maximising the expected number of unchanged components

In recent years, several ensemble-based filtering methods have been proposed and studied. The main challenge in such procedures is the updating of a prior ensemble to a posterior ensemble at every step of the filtering recursions. In the famous ensemble Kalman filter, the assumption of a linear-Gaussian state space model is introduced in order to overcome this issue, and the prior ensemble is updated with a linear shift closely related to the traditional Kalman filter equations. In the current article, we consider how the ideas underlying the ensemble Kalman filter can be applied when the components of the state vectors are binary variables. While the ensemble Kalman filter relies on Gaussian approximations of the forecast and filtering distributions, we instead use first order Markov chains. To update the prior ensemble, we simulate samples from a distribution constructed such that the expected number of equal components in a prior and posterior state vector is maximised. We demonstrate the performance of our approach in a simulation example inspired by the movement of oil and water in a petroleum reservoir, where also a more na\"{i}ve updating approach is applied for comparison. Here, we observe that the Frobenius norm of the difference between the estimated and the true marginal filtering probabilities is reduced to the half with our method compared to the na\"{i}ve approach, indicating that our method is superior. Finally, we discuss how our methodology can be generalised from the binary setting to more complicated situations

Learning from experience in the stock market

We study the dynamics of a Lucas-tree model with finitely lived agents who "learn from experience." Individuals update expectations by Bayesian learning based on observations from their own lifetimes. In this model, the stock price exhibits stochastic boom-and-bust fluctuations around the rational expectations equilibrium. This heterogeneous-agents economy can be approximated by a representative-agent model with constant-gain learning, where the gain parameter is related to the survival rate. JEL Classification: G12, D83, D84assett pricing, bubbles, Heterogeneous Agents, Learning from experience, OLG

Variational Walkback: Learning a Transition Operator as a Stochastic Recurrent Net

We propose a novel method to directly learn a stochastic transition operator whose repeated application provides generated samples. Traditional undirected graphical models approach this problem indirectly by learning a Markov chain model whose stationary distribution obeys detailed balance with respect to a parameterized energy function. The energy function is then modified so the model and data distributions match, with no guarantee on the number of steps required for the Markov chain to converge. Moreover, the detailed balance condition is highly restrictive: energy based models corresponding to neural networks must have symmetric weights, unlike biological neural circuits. In contrast, we develop a method for directly learning arbitrarily parameterized transition operators capable of expressing non-equilibrium stationary distributions that violate detailed balance, thereby enabling us to learn more biologically plausible asymmetric neural networks and more general non-energy based dynamical systems. The proposed training objective, which we derive via principled variational methods, encourages the transition operator to "walk back" in multi-step trajectories that start at data-points, as quickly as possible back to the original data points. We present a series of experimental results illustrating the soundness of the proposed approach, Variational Walkback (VW), on the MNIST, CIFAR-10, SVHN and CelebA datasets, demonstrating superior samples compared to earlier attempts to learn a transition operator. We also show that although each rapid training trajectory is limited to a finite but variable number of steps, our transition operator continues to generate good samples well past the length of such trajectories, thereby demonstrating the match of its non-equilibrium stationary distribution to the data distribution. Source Code: http://github.com/anirudh9119/walkback_nips17Comment: To appear at NIPS 201

First Passage Percolation on Inhomogeneous Random Graphs

We investigate first passage percolation on inhomogeneous random graphs. The random graph model G(n,kappa) we study is the model introduced by Bollob\'as, Janson and Riordan, where each vertex has a type from a type space S and edge probabilities are independent, but depending on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal weight path, properly normalized follows a central limit theorem. We handle the cases where lambda(n)->lambda is finite or infinite, under the assumption that the average number of neighbors lambda(n) of a vertex is independent of the type. The paper is a generalization the paper by Bhamidi, van der Hofstad and Hooghiemstra, where FPP is explored on the Erdos-Renyi graphs