ترغب بنشر مسار تعليمي؟ اضغط هنا

Numerical Probabilistic Approach to MFG

50   0   0.0 ( 0 )
 نشر من قبل Houzhi Li
 تاريخ النشر 2018
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see cite{Lasry_Lions}, and by Huang, Caines, and Malham{e}, see cite{Huang}. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.



قيم البحث

اقرأ أيضاً

171 - Xinjia Chen 2013
We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration inequalities su ch as Chernoff bounds. We discover that the proposed approach is inherently related to statistical concepts such as monotone likelihood ratio, maximum likelihood, and the method of moments for parameter estimation. A connection between the proposed approach and the large deviation theory is also established. We show that, without using moment generating functions, tightest possible concentration inequalities may be readily derived by the proposed approach. We have derived new concentration inequalities using the proposed approach, which cannot be obtained by the classical approach based on moment generating functions.
The local volatility model is a widely used for pricing and hedging financial derivatives. While its main appeal is its capability of reproducing any given surface of observed option prices---it provides a perfect fit---the essential component is a l atent function which can be uniquely determined only in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimate representations. We look at the calibration problem in a probabilistic framework with a nonparametric approach based on a Gaussian process prior. This immediately gives a way of encoding prior beliefs about the local volatility function and a hypothesis model which is highly flexible yet not prone to over-fitting. Besides providing a method for calibrating a (range of) point-estimate(s), we draw posterior inference from the distribution over local volatility. This leads to a better understanding of uncertainty associated with the calibration in particular, and with the model in general. Further, we infer dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this provides predictive distributions not only locally, but also for entire surfaces forward in time. We apply our approach to S&P 500 market data.
We discuss a general formalism for numerically evolving initial data in general relativity in which the (complex) Ashtekar connection and the Newman-Penrose scalars are taken as the dynamical variables. In the generic case three gauge constraints and twelve reality conditions must be solved. The analysis is applied to a Petrov type {1111} planar spacetime where we find a spatially constant volume element to be an appropriate coordinate gauge choice.
The forecast of the time of arrival of a coronal mass ejection (CME) to Earth is of critical importance for our high-technology society and for any future manned exploration of the Solar System. As critical as the forecast accuracy is the knowledge o f its precision, i.e. the error associated to the estimate. We propose a statistical approach for the computation of the time of arrival using the drag-based model by introducing the probability distributions, rather than exact values, as input parameters, thus allowing the evaluation of the uncertainty on the forecast. We test this approach using a set of CMEs whose transit times are known, and obtain extremely promising results: the average value of the absolute differences between measure and forecast is 9.1h, and half of these residuals are within the estimated errors. These results suggest that this approach deserves further investigation. We are working to realize a real-time implementation which ingests the outputs of automated CME tracking algorithms as inputs to create a database of events useful for a further validation of the approach.
51 - C. Landim , D. Marcondes , I. Seo 2021
We provide a necessary and sufficient condition for the metastability of a Markov chain, expressed in terms of a property of the solutions of the resolvent equation. As an application of this result, we prove the metastability of reversible, critical zero-range processes starting from a configuration.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا