2,421 research outputs found
Large deviations for Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton-Jacobi equation
We prove the large deviation principle for the trajectory of a broad class of
mean field interacting Markov jump processes via a general analytic approach
based on viscosity solutions. Examples include generalized Ehrenfest models as
well as Curie-Weiss spin flip dynamics with singular jump rates.
The main step in the proof of the large deviation principle, which is of
independent interest, is the proof of the comparison principle for an
associated collection of Hamilton-Jacobi equations.
Additionally, we show that the large deviation principle provides a general
method to identify a Lyapunov function for the associated McKean-Vlasov
equation
String and Membrane Gaussian Processes
In this paper we introduce a novel framework for making exact nonparametric
Bayesian inference on latent functions, that is particularly suitable for Big
Data tasks. Firstly, we introduce a class of stochastic processes we refer to
as string Gaussian processes (string GPs), which are not to be mistaken for
Gaussian processes operating on text. We construct string GPs so that their
finite-dimensional marginals exhibit suitable local conditional independence
structures, which allow for scalable, distributed, and flexible nonparametric
Bayesian inference, without resorting to approximations, and while ensuring
some mild global regularity constraints. Furthermore, string GP priors
naturally cope with heterogeneous input data, and the gradient of the learned
latent function is readily available for explanatory analysis. Secondly, we
provide some theoretical results relating our approach to the standard GP
paradigm. In particular, we prove that some string GPs are Gaussian processes,
which provides a complementary global perspective on our framework. Finally, we
derive a scalable and distributed MCMC scheme for supervised learning tasks
under string GP priors. The proposed MCMC scheme has computational time
complexity and memory requirement , where
is the data size and the dimension of the input space. We illustrate the
efficacy of the proposed approach on several synthetic and real-world datasets,
including a dataset with millions input points and attributes.Comment: To appear in the Journal of Machine Learning Research (JMLR), Volume
1
Modelling FX smile : from stochastic volatility to skewness
Imperial Users onl
Interest rate models with Markov chains
Imperial Users onl
Advances in stabilisation of hybrid stochastic differential equations by delay feedback control
A novel approach to design the feedback control based on past states is proposed for hybrid stochastic differential equations (HSDEs). This new theorem builds up the connection between the delay feedback control and the control function without delay terms, which enables one to construct the delay feedback control using the existing results on stabilities of HSDEs. Methods to find the upper bound of the length of the time delay are also investigated. Numerical simulations are presented to demonstrate the new theorem
LMI Approach to Exponential Stability and Almost Sure Exponential Stability for Stochastic Fuzzy Markovian-Jumping Cohen-Grossberg Neural Networks with Nonlinear p-Laplace Diffusion
The robust exponential stability of delayed fuzzy Markovian-jumping Cohen-Grossberg neural networks (CGNNs) with nonlinear p-Laplace diffusion is studied. Fuzzy mathematical model brings a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form. But stochastic mathematical formulae are always described in matrix forms. By way of some variational methods in W1,p(Ω), Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the robust exponential stability and almost sure exponential robust stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer MatLab LMI toolbox. It is worth mentioning that even corollaries of the main results of this paper improve some recent related existing results. Moreover, some numerical examples are presented to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays
- …