Search results for "brownian motion"
showing 7 items of 177 documents
Density flow over networks: A mean-field game theoretic approach
2014
A distributed routing control algorithm for dynamic networks has recently been presented in the literature. The networks were modeled using time evolution of density at network edges and the routing control algorithm allowed edge density to converge to a Wardrop equilibrium, which was characterized by an equal traffic density on all used paths. We borrow the idea and rearrange the density model to recast the problem within the framework of mean-field games. The contribution of this paper is three-fold. First, we provide a mean-field game formulation of the problem at hand. Second, we illustrate an extended state space solution approach. Third, we study the stochastic case where the density …
Geometric Brownian Motion (GBM) of Stock Indexes and Financial Market Uncertainty in the Context of Non-Crisis and Financial Crisis Scenarios
2022
The present article proposes a methodology for modeling the evolution of stock market indexes for 2020 using geometric Brownian motion (GBM), but in which drift and diffusion are determined considering two states of economic conjunctures (states of the economy), i.e., non-crisis and financial crisis. Based on this approach, we have found that the GBM proved to be a suitable model for making forecasts of stock market index values, as it describes quite well their future evolution. However, the model proposed by us, modified geometric Brownian motion (mGBM), brings some contributions that better describe the future evolution of stock indexes. Evidence in this regard was provided by analyzing …
Long-Time Behaviour for the Brownian Heat Kernel on a Compact Riemannian Manifold and Bismut’s Integration-by-Parts Formula
2007
We give a probabilistic proof of the classical long-time behaviour of the heat kernel on a compact manifold by using Bismut’s integration-by-parts formula.
Bayesian semiparametric long memory models for discretized event data
2020
We introduce a new class of semiparametric latent variable models for long memory discretized event data. The proposed methodology is motivated by a study of bird vocalizations in the Amazon rain forest; the timings of vocalizations exhibit self-similarity and long range dependence. This rules out Poisson process based models where the rate function itself is not long range dependent. The proposed class of FRActional Probit (FRAP) models is based on thresholding, a latent process. This latent process is modeled by a smooth Gaussian process and a fractional Brownian motion by assuming an additive structure. We develop a Bayesian approach to inference using Markov chain Monte Carlo and show g…
Approximation of heat equation and backward SDEs using random walk : convergence rates
2018
This thesis addresses questions related to approximation arising from the fields of stochastic analysis and partial differential equations. Theoretical results regarding convergence rates are obtained by using discretization schemes where the limiting process, the Brownian motion, is approximated by a simple discrete-time random walk. The rate of convergence is derived for a finite-difference approximation of the solution of a terminal value problem for the backward heat equation. This weak approximation result is proved for a terminal function which has bounded variation on compact sets. The sharpness of the according rate is achieved by applying some new results related to the first exit time …
Convergence of Measures
2020
One focus of probability theory is distributions that are the result of an interplay of a large number of random impacts. Often a useful approximation can be obtained by taking a limit of such distributions, for example, a limit where the number of impacts goes to infinity. With the Poisson distribution, we have encountered such a limit distribution that occurs as the number of very rare events when the number of possibilities goes to infinity (see Theorem 3.7). In many cases, it is necessary to rescale the original distributions in order to capture the behavior of the essential fluctuations, e.g., in the central limit theorem. While these theorems work with real random variables, we will a…
Liftings and extensions of operators in Brownian setting
2020
We investigate the operators T on a Hilbert space H which have 2-isometric liftings S with the property S ∗ S H ⊂ H . We show that such liftings are closely related to some extensions of T, which h...