اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe exponential Lie series for continuous semimartingales
57
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct self-contained proof of the corresponding Chen-Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.
قيم البحث
اقرأ أيضاً
Let $mathbb{hat{E}}$ be the upper expectation of a weakly compact but non-dominated family $mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $mathcal{P}$-semimartingale under $mathbb{hat{E}}$. Given an open set $Qsubsetmathbb
{R}^{d}$, the exit time of $Y$ from $Q$ is defined by [ {tau}_{Q}:=inf{tgeq0:Y_{t}in Q^{c}}. ] The main objective of this paper is to study the quasi-continuity properties of ${tau}_{Q}$ under the nonlinear expectation $mathbb{hat{E}}$. Under some additional assumptions on the growth and regularity of $Y$, we prove that ${tau}_{Q}wedge t$ is quasi-continuous if $Q$ satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we get the quasi-continuity of exit times for multi-dimensional $G$-martingales, which nontrivially generalizes the previous one-dimensional result of Song.
We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows
exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $lambda_1$ for weak survival, and the survival probability $p(lambda)$ is continuous with respect to the infection rate $lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $lambda_1<lambda_2$, which confirms a conjecture of Staceys cite{Stacey}. We also prove that if the contact process survives strongly at $lambda$ then it survives strongly at a $lambda<lambda$, which implies that the process does not survive strongly at the critical value $lambda_2$ for strong survival.
This paper establishes It^os formula along a flow of probability measures associated with gene-ral semimartingales. This generalizes existing results for flow of measures on It^o processes. Our approach is to first prove It^os formula for cylindrical
polynomials and then use function approximation and localization techniques for the general case. This general form of It^os formula enables derivation of dynamic programming equations and verification theorems for McKean-Vlasov controls with jump diffusions and for McKean-Vlasov mixed regular-singular control problems. It also allows for generalizing the classical relation between the maximum principle and the dynamic programming principle to the McKean-Vlasov singular control setting, where the adjoint process is expressed in term of the derivative of the value function with respect to probability measures.
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes a
ffine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs.
We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer li
ne, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
