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Search Results: 1 - 10 of 162473 matches for " Richard F. Bass "
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Stochastic differential equations with jumps
Bass Richard F.
Probability Surveys , 2004,
Regularity results for stable-like operators
Richard F. Bass
Mathematics , 2008,
Abstract: For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ we consider the same operator but where the $\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in $C^{\alpha+\beta}$ if $f\in C^\beta$.
A stochastic differential equation with a sticky point
Richard F. Bass
Mathematics , 2012,
Abstract: We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution exist.
The rate of escape of the most visited site of Brownian motion
Richard F. Bass
Mathematics , 2013,
Abstract: Let $\{L^z_t\}$ be the jointly continuous local times of a one-dimensional Brownian motion and let $L^*_t=\sup_{z\in \mathbb R} L^z_t$. Let $V_t$ be any point $z$ such that $L^z_t=L^*_t$, a most visited site of Brownian motion. Lifshits and Shi conjectured that if $\gamma>1$, then \[\liminf_{t\to \infty} \frac{|V_t|}{\sqrt t/(\log t)^\gamma}=\infty, \qquad {\rm a.s.}, \] with an analogous result for simple random walk. Version 1 purported to prove the conjecture; we point out an error in the proof.
Stochastic differential equations with jumps
Richard F. Bass
Mathematics , 2003, DOI: 10.1214/154957804100000015
Abstract: This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions.
The measurability of hitting times
Richard F. Bass
Mathematics , 2010,
Abstract: Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems foroptional and predictable sets are easy corollaries of the proof.
A stability theorem for elliptic Harnack inequalities
Richard F. Bass
Mathematics , 2011,
Abstract: We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.
An almost sure invariance principle for renormalized intersection local times
Richard F. Bass,Jay Rosen
Mathematics , 2004,
Abstract: Let \beta_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+\delta moments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |\beta_k(n)-\gamma_k(n)|=O(n^{-a}), a.s. for some a>0 where \gamma_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.
Pathwise uniqueness for two dimensional reflecting Brownian motion in Lipschitz domains
Richard F. Bass,Krzysztof Burdzy
Mathematics , 2005,
Abstract: We give a simple proof that in a Lipschitz domain in two dimensions with Lipschitz constant one, there is pathwise uniqueness for the Skorokhod equation governing reflecting Brownian motion.
On pathwise uniqueness for reflecting Brownian motion in $C^{1+γ}$ domains
Richard F. Bass,Krzysztof Burdzy
Mathematics , 2007, DOI: 10.1214/08-AOP390
Abstract: Pathwise uniqueness holds for the Skorokhod stochastic differential equation in $C^{1+\gamma}$ domains in $\mathbb{R}^d$ for $\gamma >1/2$ and $d\geq3$.
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