Brownian motion and random walk perturbed at extrema

Let b _t be Brownian motion. We show there is a unique adapted process x _t which satisfies dx _t = db _t except when x _t is at a maximum or a minimum, when it receives a push, the magnitudes and directions of the pushes being the parameters of the process. For some ranges of the parameters this is already known. We show that if a random walk close to b _t is perturbed properly, its paths are close to those of x _t . Received: 15 October 1997 / Revised version: 18 May 1998     

An occupation time identity for reflecting Brownian motion with drift

Summary This note is about an occupation time identity derived in [14] for reflecting Brownian motion with drift ]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>-\mu<0,$ RBM($-\mu$), for short. The identity says that for RBM($-\mu$) in stationary state ]]>(I^{+}_t, I^{-}_t) \rr (t-G_t,D_t-t),\qquad t\in \mathbb{R},$$ where $G_t$ and $D_t$ denote the starting time and the ending time, respectively, of an excursion from 0 to 0 (straddling $t$) and $I^{+}_t$ and $I^{-}_t$ are the occupation times above and below, respectively, of the observed level at time $t$ during the excursion. Due to stationarity, the common distribution does not depend on $t.$ In fact, it is proved in [9] that the identity is true, under some assumptions, for all recurrent diffusions and stationary processes. In the null recurrent diffusion case the common distribution is not, of course, a probability distribution. The aim of this note is to increase understanding of the identity by studying the RBM($-\mu$) case via Ray--Knight theorems.     

The most visited site of Brownian motion and simple random walk

Summary Let L(t, x) be the local time at x for Brownian motion and for each t, let } 0;L(t,x) \vee L(t, - x) = \mathop {\sup }\limits_y L(t,y)\} $$ " align="middle" border="0"> , the absolute value of the most visited site for Brownian motion up to time t. In this paper we prove that ¯V(t) is transient and obtain upper and lower bounds for the rate of growth of ¯V(t). The main tools used are the Ray-Knight theorems and William's path decomposition of a diffusion. An invariance principle is used to get analogous results for simple random walks. We also obtain a law of the iterated logarithm for ¯V(t).This research was partially supported by NSF Grants MCS 83-00581 and MCS 83-03297     

The local time of iterated Brownian motion

We define and study the local time process {L ^*(x,t);x¹,t0} of the iterated Brownian motion (IBM) {H(t):=W ₁(|W ₂ (t)|); t0}, whereW ₁(·) andW ₂(·) are independent Wiener processes.Research supported by Hungarian National Foundation for Scientific Research, Grant No. T 016384.Research supported by an NSERC Canada Grant at Carleton University, Ottawa.Research supported by a PSC CUNY Grant, No. 6-66364.     

The intrinsic local time sheet of Brownian motion

Summary McGill showed that the intrinsic local time process (t, x), t 0, x , of one-dimensional Brownian motion is, for fixedt>0, a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale.     

Fractional Brownian motion with complex variance via random walk in the complex plane and applications

A model of complex-valued fractional Brownian motion has been built up recently as the limit of a random walk in the complex plane, but this model involves radial steps only. It is shown that, by using non-radial steps, this model can be easily extended to define a fractional Brownian motion with complex-valued variance. The relations between complex-valued Brownian motion and the heat equation of order n is clarified and mainly one obtains the general expression of the probability density functions for these processes. One shows that the maximum entropy principle (MPE) provides the probability density of the complex-valued fractional Brownian motion, exactly like for the standard Brownian motion. And lastly, one shows that the heat equation of order 2n (which is the Fokker–Planck equation (FPE) of the complex-valued Brownian motion) has a solution which is similar to that of the FPE of fractional order introduced before by the author, therefore, to some extent, an identification between the complex-valued model via random walk in the complex plane and the model involving a derivative of fractional order.     

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

Small deviation of Brownian motion with large drift

In this article, we obtain exact asymptotics of the sojourn probability of Brownian motion with large drift on a small curvilinear strip in a finite time interval.     

Brownian motion with drift on spaces with varying dimension

Many properties of Brownian motion on spaces with varying dimension (BMVD in abbreviation) have been explored in Chen and Lou (2018). In this paper, we study Brownian motion with drift on spaces with varying dimension (BMVD with drift in abbreviation). Such a process can be conveniently defined by a regular Dirichlet form that is not necessarily symmetric. Through the method of Duhamel’s principle, it is established in this paper that the transition density of BMVD with drift has the same type of two-sided Gaussian bounds as that for BMVD (without drift). As a corollary, we derive Green function estimate for BMVD with drift.     

Zeno's walk: A random walk with refinements

Summary. A self-modifying random walk on is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which is supported on a subset of having Hausdorff dimension less than , which we calculate by a theorem of Billingsley. By generating function techniques we then calculate the exponential rate of convergence of the process to its limit point, which may be taken as a bound for the convergence of the measure in the Wasserstein metric. We describe how the process may viewed as a random walk on the space of monotone piecewise linear functions, where moves are taken by successive compositions with a randomly chosen such function. Received: 20 November 1995 / In revised form: 14 May 1996     

Filtering of a reflected Brownian motion with respect to its local time

We consider a filtering problem when the state process is a reflected Brownian motion _Xt$X_t$ and the observation process is its local time _Λs${\Lambda }_s$, for s≤t$s\le t$. For this model we derive an approximation scheme based on a suitable interpolation of the observation process _Λt${\Lambda }_t$. The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azéma martingale.     

Brownian motion with a parabolic drift and airy functions

Summary Let {W(t): t } be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)=sup}t : W(t)–(t–a)² is maximal}. Then {V(a): a } is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas f _a(t)=(t–a)². We give an analytic expression for the infinitesimal generators of the processes a , in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in Groeneboom (1985), where the asymptotic distribution of the (standardized) L ₁-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process {W(t)–ct ²: t }, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods.This paper was awarded the Rollo Davidson prize 1985 (Cambridge, UK)