Local Time Of Brownian Motion Research Articles

The Markov property of local times of Brownian motion indexed by the Brownian tree

The Markov property of local times of Brownian motion indexed by the Brownian tree

On local times of Martin-Löf random Brownian motion

In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-Löf random with respect to the Wiener measure has continuous effective local times at every computable point. Finally we obtain a new simple representation of classical Brownian local times, computationally expressed.

Calculation of Local Times of Brownian Motion in Pricing of Derivative Instruments

Calculation of Local Times of Brownian Motion in Pricing of Derivative Instruments

Distributions of Functionals of the Local Time of Brownian Motion with Discontinuous Drift

Distributions of Functionals of the Local Time of Brownian Motion with Discontinuous Drift

Symmetry in the Green’s Function for Birth-death Chains

A symmetric relation in the probabilistic Green’s function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green’s functions in other contexts. The first uses as primary tool the local time of Brownian motion, while the second uses the reciprocity principle from electric network theory. We also show that the the second proof extends easily to cover birth-death chains (a.k.a. state-dependent random walks) on trees, and can be adapted in order to derive hitting times on trees.

Stochastic differential equation for Brox diffusion

This paper studies the weak and strong solutions to the stochastic differential equation dX(t)=−12Ẇ(X(t))dt+dB(t), where (B(t),t≥0) is a standard Brownian motion and W(x) is a two sided Brownian motion, independent of B. It is shown that the Itô–McKean representation associated with any Brownian motion (independent of W) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ∫0TẆ(X(t))dt, the main idea is to use the concept of local time together with the polygonal approximation Wπ. Some new results on the local time of Brownian motion needed in our proof are established.

A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME INCREMENTS

Let [Formula: see text] denote the local time of Brownian motion. Our main result is to show that for each fixed t[Formula: see text] as h → 0, where η is a normal random variable with mean zero and variance one, that is independent of [Formula: see text]. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments.

Continuous differentiability of renormalized intersection local times in R1

We study $\gamma_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $\gamma_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.

The Integral of the Supremum Process of Brownian Motion

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

The Integral of the Supremum Process of Brownian Motion

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

Renormalization and convergence in law for the derivative of intersection local time in [formula omitted

In this paper we will examine the derivative of intersection local time of Brownian motion and symmetric stable processes in R 2 . These processes do not exist when defined in the canonical way. The purpose of this paper is to exhibit the correct rate for renormalization of these processes.

A stochastic flow arising in the study of local times

A stochastic flow of homeomorphisms of previously studied by Bass and Burdzy [2] and Hu and Warren [4] is shown to arise in the study of the local times of Brownian motion. This leads to a new proof of the Ray-Knight theorems for the flow via the classical Ray-Knight theorems for Brownian motion.

Pricing path-dependent options in a Black-Scholes market from the distribution of homogeneous Brownian functionals

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.

Pricing path-dependent options in a Black-Scholes market from the distribution of homogeneous Brownian functionals

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.

Unrenormalized intersection local time of Brownian motion and its local time representation

We consider the intersection local time of Brownian motion without renormalization through Itô-Wiener expansions. In order to recognize the existence, we extend the Watanabe space. We also discuss how to substitute Wiener functionals for parameters of a generalized Wiener functional. As a consequence a relationship between the unrenormalized intersection local time and the local time is clarified.

Small Scale Limit Theorems for the Intersection Local Times of Brownian Motion

In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the difference in the small scale behaviour of the intersection local times in three-dimensional space and in the plane by studying almost sure limit theorems motivated by the notion of average densities introduced by Bedford and Fisher. We show that in 3-space the intersection local time measure of two paths has an average density of order two with respect to the gauge function $\varphi(r)=r$, but in the plane, for the intersection local time measure of p Brownian paths, the average density of order two fails to converge. The average density of order three, however, exists for the gauge function $\varphi_p(r)=r^2[\log(1/r)]^p$. We also prove refined versions of the above results, which describe more precisely the fluctuations of the volume of small balls around these gauge functions by identifying the density distributions, or lacunarity distributions, of the intersection local times.

Anomalies in Multifractal Formalism for Local Time of Brownian Motion

The Renyi function for the logical time measure μ of Brownian motion is found. It is shown that this function, the Legendre transform of the multifractal spectrum of μ and the τ-function derived by the reciprocal measure formalism are not identical. More examples of μ having similar anomalies are discussed.

On the self intersection local time of Brownian motion-via chaos expansion

We discuss the weak compactness problem related to the self-intersection local time of Brownian motion. We also propose a regular renormalization for self-intersection local time of higher dimensional Brownian motion.

Chaos expansions of double intersection local time of Brownian motion in [formula omitted] and renormalization

Double intersection local times α( x,.) of Brownian motion W if R d which measure the size of the set of time pairs ( s, t), s ≠ t, for which W t and W s + x coincide can be developed into series of multiple Wiener-Ito integrals. These series representations reveal on the one hand the degree of smoothness of α( x,.) in terms of eventually negative order Sobolev spaces with respect to the canonical Dirichlet structure on Wiener space. On the other hand, they offer an easy access to renormalization of α( x,.) as | x| → 0. The results, valid for any dimension d, describe a pattern in which the well known cases d = 2, 3 are naturally embedded.

Quadratic Covariation and an Extension of Itô's Formula

Let X be a standard Brownian motion. We show that for any locally square integrable function f the quadratic covariation [ f(X),X] exists as the usual limit of sums converging in probability. For an absolutely continuous function F with derivative f , Ito's formula takes the form F (X t)=F(X 0)+∫ 0 tf(X s)dX s+1 2 [f(X),X] t . This is extended to the time-dependent case. As an example, we introduce the local time of Brownian motion at a continuous curve.