Rates of convergence for increments of Brownian motion

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

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Discrete-type approximations for non-Markovian optimal stopping problems: Part I

Journal of Applied Probability ◽

10.1017/jpr.2019.57 ◽

2019 ◽

Vol 56 (4) ◽

pp. 981-1005 ◽

Cited By ~ 1

Author(s):

Dorival Leão ◽

Alberto Ohashi ◽

Francesco Russo

Keyword(s):

Brownian Motion ◽

Optimal Stopping ◽

Stopping Time ◽

Rates Of Convergence ◽

Stopping Times ◽

Full Generality ◽

Optimal Values ◽

Optimal Stopping Problems ◽

Optimal Stopping Times ◽

Discrete Type

AbstractWe present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

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Feeling boundary by Brownian motion in a ball

Asymptotic Analysis ◽

10.3233/asy-211734 ◽

2021 ◽

pp. 1-34

Author(s):

G. Serafin

Keyword(s):

Brownian Motion ◽

Heat Kernel ◽

Rates Of Convergence ◽

Half Line ◽

Short Time Asymptotics ◽

Time Asymptotics ◽

Explicit Formulae ◽

Short Time ◽

Dirichlet Heat Kernel

We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous “principle of not feeling the boundary” in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.

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Rates of Convergence to Stationarity for Reflected Brownian Motion

Mathematics of Operations Research ◽

10.1287/moor.2019.1006 ◽

2020 ◽

Vol 45 (2) ◽

pp. 660-681

Author(s):

Jose Blanchet ◽

Xinyun Chen

Keyword(s):

Brownian Motion ◽

Rates Of Convergence ◽

Reflected Brownian Motion

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Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200003041 ◽

2007 ◽

Vol 44 (02) ◽

pp. 393-408 ◽

Cited By ~ 1

Author(s):

Allan Sly

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

White Noise ◽

Gaussian Process ◽

Discrete Data ◽

Hölder Exponent ◽

Scaling Properties ◽

Multifractional Brownian Motion ◽

Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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Brownian motion with quadratic killing and some implications

Journal of Applied Probability ◽

10.1017/s0021900200116079 ◽

1986 ◽

Vol 23 (04) ◽

pp. 893-903 ◽

Cited By ~ 2

Author(s):

Michael L. Wenocur

Keyword(s):

Brownian Motion ◽

Weibull Distribution ◽

Special Function ◽

Function Theory ◽

Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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