OCCUPATION TIMES OF BROWNIAN SEGMENTS AND THE σ-FINITE WIENER MEASURE

2005 ◽  
Vol 08 (02) ◽  
pp. 199-213
Author(s):  
TOMASZ BOJDECKI ◽  
LUIS G. GOROSTIZA
Keyword(s):  
Brownian Motion ◽  
Polish Space ◽  
Wiener Measure ◽  
Asymptotic Result ◽  
Main Role ◽  
Occupation Times ◽  
Borel Sets ◽  
New Information ◽  
Strengthened Form ◽  
Consecutive Time

We give an asymptotic result for the occupation of Borel sets of functions by the segments of recurrent Brownian motion on consecutive time intervals [n, n +1], n =0, 1, 2, …. This result provides new information on the behavior of Brownian motion, which is illustrated by examples. A formulation in terms of weak convergence of random measures on Polish space is also given. The proof is based on (a strengthened form of) the Darling–Kac occupation time theorem for Markov chains, and our result can be viewed as a "trajectorial" extension of that theorem. The main role in the occupation limit for Brownian segments is played by the σ-finite Wiener measure, which first appeared in a different context. An extension for segments of symmetric α-stable Lévy processes is also given.

scholarly journals Condensation of SIP Particles and Sticky Brownian Motion

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Mario Ayala ◽  
Gioia Carinci ◽  
Frank Redig
Keyword(s):  
Brownian Motion ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Self Duality ◽  
Inclusion Process ◽  
Interacting Particle ◽  
New Information ◽  
Homogeneous Product ◽  
Coarsening Dynamics ◽  
The Difference

AbstractWe study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP particles to be together at time t. This, combined with self-duality, allows us to obtain the explicit scaling for the variance of the fluctuation field.

THE FEYNMAN–KAC FORMULA AND PRICING OCCUPATION TIME DERIVATIVES

1999 ◽  
Vol 02 (02) ◽  
pp. 153-178 ◽  
Author(s):  
JULIEN-N. HUGONNIER
Keyword(s):  
Brownian Motion ◽  
Time Derivative ◽  
Occupation Time ◽  
Barrier Options ◽  
Occupation Times

In this paper, we undertake a study of occupation time derivatives that is derivatives for which the pay-off is contingent on both the terminal asset's price and one of its occupation times. To this end we use a formula of M. Kac to compute the joint law of Brownian motion and one of its occupation times. General pricing formulas for occupation time derivatives are established and it is shown that any occupation time derivative can be continuously hedged by a controlled portfolio of the basic securities. We further study some examples of interest including cumulative barrier options and discuss some numerical implementations.

scholarly journals Second-order limit laws for occupation times of fractional Brownian motion

2017 ◽  
Vol 54 (2) ◽  
pp. 444-461 ◽  
Author(s):  
Fangjun Xu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Method Of Moments ◽  
Second Order ◽  
Hurst Index ◽  
Occupation Times ◽  
Limit Laws ◽  
Additive Functionals ◽  
Finite Dimensional ◽  
Functional Version

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

scholarly journals Generalized functionals of Brownian motion

1994 ◽  
Vol 7 (3) ◽  
pp. 247-267
Author(s):  
N. U. Ahmed
Keyword(s):  
Brownian Motion ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Broad Class ◽  
Wiener Measure ◽  
Multiple Stochastic Integrals ◽  
Finite Dimensional ◽  
Nonlinear Functionals ◽  
Recent Developments ◽  
Locally Convex

In this paper we discuss some recent developments in the theory of generalized functionals of Brownian motion. First we give a brief summary of the Wiener-Ito multiple Integrals. We discuss some of their basic properties, and related functional analysis on Wiener measure space. then we discuss the generalized functionals constructed by Hida. The generalized functionals of Hida are based on L2-Sobolev spaces, thereby, admitting only Hs, s∈R valued kernels in the multiple stochastic integrals. These functionals are much more general than the classical Wiener-Ito class. The more recent development, due to the author, introduces a much more broad class of generalized functionals which are based on Lp-Sobolev spaces admitting kernels from the spaces 𝒲p,s, s∈R. This allows analysis of a very broad class of nonlinear functionals of Brownian motion, which can not be handled by either the Wiener-Ito class or the Hida class. For s≤0, they represent generalized functionals on the Wiener measure space like Schwarz distributions on finite dimensional spaces. In this paper we also introduce some further generalizations, and construct a locally convex topological vector space of generalized functionals. We also present some discussion on the applications of these results.

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

scholarly journals Occupation Times for Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (02) ◽  
pp. 549-565 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Laplace Transforms ◽  
Occupation Times ◽  
First Passage ◽  
Scale Functions ◽  
Positive Variation ◽  
Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

scholarly journals Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
Author(s):  
Hongzhong Zhang
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Time Horizon ◽  
Three Dimensional ◽  
Bessel Processes ◽  
Occupation Times ◽  
Exponential Time ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

scholarly journals Some Measure-Valued Markov Processes Attached to Occupation Times of Brownian Motion

Bernoulli ◽  
2000 ◽  
Vol 6 (1) ◽  
pp. 63 ◽  
Author(s):  
Catherine Donati-Martin ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Markov Processes ◽  
Occupation Times

Modeling liquidation risk with occupation times

2016 ◽  
Vol 03 (04) ◽  
pp. 1650028 ◽  
Author(s):  
Roman N. Makarov
Keyword(s):  
Brownian Motion ◽  
Structural Model ◽  
Cox Model ◽  
Analytical Formula ◽  
Diffusion Models ◽  
Geometric Brownian Motion ◽  
Occupation Times ◽  
Proposed Model ◽  
Asset Value ◽  
Limiting Case

In this paper, we develop a new structural model that allows for a distinction between default and liquidation to be made. Default occurs when firm’s asset value process crosses a bankruptcy barrier. Here, we do not assume that default immediately triggers liquidation. Instead, the firm is allowed to continue operating even if it is in default. Liquidation is triggered as soon as the firm’s asset value has cumulatively spent a prespecified amount of time below the default barrier or has dropped below the liquidation barrier. The proposed model includes the Black–Cox model as a limiting case. A semi-analytical formula of the liquidation probability is derived for the case where firm’s asset value follows a geometric Brownian motion. Nonlinear volatility diffusion models are discussed as well.

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