Time changes of Brownian motion and the conditional excursion theorem

1984 ◽  
pp. 83-99
Author(s):  
Paul McGill
Keyword(s):  
Brownian Motion ◽  
Time Changes

scholarly journals Strong approximation of continuous local martingales by simple random walks

2004 ◽  
Vol 41 (1) ◽  
pp. 101-126
Author(s):  
B. Székely ◽  
T. Szabados
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Rates Of Convergence ◽  
Quadratic Variation ◽  
Local Martingale ◽  
Motion Time ◽  
Time Changes ◽  
Necessary And Sufficient ◽  
Nested Sequence ◽  
The Given

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].

scholarly journals On fractional heat equation

2021 ◽  
Vol 24 (1) ◽  
pp. 73-87
Author(s):  
Anatoly N. Kochubei ◽  
Yuri Kondratiev ◽  
José Luís da Silva
Keyword(s):  
Brownian Motion ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Convolution Operators ◽  
Fractional Heat Equation ◽  
Long Time ◽  
Time Changes ◽  
Distributed Order

Abstract In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

scholarly journals Quadratic variation, models, applications and lessons

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dilip B. Madan ◽  
King Wang
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Physical Process ◽  
Continuous Time ◽  
Fourier Transforms ◽  
Quadratic Variation ◽  
Market Prices ◽  
Variance Swap ◽  
Time Changes ◽  
Quadratic Variations

<p style="text-indent:20px;">Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.</p>

scholarly journals CREDIT RISK MODELING USING TIME-CHANGED BROWNIAN MOTION

2009 ◽  
Vol 12 (08) ◽  
pp. 1213-1230 ◽  
Author(s):  
T. R. HURD
Keyword(s):  
Brownian Motion ◽  
Firm Value ◽  
Credit Derivatives ◽  
Distribution Functions ◽  
Reduced Form ◽  
Risk Modeling ◽  
First Passage ◽  
Wide Range ◽  
Time Changes ◽  
First Passage Problem

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard first passage problem for stochastic processes to capitalize on this time change structure and find that the distribution functions of such "first passage times of the second kind" are efficiently computable in a wide range of useful examples. Thus this new notion of first passage can be used to define the time of default in generalized structural credit models. Formulas for defaultable bonds and credit default swaps are given that are both efficiently computable and lead to realistic spread curves. Finally, we show that by treating joint firm value processes as dependent time changes of independent Brownian motions, one can obtain multifirm credit models with rich and plausible dynamics and enjoying the possibility of efficient valuation of portfolio credit derivatives.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
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.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
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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