scholarly journals Conditional Distributions of Processes Related to Fractional Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 166-183 ◽  
Author(s):  
Holger Fink ◽  
Claudia Klüppelberg ◽  
Martina Zähle
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
Bond Market ◽  
Market Model ◽  
Standard Brownian Motion ◽  
Conditional Distributions ◽  
Prediction Formula ◽  
Defaultable Bond ◽  
The Core

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

Conditional Distributions of Processes Related to Fractional Brownian Motion

2013 ◽  
Vol 50 (01) ◽  
pp. 166-183 ◽  
Author(s):  
Holger Fink ◽  
Claudia Klüppelberg ◽  
Martina Zähle
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
Bond Market ◽  
Market Model ◽  
Standard Brownian Motion ◽  
Conditional Distributions ◽  
Prediction Formula ◽  
Defaultable Bond ◽  
The Core

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

scholarly journals Prediction of fractional Brownian motion with Hurst index less than 1/2

2004 ◽  
Vol 70 (2) ◽  
pp. 321-328 ◽  
Author(s):  
V. V. Anh ◽  
A. Inoue
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Index ◽  
Prediction Formula

We give a proof based on an integral equation for an explicit prediction formula for fractional Brownian motion with Hurst index less than 1/2.

ON BOUNCING GEOMETRIC BROWNIAN MOTIONS

2018 ◽  
Vol 33 (4) ◽  
pp. 591-617
Author(s):  
Xin Liu ◽  
Vidyadhar G. Kulkarni ◽  
Qi Gong
Keyword(s):  
Brownian Motion ◽  
Standard Brownian Motion ◽  
Prediction Formula ◽  
Limit Order ◽  
Brownian Motions ◽  
Meeting Point ◽  
Order Books ◽  
Geometric Brownian Motions ◽  
Limit Order Books

A pair of bouncing geometric Brownian motions (GBMs) is studied. The bouncing GBMs behave like GBMs except that, when they meet, they bounce off away from each other. The object of interest is the position process, which is defined as the position of the latest meeting point at each time. We study the distributions of the time and position of their meeting points, and show that the suitably scaled logarithmic position process converges weakly to a standard Brownian motion as the bounce size δ→0. We also establish the convergence of the bouncing GBMs to mutually reflected GBMs as δ→0. Finally, applying our model to limit order books, we derive a simple and effective prediction formula for trading prices.

scholarly journals On the Hurst exponents, Markov processes, and fractional Brownian motion

2021 ◽  
Author(s):  
Ginno Millán
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
The Other ◽  
Time Interval ◽  
Infinite Time Interval ◽  
Point Density ◽  
The Difference ◽  
The One ◽  
Hurst Exponents

There is much confusion in the literature over Hurst exponent (H). The purpose of this paper is to illustrate the difference between fractional Brownian motion (fBm) on the one hand and Gaussian Markov processes where H is different to 1/2 on the other. The difference lies in the increments, which are stationary and correlated in one case and nonstationary and uncorrelated in the other. The two- and one-point densities of fBm are constructed explicitly. The two-point density does not scale. The one-point density for a semi-infinite time interval is identical to that for a scaling Gaussian Markov process with H different to 1/2 over a finite time interval. We conclude that both Hurst exponents and one-point densities are inadequate for deducing the underlying dynamics from empirical data. We apply these conclusions in the end to make a focused statement about nonlinear diffusion.

On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions

2020 ◽  
pp. 1-32
Author(s):  
Nguyen Huy Tuan ◽  
Tomás Caraballo ◽  
Tran Ngoc Thach
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Wiener Process ◽  
Mild Solution ◽  
Parabolic Equations ◽  
Regularization Method ◽  
Hurst Parameter ◽  
Standard Brownian Motion ◽  
Brownian Motions ◽  
Fractional Brownian Motions

In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h ∈ ( 1 2 , 1 ) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guaranteed. Additionally, we show clearly that the solution of each problem is not stable, which leads to the ill-posedness of each problem. Finally, we propose two regularization results for both considered problems by using the filter regularization method.

scholarly journals A Bayesian sequential test for the drift of a fractional Brownian motion

2020 ◽  
Vol 52 (4) ◽  
pp. 1308-1324
Author(s):  
Alexey Muravlev ◽  
Mikhail Zhitlukhin
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Optimal Stopping ◽  
Exit Time ◽  
Sequential Test ◽  
Standard Brownian Motion ◽  
Optimal Stopping Problem ◽  
First Exit Time ◽  
Bayesian Sequential Test ◽  
Drift Coefficient

AbstractWe consider a fractional Brownian motion with linear drift such that its unknown drift coefficient has a prior normal distribution and construct a sequential test for the hypothesis that the drift is positive versus the alternative that it is negative. We show that the problem of constructing the test reduces to an optimal stopping problem for a standard Brownian motion obtained by a transformation of the fractional Brownian motion. The solution is described as the first exit time from some set, and it is shown that its boundaries satisfy a certain integral equation, which is solved numerically.

scholarly journals Hurst exponents, Markov processes, and fractional Brownian motion

2007 ◽  
Vol 379 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Joseph L. McCauley ◽  
Gemunu H. Gunaratne ◽  
Kevin E. Bassler
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
Hurst Exponents
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