scholarly journals On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion

2006 ◽  
Vol 76 (9) ◽  
pp. 907-912 ◽  
Author(s):  
Ivan Nourdin ◽  
Thomas Simon
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Absolute Continuity ◽  
One Dimensional ◽  
The Absolute

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (02) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

scholarly journals Generalized Mean-Field Fractional BSDEs With Non-Lipschitz Coefficients

2021 ◽  
Vol 10 (3) ◽  
pp. 77
Author(s):  
Qun Shi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Comparison Theorem ◽  
Lipschitz Condition ◽  
Linear Growth ◽  
Mean Field ◽  
One Dimensional ◽  
Generalized Mean

In this paper we consider one dimensional generalized mean-field backward stochastic differential equations (BSDEs) driven by fractional Brownian motion, i.e., the generators of our mean-field FBSDEs depend not only on the solution but also on the law of the solution. We first give a totally new comparison theorem for such type of BSDEs under Lipschitz condition. Furthermore, we study the existence of the solution of such mean-field FBSDEs when the coefficients are only continuous and with a linear growth.

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (2) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

scholarly journals On the Solution of Fractional Option Pricing Model by Convolution Theorem

2019 ◽  
pp. 143-157
Author(s):  
A. I. Chukwunezu ◽  
B. O. Osu ◽  
C. Olunkwa ◽  
C. N. Obi
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Long Memory ◽  
Hurst Exponent ◽  
Convolution Theorem ◽  
Pricing Model ◽  
One Dimensional ◽  
Black Scholes ◽  
Option Pricing Model

The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.

scholarly journals A Variant of the Necessary Condition for the Absolute Continuity of Symmetric Multivariate Mixture

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1505
Author(s):  
Evgeniy Anatolievich Savinov
Keyword(s):  
Absolute Continuity ◽  
Joint Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Necessary Condition ◽  
Conditional Distributions ◽  
One Dimensional ◽  
The Absolute

Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension n>1 this occurs for a sufficiently large number of discontinuity points of one-dimensional conditional distributions.

Absolute continuity of distributions of one-dimensional Lévy processes

2017 ◽  
Vol 54 (3) ◽  
pp. 852-872
Author(s):  
Tongkeun Chang
Keyword(s):  
Absolute Continuity ◽  
Lévy Processes ◽  
Levy Processes ◽  
Previous Literature ◽  
One Dimensional ◽  
The Absolute

Abstract In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.

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