scholarly journals Applying the IR statistic to estimate the Hurst index of the fractional geometric Brownian motion

2010 ◽  
Vol 51 ◽  
Author(s):  
Dimitrij Melichov
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Lévy Process ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Levy Process ◽  
Hurst Index ◽  
Standard Brownian Motion ◽  
Increment Ratio

In 2010 J.M. Bardet and D. Surgailis [1] have introduced the increment ratio (IR) statistic which measures the roughness of random paths. It was shown that this statistic was applicable in the cases of diffusion processes driven by the standard Brownian motion, certain Gaussian processes and the Lévy process. This paper shows that the IR statistic can be applied to estimate the Hurst index H of the fractional geometric Brownian motion.

scholarly journals Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

2012 ◽  
Vol 153 (2) ◽  
pp. 215-234 ◽  
Author(s):  
YUVAL PERES ◽  
PERLA SOUSI
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Lévy Process ◽  
Dirichlet Space ◽  
Levy Process ◽  
Standard Brownian Motion ◽  
Double Points ◽  
Hitting Probabilities

AbstractBy the Cameron–Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. We start by establishing a general 0-1 law, which in particular implies that for any fixed f, the Hausdorff dimension of the image and the graph of B + f are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 2, the Hausdorff dimension of the image of B + f is a.s. at least the maximum of 2 and the dimension of the image of f.

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

scholarly journals Research on Systemic Risk of the Turkish Banking Industry Based on a Systemic Risk Measurement Framework of the Fractional Brownian Motion

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Hong Fan ◽  
Lingli Feng ◽  
Ruoyu Zhou
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Systemic Risk ◽  
Banking System ◽  
Risk Measurement ◽  
Geometric Brownian Motion ◽  
Hurst Index ◽  
Measurement Framework ◽  
Asset Value

Since the 2008 financial crisis, it is an important issue to assess the systemic risk of banks, but there is a lack of research on the assessment of the systemic risk of Turkey’s financial system. In addition, geometric Brownian motion is used in most of the assessment frameworks of systemic risk under the normal financial market state, while the Turkish financial market has the situation of spike and thick tail. Therefore, this paper proposes a fractional Brownian motion measurement framework of systemic risk to study the systemic risk of the Turkish financial system. Firstly, this paper uses the data of 11 Turkish listed banks from 2014 to 2019 to conduct a normality test and demonstrate that its market has the characteristics of a fractal market; that is, there is a spike and thick tail distribution phenomenon in the stock price trend. Then, this paper proposes a fractional Brownian motion systemic risk measurement framework (fBSM). Based on the proposed theoretical framework and the actual data of Turkish listed banks from 2014 to 2019, a dynamically evolving Turkish banking network system is constructed to measure the systemic risk in the Turkish banking system. The research results find that the systemic risk is the highest in 2017, which then improved and gradually recovered. In addition, when analyzing the sensitivity of the Hurst index, it shows that with the increase in Hurst index, the Hurst index elasticity of Turkish banks’ asset value increases gradually and the asset value also increases continuously. Hence, the Hurst index has a greater impact on asset value. Therefore, the measurement framework of systemic risk based on the fBSM can better monitor the systemic risk than the traditional geometric Brownian motion in the Turkish banking system.

scholarly journals LÉVY FLIGHT SUPERDIFFUSION: AN INTRODUCTION

2008 ◽  
Vol 18 (09) ◽  
pp. 2649-2672 ◽  
Author(s):  
A. A. DUBKOV ◽  
B. SPAGNOLO ◽  
V. V. UCHAIKIN
Keyword(s):  
Brownian Motion ◽  
Characteristic Function ◽  
Lévy Process ◽  
Levy Process ◽  
Lévy Flight ◽  
Lévy Flights ◽  
Levy Flight ◽  
Barrier Crossing ◽  
Lévy Motion ◽  
Levy Flights

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

ORTHOGONAL DECOMPOSITIONS FOR LÉVY PROCESSES WITH AN APPLICATION TO THE GAMMA, PASCAL, AND MEIXNER PROCESSES

2003 ◽  
Vol 06 (01) ◽  
pp. 73-102 ◽  
Author(s):  
EUGENE LYTVYNOV
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Lévy Process ◽  
Lévy Processes ◽  
Random Variable ◽  
Levy Processes ◽  
Levy Process ◽  
Nuclear Space ◽  
Multiple Stochastic Integrals ◽  
Orthogonal Decompositions

It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists of representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of this paper are to develop the three approaches in the case of a general (ℝ-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

Reflected generalized BSDEs with discontinuous barriers driven by a Lévy process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed El Otmani
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Backward Stochastic Differential Equations ◽  
Levy Process ◽  
Uniqueness Of The Solution ◽  
Teugels Martingales

Abstract This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles.

Rare event simulation for diffusion processes via two-stage importance sampling

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Adam Metzler ◽  
Alexandre Scott
Keyword(s):  
Brownian Motion ◽  
Importance Sampling ◽  
Diffusion Processes ◽  
Practical Importance ◽  
Rare Event ◽  
Region Of Interest ◽  
Sampling Procedure ◽  
Importance Measure ◽  
Standard Brownian Motion ◽  
Two Stage

Abstract.We consider the problem of estimating expected values of functionals of real-valued diffusions over regions in path space that have very small probability. We propose a two-stage importance sampling procedure that first converts the problem into one involving standard Brownian motion and then addresses the rare event problem in this simpler setting. In order to identify an effective yet practical importance measure we propose using a time-dependent deterministic drift that minimizes the relative entropy between the corresponding importance measure and the conditional law of the standard Brownian motion, given that its trajectory lies in the region of interest. We provide numerical evidence that (i) our entropy-based criteria performs favourably with an alternative, but less general and less practical, criteria based on large deviations and (ii) our two-stage procedure performs admirably in cases where the region of interest is so rare that crude estimators fail completely.

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