scholarly journals Collision of eigenvalues for matrix-valued processes

2019 ◽  
Vol 09 (04) ◽  
pp. 2030001
Author(s):  
Arturo Jaramillo ◽  
David Nualart
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Gaussian Process ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Hermitian Matrix ◽  
Symmetric Matrix ◽  
Hurst Parameter ◽  
Real Symmetric Matrix ◽  
Hitting Probabilities

We examine the probability that at least two eigenvalues of a Hermitian matrix-valued Gaussian process, collide. In particular, we determine sharp conditions under which such probability is zero. As an application, we show that the eigenvalues of a real symmetric matrix-valued fractional Brownian motion of Hurst parameter [Formula: see text], collide when [Formula: see text] and do not collide when [Formula: see text], while those of a complex Hermitian fractional Brownian motion collide when [Formula: see text] and do not collide when [Formula: see text]. Our approach is based on the relation between hitting probabilities for Gaussian processes with the capacity and Hausdorff dimension of measurable sets.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (02) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (2) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

scholarly journals On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 991 ◽  
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Brownian Motion ◽  
Markov Process ◽  
Gaussian Process ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Statistical Parameters ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Over Time

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

Malliavin calculus used to derive a stochastic maximum principle for system driven by fractional Brownian and standard Wiener motions with application

2020 ◽  
Vol 28 (4) ◽  
pp. 291-306
Author(s):  
Tayeb Bouaziz ◽  
Adel Chala
Keyword(s):  
Brownian Motion ◽  
Maximum Principle ◽  
Control Problem ◽  
Fractional Brownian Motion ◽  
Stochastic Control ◽  
Malliavin Calculus ◽  
Stochastic Maximum Principle ◽  
Hurst Parameter ◽  
Principle Of Optimality ◽  
Initial Cost

AbstractWe consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter {H\in(\frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.

scholarly journals Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[

2019 ◽  
Vol 11 (1) ◽  
pp. 76
Author(s):  
Eric Djeutcha ◽  
Didier Alain Njamen Njomen ◽  
Louis-Aimé Fono
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Existence And Uniqueness ◽  
Hurst Parameter ◽  
Existence And Uniqueness Theorem ◽  
Martingale Approximation ◽  
Arbitrage Opportunities ◽  
Mixed Fractional Brownian Motion ◽  
Geometric Fractional Brownian Motion

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

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.

Harnack inequalities and applications for functional SDEs driven by fractional Brownian motion

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
Zhi Li ◽  
Jiaowan Luo
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Functional Differential Equations ◽  
Hurst Parameter ◽  
Stochastic Functional Differential Equations ◽  
Harnack Inequalities ◽  
Functional Differential ◽  
Stochastic Functional

AbstractIn this paper, Harnack inequalities are established for stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter

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