Independent Fractional Brownian Motions Research Articles

Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations

Abstract Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.

Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

The object of investigation is the mixed fractional Brownian motion of the form ${X_{t}}=\kappa {B_{t}^{{H_{1}}}}+\sigma {B_{t}^{{H_{2}}}}$, driven by two independent fractional Brownian motions ${B_{1}^{H}}$ and ${B_{2}^{H}}$ with Hurst parameters ${H_{1}}\lt {H_{2}}$. Strongly consistent estimators of unknown model parameters ${({H_{1}},{H_{2}},{\kappa ^{2}},{\sigma ^{2}})^{\top }}$ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for $0\lt {H_{1}}\lt {H_{2}}\lt \frac{3}{4}$.

On the Sum of Gaussian Martingale and an Independent Fractional Brownian Motion

On the Sum of Gaussian Martingale and an Independent Fractional Brownian Motion

Correlation Structure of Time-Changed Generalized Mixed Fractional Brownian Motion

The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. In this paper, we investigate the long-time behavior of gmfBm when it is time-changed by a tempered stable subordinator or a gamma process. As a main result, we show that the time-changed process exhibits a long-range dependence property under some conditions on the Hurst indices. The time-changed gmfBm can be used to model natural phenomena that exhibit long-range dependence, even when the underlying process is not itself long-range dependent.

On the sum of Gaussian martingale and an independent fractional Brownian motion

В том же контексте, что и в основополагающей работе П. Черидито "Mixed fractional Brownian motion" (2001 г.), мы изучаем семимартингальное свойство суммы некоторого гауссовского мартингала и независимого дробного броуновского движения с параметром Хeрста $H\in(0,1)$. В то же время мы подчеркиваем, что марковское свойство утрачивается, даже если мартингал им обладает.

Multi-mixed fractional Brownian motions and Ornstein–Uhlenbeck processes

The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.

Weak solutions for singular multiplicative SDEs via regularization by noise

We study multiplicative SDEs perturbed by an independent additive fractional Brownian motion. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients.

Large deviation principle for a mixed fractional and jump diffusion process

Abstract We study the asymptotic behavior of a solution of a mixed differential equation driven by independent fractional Brownian motion with Hurst index H ∈ ( 0 ; 1 ) {H\in(0;1)} and compensated Poisson process. This study consists in determining the uniform Freidlin–Wentzell estimates in a temporal distribution space.

Stochastic mortality dynamics driven by mixed fractional Brownian motion

Recently, the long-range dependence (LRD) of mortality dynamics has been identified and studied in the actuarial literature. The non-Markovian feature caused by LRD can raise new challenges in actuarial valuation and risk management. This paper proposes a new modeling approach that uses a combination of independent Brownian motion and fractional Brownian motion to achieve a flexible setting on capturing the LRD in mortality dynamics. The closed-form solutions of survival probabilities are derived for valuation and hedging purposes. To obtain mortality sensitivity measures in the presence of LRD, we develop a novel derivation method using directional derivatives. Our method is flexible in the sense that it can not only reflect the effect of LRD on mortality sensitivities, but also include some existing sensitivity measures as a special case. Finally, we provide a numerical illustration to analyze the performance of different sensitivity measures in a natural hedge of mortality risk.

Numerical approach to the drift parameter estimation in the model with two fractional Brownian motions

The article deals with numerical estimation of the drift parameter in the continuous-time linear model with two independent fractional Brownian motions. The main focus is given to the computational difficulties of the maximum likelihood approach, in particular, to the construction of the approximate solution to the Fredholm integral equation of the second kind with a singular kernel. We also introduce two alternative estimators and investigate their asymptotic properties. The performance of all estimators is studied numerically.

Linear Filtering with Fractional Noises: Large Time and Small Noise Asymptotics

The classical state-space approach to optimal estimation of stochastic processes is efficient when the driving noises are generated by martingales. In particular, the weight function of the optimal linear filter, which solves a complicated operator equation in general, simplifies to the Riccati ordinary differential equation in the martingale case. This reduction lies in the foundations of the Kalman--Bucy approach to linear optimal filtering. In this paper we consider a basic Kalman--Bucy model with noises, generated by independent fractional Brownian motions, and develop a new method of asymptotic analysis of the integro-differential filtering equation arising in this case. We establish existence of the steady-state error limit and find its asymptotic scaling in the high signal-to-noise regime. Closed form expressions are derived in a number of important cases.

Call warrants pricing formula under mixed-fractional Brownian motion with Merton jump-diffusion

Mixed fractional Brownian motion (MFBM) is a linear combination of a Brownian motion and an independent fractional Brownian motion which may overcome the problem of arbitrage, while a jump process in time series is another problem to be address in modeling stock prices. This study models call warrants with MFBM and includes the jump process in its dynamics. The pricing formula for a warrant with mixed-fractional Brownian motion and jump, is obtained via quasi-conditional expectation and risk-neutral valuation.

Averaging principle for BSDEs driven by two mutually independent fractional Brownian motions

In this paper, we study the stochastic averaging principle for backward stochastic differential equations driven by two mutually independent fractional Brownian motions (FrBSDEs in short). An averaged FrBSDEs for the original equations is proposed and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square.

Derivative for the intersection local time of two independent fractional Brownian motions

Let and be two independent fractional Brownian motions with respective indices and . In this paper, we consider the derivative of their intersection local time . We show that the derivative exists in for all and it is joint Hölder continuous in space and time if . In particular, we show that is differentiable in the spatial variable a under the condition, and we introduce the so-called hybrid fractional quadratic covariation . When , we construct a Banach space of measurable functions such that the quadratic covariation exists in and for all . When a similar result is proved to hold for all Hölder functions f of order .

Anticipated BSDEs driven by two mutually independent fractional Brownian motions with non-Lipschitz coefficients

Abstract This paper deals with a class of anticipated backward stochastic differential equations driven by two mutually independent fractional Brownian motions. We essentially establish the existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion

We obtain the high-order asymptotic expansion for the distribution of the quadratic variation of the mixed fractional Brownian motion, which is defined as the sum of a Brownian motion and an independent fractional Brownian motion. Our approach is based on the analysis of the cumulants of this sequence. We show that both the Brownian and fractional Brownian parts contribute to the asymptotic expansion.

Maximum likelihood estimators from discrete data modeled by mixed fractional Brownian motion with application to the Nordic stock markets

Mixed fractional Brownian motion is a linear combination of Brownian motion and independent Fractional Brownian motion that is extensively used for option pricing. The consideration of the mixed process is able to capture the long–range dependence property that financial time series exhibit. This paper examines the problem of deriving simultaneously the estimators of all the unknown parameters for a model driven by the mixed fractional Brownian motion using the maximum likelihood estimation method. The consistency and asymptotic normality properties of these estimators are provided. The performance of the methodology is tested on simulated data sets, and the outcomes illustrate that the maximum likelihood technique is efficient and reliable. An empirical application of the proposed method is also made to the real financial data from four Nordic stock market indices.

A Limit Law for Functionals of Multiple Independent Fractional Brownian Motions

A Limit Law for Functionals of Multiple Independent Fractional Brownian Motions

Impulsive stochastic fractional differential equations driven by fractional Brownian motion

In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index 1/2< H<1 under a non-Lipschitz condition with the Lipschitz one as a particular case. Our analysis depends on an approximation scheme of Carathéodory type. Some previous results are improved and extended.

Large deviation principle applied for a solution of mixed stochastic differential equation involving independent standard Brownian motion and fractional Brownian motion

Large deviation principle applied for a solution of mixed stochastic differential equation involving independent standard Brownian motion and fractional Brownian motion