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Stochastic differential equations driven by an additive fractional Brownian sheet
Oussama El Barrimi,Youssef Ouknine 대한수학회 2019 대한수학회보 Vol.56 No.2
In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian sheet with Hurst parameters $H,H' > 1/2$, and a drift coefficient satisfying the linear growth condition. The result is obtained using a suitable Girsanov theorem for the fractional Brownian sheet.
Some stability results for semilinear stochastic heat equation driven by a fractional noise
Oussama El Barrimi,Youssef Ouknine 대한수학회 2019 대한수학회보 Vol.56 No.3
In this paper, we consider a semilinear stochastic heat equation driven by an additive fractional white noise. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application to the convergence of the Picard successive approximation.
SOME STABILITY RESULTS FOR SEMILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE
El Barrimi, Oussama,Ouknine, Youssef Korean Mathematical Society 2019 대한수학회보 Vol.56 No.3
In this paper, we consider a semilinear stochastic heat equation driven by an additive fractional white noise. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application to the convergence of the Picard successive approximation.
STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY AN ADDITIVE FRACTIONAL BROWNIAN SHEET
El Barrimi, Oussama,Ouknine, Youssef Korean Mathematical Society 2019 대한수학회보 Vol.56 No.2
In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian sheet with Hurst parameters H, H' > 1/2, and a drift coefficient satisfying the linear growth condition. The result is obtained using a suitable Girsanov theorem for the fractional Brownian sheet.
Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes
Mohamed El Machkouri,Khalifa Es-Sebaiy,Youssef Ouknine 한국통계학회 2016 Journal of the Korean Statistical Society Vol.45 No.3
The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein–Uhlenbeck process defined as dXt = θXt dt + dGt , t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimatorθt of θ based on the observation {Xs, s ∈ [0, t]} as t → ∞. Our approach offers an elementary, unifying proof of Belfadli (2011), and it allows to extend the result of Belfadli (2011) to the case when G is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.