Youssef Ouknine

Regularization of differential equations by fractional noise

Let {BtH,t[set membership, variant][0,T]} be a fractional Brownian motion with Hurst parameter H. We prove the existence and uniqueness of a strong solution for a stochastic differential equation of the form , where b(s,x) is a bounded Borel function with linear growth in x (case ) or a Holder continuous function of order strictly larger than 1-1/2H in x and than in time (case ).

REGULARIZATION OF QUASILINEAR HEAT EQUATIONS BY A FRACTIONAL NOISE

We show the existence and uniqueness of a solution for a quasilinear parabolic equation in one dimension driven by an additive fractional white noise, assuming that the drift is measurable and satisfies a suitable integrability condition. The proof is based on Girsanov theorem and lower estimates of the density of the solution of the equation without drift.

Reflected backward stochastic differential equations with jumps

A backward stochastic differential equation of the Wiener -Poisson type is considered in a d-dimensional convex and bounded region. By using a penalization argument on the domain, we are able to prove the existence and uniqueness of solutions. Moreover, the reflecting process is absolutely continuous

Hyperbolic Stochastic Partial Differential Equations with Additive Fractional Brownian Sheet

Let be a fractional Brownian sheet with Hurst parameters H, H′ ≤ 1/2. We prove the existence and uniqueness of a strong solution for a class of hyperbolic stochastic partial differential equations with additive fractional Brownian sheet of the form , where b(ζ, x) is a Borel function satisfying some growth and monotonicity assumptions. We also prove the convergence of Eulers approximation scheme for this equation.

Backward stochastic differential equation with local time

In this paper we deal with the following backward stochastic differential equation: where W is a d-dimensional Brownian motion is the symmetric local time of Fat the level a, v is a signed measure on is a -measurable random variable in and is an adapted map from to . If h is continuous with linear growth, we show the existence of a solution (Y,Z) for this backward equation. Some applications of this result, in connection with partial differential equations, and with linear quadratic stochastic control problem, are also given

Stochastic Differential Equations with Additive Fractional Noise and Locally Unbounded Drift

Let \(\{ B_{t}^{{H,}}t \in [0,T]\}\) be a fractional Brownian motion with Hurst parameter \(H < \tfrac{1}{2}\). We prove the existence and uniqueness of a strong solution for a stochastic differential equation of the form \({{X}_{t}} = {{x}_{0}} + B_{t}^{H} + \smallint _{0}^{t}b(s,{{X}_{s}})ds\), where b(s, x) is not locally bounded and satisfies a suitable integrability condition.

Local times of functions of continuous semimartingales

The paper provides a review of formulae related to the local times of functions of continuous semimartingales. We present a unified approach to this problem based on the Ito-Tanaka formula and the density of occupation times formula. In the appendix an application to a problem of mathematical finance is given

On a general result for backward stochastic differential equations

The existence of the solution of a general infinite dimensional backward stochastic differential equation is discussed. In our setting, we generalize many works concerning the existence problem (by a new approach).

On countably skewed Brownian motion with accumulation point

In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in

Infinite dimensional BSDE with jumps

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