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Y. Ouknine

Publications and source records attributed to Y. Ouknine.

8 recordsLinked to original sources
arXiv2025

On Carath\'eodory approximate scheme for a class of one-dimensional doubly perturbed diffusion processes

In this paper, we introduce and study the convergence of new Carath\'eodory's approximate solution for one-dimensional $\alpha, \beta$-doubly perturbed stochastic differential equations (DPSDEs) with parameters $\alpha <1$ and $\beta <1$ such that $|\rho| < 1$, where $ \rho : = \frac{\alpha\beta}{(1-\alpha)(1-\beta)}$. Under Lipschitz's condition on the coefficients, we establish the $L^{p}$-convergence of the Carath\'eodory approximate solution uniformly in time, for all $p\geq 2$. As a consequence, and relying only on our scheme, we obtain the existence and uniqueness of strong solution for $\alpha, \beta$-DPSDEs. Furthermore, an extension to non-Lipschitz coefficients are also studied. Our results improve earlier work by Mao and al. (2018).

math.PR↗
arXiv2018

Non linear optimal stopping problem and Reflected BSDEs in the predictable setting

In the first part of this paper, we study RBSDEs in the case where the filtration is not quasi-left continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from the general theory of processes as the Merten's decomposition of predictable strong supermartingale. In the second part we introduce an optimal stopping problem indexed by predictable stopping times with the non linear predictable $g$ expectation induced by an appropriate BSDE. We establish some useful properties of ${\cal{E}}^{p,g}$-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

math.PR↗
arXiv2012

On semimartingale local time inequalities and applications in SDE's

Using the balayage formula, we prove an inequality between the measures associated to local times of semimartingales. Our result extends the "comparison theorem of local times" of Ouknine $(1988)$, which is useful in the study of stochastic differential equations. The inequality presented in this paper covers the discontinuous case. Moreover, we study the pathwise uniqueness of some stochastic differential equations involving local time of unknown process.

math.PR↗
arXiv2012

On the time inhomogeneous skew Brownian motion

This paper is devoted to the construction of a solution for the "Inhomogenous skew Brownian motion" equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Étoré and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.

math.PR↗
arXiv2011

Reflected Backward SDEs with General Jumps

In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. The reflecting process is right continuous with left limits (rcll for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the result in the general framework we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove the existence and uniqueness of the solution of the BSDE.

math.PR↗
arXiv2011

On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations with jumps

We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy to handled and don't need any approximation approach. Similar equations without jumps were studied in the same context by \cite{Le Gall}, \cite{Ouknine} and others authors. As an application we get a new condition on the pathwise uniqueness for the solutions to stochastic differential equations driven by a symmetric stable Lévy processes.

math.PR↗
arXiv2011

Stochastic Quadratic BSDE With Two RCLL Obstacles

We study the problem of existence of solutions for generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under weaker assumptions on the data. Roughly speaking we show the existence of a maximal solution for GRBSDE when the terminal condition ξis F_T-measurable, the coefficient f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and the reflecting barriers L and U are just right continuous left limited. The result is proved without assuming any P-integrability conditions.

math.PR↗