arXiv subjects

Youssef Ouknine

Publications and source records attributed to Youssef Ouknine.

At least 19 recordsLinked to original sources

arXiv2026

Generalized reflected BSDEs with irregular obstacles driven by RCLL increasing processes on general filtered space

We study generalized backward stochastic differential equations (GBSDEs) and generalized reflected backward stochastic differential equations (GRBSDEs) on a general filtered probability space satisfying the usual conditions, without assuming that the underlying filtration is quasi-left-continuous. The equations are driven by a prescribed predictable, bounded, nondecreasing RCLL process $A$, which acts as a possibly discontinuous stochastic clock, which we call a driver. We first establish a priori estimates, stability, existence, and uniqueness results for GBSDEs whose generator is Lipschitz continuous with respect to the state variable. Since $A$ may have jumps, the analysis is carried out in weighted spaces defined through the stochastic exponential $\mathcal{E}(\beta A)$. We then investigate GRBSDEs with an optional regulated lower obstacle. When the generator is independent of the state variable, we develop two complementary approaches. The first relies on a Snell-envelope representation and optimal stopping arguments, while the second is based on a modified penalization procedure adapted to the discontinuities of the right jumps of the obstacle. The general Lipschitz case is subsequently obtained through a fixed-point argument in an appropriate weighted Banach space.

math.PR↗

arXiv2026

On the Besov-Orlicz path regularity of some Gaussian processes

In this paper, we rely on the additive decomposition in law satisfied by a class of stochastic processes, combined with the well-known regulariy properties of fractional Brownian motion, to establish Besov-Orlicz regularity of their sample paths. This provides a unified and direct proof for a broad class of processes, including bifractional Brownian motion with parameters $H\in (0, 1]$, $ K\in (0, 2)$ such that $HK \in (0, 1)$, subfractional Brownian motion with Hurst parameter $H\in (0, 1)$, and certain class of self-similar processes. %associated with the stochastic heat equation.

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arXiv2026

Regularization of Hyperbolic Stochastic Partial Differential Equations By Two Fractional Brownian Sheets

In this paper, we establish existence and uniqueness of strong solutions for a stochastic differential equation driven by an additive noise given by the sum of two correlated fractional Brownian sheets with different Hurst parameters. Our analysis relies on techniques from two-parameter fractional calculus and a tailored version of Girsanov's theorem. The main challenge arises from the correlation between the two noises and the technical requirements for applying Girsanov's theorem in this setting. We show that, despite these difficulties, the additive noise regularizes the equation, allowing well-posedness under weak assumptions on the drift.

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arXiv2025

Multivalued backward stochastic differential equations with jumps and moving boundary

We prove existence and uniqueness for a one-dimensional multivalued backward stochastic differential equation with jumps. The equation involves a time-indexed family of maximal monotone operators $k_t(\cdot)$ associated with increasing functions $k(t,\cdot)$ taking values in $\mathbb{R}_-$ and having domains that are intervals with time-dependent boundaries. Existence is obtained by a penalization method under a Lipschitz condition on the driver in $(y,z)$, a monotonicity condition in the jump parameter $ψ$, square-integrability of the terminal condition and the driver, and local-in-time integrability conditions on $k(\cdot,y)$. We also address the extension to the case where the operators $k_t(\cdot)$ act on unbounded intervals.

math.PR↗

arXiv2025

On Malliavin differentiability and absolute continuity of one-dimensional doubly perturbed diffusion processes

In this paper, we establish Malliavin differentiability and absolute continuity for $α, β$-doubly perturbed diffusion process with parameters $α<1$ and $β<1$ such that $|ρ| < 1$, where $ ρ: = \frac{αβ}{(1-α)(1-β)}$. Furthermore, under some regularity conditions on the coefficients, we prove that the solution $X_t$ has a smooth density for all $t\in(0, t_0)$ for some finite number $t_0>0$. Our results recover earlier works by Yue and Zhang (2015) and Xue, Yue and Zhang (2016), and the proofs are based on the techniques of the Malliavin calculus.

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arXiv2024

Reflected Mckean-Vlasov stochastic differential equations with jumps in time-dependent domains

In this paper, we investigate the deterministic multidimensional Skorokhod problem with normal reflection in a family of time-dependent convex domains that are càdlàg with respect to the Hausdorff metric. We then show the existence and uniqueness of solutions to multidimensional McKean-Vlasov stochastic differential equations reflected in these time-dependent domains. Additionally, we derive stability properties with respect to the initial condition and the coefficients. Finally, we establish a propagation of chaos result.

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arXiv2024

Optimal Stopping Under Model Uncertainty in a General Setting

We consider the optimal stopping time problem under model uncertainty $R(v)= {\text{ess}\sup\limits}_{ \mathbb{P} \in \mathcal{P}} {\text{ess}\sup\limits}_{τ\in \mathcal{S}_v} E^\mathbb{P}[Y(τ) \vert \mathcal{F}_v]$, for every stopping time $v$, set in the framework of families of random variables indexed by stopping times. This setting is more general than the classical setup of stochastic processes, and particularly allows for general payoff processes that are not necessarily right-continuous. Under weaker integrability, and regularity assumptions on the reward family $Y=(Y(v), v\in \mathcal{S})$, we show the existence of an optimal stopping time. We then proceed to find sufficient conditions for the existence of an optimal model. For this purpose, we present a universal Doob-Meyer-Mertens's decomposition for the Snell envelope family associated with $Y$ in the sense that it holds simultaneously for all $\mathbb{P} \in \mathcal{P}$. This decomposition is then employed to prove the existence of an optimal probability model and study its properties.

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arXiv2024

Intrinsic regularization by noise for $1d$ mean field games

The purpose of this article is to show that an intrinsic noise with values in the space ${\mathcal P}({\mathbb R})$ of $1d$ probability measures may force uniqueness to first order mean field games. The structure of the noise is inspired from an earlier work [arXiv:2210.01239]. It reads as a coloured Ornstein-Uhlenbeck process with reflection on the boundary of quantile functions on the $1d$ torus, with the elements of the latter playing the role of indices for the continuum of players underpinning the game. In [arXiv:2210.01239], the semi-group generated by the noise is shown to enjoy smoothing properties that become key in the study carried out here. Although the analysis is limited to the 1d setting, this is the first example of uniqueness forcing for generic mean field games set over an infinite dimensional set of probability measures and this may be one step forward towards a more systematic regularization by noise theory for mean field games.

math.PR↗

arXiv2023

Doubly Reflected BSDEs in the predictable setting

In this paper, we introduce a specific kind of doubly reflected Backward Stochastic Differential Equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki's condition, we show the existence of the solution (in consideration of the driver's nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Itô's formula due to Gal'chouk and Lenglart, we provide a suitable a priori estimates which immediately implies the uniqueness of the solution.

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arXiv2023

Reflected and Doubly RBSDEs with Irregular Obstacles and a Large Set of Stopping Strategies

We introduce a new formulation of reflected BSDEs and doubly reflected BSDEs associated with irregular obstacles. In the first part of the paper, we consider an extension of the classical optimal stopping problem over a larger set of stopping systems than the set of stopping times (namely, the set of split stopping times), where the payoff process $ξ$ is irregular and in the case of a general filtration. Split stopping times are a powerful tool for modeling financial contracts and derivatives that depend on multiple conditions or triggers, and for incorporating stochastic processes with jumps and other types of discontinuities. We show that the value family can be aggregated by an optional process $v$, which is characterized as the Snell envelope of the reward process $ξ$ over split stopping times. Using this, we prove the existence and uniqueness of a solution $Y$ to irregular reflected BSDEs. In the second part of the paper, motivated by the classical Dynkin game with completely irregular rewards considered by Grigorova et al. (2018), we generalize the previous equations to the case of two reflecting barrier processes.

math.PR↗

arXiv2022

Stochastic differential equations with respect to optional semimartingales and two reflecting regulated barriers

In this work, we introduce a new Skorokhod problem with two reflecting barriers when the trajectories of the driven process and the barriers are right and left limited. We show that this problem has an explicit unique solution in a deterministic case. Then, we apply our result to study the existence and uniqueness of solutions of reflected stochastic differential equations with respect to optional semimartingales. The study is carried out on a probability space that does not necessarily satisfy the usual conditions.

math.PR↗

arXiv2021

Reflected backward stochastic differential equations with optional barriers: monotone approximation

In this short note we consider RBSDE with Lipschitz drivers and barrier processes that are optional and right upper semicontinuous. We treat the case when the barrier can be represented as a decreasing limit of cadlag barriers. We combine well known existence results for cadlag barriers with comparison arguments for the control process to construct solutions. Finally, we highlight the connection of such RBSDEs with usual cadlag BSDEs.

math.PR↗

arXiv2020

An ideal class to construct solutions for skew Brownian motion equations

This paper contributes to the study of stochastic processes of the class $(Σ)$. First, we extend the notion of the above-mentioned class to càdlàg semi-martingales, whose finite variational part is considered càdlàg instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class $(Σ)$. More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class $(Σ)$.

math.PR↗

arXiv2019

On Skorokhod Problem with Two RCLL Reflecting Completely Separated Barriers

In this paper we deal with Skorokhod problem for right continuous left limited (rcll) barriers. We prove existence and uniqueness of the solution when the barriers are only supposed to be rcll and completely separated. Then, we apply our results to prove existence and uniqueness of the solution of a reflected stochastic differential equation (SDE).

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arXiv2018

Reflected BSDEs when the obstacle is not right-continuous in a general filtration

We prove existence and uniqueness of the reflected backward stochastic differential equation's (RBSDE) solution with a lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous in a filtration that supports a Brownian motion $W$ and an independent Poisson random measure $π$. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right continuous) supermartingales and some tools from optimal stopping theory, as well as an appropriate generalization of Itô's formula due to Gal'chouk and Lenglart. Two applications on dynamic risk measure and on optimal stopping will be given.

math.PR↗

arXiv2018

Optimal Stopping in General Predictable Framework

In this paper, we study the optimal stopping problem in the case where the reward is given by a family $(ϕ(τ),\;\;τ\in \stopo)$ of non negative random variables indexed by predictable stopping times. We treat the problem by means of Snell's envelope techniques. We prove some properties of the value function family associated to this setting.

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arXiv2018

Pathwise uniqueness of non-uniformly elliptic SDEs with rough coefficients

In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.

math.PR↗

arXiv2018

Optimal stopping with f -expectations: the irregular case

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $ξ$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $ξ$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $ξ$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.

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