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Auguste Aman

Publications and source records attributed to Auguste Aman.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic properties for fully coupled delayed forward-backward stochastic differential equations

We investigate the asymptotic behavior of solutions to a class of fully coupled forward-backward stochastic differential equations with time-delayed generators. Such systems arise naturally in stochastic models with memory effects and constitute a significant extension of the classical fully coupled FBSDE framework. The presence of delay introduces additional analytical difficulties due to the dependence of the coefficients on the past trajectories of the solution processes and the resulting non-Markovian structure. Under suitable assumptions on the coefficients, we study the asymptotic properties of a perturbed delayed FBSDE driven by a small noise parameter. We first establish the convergence in distribution of the associated solution processes as the perturbation parameter tends to zero. We then prove almost sure convergence towards the solution of the corresponding deterministic limiting system. As a consequence of these asymptotic results, we derive a large deviation principle for the solution processes. Our results extend the asymptotic analysis of Cruzeiro, Gomes and Zhang (2014) from the classical fully coupled FBSDE setting to the delayed framework, and complement existing works on weakly coupled delayed forward-backward systems. They provide, to the best of our knowledge, the first large deviation principle for fully coupled forward-backward stochastic differential equations with delayed generators.

math.PR↗
arXiv2026

New approach to optimal control of delayed stochastic Volterra integral equations

We address the optimal control of stochastic Volterra integral equations with delay through the lens of Hida-Malliavin calculus. We show that the corresponding adjoint processes satisfy an anticipated backward stochastic Volterra integral equation (ABSVIE), and, exploiting this structure, we establish both necessary and sufficient stochastic maximum principles. Our results provide a comprehensive and rigorous framework for characterizing optimal controls in delayed stochastic systems.

math.PR↗
arXiv2025

Generalized delayed Black and Scholes Formula

The mean objective of this paper is to derive an explicit formula for a price of an European option associated to the underlying delayed stock price which follows a linear differential equation with a general delay in the drift term. We use an equivalent martingale measure method based on Girsanov's property. Two of our model maintains the no-arbitrage property and the completeness of the market and can be considered as an extension some previous model introduced by Arriojas et al. in \cite{Aal}. The last one has a possible arbitrage property such that we can not obtain an unique price of an European option associated.

math.PR↗
arXiv2021

General Fully Coupled Forward Backward Stochastic Differential Equations with delayed generator

This paper is devoted to study different type of BSDE with delayed generator. We first establish an existence and uniqueness result under delayed Lipschitz condition for non homogenous backward stochastic differential equation with delayed generator. Next, existence and uniqueness result for general fully coupled forward backward stochastic differential equation with delayed coefficients has been derived.

math.PR↗
arXiv2021

Backward stochastic Volterra integral equations with time delayed generators

In this paper, we study backward stochastic Volterra integral equations of type-I with time delayed generators. Under some condition (small time horizon or a Lipschitz constant), we derive an existence and uniqueness results. Next, with the help of two examples of this BSVIEs we provide this condition is necessary. However, for special class of delayed generators, the previous existence and uniqueness result hold for an arbitrary time horizon and Lipschitz constant. We end this paper by establishing under an additive assumptions, a path continuity property.

math.PR↗
arXiv2021

$L^{p}$-solutions of backward stochastic differential equations with time-delayed generators

This article is devoted to study the class of backward stochastic differential equation with delayed generator. We suppose the terminal value and the generator to be $L^{p}$-integrable with $p>1$. We derive a new type of estimation related to this BSDE. Next, we establish the existence and uniqueness result in two ways. First, an approximation technics used by Briand et al. (Stochastic Process. Appl. 108 (2003) 109-129) and hence the well-know Picard iterative procedure. Using Picard iterative procedure, we revisit the result of Dos Reis et al. (Stochastic Process. Appl. 121 (9) (2011) 2114-2150), simplifying the proof and give an explicit existence and uniqueness condition related to the Lipschitz constant $K$ and the terminal time $T$.

math.PR↗
arXiv2021

Stochastic viscosity solutions of reflected stochastic partial differential equations with non-Lipschitz coefficients

This paper, is an attempt to extend the notion of stochastic viscosity solution to reflected semi-linear stochastic partial differential equations (RSPDEs, in short) with non-Lipschitz condition on the coefficients. Our method is fully probabilistic and use the recently developed theory on reflected backward doubly stochastic differential equations (RBDSDEs, in short). Among other, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula to reflected SPDEs, like one appear in \cite{2}. Indeed, in their recent work, Aman and Mrhardy \cite{2} established a stochastic viscosity solution for semi-linear reflected SPDEs with nonlinear Neumann boundary condition by using its connection with RBDSDEs. However, even Aman and Mrhardy consider a general class of reflected SPDEs, all their coefficients are at least Lipschitz. Therefore, our current work can be thought of as a new generalization of a now well-know Feymann-Kac formula to SPDEs with large class of coefficients, which does not seem to exist in the literature. In other words, our work extends (in non boundary case) Aman and Mrhardy's paper.

math.PR↗
arXiv2021

Robust optimal problem for dynamic risk measures governed by BSDEs with jumps and delayed generator

The aim of this paper is to study an optimal stopping problem for dynamic risk measures induced by backward stochastic differential equations with jumps and delayed generator. Firstly, we connect the value function of this problem to reflected BSDEs with jump and delayed generator. Furthermore, after establishing existence and uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal stopping game problem for the previous dynamic risk measure in ambiguity case.

math.PR↗
arXiv2021

Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with sub-differential of a lower semi-continuous convex function under a non Lipschitz condition. This study allows us give a probabilistic representation (in stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet-Neumann conditions.

math.PR↗
arXiv2011

L^{p}-solutions of backward doubly stochastic differential equations

The goal of this paper is to solve backward doubly stochastic differential equation (BDSDE, in short) under weak assumptions on the data. The first part is devoted to the development of some new technical aspects of stochastic calculus related to BDSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in Lp, p\in (1,2), extending the work of pardoux and Peng (see Probab. Theory Related Fields 98 (1994), no. 2).

math.PR↗
arXiv2011

Multivalued stochastic partial differential-integral equations via backward doubly stochastic differential equations driven by a Lévy process

In this paper, we deal with a class of backward doubly stochastic differential equations (BDSDEs, in short) involving subdifferential operator of a convex function and driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness result by means of Yosida approximation. As an application, we give the existence of stochastic viscosity solution for a class of multivalued stochastic partial differential-integral equations (MSPIDEs, in short).

math.PR↗
arXiv2011

Multivalued stochastic Dirichlet-Neumann problems and generalized backward doubly stochastic differential equations

In this paper, a class of generalized backward doubly stochastic differential equations whose coefficient contains the subdifferential operators of two convex functions (also called generalized backward doubly stochastic variational inequalities) are considered. By means of a penalization argument based on Yosida approximation, we establish the existence and uniqueness of the solution. As an application, this result is used to derive existence result of stochastic viscosity solution for a class of multivalued stochastic Dirichlet-Neumann problems.

math.PR↗