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Auguste Aman

Publications and source records attributed to Auguste Aman.

33 records · Page 2Linked to original sources
arXiv2011

A Numerical scheme for backward doubly stochastic differential equations

In this paper we propose a numerical scheme for the class of backward doubly stochastic (BDSDEs) with possible path-dependent terminal values. We prove that our scheme converge in the strong $L^2$-sense and derive its rate of convergence. As an intermediate step we derive an $L^2$-type regularity of the solution to such BDSDEs. Such a notion of regularity which can be though of as the modulus of continuity of the paths in an $L^2$-sense, is new.

math.PR↗
arXiv2011

A new type of reflected backward doubly stochastic differential equations

In this paper, we introduce a new kind of "variant" reflected backward doubly stochastic differential equations (VRBDSDEs in short), where the drift is the nonlinear function of the barrier process. In the one stochastic case, this type of equations have been already studied by Ma and Wang. They called it as "variant" reflected BSDEs (VRBSDEs in short) based on the general version of the Skorohod problem recently studied by Bank and El Karoui. Among others, Ma and Wang showed that VRBSDEs is a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable. Since more of those models have their stochastic counterpart, it is very useful to transpose the work of Ma and Wang to doubly stochastic version. In doing so, we firstly establish the stochastic variant Skorohod problem based on the stochastic representation theorem, which extends the work of Bank and El Karoui. We prove the existence and uniqueness of the solution for VRBDSDEs by means of the contraction mapping theorem. By the way, we show the comparison theorem and stability result for the solutions of VRBDSDEs.

math.PR↗
arXiv2010

Stochastic viscosity solution for stochastic PDIEs with nonlinear Neumann boundary condition

This paper is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential integral equations with nonlinear Neumann boundary condition. Using the recently developed theory on generalized backward doubly stochastic differential equations driven by a Lévy process, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula.

math.PR↗
arXiv2010

Reflected generalized BSDEs with random time and applications

In this paper, we aim to study solutions of reflected generalized BSDEs, involving the integral with respect to a continuous process, which is the local time of the diffusion on the boundary. We consider both a finite random terminal and a infinite horizon. In both case, we establish an existence and uniqueness result. Next, as an application, we get an American pricing option in infinite horizon and we give a probabilistic formula for the viscosity solution of an obstacle problem for elliptic PDEs with a nonlinear Neumann boundary condition.

math.PR↗
arXiv2010

Reflected Generalized Backward Doubly SDEs driven by Lévy processes and Applications

In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process (RGBDSDELs, in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove existence and uniqueness result by means of the penalization method and the fixed point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs, in short) with a nonlinear Neumann boundary condition.

math.PR↗
arXiv2009

L$^{p}$-solution of reflected generalized BSDEs with non-Lipschitz coefficients

In this paper, we continue in solving reflected generalized backward stochastic differential equations (RGBSDE for short) and fixed terminal time with use some new technical aspects of the stochastic calculus related to the reflected generalized BSDE. Here, existence and uniqueness of solution is proved under a non-Lipschitz condition on the coefficients.

math.PR↗
arXiv2009

Numerical scheme for backward doubly stochastic differential equations

We study a discrete-time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations (FBDSDEs). Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the step of time discretization, $|π|$ goes to zero. The rate of convergence is exactly equal to $|π|^{1/2}$. The proof is based on a generalization of a remarkable result on the $^{2}$-regularity of the solution of the backward equation derived by J. Zhang

math.PR↗
arXiv2009

Homeomorphism of solutions to backward doubly SDEs and applications

In this paper we study the homeomorphic properties of the solutions to one dimensional backward doubly stochastic differential equations under suitable assumptions, where the terminal values depend on a real parameter. Then, we apply them to the solutions for a class of second order quasilinear parabolic stochastic partial differential equations.

math.PR↗
arXiv2009

Reflected generalized backward doubly SDEs driven by Lévy processes and Applications

In this paper, a class of reflected generalized backward doubly stochastic differential equations (reflected GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of reflected stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.

math.PR↗
arXiv2009

Lp-solution of backward doubly stochastic differential equations

In this paper, our goal is solving backward doubly stochastic differential equation (BDSDE for short) under weak assumptions on the data. The first part of the paper 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, extending the results of Pardoux and Peng \cite{PP1} to the case where the solution is taked in $L^{p}, p>1$ and the monotonicity conditions are satisfied. This study is limited to deterministic terminal time.

math.PR↗
arXiv2009

Homogenization of reflected semilinear PDE with nonlinear Neumann boundary condition

We study the homogenization problem of semi linear reflected partial differential equations (reflected PDEs for short) with nonlinear Neumann conditions. The non-linear term is a function of the solution but not of its gradient. The proof are fully probabilistic and uses weak convergence of associated reflected generalized backward differential stochastic equations (reflected GBSDEs in short).

math.PR↗
arXiv2008

Representation theorems for backward doubly stochastic differential equations

In this paper we study the class of backward doubly stochastic differential equations (BDSDEs, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial gradient of the stochastic viscosity solution to a quasilinear parabolic SPDE in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding BDSDE. Then such a representation leads to a closed-form representation of the martingale integrand of BDSDE, under only standard Lipschitz condition on the coefficients.

math.PR↗