Lp solutions of backward stochastic differential equations

In this paper, we are interested in solving backward stochastic differential equations (BSDEs 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 BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in L^p p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503–549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L¹.     

Continuous dependence on geometry for the backward heat equation

In this paper we are concerned with the development of criteria for stabilizing inherently unstable initial-boundary value problems under small errors in the geometry of the underlying domain. We consider in particular the initial-boundary-value problem for the backward heat equation assuming that some error has been made in characterizing the geometry of the domain under consideration. It is shown that solutions which belong to an appropriately defined constraint set depend continuously in L₂ on errors in the geometry.     

On measure solutions of backward stochastic differential equations

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions.     

Lp solutions of backward stochastic differential equations with jumps

Given $p\in (1,2)$, we study ${\mathbb{L}}^p$ solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)$-variables. We show that such a BSDEJ with $p$-integrable terminal data admits a unique ${\mathbb{L}}^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.     

Path regularity for solutions of backward stochastic differential equations

 In this paper we study the path regularity of the adpated solutions to a class of backward stochastic differential equations (BSDE, for short) whose terminal values are allowed to be functionals of a forward diffusion. Using the new representation formula for the adapted solutions established in our previous work [7], we are able to show, under the mimimum Lipschitz conditions on the coefficients, that for a fairly large class of BSDEs whose terminal values are functionals that are either Lipschitz under the L ^∞ -norm or under the L ¹ -norm, then there exists a version of the adapted solution pair that has at least càdlàg paths. In particular, in the latter case the version can be chosen so that the paths are in fact continuous. Received: 26 May 2000 / Revised version: 1 December 2000 / Published online: 19 December 2001     

Reflected backward stochastic differential equation with jumps and RCLL obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left limits obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.     

Higher order differentiability of solutions to backward stochastic differential equations

In this paper, we consider the differentiability in the sense of the Malliavin calculus of solutions to backward stochastic differential equations (BSDEs for short). It is known that a solution is differentiable in the sense of the Malliavin calculus and the derivative is also a solution to a linear BSDE. Under additional conditions, we will show that the higher order differentiability of a solution to a BSDE and that it also becomes a solution to a linear BSDE.     

The uniqueness of solutions of perturbed backward stochastic differential equations

We prove a result on the preservation of the pathwise uniqueness property for the adapted solution to backward stochastic differential equation under perturbations.     

Reflected solutions of backward stochastic differential equations with continuous coefficient

We prove the existence of a reflected solution of one-dimensional backward stochastic differential equations with continuous and linear growth coefficient and squared integrable terminal condition.     

Continuous dependence theorems on solutions of uncertain differential equations

In ordinary differential equation (ODE) and stochastic differential equation (SDE), the solution continuously depends on initial value and parameter under some conditions. This paper investigates the analogous continuous dependence theorems in uncertain differential equation (UDE). It proves two continuous dependence theorems, a basic one and a general one.     

Generalized solutions of a stochastic partial differential equation

We discuss the Cauchy problem of a certain stochastic parabolic partial differential equation arising in the nonlinear filtering theory, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The generalized (distributional) solution is represented by a partial (conditional) generalized expectation ofT(t)° _0,t ^–1 , whereT(t) is a stochastic process with values in distributions and _s,t is a stochastic flow generated by a certain stochastic differential equation. The representation is used for getting estimates of the solution with respect to Sobolev norms.Further, by applying the partial Malliavin calculus of Kusuoka-Stroock, we show that any generalized solution is aC -function under a condition similar to Hörmander's hypoellipticity condition.     

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.     

Viability property for a backward stochastic differential equation and applications to partial differential equations

In the present paper, we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints. This property is the so-called “viability property”. In a separate section, this condition is translated to a class of partial differential equations. Received: 23 April 1998 / Published online: 14 February 2000     

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??A class of backward doubly stochastic differential equations driven by white noises and Poisson random measures are studied in this paper. The definitions of solutions and Yamada-Watanabe type theorem to this equation are established.     

On solutions set of a multivalued stochastic differential equation

We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.     

Large deviation principle for a backward stochastic differential equation with subdifferential operator

In this note, we prove that the solution of a backward stochastic differential equation, which involves a subdifferential operator and associated to a family of reflecting diffusion processes, converges to the solution of a deterministic backward equation and satisfies a large deviation principle. To cite this article: E.H. Essaky, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Numerical solutions of backward stochastic differential equations: A finite transposition method

In this Note, we present a new numerical method for solving backward stochastic differential equations. Our method can be viewed as an analogue of the classical finite element method solving deterministic partial differential equations.