Multidimensional BSDEs Research Articles

Multi-dimensional BSDEs with mean reflection

Multi-dimensional BSDEs with mean reflection

Representation theorem and viability property for multidimensional BSDEs and their applications

Representation theorem and viability property for multidimensional BSDEs and their applications

A Global Stochastic Maximum Principle for Forward-Backward Stochastic Control Systems with Quadratic Generators

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.

Multi-dimensional BSDEs driven by G-Brownian motion and related system of fully nonlinear PDEs

In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). The existence and uniqueness of solutions are obtained via a contraction argument for Y component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional G-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differential equations.

Existence and uniqueness results for BSDE with jumps: the whole nine yards

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.

Representation and converse comparison theorems for multidimensional BSDEs

In this paper, we consider a certain kind of multidimensional BSDEs. Under the assumption that the generators satisfy monotonicity condition in y, we investigate representation and converse comparison theorems for these BSDEs.

BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions in provided that the generator and the terminal condition satisfy appropriate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principle in dimension one.

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose ith component depends only on the ith row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John–Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in El Karoui and Hamadène [Stochastic Process. Appl. 107 (2003), page 164].

A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients

The purpose of this paper is to study some properties of solutions to one-dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth and non-Lipschitz conditions on the coefficients. Taking inspiration from [K. Bahlali, E.H. Essaky, M. Hassani, and E. Pardoux Existence, uniqueness and stability of backward stochastic differential equation with locally monotone coefficient, C.R.A.S. Paris. 335(9) (2002), pp. 757–762; K. Bahlali, E. H. Essaky, and H. Hassani, Multidimensional BSDEs with super-linear growth coefficients: Application to degenerate systems of semilinear PDEs, C. R. Acad. Sci. Paris, Ser. I. 348 (2010), pp. 677-682; K. Bahlali, E. H. Essaky, and H. Hassani, p-Integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities, (2010). Available at arXiv:1007.2388v1 [math.PR]], we introduce a new local condition which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin–Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals.

Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the formwith bounded terminal conditions and drivers that grow at most quadratically in . We consider three different cases. In the first case, the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver in is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending it recursively.

BSDE representations for optimal switching problems with controlled volatility

This paper provides two different strong BSDE representations for optimal switching problems in the case where the dynamics of the underlying diffusion process depends on the current value of the switching mode. These new representations are valid in a non-Markovian framework and make use of either one-dimensional constrained BSDEs with jumps or multidimensional BSDEs with oblique reflections, thus extending the framework considered by Hu and Tang [12]. In particular, the numerical resolution of the corresponding switching problem can therefore be treated via the entirely probabilistic schemes presented in [4] or [8].

Multidimensional BSDEs with uniformly continuous coefficients: The general result

In this paper, we prove that the multidimensional backward stochastic differential equations (BSDEs for short) with the more general uniformly continuous coefficients are uniquely solvable, and accordingly we solve the open problem of multidimensional BSDEs with uniformly continuous coefficients.

Comparison theorem for Brownian multidimensional BSDEs via jump processes

In this Note, we provide an original proof of the comparison theorem for multidimensional Brownian BSDEs in the case where at each line k the generator depends on the matrix variable Z only through its row k.

Discrete-time approximation of multidimensional BSDEs with oblique reflections

In this paper, we study the discrete-time approximation of multidimensional reflected BSDEs of the type of those presented by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89–121] and generalized by Hamadène and Zhang [Stochastic Process. Appl. 120 (2010) 403–426]. In comparison to the penalizing approach followed by Hamadène and Jeanblanc [Math. Oper. Res. 32 (2007) 182–192] or Elie and Kharroubi [Statist. Probab. Lett. 80 (2010) 1388–1396], we study a more natural scheme based on oblique projections. We provide a control on the error of the algorithm by introducing and studying the notion of multidimensional discretely reflected BSDE. In the particular case where the driver does not depend on the variable Z, the error on the grid points is of order ½ − ε, ε > 0.