Quadratic BSDEs Research Articles

Diagonally quadratic BSDE with oblique reflection and optimal switching

The present paper is devoted to the study of diagonally quadratic backward stochastic differential equation with oblique reflection. Using a penalization approach, we show the existence of a solution by providing some delicate a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching problem for quadratic BSDEs. Moreover, we provide an extension for the solvability and apply our results to study a risk-sensitive switching problem for functional stochastic differential equations.

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

Invariant representation for generators of general time interval quadratic BSDEs under stochastic growth conditions

This paper is devoted to proving a general invariant representation theorem for generators of general time interval backward stochastic differential equations, where the generator g has a quadratic growth in the unknown variable z and satisfies some stochastic growth conditions in the unknown variable y. This unifies and strengthens some known results. A natural and innovative idea is used to prove the representation theorem.

Quadratic BSDEs with mean reflection driven by G-brownian motion

In this paper, the well-posedness of a kind of backward stochastic differential equation driven by [Formula: see text]-Brownian motions ([Formula: see text]-BSDEs for short) with mean reflection is investigated. In particular, the generator is allowed having quadratic growth in [Formula: see text]. Combining a representation of the solution with [Formula: see text]-BMO martingale techniques, fixed point argument, and [Formula: see text]-method, existence and uniqueness results for such [Formula: see text]-BSDEs are provided under bounded and unbounded terminal condition.

On quadratic multidimensional type-I BSVIEs, infinite families of BSDEs and their applications

This paper investigates multidimensional extended type-I BSVIEs and infinite families of BSDEs in the case of quadratic generators. We establish existence and uniqueness results in the case of fully quadratic as well as Lipschitz-quadratic generators. As a preliminary step, we establish the well-posedness of a class of infinite families of BSDEs, as introduced in Hernández and Possamaï (2021), which are of interest in their own right. Our approach relies on the strategy developed by Tevzadze (2008) for quadratic BSDEs and the treatment of Lipschitz extended type-I BSVIEs in Hernández and Possamaï (2021). We also present and discuss a type of flow property satisfied by this family of BSVIEs. We motivate the analysis of both of these objects by a series of practical applications.

Radner equilibrium and systems of quadratic BSDEs with discontinuous generators

Motivated by an equilibrium problem, we establish the existence of a solution for a family of Markovian backward stochastic differential equations with quadratic nonlinearity and discontinuity in Z. Using unique continuation and backward uniqueness, we show that the set of discontinuity has measure zero. In a continuous-time stochastic model of an endowment economy, we prove the existence of an incomplete Radner equilibrium with nondegenerate endogenous volatility.

Local existence and uniqueness of solutions to quadratic BSDEs with weak monotonicity and general growth generators

In this paper, we study quadratic backward stochastic differential equations (QBSDEs for short) with generators satisfying a weak monotonicity and a general growth condition with respect to the first variable. We provide the local existence and uniqueness of solutions to multi-dimensional QBSDEs. We use the convolution approach, the Picard iteration, the truncation and the Bihari inequality.

A characterization of solutions of quadratic BSDEs and a new approach to existence

We provide a novel characterization of the solutions of a quadratic BSDE, which is analogous to the characterization of local martingales by convex functions. We then use our main result to show that BSDE solutions are closed under ucp convergence. Finally, we use our closure result obtain a sufficient condition for existence, and discuss specific cases in which this sufficient condition can be verified.

Quadratic BSDEs with jumps and related PIDEs

In this paper we are interested to solve a class of quadratic BSDEs with jumps (QBSDEJs for short) of the following form: Herein, the terminal data ξ will be assumed to be square integrable. Our study covers the following cases where f is a measurable and integrable function, is a functional of the unknown processes and to be defined later and h and enjoy some classical assumptions. The generators show quadratic growth in the Brownian component and non-linear functional form with respect to the jump term. Existence or uniqueness of solutions as well as a comparison and strict comparison principles are established under no monotonicity condition in the third argument of the generator. Probabilistic representations of solutions to some classes of quadratic PIDE are given by means of solutions of these QBSDEJs. The main idea is to use a phase space transformation to transform our initial QBSDEJ to a standard BSDEJ without quadratic term.

The Convergence Rate from Discrete to Continuous Optimal Investment Stopping Problem

The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth. Based on the work of Hu et al. (2018) with an additional stochastic payoff function, the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations (BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.

Comparison theorem for diagonally quadratic BSDEs

<p style='text-indent:20px;'>The present work is devoted to study comparison and converse comparison theorems for diagonally quadratic BSDEs. We give sufficient and necessary conditions under which the comparison holds. Sufficient and necessary conditions for non-positive and non-negative solutions are presented.

Reflected Quadratic BSDEs Driven by G-Brownian Motions

In this paper, the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion (G-BSDE for short), with the generator growing quadratically in the second unknown. The authors obtain the existence by the penalty method, and some a priori estimates which imply the uniqueness, for solutions of the G-BSDE. Moreover, focusing their discussion at the Markovian setting, the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.

A type of globally solvable BSDEs with triangularly quadratic generators

The present paper is devoted to the study of the well-posedness of a type of BSDEs with triangularly quadratic generators. This work is motivated by the recent results obtained by Hu and Tang [14] and Xing and Žitković [28]. By the contraction mapping argument, we first prove that this type of triangularly quadratic BSDEs admits a unique local solution on a small time interval whenever the terminal value is bounded. Under additional assumptions, we build the global solution on the whole time interval by stitching local solutions. Finally, we give solvability results when the generators have path dependence in value process.

On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions

We prove a uniqueness result of the unbounded solution for a quadratic backward stochastic differential equation whose terminal condition is unbounded and whose generator $g$ may be non-Lipschitz continuous in the state variable $y$, non-convex (non-concave) in the state variable $z$, and instead satisfies a strictly quadratic condition and an additional assumption. The key observation is that if the generator is strictly quadratic, then the quadratic variation of the first component of the solution admits an exponential moment. Typically, a Lipschitz perturbation of some convex (concave) function satisfies the additional assumption mentioned above. This generalizes some results obtained in \cite{BriandHu2006PTRF} and \cite{BriandHu2008PTRF}.

Contracting Theory with Competitive Interacting Agents

In a framework close to the one developed by Holmström and Milgrom [Econometrica, 55 (1987), pp. 303--328], we study the optimal contracting scheme between a principal and several agents. Each hired agent is in charge of one project and can make efforts toward managing his own project, as well as impact (positively or negatively) the projects of the other agents. Considering economic agents in competition with relative performance concerns, we derive the optimal contracts in both first-best and moral hazard settings. The enhanced resolution methodology relies heavily on the connection between Nash equilibria and multidimensional quadratic BSDEs. The optimal contracts are linear and each agent is paid a fixed proportion of the terminal value of all the projects of the firm. Besides, each agent receives his reservation utility, and those with high competitive appetite are assigned less volatile projects, and shall even receive help from the other agents. From the principal point of view, it is in the firm's interest in our model to strongly diversify the competitive appetite of the agents.

Quadratic BSDEs with mean reflection

The present paper is devoted to the study of the well-posedness of BSDEs with mean reflection whenever the generator has quadratic growth in the $z$ argument. This work is the sequel of Briand et al. [BSDEs with mean reflection, arXiv:1605.06301] in which a notion of BSDEs with mean reflection is developed to tackle the super-hedging problem under running risk management constraints. By the contraction mapping argument, we first prove that the quadratic BSDE with mean reflection admits a unique deterministic flat local solution on a small time interval whenever the terminal value is bounded. Moreover, we build the global solution on the whole time interval by stitching local solutions when the generator is uniformly bounded with respect to the $y$ argument.

On the Malliavin differentiability of BSDEs

Dans cet article, nous donnons de nouvelles conditions nous assurant que les solutions d’EDSR à générateurs lipschitziens ou à croissance quadratique sont différentiables au sens de Malliavin, en utilisant l’interprétation de la dérivée de Malliavin comme dérivée de Gâteaux directionnelle par rapport à l’espace de Cameron–Martin. Ce résultat est en outre basé sur une nouvelle caractérisation des espaces de Malliavin–Sobolev $\mathbb{D}^{1,p}$ que nous fournissons.

Multidimensional quadratic BSDEs with separated generators

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.

Quadratic BSDEs with jumps: Related nonlinear expectations

In this paper, we continue the study of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables [Formula: see text], started in our accompanying paper [12]. Relying on the existence and uniqueness result of [12], we define the corresponding [Formula: see text]-expectations and study some of their properties. We obtain in particular a nonlinear Doob–Meyer decomposition for [Formula: see text]-submartingales and a downcrossing inequality which implies their regularity in time. As a consequence of these results, we also obtain a converse comparison theorem for our class of BSDEs. Finally, we provide a dual representation for the corresponding dynamic risk measures, and study the properties of their inf-convolution, giving several explicit examples.

Stability and Analytic Expansions of Local Solutions of Systems of Quadratic BSDEs with Applications to a Price Impact Model

We obtain stability estimates and derive analytic expansions for local solutions of multi-dimensional quadratic BSDEs. We apply these results to a financial model where the prices of risky assets are quoted by a representative dealer in such a way that it is optimal to meet an exogenous demand. We show that the prices are stable under the demand process and derive their analytic expansions for small risk aversion coefficients of the dealer.