A characterization of solutions of quadratic BSDEs and a new approach to existence
Joseph Jackson, Gordan Žitković
备注：
The discussion of the one-dimensional case in Section 3 has been replaced with a more general "template for existence", i.e. a sufficient condition for the existence of a solution to a quadratic BSDE. Some situations in which this sufficient condition can be verified are discussed in a remark
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.
We refine the solvability of quadratic semimartingale BSDEs by employing a Lipschitz-quadratic regularization procedure. In the first step, we prove an existence and uniqueness result for a class of Lipschitz-quadratic BSDEs. A corresponding stability theorem and a Lipschitz-quadratic regularization are developed to solve quadratic BSDEs. The advantage of our approach is that much weaker conditions ensure the existence and uniqueness results.
In this paper, we study a Backward Stochastic Differential Equation with Jumps (BSDEJs in short) where the jumps have infinite activity. Following a forward approach based on Exponential Quadratic semimartingale, we prove the existence of solution of Quadratic BSDEJs with unbounded terminal condition and quadratic growth in z.
In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in $z$. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic $g$-expectation.
We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDE). This class is characterized by constraints on some uniform a priori estimate on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921--2939] concerning scalar quadratic BSDEs.
This thesis consists of three parts. In the first part, we study $\mathbb{L}^p$ solutions of a large class of BSDEs. Existence, comparison theorem, uniqueness and a stability result are proved. In the second part, we establish the solvability of quadratic semimartingale BSDEs. In contrast to current literature, we use Lipschitz-quadratic regularization and obtain the existence and uniqueness results with minimal assumptions. The third part is a brief summary of quadratic semimartingales and the monotone stability result. This provides an alternative proof of monotone stability result for quadr…
In this paper, we study the existence of solution to BSDE with quadratic growth and unbounded terminal value. We apply a localization procedure together with a priori bounds. As a byproduct, we apply the same method to extend a result on BSDEs with integrable terminal condition.
We study a general class of quadratic BSDEs with terminal value in Lp for p > 1. First of all, we give an Lp-type estimate and existence result. Under the additional assumption of monotonicity and convexity, we derive the comparison theorem, uniqueness and stability result via θ-technique (Briand and Hu [7]). The assumptions employed throughout this paper are rather weak and extend the quadratic BSDE literature. Finally, a probabilistic representation for the viscosity solution to the associated quadratic PDEs is given.
In this study, we consider the exponential utility maximization problem in the context of a jump-diffusion model. To solve the problem, we rely on the dynamic programming principle and we derive from it a quadratic BSDE with jumps. Since this quadratic BSDE is driven both by a Wiener process and by a Poisson random measure having a Levy measure with infinite mass, our main task consists in establishing a new existence result for the specific BSDE introduced.
In this paper, we study multi-dimensional reflected backward stochastic differential equations with diagonally quadratic generators. Using the comparison theorem for diagonally quadratic BSDEs which is established recently in [14], we obtain the existence and uniqueness of a solution by a penalization method. Moreover, we provide a comparison theorem.
In this paper we study a multidimensional quadratic BSDE with a particular class of product generators and give a result of existence of solution in a suitable complete metric space under some constraints on parameters. We also use that result to derive the existence and uniqueness of solution to the one dimensional case with bounded terminal values and show the existence of solution to a lower triangular quadratic BSDE with certain bounded terminal values.
In this paper, we study a class of Quadratic Backward Stochastic Differential Equations (QBSDE in short) with jumps and unbounded terminal condition. We extend the class of quadratic semimartingales introduced by Barrieu and El Karoui (2013) in the jump diffusion model. The properties of these class of semimartingales lead us to prove existence result for the solution of a quadratic BSDEs.
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.
This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations.
This article deals with the numerical resolution of Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth with respect to $z$ and bounded terminal conditions. We first show some bound estimates on the process $Z$ and we specify the Zhang's path regularity theorem. Then we give a new time discretization scheme with a non uniform time net for such BSDEs and we obtain an explicit convergence rate for this scheme.
In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean,…
In this paper, we study the connections between three concepts - the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo (bounded mean oscillation) coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these two equivalent conditions are satisfied. Finally, we apply our results on l…
In this paper, we consider a reflected backward stochastic differential equation driven by a $G$-Brownian motion ($G$-BSDE), with the generator growing quadratically in the second unknown. We obtain the existence by the penalty method, and a priori estimates which implies the uniqueness, for solutions of the $G$-BSDE. Moreover, focusing our discussion at the Markovian setting, we give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.
In this paper, we study a class of second order backward stochastic differential equations (2BSDEs) with quadratic growth in coefficients. We first establish solvability for such 2BSDEs and then give their applications to robust utility maximization problems.
A note on convergence rate for reflected BSDEs with quadratic generators by penalization method
Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
This article deals with the existence and the uniqueness of solutions to quadratic and superquadratic Markovian backward stochastic differential equations (BSDEs for short) with an unbounded terminal condition. Our results are deeply linked with a strong a priori estimate on $Z$ that takes advantage of the Markovian framework. This estimate allows us to prove the existence of a viscosity solution to a semilinear parabolic partial differential equation with nonlinearity having quadratic or superquadratic growth in the gradient of the solution. This estimate also allows us to give explicit conve…
In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs) with unbounded terminal values. With the help of a $θ$-method of Briand and Hu [4] and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.
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. M…
We study multidimensional BSDEs of the form $$ Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s $$ with bounded terminal conditions $ξ$ and drivers $f$ that grow at most quadratically in $Z_s$. We consider three different cases. In the first one 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 $f$ in $Z_s$ is stric…
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.
In the present article we provide existence, uniqueness and stability results under an exponential moments condition for quadratic semimartingale backward stochastic differential equations (BSDEs) having convex generators. We show that the martingale part of the BSDE solution defines a true change of measure and provide an example which demonstrates that pointwise convergence of the drivers is not sufficient to guarantee a stability result within our framework.
We consider Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial…
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 pa…
In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces.
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 fo a solution by providing some delicated a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching probelm 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.
We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds.
In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from $\mathbb{H}^1$ to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied …
In this paper, we study the multi-dimensional mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. Under small terminal value, the existence and uniqueness are proved for the multi-dimensional situation when the generator f(t,Y,E[Y],Z,E[Z]) is of quadratic growth with respect to the last four items, using some new methods. Besides, a kind of comparison theorem is obtained.
In this paper we prove that every random variable of the form $F(M_T)$ with $F:\real^d \to\real$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $\mathcal{F}$ admits an exact integral representation with respect to $M$, that is, without any orthogonal component. This representation holds true regardless any regularity assumption on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$ and show they can be solved without an orthogonal component. To this end, we extend first existence results for such BSDEs under a …
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. And, a natural and innovative idea is used to prove the representation theorem.
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz contin…
Over the past few years quadratic Backward Stochastic Differential Equations (BSDEs) have been a popular field of research. However there are only very few examples where explicit solutions for these equations are known. In this paper we consider a class of quadratic BSDEs involving affine processes and show that their solution can be reduced to solving a system of generalized Riccati ordinary differential equations. In other words we introduce a rich and flexible class of quadratic BSDEs which are analytically tractable, i.e. explicit up to the solution of an ODE. Our results also provide ana…
We consider a backward stochastic differential equation in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. Under non-degeneracy assumptions, we prove an analogue of the well-known Bismut-Elworty formula when the generator has quadratic growth with respect to $Z$. Applications to the solution of a semilinear Kolmogorov equation for the unknown $v$ with nonlinear term with quadratic growth with respect to $\nabla v$ and final condition only bounded and continuous are given, as well as applications to stochastic optimal control pr…
In [3], the authors proved that uniqueness holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $γ$ ($p\textgreater{}γ$). In this paper, we consider the critical case: $p=γ$. We prove that the uniqueness holds among solutions whose exponentials are $L^γ$ under the additional assumption that the generator is strongly convex.
In a previous work, we proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions. However, no uniqueness result was stated in that work. The main goal of this paper is to fill this gap. In order to obtain a comparison theorem for this kind of BSDEs, we assume that the generator is convex with respect to the variable z. Under this assumption of convexity, we are also able to prove a stability result in the spirit of the a priori estimates stated in the article of N. El Karoui, S. Peng and M.-C. Quenez. With these tools i…
In this paper, we study the existence and uniqueness of solutions to quadratic Backward Stochastic Differential Equations (QBSDEs for short) with rough driver and square integrable terminal condition. The main idea consists in using both Doss-Sussman and Zvonkin type transformations. As an application we study connection between QBSDEs and quadratic PDEs with rough drivers. We also obtain Backward Doubly SDEs and QBSDEs driven by Fractional Brownian with Hurst parameter greater than $\frac{1}{4}$ as particular cases of our QBSDEs with rough drivers. A probabilistic representation of a class of…
With an emphasis on generators with quadratic growth in the control variable we consider measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces to martingale representation with respect to an underlying filtration. Measure solutions related to measures equivalent to the historical one provide classical solutions. We derive the existence of measure solutions in scenarios in which the generating functions are just continuous, of at most linear growth in the control variable (…
In a previous work, P. Briand and Y. Hu proved the uniqueness among the solutions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman-Kac formula proved by P. Briand and Y. Hu.
In this paper, we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown $z$. Using linearization technique and BMO martingale theory, we first apply fixed point argument to establish uniqueness and existence result for the case with bounded terminal condition and obstacle. Then, with the help of a $θ$-method, we develop a successive approximation procedure to remove the boundedness condition on the terminal condition and obstacle when the generator is concave (or convex) with respect to the 2nd unknown $z$
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) func…
We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f of the driver has at most quadratic growth in the control variable Z, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting : comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs.…
We study the problem of existence of solutions for generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under weaker assumptions on the data. Roughly speaking we show the existence of a maximal solution for GRBSDE when the terminal condition ξis F_T-measurable, the coefficient f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and the reflecting barriers L and U are just right continuous left limited. The result is proved without assuming any P-integrability condition…
In this paper we prove some uniqueness results for quadratic backward stochastic differential equations without any convexity assumptions on the generator. The bounded case is revisited while some new results are obtained in the unbounded case when the terminal condition and the generator depend on the path of a forward stochastic differential equation. Some of these results are based on strong estimates on $Z$ that are interesting on their own and could be applied in other situations.
In this paper we prove the existence of a solution for reflected BSDE's\ whose coefficient is of quadratic growth in $z$ and of linear growth in $y$, with an unbounded terminal value.
We study the existence of a solution for a one-dimensional generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under assumptions on the input data which are weaker than that on the current literature. In particular, we construct a maximal solution for such a GRBSDE when the terminal condition ξis only F_T-measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any P-integrability conditions. The work is suggested by the interest …
In this paper, we study the solvability of anticipated backward stochastic differential equations (BSDEs, for short) with quadratic growth for one-dimensional case and multi-dimensional case. In these BSDEs, the generator, which is of quadratic growth in Z, involves not only the present information of solution (Y, Z) but also its future one. The existence and uniqueness of such BSDEs, under different conditions, are derived for several terminal situations, including small terminal value, bounded terminal value and unbounded terminal value.
We consider a class of multi-dimensional BSDEs on a finite time horizon (containing in particular Lipschitzian-quadratic BSDEs), whose terminal values are bounded as well as their corresponding Malliavin derivatives. We prove two results. The first one is an exponential integrability condition which determines when a BSDE in this class has a solution up to a given time horizon. In the second result, via an ordinary differential equation, we compute a minimum horizon up to which any BSDE of this class has a solution. The combination of these two results leads to a new scheme to solve quadratic …
We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator $g$ allowing a general growth in the state variable $y$ and a sub-quadratic growth in the state variable $z$, like $|z|^α$ for some $α\in (1,2)$, when the terminal condition satisfies a sub-exponential moment integrability condition like $\exp\left(μL^{2/α^*}\right)$ for the conjugate $α^*$ of $α$ and a positive parameter $μ>μ_0$ with a certain value $μ_0$, which is clearly weaker than the usual $\exp(μL)$ integrability and stronger than $L^p\ (p>1)$ integrability. The…
This paper establishes a new existence and uniqueness result of solutions for multidimensional backward stochastic differential equations (BSDEs) whose generators satisfy a weak monotonicity condition and a general growth condition in $y$, which generalizes the corresponding results in [2], [3] and [5].
The representation of the solution of some Backward Stochastic Differential Equation as an infinite series is obtained. Some exactly solvable examples are considered.
In this paper, we study a class of quadratic Backward Stochastic Differential Equations (BSDEs) which arises naturally when studying the problem of utility maximization with portfolio constraints. We first establish existence and uniqueness results for such BSDEs and then, we give an application to the utility maximization problem. Three cases of utility functions will be discussed: the exponential, power and logarithmic ones.
In this paper, we study a scalar linearly growing BSDE with a weakly $L^{1+}$-integrable terminal value. We prove that the BSDE admits a solutionif the terminal value satisfies some $Ψ$-integrability condition, which is weaker than the usual $L^p$ ($p>1$) integrability and stronger than $L\log L$ integrability. We show by a counterexample that $L\log L$ integrability is not sufficient for the existence of solutionto a BSDE of a linearly growing generator.
