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 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.
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.
In [Stochastc Process. Appl., 122(9):3173-3208], the author proved the existence and the uniqueness of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition when the generator and the terminal condition are locally Lipschitz. In this paper, we prove that the existence result remains true for these BSDEs when the regularity assumptions on the terminal condition is weakened.
This paper considers a class of scalar backward stochastic differential equations (BSDEs) with $L\exp(μ\sqrt{2\log(1+L)})$-integrable terminal values. We associate these BSDEs with BSDEs with integrable parameters through Girsanov change. Using this technique, we prove uniqueness, comparisons and stability for them under an extended monotonicity condition (more precisely one sided Osgood condition).
In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if $(W^n)$ is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion $W$ and if $(τ^n)$ is a sequence of stopping times that converges to a stopping time $τ$, then the solution of the BSDE driven by $W^n$ with random terminal time $τ^n$ converges to the solution of the BSDE driven by $W$ with random terminal time $τ$.
In [4], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is $L\exp{\left(μ\sqrt{2\log{(1+L)}}\,\right)}$-integrable with the positive parameter $μ$ being bigger than a critical value $μ\_0$. In this note, we give the uniqueness result for the preceding BSDE.
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.
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 relate the $L_p$-variation, $2\le p < \infty$, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.
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.
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…
We use the functional It{ô} calculus to prove that the solution of a BSDE with singular terminal condition is continuous at the terminal time. Hence we extend known results for a non-Markovian terminal condition.
In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally H{ö}lder continuous function of the Brownian motion. We state the rate of the L 2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.
We consider the $L_2$-regularity of solutions to backward stochastic differential equations (BSDEs) with Lipschitz generators driven by a Brownian motion and a Poisson random measure associated with a Lévy process $(X_t)_{t\in[0,T]}$. The terminal condition may be a Borel function of finitely many increments of the Lévy process which is not necessarily Lipschitz but only satisfies a fractional smoothness condition. The results are obtained by investigating how the special structure appearing in the chaos expansion of the terminal condition is inherited by the solution to the BSDE.
We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with constraint in the martingale part. We compare our result with the classical representation in terms of (super)quadratic BSDE, and show in particular that existence of a solution to the viscous HJ equation can be obtained under more general growth assumptions on the coefficients, including both unbounded diffusion coefficient and terminal data.
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 \cite{HuTang2018ECP}, the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) when the terminal value is $L\exp\left(μ\sqrt{2\log(1+L)}\right)$-integrable for a positive parameter $μ>μ_0$ with a critical value $μ_0$, and a counterexample is provided to show that the preceding integrability for $μ<μ_0$ is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with $μ>μ_0$) is also given in \cite{BuckdahnHuTang2018ECP} for the preceding BSDE under the uniformly Lipschitz condition of the gene…
We show existence and uniqueness of solutions to BSDEs of the form $$ Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ in the case where the terminal condition $ξ$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $ξ$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for se…
We study the behaviour at the terminal time T of the minimal solution of a backward stochastic differential equation when the terminal data can take the value +$\infty$ with positive probability. In a previous paper, we have proved existence of this minimal solution (in a weak sense) in a quite general setting. But two questions arise in this context and were still open: is the solution c{à}d{ì}{à}g on [0,T] ? In other words does the solution have a left limit at time T ? The second question is: is this limit equal to the terminal condition? In this paper, under additional conditions on the ge…
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…
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 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 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…
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…
We study the existence of solutions to backward stochastic differential equations with drivers f(t,W,y,z) that are convex in z. We assume f to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y,Z) with bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W, we show the existence of a minimal continuous supersolution by uniformly approximating the terminal condition wit…
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.
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 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 this paper, we study ergodic backward stochastic differential equations (EBSDEs for short), for which the underlying diffusion is assumed to be multiplicative and of at most linear growth. The fact that the forward process has an unbounded diffusion is balanced with an assumption of weak dissipativity for its drift. Moreover, the forward equation is assumed to be non-degenerate. Like in [HMR15], we show that the solution of a BSDE in finite horizon T behaves basically as a linear function of T, with a shift depending on the solution of the associated EBSDE, with an explicit rate of converge…
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.
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 provide a one-to-one correspondence between the solution Y of a BSDE with singular terminal condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behavior of Y close to the final time and enlarges the uniqueness result to a wider class of generators.
In a first step, we establish the existence (and sometimes the uniqueness) of solutions for a large class of quadratic backward stochastic differential equations (QBSDEs) with continuous generator and a merely square integrable terminal condition. Our approach is different from those existing in the literature. Although we are focused on QBSDEs, our existence result also covers the BSDEs with linear growth, keeping $ξ$ square integrable in both cases. As byproduct, the existence of viscosity solutions is established for a class of quadratic partial differential equations (QPDEs) with a square …
This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables $y$ and $z$) in a super-linear way like $|y||\ln |y||^{(λ+1/2)\wedge 1}+|z||\ln |z||^λ$ for some $λ\geq 0$. For the following four different ranges of the growth power parameter $λ$: $λ=0$, $λ\in (0,1/2)$, $λ=1/2$ and $λ>1/2$, we give reasonably weakest possible different integrability conditions of the terminal value for the existence of an unbounded solution to the BSDE. In t…
In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differentia…
We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [7] and [8]. We perform a verification and discuss special cases for which the control problem has explicit solutions.
We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $ξ$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $ξ$ is of the form $ξ$ = g($Ξ$ S) where $Ξ$ is a continuous Markovian di…
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.
We introduce a domination argument which asserts that: if we can dominate theparameters of a quadratic backward stochastic differential equation (QBSDE) with continuousgenerator from above and from below by those of two BSDEs having ordered solutions, thenalso the original QBSDE admits at least one solution. This result is presented in a generalframework: we do not impose any integrability condition on none of the terminal data of thethree involved BSDEs, we do not require any constraint on the growth nor continuity of thetwo dominating generators. As a consequence, we establish the existence …
We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward pr…
In this paper, we study one-dimensional backward stochastic differential equation (BSDE, for short), whose coefficient $f$ is Lipschitz in $y$ but only continuous in $z$. In addition, if the terminal condition $ξ$ has bounded Malliavin derivative, we prove some uniqueness results for the BSDE with quadratic and linear growth in $z$, respectively.
We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[Ψ(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $Ψ$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained fo…
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 $z$
In this paper, we introduce and prove a stochastic Gronwall's inequality in (unbounded) random time horizon. As an application, we prove a comparison theorem for backward stochastic differential equation (BSDE for short) with random terminal time under stochastic monotonicity condition.
Two discretizations of a class of locally Lipschitz Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes Z, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process Z directly. Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs. These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in …
We study multidimensional backward stochastic differential equations (BSDEs) which cover the logarithmic nonlinearity u log u. More precisely, we establish the existence and uniqueness as well as the stability of p-integrable solutions (p > 1) to multidimensional BSDEs with a p-integrable terminal condition and a super-linear growth generator in the both variables y and z. This is done with a generator f(y, z) which can be neither locally monotone in the variable y nor locally Lipschitz in the variable z. Moreover, it is not uniformly continuous. As application, we establish the existence and …
In this paper, we will prove that, if the coefficient $g=g(t,y,z)$ of a BSDE is assumed to be continuous and linear growth in $(y,z)$, then the uniqueness of solution and continuous dependence with respect to $g$ and the terminal value $ξ$ are equivalent.
We give a dual representation of minimal supersolutions of BSDEs with non-bounded, but integrable terminal conditions and under weak requirements on the generator which is allowed to depend on the value process of the equation. Conversely, we show that any dynamic risk measure satisfying such a dual representation stems from a BSDE. We also give a condition under which a supersolution of a BSDE is even a solution.
