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.
We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the drift is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset of $\R^d$ not necessary bounded. We study the link of such EBSDEs with PDEs and we apply our results to an ergodic optimal control problem.
We focus on a class of BSDEs driven by a cadlag martingale and corresponding Markov type BSDE which arise when the randomness of the driver appears through a Markov process. To those BSDEs we associate a deterministic problem which, when the Markov process is a Brownian diffusion, is nothing else but a parabolic type PDE. The solution of the deterministic problem is intended as decoupled mild solution, and it is formulated with the help of a time-inhomogeneous semigroup.
In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.
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.
We are interested in path-dependent semilinear PDEs, where the derivatives are of G{â}teaux type in specific directions k and b, being the kernel functions of a Volterra Gaussian process X. Under some conditions on k, b and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is X.
In this paper, we study the reflected BSDE with one continuous barrier, under the monotonicity and general increasing condition on $y$ and non Lipschitz condition on $z$. We prove the existence and uniqueness of the solution to these equation by approximation method.
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.
In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of solutions in $L^p\ (p>1)$, which includes some known results as its particular cases.
In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators, the $z$ parts of whose $l$-th components only depend on the $l$-th columns. 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 differ…
In this paper, we deal with a class of one-dimensional reflected backward stochastic differential equations with stochastic Lipschitz coefficient. We derive the existence and uniqueness of the solutions for those equations via Snell envelope and the fixed point theorem.
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 continue in solving reflected generalized backward stochastic differential equations (RGBSDE for short) and fixed terminal time with use some new technical aspects of the stochastic calculus related to the reflected generalized BSDE. Here, existence and uniqueness of solution is proved under a non-Lipschitz condition on the coefficients.
We study a new class of ergodic backward stochastic differential equations (EBSDEs for short) which is linked with semi-linear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in Neumann boundary conditions. We study the existence and uniqueness of solutions to EBSDEs and the link with partial differential equations. Then we apply these results to optimal ergodic control problems.
In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) in $L^p\ (p>1)$ under weaker assumptions on the coefficients, considering both a finite and an infinite time interval. We establish a general existence and uniqueness result of solutions in $L^p\ (p>1)$ to finite and infinite time interval BSDEs with non-Lipschitz coefficients, which includes the corresponding results in \citet{Par90}, \citet{Mao95}, \citet{Chen97}, \citet{Cons01}, \citet{Wang03}, \citet{Chen00} and \citet{Wang09} as its particular cases.
In this paper, we consider the reflected backward stochastic differential equations driven by G-Brownian motion (reflected G-BSDEs) whose coefficients satisfy the beta-order Mao's condition. The uniqueness is obtained by some a priori estimates and the existence can be proved by two different methods. The first one is Picard iteration and the second one is approximation via penalization. The latter construction is useful to get the comparison theorem.
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 study, we consider a class of backward SDE driven by jump Markov process. An existence and uniqueness result to this kind of equations is obtained in a locally Lipschitz case. We essentially approximate the initial problem by constructing a convenient sequence of globally Lipschitz BSDEs having the existence and the uniqueness propriety. Then, we show, by passing to the limits, the existence, and uniqueness of a solution to the initial problem. After that, a stability theorem is also proved in the local Lipschitz setting. Applying the aforementioned result, we give an application to Eu…
This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.
We focus on a class of path-dependent problems which include path-dependent (possibly Integro) PDEs, and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the afore mentioned BSDEs. This concept generalizes a similar notion introduced by the authors in previous papers in the framework of classical PDEs and IPDEs. For every initial condition (s, $η$), where s is an initial time and $η$ an initial path, the solution …
Mean-field BSDEs with non-Lipschitz coefficients and double mean reflections
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究院）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性预期前沿科学中心）
We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a locally Lipschitz condition in the $Z$ and $U$ variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these re…
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Converg…
Using a new notion of path-derivative, we study well-posedness of backward stochastic differential equation driven by a continuous martingale $M$ when $f(s,γ,y,z)$ is locally Lipschitz in $(y,z)$: \[Y_{t}=ξ(M_{[0,T]})+\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\rm tr}[M,M]_{s}-\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t}\] Here, $M_{[0,t]}$ is the path of $M$ from $0$ to $t$ and $m$ is defined by $[M,M]_{t}=\int_{0}^{t}m_{s}m_{s}^{*}d{\rm tr}[M,M]_{s}$. When the BSDE is one-dimensional, we could show the existence and uniqueness of solution. On the contrary, when the BSDE is multidimensional, we sho…
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…
In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.
The present paper is devoted to investigating the existence and uniqueness of solutions to a class of non-Lipschitz scalar valued backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs). In fact, when the generators are Lipschitz continuous in $y$ and uniformly continuous in $z$, we construct the unique solution to such equations by monotone convergence argument. The comparison theorem and related Feynman-Kac formula are stated as well.
In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in $y$ and a Lipschitz condition in $z$. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in $y$. Then, such representation theorem is adopted to prove a probabilistic formula, in vis…
$\mathbb{L}^p$-solutions for Reflected BSDEs with jumps in a general filtration under stochastic Lipschitz coefficients
National Institute of Statistics and Applied Economics (INSEA)（国立统计与应用经济研究所）；Cadi Ayyad University (UCA), National School of Applied Sciences of Marrakech (ENSA-M)（卡迪·阿亚德大学，马拉喀什国立应用科学学院）
Existence and uniqueness is established for a large class of backward stochastic differential equations which contain singular terms of the form $\pm|z|^2/y$. The results are applied to investigate singular partial differential equations (PDEs) and to decision theory problems that cannot be studied using classical regular BSDEs. The application to PDEs concerns the existence of viscosity solutions to PDEs containing a singular term of the form $\pm|\nabla v|^2/v$ with rather weak assumptions on the regularity of the coefficients. Such PDEs with singularity in the value process appear in severa…
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the eqution and mild conditions on the obstacle a unique continuous solution of the problem admits a stochastic representation in terms of reflected backward stochastic differential equations. We derive also some regularity properties of so- lutions and prove useful approximation results.
In this paper, we study the existence of densities (with respect to the Lebesgue measure) for marginal laws of the solution $(Y,Z)$ to a quadratic growth BSDE. Using the (by now) well-established connection between these equations and their associated semi-linear PDEs, together with the Nourdin-Viens formula, we provide estimates on these densities.
This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time adva…
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…
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.
In this paper, we study backward stochastic differential equations (BSDEs shortly) with jumps that have Lipschitz generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. Under just integrability on the data we show that such equations admits a unique solution which belongs to class D.
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.
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.
In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed point problem in a space of random vectors. This makes it possible to employ general fixed point arguments to find solutions. For instance, Banach's contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed point arguments can be applied…
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 …
Stemmed from the derivation of the optimal control to a stochastic linear-quadratic control problem with Markov jumps, we study one kind of backward stochastic differential equations (BSDEs) that the generator f is affected by a Markovian switching. Then, the case that the Markov chain is involved in a large state space is considered. Following the classical approach, a hierarchical approach is adopted to reduce the complexity and a singularly perturbed Markov chain is involved. We will study the asymptotic property of BSDE with the singularly perturbed Markov chain. At last, as an application…
In this paper, a probabilistic interpretation for the viscosity solution of a parabolic partial differential equation is obtained by virtue of the solution of a class of quadratic backward stochastic differential equations (BSDEs, for short). Furthermore, we prove the convergence and the large deviation principle for the solution of this class of quadratic BSDEs, which is associated with a family of Markov processes with the diffusion coefficients that tend to be zero.
This paper is dedicated to solving high-dimensional coupled FBSDEs with non-Lipschitz diffusion coefficients numerically. Under mild conditions, we provided a posterior estimate of the numerical solution that holds for any time duration. This posterior estimate validates the convergence of the recently proposed Deep BSDE method. In addition, we developed a numerical scheme based on the Deep BSDE method and presented numerical examples in financial markets to demonstrate the high performance.
In two preceding articles, we studied the problem of the existence and uniqueness of a solution to some general BSDE on manifolds. In these two articles, we assumed some Lipschitz conditions on the drift $f(b,x,z)$. The purpose of this article is to extend the existence and uniqueness results under weaker assumptions, in particular a monotonicity condition in the variable $x$. This extends well-known results for Euclidean BSDE.
We give necessary and sufficient condition for existence and uniqueness of $\mathbb{L}^{p}$-solutions of reflected BSDEs with continuous barrier, generator monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$, $p\ge 1$. We also prove that the solutions may be approximated by the penalization method.
Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretat…
In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a G{â}teaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $D^{1,p}$ .
The present paper considers a new kind of backward stochastic differential equations driven by G-Brownian motion, which is called ergodic G-BSDEs. Firstly, the well-posedness of G-BSDEs with infinite horizon is given by a new linearization method. Then, the Feynman-Kac formula for fully nonlinear elliptic partial differential equations is established. Moreover, a new probabilistic approach is introduced to prove the uniqueness of viscosity solution to elliptic PDEs in the whole space. Finally, we obtain the existence of solution to G-EBSDE and some applications are also stated.
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.
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 consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of $\mathbb{L}^{p}$ solutions for equations with generators monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$ spaces for $p\ge 1$. We also prove that the solutions can be approximated via penalization method.
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…
