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 $|y||ln |y||^{(lambda+1/2)wedge 1}+|z||ln |z||^{lambda}$ for some $lambdageq 0$. For the following four different ranges of the growth power parameter $lambda$: $lambda=0$, $lambdain (0,1/2)$, $lambda=1/2$ and $lambda>1/2$, we give reasonably weakest possible different integrability conditions of the terminal value for the existence of an unbounded solution to the BSDE. In the first two cases, they are stronger than the $Lln L$-integrability and weaker than any $L^p$-integrability with $p>1$; in the third case, the integrability condition is just some $L^p$-integrability for $p>1$; and in the last case, the integrability condition is stronger than any $L^p$-integrability with $p>1$ and weaker than any $exp(L^epsilon)$-integrability with $epsilonin (0,1)$. We also establish the comparison theorem, which yields naturally the uniqueness, when either generator of both BSDEs is convex (concave) in both unknown variables $(y,z)$, or satisfies a one-sided Osgood condition in the first unknown variable $y$ and a uniform continuity condition in the second unknown variable $z$.
This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator $g(t,y,z)$, by relaxing the assumptions of citet{HuTang2016SPA} on the generator and terminal v
alue. More precisely, the generator $g(t,y,z)$ can have more general growth and continuity in $y$ in the local solution; while in the global solution, the generator $g(t,y,z)$ can have a skew sub-quadratic but in addition strictly and diagonally quadratic growth in the second unknown variable $z$, or the terminal value can be unbounded but the generator $g(t,y,z)$ is diagonally dependent on the second unknown variable $z$ (i.e., the $i$-th component $g^i$ of the generator $g$ only depends on the $i$-th row $z^i$ of the variable $z$ for each $i=1,cdots,n$ ). Three new results are established on the local and global solutions when the terminal value is bounded and the generator $g$ is subject to some general assumptions. When the terminal value is unbounded but is of exponential moments of arbitrary order, an existence and uniqueness result is given under the assumptions that the generator $g(t,y,z)$ is Lipschitz continuous in the first unknown variable $y$, and varies with the second unknown variable $z$ in a diagonal , component-wisely convex or concave, and quadratically growing way, which seems to be the first general solvability of systems of quadratic BSDEs with unbounded terminal values. This generalizes and strengthens some existing results via some new ideas.
The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of back
ward stochastic differential equations. The key idea of our approach for proving this interpretation consists of transmitting the signs between the solution and generator via the identity given by representation theorem. Compared with existing methods, our approach seems to be more applicable for general settings. This can also be regarded as a new application of such representation theorem.
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 viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.
This paper is devoted to solving a multidimensional backward stochastic differential equation with a general time interval, where the generator is uniformly continuous in $(y,z)$ non-uniformly with respect to $t$. By establishing some results on dete
rministic backward differential equations with general time intervals, and by virtue of Girsanovs theorem and convolution technique, we establish a new existence and uniqueness result for solutions of this kind of backward stochastic differential equations, which extends the results of Hamadene (2003) and Fan, Jiang, Tian (2011) to the general time interval case.
In this paper, we are interested in solving general time interval multidimensional backward stochastic differential equations in $L^p$ $(pgeq 1)$. We first study the existence and uniqueness for $L^p$ $(p>1)$ solutions by the method of convolution an
d weak convergence when the generator is monotonic in $y$ and Lipschitz continuous in $z$ both non-uniformly with respect to $t$. Then we obtain the existence and uniqueness for $L^1$ solutions with an additional assumption that the generator has a sublinear growth in $z$ non-uniformly with respect to $t$.
