Multidimensional Markovian FBSDEs with super-quadratic growth

A study of backward stochastic differential equation on a Riemannian manifold

“…According to (2.5) and (2.8) we have for every y 1 , y 2 ∈ R L and z 1 , z 2 ∈ R mL , 2) for some C 2 > 0. In fact, although (3.4) is slightly different from those in [24] where associated coefficients are required to be globally Lipschitz continuous with respect to variable y, following the same procedure in the proof of [24, Theorem 3.1] we can still obtain the desired conclusion here, see also the arguments in [24, Example 2.2]. Then applying Theorem 3.2 we obtain the desired conclusion immediately.…”

“…Xing and Zitković [40] proved global existence of a unique Markovian solution of (1.1) based on the existence of a single convex function. We also refer readers to [19,18,24,37] and reference therein for various results concerning the local existence of a solution to (1.1) in R n with the generator having quadratic growth under different conditions, including the boundness for Malliavin derivatives of terminal value(see Kupper, Luo and Tangpi [24]), small L ∞ norm of terminal value (see Harter and Richou [18] or Tevzadze [37]) and the special diagonal structure of the generator(see Hu and Tang [19]).…”

Equilibrium asset pricing with transaction costs

“…Evidently, settings that combine aspects (i) and (ii) above are even more challenging to deal with. As far as we know, the only works addressing multidimensional fully coupled quadratic FBSDEs are the references [6,24] already mentioned above, Luo and Tangpi [45] which considers diagonally quadratic generators (an assumption not satisfied in our context), as well as Kupper et al [41] which considers the Markovian case and obtains global existence under a uniform non-degeneracy assumption for the volatility of the forward process (that does not hold for our system).…”

Solvability of coupled FBSDEs with diagonally quadratic generators

“…We further show using similar estimates that the solution (X, Y, Z) is continuous and differentiable with respect to the initial value x. To the best of our knowledge, the only works considering existence of coupled FBSDEs with quadratic growth are the article of Antonelli and Hamadène [1] and the Ph.D. thesis of Fromm [8] and the preprint by Kupper et al [18]. In [1] the focus is on global solvability.…”

“…See also [8,Chapter 4] for an extension of this result to multi-dimension and locally Lipschitz generator in Z, for small time horizons. The main idea of [18] rests on Malliavin differentiability of solutions in the Lipschitz setting and the focus is on global existence of multidimensional superquadratic equations in the Markovian case. The structure of the rest of the paper is the following: In the next section we make precise the probabilistic setting, introduce some notation and state our main existence and uniqueness result.…”