Solvability of coupled FBSDEs with diagonally quadratic generators

Abstract

We study the well-posedness for multi-dimensional and coupled systems of forward–backward SDEs when the generator can be separated into a quadratic and a subquadratic part. We obtain the existence and uniqueness of the solution on a small time interval. Moreover, the continuity and differentiability with respect to the initial value are presented.

Contracting Theory with Competitive Interacting Agents

“…that even in seemingly benign situations, existence of global solutions could fail. Later on, Cheredito and Nam[9], Kardaras et al[49], Kramkov and Pulido[54,55], Hu and Tang[45], Jamneshan et al[47], or more recently Luo and Tangpi[57] all obtained some positive results, but only in particular instances, which do not readily apply in our setting. Recently, Xing and Žitković[86] obtained quite general existence and uniqueness results, but in a Markovian framework.…”

BSDEs on Finite and Infinite Horizon with Time-Delayed Generators

“…Moreover, we prove that in our setting existence and uniqueness also hold in the case of reflexion on a càdlàg barrier. We observe a link between delay BSDEs and coupled FBSDE and, based on the findings in Luo and Tangpi [13], we derive existence of delay quadratic BSDEs in the case where only the value process is subjected to delay. We refer to Briand and Elie [5] for a similar result, again for a different type of delay and in the one-dimensional case.…”

“…Put u(s) = v(s) = 1, for s ∈ [0, T ]. We are considering the following BSDE with time delay only in the value process: so that 1. and 2. follow from [13], and 3. and 4. from [3].…”

Multidimensional Markovian FBSDEs with super-quadratic growth

“…Some important progress have been achieved recently for BSDEs with small terminal conditions, see Tevzadze [26] and for more recent development, see Hu and Tang [17], Luo and Tangpi [22], Jamneshan et al [19], Cheridito and Nam [6], Frei [12] and Xing and Žitkovic [27]. To the best of our knowledge, the only works studying well-posedness of coupled FBSDEs with quadratic growth are the article of Antonelli and Hamadène [1] and the preprints of Luo and Tangpi [22] and Fromm and Imkeller [13]. In [1] the authors consider a one-dimensional equation with one dimensional Brownian motion and impose monotonicity conditions on the coefficient so that comparison principles for SDEs and BSDEs can be applied.…”

“…In [13], a fully coupled Markovian FBSDE is considered with multidimensional forward and value processes and locally Lipschitz generator in (Y, Z) and a existence of a unique local solution is obtained using the technique of decoupling fields and an extension to global solutions is proposed. Although the (non-Markovian) system studied in Luo and Tangpi [22] is the same as the one considered in the present paper, the techniques here are essentially different. Furthermore, the main results we present here can be extended to the non-Markovian setting and to random diffusion coefficient (when σ depends on X and Y ) under stronger assumptions involving the Malliavin derivatives of g and h. We refer to the Ph.D. thesis of Luo [21] for details.…”