On solutions of a class of infinite horizon FBSDEs

How Do Future Expectations Affect the Financial Sector? Expectations Modeling, Infinite Horizon (Controlled) Random FBSDEs and Stochastic Viscosity Solutions

“…In order for this system not to explode but reach eventually equilibrium after long enough time, we also impose a square-integrability condition to its solution with respect to a scalar discount rate of a given range. Then, making use of the contraction mapping principle, we extend Yin's [55] result (cf. Theorem 3.1 therein) on existence and uniqueness of solutions of such FBSDEs by providing a more general lower bound for the above range.…”

“…In particular, Theorem 3.7 below extends Theorem 3.1 in Yin [55] by providing the same upper bound but a more general lower bound for λ via (13), subject to which the infinite horizon FBSDEs system of (11) admits a unique and adapted solution. To see this, according to Remark 3.2 the best possible lower bound allowed for λ is given by (14), which thanks to 0 ≤ γ ≤ 1 is strictly smaller than 2µ 2 + 4c 2 1 + 2c 2 2 + 1 that was proposed by Yin [55]. First, we quote two lemmata which will be used in the subsequent analysis.…”

Sufficient Stochastic Maximum Principle for Discounted Control Problem

“…There are several papers (e.g. Pardoux [19], Peng and Shi [23], Yin [26], Wu [27]) answering this question under different sets of assumptions both on the coefficients of the FBSDE and on the terminal condition. Some of these papers naturally require that the terminal condition of the backward equation at infinity (in certain sense) is given in advance with suitable properties that consequently determine the space for the solution processes.…”

“…In the paper by Wu [27], a different monotonicity condition is assumed to obtain a solution process with non zero (in general) yet still a.s. constant terminal condition. The most general result is due to Yin [26] who weakens the assumptions to obtain the solution in some exponentially-weighted L 2 space for some suitable discount factor. Nevertheless, in the latter paper, existence of the solution to the backward part of the system employs the result by Pardoux [19] based on the knowledge of the terminal condition.…”

A Note on FBSDE characterization of mean exit times

“…Since then, there has been a lot of interest in the study of both BSDEs ( [3][4][5][10][11][12]19]) and FBSDEs ( [1,2,9,16,21,22]), under various assumptions. The motivation for the study of BSDEs and FBSDEs is their wider applications in mathematical finance/stochastic control ( [6][7][8]17,20], etc.).…”