Generalized fractional BSDE with non Lipschitz coefficients

Abstract

In this work, we deal with a generalized backward stochastic differential equation driven by a fractional Brownian motion. We essentially prove an existence and uniqueness result under non-Lipschitz condition on the generator by help of an iterated scheme on a suitable sequence.

BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients

“…Exploiting the argument developed in [ [1], Theorem 3.9] we prove that the sequence (Y n , Z n 1 , Z n 2 ) is a Cauchy sequence in B 2 ([T 1 , T ], R). Letting n → +∞ in eq.…”

“…As a consequence, we have Y 1 t = Y 2 t for t ∈ [0, T ]. From (3.19), we deduce Z 1 1,t , Z 1 2,t = Z 2 1,t , Z 2 2,t for t ∈ [0, T ]. This completes the proof.…”

Fractional backward stochastic variational inequalities with non-Lipschitz coefficient

“…Proof. We repeat here the arguments from the proof of Theorem 3.9 in Aïdara, S. and Sow, A.B. [2016] to show that (Y p , Z p ) is a Cauchy sequence in the Banach spaceṼ T ] .…”

“…where ρ is a continuous, concave and nondecreasing function satisfying some technical conditions (see assumption (H 3 )) and proved the existence and uniqueness of the solution of BSDE with respect to Wiener process. In Aïdara, S. and Sow, A.B. [2016] the non Lipschitz assumption (1.1) was considered to show the existence and uniqueness of the solution to fractional generalized BSDE.…”

backward stochastic differential equations driven by both standard and fractional Brownian motions with time deplayed generators

“…Since this first result, it has been widely recognized that BSDEs provide a useful framework for formulating a lot of mathematical problems such as used in financial mathematics, optimal control, stochastic games and partial differential equations. We also mention that, following Pardoux and Peng [12], many papers were devoted to improving the results of Pardoux and Peng [12] by weakening the Lipschitz conditions on coefficients (for example, see Aidara [2], Wang and Huang [13], Aidara and Sow [1], Mao [10], Aidara and Sagna [3]). Based on the above important applications, specially in the field of Finance, and optimal control, recently in [6], Delong and Imkeller introduced BSDEs with time delayed generators defined by…”