A solution is given to generalized backward stochastic differential equations driven by a real-valued RCLL martingale on an arbitrary filtered probability space. The existence and uniqueness of a solution are proved via the Yosida approximation method when the generators are only stochastic monotone with respect to the

Backward stochastic differential equations (BSDEs, for short) were initially investigated as adjoint processes under the maximum stochastic control principle by Bismut [

In other context, Pardoux and Zhang [

Following this work, Pardoux [

In this paper, we are interested in exploring, on a complete probability space

As opposed to (

In comparison to existing literature, our work reexamines and generalizes papers by Barles et al. [

The main difficulties of our problem lies in the fact that, firstly, the GBSDE (

The article is structured as follows. The notations, assumptions, definitions and other properties are covered in Section

Let

We assume given an

Now, we present the conditions imposed on the data of the generalized BSDE (

The process

for all

for all

for all

Note that, since

We adopt the following definition of a solution to the GBSDE (

A solution to GBSDE associated with parameters

First, we give some remarks that will be used subsequently.

Recall that

For any process

We point out that, since

The jump part of the process

Let

It suffices to prove the result in the case where

Note that from the integrability condition satisfied by the solution of GBSDE (

After that, taking this into account with (

Now coming back to (

The result is a direct consequence of Proposition

We are going to show that equation (

Assume for the moment that the driver

To prove that the solution exists, we first rewrite the equation as follows:

The proof is performed in 4 steps.

From Annex B, Proposition 6.7 in [

for every

Let

By applying the Itô formula, we have, for all

For simplicity of notation, we denote

Also

Using the continuity of the driver

Then, passing to the limit term by term in

We are now prepared to present the paper’s main result.

All that is left is the proof of existence, which will be made by a fixed point reasoning. To this end, let

In this section, we will prove that the GBSDE (

The remark that follows proves that investigating the generalized BSDE (

Set

Linear growth of

Now consider the GBSDE

Let

Thus,

Next, we endow the space

Using the first part of the current proof, we may define a map Ψ from