Existence and uniqueness of solutions of the following backward stochastic differential equation are studied:
Here is a fixed time, a -dimensional Brownian motion on a complete probability space () where the natural filtration generated by is considered, an -measurable random variable, maps into and is progressively measurable in first two variables, continuous and uniformly monotone in the third and uniformly Lipschitz in the fourth one. A solution of (1) is a pair of progressively measurable processes with values in such that paths of and are integrable and square integrable respectively almost surely.
If the terminal condition belongs to for some then there exists a unique solution of (1) in the class of progressively measurable processes in . This result weakens the assumptions imposed on in the analogous Theorem 2.2 in E. Pardoux [Math. Phys. Sci. 528, 503–549 (1999; Zbl 0959.60049)]. This result (Theorem 2.2) is the starting point in the proof of the present theorem: is approximated by a suitable sequence of more regular ’s for which existence and uniqueness of solutions with the terminal condition is known by Theorem 2.2 in the mentioned Pardoux’s paper, and this sequence is Cauchy, hence convergent to a solution of the equation (1). To show that are Cauchy, the authors lean upon apriori estimates which they have developed in the preliminary-part of this reviewed paper, and which yield, on the same hand, uniqueness of the constructed solution.
The second part of the paper deals with existence and uniqueness of solutions of the equation (1) where is a stopping time (including the possibility ) and is measurable. In this case, the equation (1) has a form
and and on the set almost surely. Existence and uniqueness of the equation (2) is shown under additional, yet fairly general set of assumptions upon integrability of and form and integrability of .
The third part of the paper concerns existence and uniqueness of solutions of the equation (1) when the terminal time is fixed, positive and finite, and the terminal condition belongs to . Growth and integrability assumptions on additional to those assumed in the first part of the paper are shown to yield a solution of (1) unique in a certain class of processes.