N ov 2 00 7 Mean - Field Backward Stochastic Differential Equations . A Limit Approach ∗ Rainer

Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to study a special mean-field problem in a purely stochastic approach. We consider a stochastic differential equation that describes the dynamics of a particle X(N) influenced by the dynamics of N other particles, which are supposed to be independent identically distributed and of the same law as X(N). This equation (of rank N) is then associated with a backward stochastic differential equation (BSDE). After proving the existence and the uniqueness of a solution (X(N), Y (N), Z(N)) for this couple of equations we investigate its limit behavior. With an approach which uses the tightness of the laws of the above sequence of triplets in a suitable space, and combines it with BSDE methods and the Law of Large Numbers, it is shown that (X(N), Y (N), Z(N)) converges in L2 to the unique solution of a limit equation, formed by a Mean-Field forward and a Mean-Field backward equation. AMS Subject classification: 60H10; 60B10.