We formulate and solve the martingale problem in a nonlinear expectation
space. Unlike the classical work of Stroock and Varadhan (1969) where the
linear operator in the associated PDE is naturally defined from the
corresponding diffusion process, the main difficulty in the nonlinear setting
is to identify an appropriate class of nonlinear operators for the associated
fully nonlinear PDEs.
Based on the analysis of the martingale problem, we introduce the notion of
weak solution for stochastic differential equations under nonlinear
expectations and obtain an existence theorem under the H\"older continuity
condition of the coefficients. The approach to establish the existence of weak
solutions generalizes the classical Girsanov transformation method in that it
no longer requires the two (probability) measures to be absolutely continuous.Comment: The new version simplifies some proofs for the main theorems and
generalizes some result